The problems of the AllSovietUnion mathematical competitions 19611986
This file contains the problems, suggested for solving on the Russian
national mathematical competitions (final part). It was semi automatically
converted from the plain text with the help of the powerful GNU emacs.
Since only a few manual editing was applied (to the fractions and
pseudographics), there can be some stupid mistakes. Please, feel free to
inform me about them.
I've posted this stuff in a number of articles in rec.puzzles. but
I have got many requests for the missing parts. So I have decided to
put this material here, having provided it with the answers on the
common questions.
collected articles start
I'm going to send some problems from the book
Vasil'ev N.B, Egorov A.A. "The problems of the AllSovietUnion
mathematical competitions",Moscow.:Nauka. 1988 ISBN 5020137308.
(in Russian).
Those problems were submitted for the solving on the competition
between the pupils of 8, 9, or 10 forms for 4 hours. So they do not
contain something of the advanced topics,  all of them can be solved
by a schoolboy. They can not go out of the common school plan bounds.
Most of the problems are original and the book contains all the
necessary references. I am not going to translate all the book, so I
shall not send the solutions. Please, accept those messages as they
are, to say more exactly  as I can. I have to do my job, and this is
hobby only, but nevertheless, that should be enjoyable to solve those
problems.
"Nobody can embrace the unembraceable."
Kozma Prutkov. (beginning of the XIX c.)
May be those postings are just a harassment, but I hope, that most
of You will not only enjoy the problems solving, but will be able
to use them in Your work with the students.
"Zeal overcomes everything."
"Sometimes zeal overcomes even the common sense."
Kozma Prutkov.
There are some wonderful books in Russian, that have not been
translated into English yet, for example,
"Problems of the Moscow mathematical competitions",compiled
by G.A.Galperin, A.K.Tolpygo.,Moscow, Prosveshchenie, 1986.
A.A.Leman "Collection of Moscow mathematical competitions problems",
Moscow, Prosveshchenie, 1965.
answers for the common questions

WHAT IS THE AGE OF THE PARTICIPANTS?

The Russian pupils start studying being 67 years old, so the
pupils of the 8th form are about 14.

WHO PARTICIPATE?

The competition is held in 3  4 stages
1. At school  if there are many volunteers.
2. Subregional  if the region is big enough.
3. Regional (in some regions, as in Moscow, Leningrad=SanktPetersburg,
Sverdlovsk=Yekaterinburg or Novosibirsk they are even more
interesting and more difficult)
4. Final part, considered in the report.

WHAT ARE THE BEST AND AVERAGE RESULTS?

The winners (25) usually give the perfect solution of all the problems
with some shortages.
My personal experience refers to that times, when there were two days
of the final competition. Than the winners solved all the problems of
two days except one problem.
It is very difficult to speak about the average level, because it
depends very much on a region, but most of the participants of that
time solved at least one problem. The problem is not only the
difficulties in the problems themselves, but also in the shortage of
time. They successfully solved the problems before the official
explanation two days later.

MAY I USE THOSE PROBLEMS IN THE SCHOOL PROJECTS?

You don't need MY permission for using those problems. As concerns the
copyright, the usage of all the information in the noncommercial
purposes was never restricted in Russia if it is not related to the
state security. Moreover, the spirit of the competition encourages
everybody to distribute those problems in order to enhance the
mathematical culture of the pupils.
And You only are able to decide whether Your students can solve them.

ARE THERE THE TRANSLATED SOLUTIONS OF THOSE PROBLEMS ELSEWHERE?

Sorry, there exists no complete translation of the cited book.
Besides, the solutions (in comparison with the problems themselves)
belong to the authors, and the translation without their explicit
permission would be their copyright violation.
So...
The first one  Moscow, 1961.
form
8 001 002 003 004 005a
9 006a 007 008 009 010
10 011 012 007 006b 005b
001.
+++
  
  
+++++
   
   
++++
Given a figure, containing 16 segments.
You should prove that there is no curve, that intersect each segment
exactly once. The curve may be not closed, may intersect itself, but it
is not allowed to touch the segments or to pass through the vertices.
002.
Given a rectangle A_{1}A_{2}A_{3}A_{4}.
Four circles with A_{i} as their centres
have their radiuses r_{1}, r_{2}, r_{3}, r_{4};
and r_{1}+r_{3}=r_{2}+r_{4}<d,
where d is a diagonal of the rectangle. Two pairs of the outer common
tangents to {the first and the third} and {the second and the fourth}
circumferences make a quadrangle.
Prove that You can inscribe a circle into that quadrangle.
003.
Prove that among 39 sequential natural numbers there always is a
number with the sum of its digits divisible by 11.
004.
Given a table 4x4.
a)
Find, how 7 stars can be put in its fields in such a way, that
erasing of two arbitrary lines and two columns will always leave at
list one of the stars.
b)
Prove that if there are less than 7 stars, You can always find two
columns and two rows, such, that if You erase them, no star will remain in
the table.
005.
a)
Given a quartet of positive numbers. (a,b,c,d). It is transformed to
the new one according to the rule:
a'=ab; b' =bc; c'=cd; d'=da.
The second one is transformed to the third according to the same rule
and so on.
Prove that if at least one initial number does not
equal 1, than You can never obtain the initial set.
b)
Given a set of 2^{k} (kth power of two) numbers, equal either to 1 or
to 1. It is transformed as that was in the a) problem (each one is
multiplied by the next, and the last  by the first.
Prove that You will always finally obtain the set of positive units.
006.
a)
Points A and B move uniformly and with equal angle speed along the
circumferences with O_{a} and O_{b} centres
(both clockwise).
Prove that a vertex C of the equilateral triangle ABC also moves
along a certain circumference uniformly.
b)
The distance from the point P to the vertices of the equilateral
triangle ABC equal AP=2, BP=3.
Find the maximal value of CP.
007.
Given some mxn table, and some numbers in its fields. You are allowed
to change the sign in one row or one column simultaneously.
Prove that You can obtain a table, with nonnegative sums over each row
and over each column.
008.
Given n points, some of them connected by nonintersecting segments.
You can reach every point from every one, moving along the
segments, and there is no couple, connected by two different ways.
Prove that the total number of the segments is (n1).
009.
Given a, b, p arbitrary integers.
Prove that there always exist relatively prime (i.e. that have no
common divisor) k and l, that (ak + bl) is divisible by p.
010.
Nicholas and Peter are dividing (2n+1) nuts. Each wants to get more.
Three ways for that were suggested. (Each consist of three stages.)
First two stages are common.
 1 stage:
 Peter divides nuts onto 2 heaps,
each contain not less than 2 nuts.
 2 stage:
 Nicholas divides both heaps onto 2 heaps,
each contain not less than 1 nut.
 3 stage:
 1 way:
 Nicholas takes the biggest and the least heaps.
 2 way:
 Nicholas takes two middle size heaps.
 3 way:
 Nicholas takes either the biggest and the least heaps
or two middle size heaps, but gives one nut to the
Peter for the right of choice.
Find the most and the least profitable method for the Nicholas.
011.
Prove that for three arbitrary infinite sequences, of natural numbers
a_{1},a_{2},...,a_{n},...
b_{1},b_{2},...,b_{n},...
c_{1},c_{2},...,c_{n},...
there exist numbers p and q such, that
a_{p}>=a_{q};
b_{p}>=b_{q}
and c_{p}>=c_{q}.
012.
Given 120 unit squares arbitrarily situated in the 20x25 rectangle.
Prove that You can place a circle with the unit
diameter without intersecting any of the squares.
The second competition  Moscow, 1962.
form
8 013 014 015 016 017
9 018 019 020 021 017
10 022 023 024 025 026
013.
Given points A' ,B' ,C' ,D', on the continuation of the [AB], [BC], [CD], [DA]
sides of the convex quadrangle ABCD, such, that the following pairs of
vectors are equal:
[BB'[=[AB[,
[CC'[=[BC[,
[DD'[=[CD[,
[AA'[=[DA[.
Prove that the quadrangle A'B'C'D' area is five times more than the
quadrangle ABCD area.
014.
Given the circumference s and the straight line l, passing through the
centre O of s. Another circumference s' passes through the point O and
has its centre on the l.
Describe the set of the points M, where the
common tangent of s and s' touches s'.
015.
Given positive numbers
a_{1},a_{2},...,a_{99},a_{100}.
It is known, that
a_{1}>a_{0};
a_{2}=3a_{1}2a_{0},
a_{3}=3a_{2}2a_{1},
...,
a_{100}=3a_{99}2a_{98}.
Prove that a_{100}>2^{99}.
016.
Prove that there are no integers a,b,c,d such that the polynomial
ax^{3}+bx^{2}+cx+d equals 1 at x=19, and equals
2 at x=62.
017.
Given a nxn table, where n is odd. There is either 1 or 1 in its every
field. A product of the numbers in the column is written under every
column. A product of the numbers in the row is written to the right of
every row.
Prove that the sum of 2n products doesn't equal to 0.
018.
Given two sides of the triangle.
Build that triangle, if medians to those sides are orthogonal.
019.
Given a quartet of positive numbers a,b,c,d, and is known, that abcd=1.
Prove that
a^{2}+b^{2}+c^{2}+d^{2}+ab+ac+ad+bc+bd+dc>=10
020.
Given right pentagon ABCDE. M is an arbitrary point inside ABCDE or on
its side. Let the distances MA, MB, ... , ME be renumerated and
denoted with
r_{1}<=r_{2}<=r_{3}<=r_{4}<=r_{5}.
Find all the positions of the M, giving r_{3}
the minimal possible value.
Find all the positions of the M, giving r_{3}
the maximal possible value.
021.
Given 1962 digit number. It is divisible by 9. Let x be the sum of its
digits. Let the sum of the digits of x be y. Let the sum of the digits
of y be z.
Find z.
022.
The M point is a middle of a isosceles triangle base [AC]. [MH] is
orthogonal to [BC] side. Point P is the middle of the segment [MH].
Prove that [AH] is orthogonal to [BP].
023.
What maximal area can have a triangle if its sides a,b,c satisfy inequality
0<=a<=1<=b<=2<=c<=3?
024.
Given x,y,z, three different integers.
Prove that ((xy)^{5}+(yz)^{5}+(zx)^{5}) is
divisible by 5(xy)(yz)(zx).
025.
Given a_{0}, a_{1}, ... , a_{n}.
It is known that

a_{0}=a_{n}=0;

a_{k1}2a_{k}+a_{k+1}>=0
for all k = 1, 2, ... , k1.
Prove that all the numbers are nonnegative.
026.
Given positive numbers a_{1}, a_{2}, ..., a_{m};
b_{1}, b_{2}, ..., b_{n}.
Is known that
a_{1}+a_{2}+...+a_{m}=b_{1}+b_{2}+...+b_{n}.
Prove that You can fill an empty table with m rows and n columns with
no more than (m+n1) positive number in such a way, that for all i,j
the sum of the numbers in the ith row will equal to a_{i}, and the sum
of the numbers in the jth column  to b_{j}.
The third competition  Moscow, 1963.
form
8 027 028 029a 030 031a
9 032 033 034 031b 028
10 035 036 037 029b 028
11 038 028 039 040 029b
027.
Given 5 circumferences, every four of them have a common point.
Prove that there exists a point that belongs to all five circumferences.
028.
Eight men had participated in the chess tournament. (Each meets each;
draws are allowed, giving 1/2 of pont; winner gets 1.) Everyone has
different number of points. The second one has got as many points as
the four weakest participants together.
What was the result of the play between the third prizer and
the chessplayer that have occupied the seventh place?
029.
a)
Each diagonal of the quadrangle halves its area.
Prove that it is a parallelogram.
b)
Three main diagonals of the hexagon halve its area.
Prove that they intersect in one point.
030.
Natural numbers a and b are relatively prime.
Prove that the greatest common divisor of (a+b) and
(a^{2}+b^{2}) is either 1 or 2.
031.
Given two fixed points A and B .The point M runs along the
circumference containing A and B. K is the middle of the segment [MB].
[KP] is a perpendicular to the line (MA).
a)
Prove that all the possible lines (KP) pass through one point.
b)
Find the set of all the possible points P.
032.
Given equilateral triangle with the side l.
What is the minimal length d of a brush (segment), that will paint all
the triangle, if its ends are moving along the sides of the triangle.
033.
A chessboard 6x6 is tiled with the 2x1 dominos.
Prove that You can cut the board onto two parts by
a straight line that does not cut dominos.
034.
Given n different positive numbers
a_{1},a_{2},...,a_{n}.
We construct all the possible sums (from 1 to n terms).
Prove that among those sums there are at least n(n+1)/2 different ones.
035.
Given a triangle ABC. We build two angle bisectors in the corners A and
B. Than we build two lines parallel to those ones through the point C.
D and E are intersections of those lines with the bisectors. It happens,
that (DE) line is parallel to (AB).
Prove that the triangle is isosceles.
036.
Given the endless arithmetic progression with the positive integer
members. One of those is an exact square.
Prove that the progression
contain the infinite number of the exact squares.
037.
Given right 45angle. Can You mark its corners with the digits
{0,1,...,9} in such a way, that for every pair of digits there would be
a side with both ends marked with those digits?
038.
Find such real p, q, a, b, that for all x an equality is held:
(2x1)^{20}  (ax+b)^{20} =
(x^{2}+px+q)^{10}.
039.
On the ends of the diameter two "1"s are written. Each of the
semicircles is divided onto two parts and the sum of the numbers of its
ends (i.e. "2") is written at the middle point.
Then every of the four
arcs is halved and in its middle the sum of the numbers on its ends is
written.
Find the total sum of the numbers on the circumference after n steps.
040.
Given an isosceles triangle. Find the set of the points inside the
triangle such, that the distance from that point to the base equals to
the geometric mean of the distances to the sides.
The fourth competition  Moscow, 1964.
form
8 041 042 043 044 045a
9 041 046 047 048 049
10 050 051 045ab 052 053
11 054 055 052 053 054
041.
The two heights in the triangle are not less than the respective sides.
Find the angles.
042.
Prove that for no natural m a number m(m+1) is a power of an integer.
043.
Given 1000000000 first natural numbers. We change each number with the
sum of its digits an repeat this procedure until there will remain
1000000000 one digit numbers.
Is there more "1"s or "2"s?
044.
Given an arbitrary set of 2k+1 integers:
{a_{1},a_{2},...,a_{2k+1}}.
We make a new set:
{(a_{1}+a_{2})/2, (a_{2}+a_{3})/2,
(a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_{1})/2};
and a new one, according to the same rule, and so on...
Prove that if we obtain integers only, the initial set consisted of
equal integers only.
045.
a)
Given a convex hexagon ABCDEF with all the equal angles.
Prove that ABDE = EFBC = CDFA.
b)
The opposite problem: Prove that it is possible to build a convex
hexagon with equal angles of six segments
a_{1},a_{2},...,a_{6},
whose lengths satisfy the condition
a_{1}a_{4} =
a_{5}a_{2} =
a_{3}a_{6}.
046.
Find integer solutions (x,y) of the equation (1964 times "sqrt"):
sqrt(x+sqrt(x+sqrt(....(x+sqrt(x))....)))=y.
047.
Four perpendiculars are drawn from the vertices of a convex quadrangle
to its diagonals.
Prove that their bases make a quadrangle similar to the given one.
048.
Find all the natural n such that n! is not divisible by n^{2}.
049.
A honeybug crawls along the honeycombs with the unite length of their
hexagons. He has moved from the node A to the node B along the shortest
possible trajectory.
Prove that the half of his way he moved in one direction.
050.
The quadrangle ABCD is outscribed around the circle with the centre O.
Prove that the sum of AOB and COD angles equals 180 degrees.
051.
Given natural a,b,n. It is known, that for every natural k (k<>b) the
number ak^{n} is divisible by bk.
Prove that a=b^{n}.
052.
Given an expression x_{1}:x_{2}:...:x_{n}
( : means division).
We can put the braces as we want.
How many expressions can we obtain?
053.
We have to divide a cube onto k nonoverlapping tetrahedrons.
For what smallest k is it possible?
054.
Find the smallest exact square with last digit not 0, such that after
deleting its last two digits we shall obtain another exact square.
055.
Let ABCD be an outscribed trapezoid; E is a point of its diagonals
intersection; r_{1},r_{2},r_{3},r_{4} 
the radiuses of the circles
inscribed in the triangles ABE, BCE, CDE, DAE respectively.
Prove that
1/(r_{1})+1/(r_{3}) = 1/(r_{2})+1/(r_{4}).
The fifth competition  Moscow, 1965.
form
8 056a 057 058 059 060
9 061 062 063 064 065
10 056b 066 067a 068a 069
11 063 067b 070 068b 071
056.
a)
Each of the numbers x_{1},x_{2},...,x_{n}
can be 1, 0, or 1.
What is the minimal possible value of the sum of all products of
couples of those numbers.
b)
Each absolute value of the numbers
x_{1},x_{2},...,x_{n}
is less than 1.
What is the minimal possible value of the sum of all
products of couples of those numbers.
057.
Given a board 3x3 and 9 cards with some numbers (known to the
players). Two players, in turn, put those cards on the board.
The first wins if the sum of the numbers in the first and the
third row is greater than in the first and the third column.
Prove that it doesn't matter what numbers are on the cards,
but if the first plays the best way, the second can not win.
058.
A circle is outscribed around the triangle ABC. Chords, from the middle
of the arc AC to the middles of the arcs AB and BC, intersect sides
[AB] and [BC] in the points D and E.
Prove that (DE) is parallel to
(AC) and passes through the centre of the inscribed circle.
059.
A bus ticket is considered to be lucky if the sum of the first three
digits equals to the sum of the last three (6 digits in Russian buses).
Prove that the sum of all the lucky numbers is divisible by 13.
060.
There is a lighthouse on a small island. Its lamp enlights a segment of
a sea to the distance a. The light is turning uniformly, and the end of
the segment moves with the speed v.
Prove that a ship, whose speed doesn't exceed v/8 cannot arrive to the
island without being enlightened.
061.
A society created in the help to the police contains 100 man exactly.
Every evening 3 men are on duty.
Prove that You can not organise
duties in such a way, that every couple will meet on duty once exactly.
062.
What is the maximal possible length of the segment, being cut out by
the sides of the triangle on the tangent to the inscribed circle, being
drawn parallel to the base, if the triangle's perimeter equals 2p?
063.
Given n^{2} numbers x_{i,j} (i,j=1,2,...,n)
satisfying the system of n^{3} equations:
x_{i,j}+x_{j,k}+x_{k,i}=0 (i,j,k = 1,...,n).
Prove that there exist such numbers
a_{1},a_{2},...,a_{n};
that for all i,j=1,...n x_{i,j}=a_{i}a_{j}.
064.
Is it possible to pose 1965 points inside the unit square to make every
rectangle with the square 0.005 and sides parallel to the side of the
square to contain at least one of those points?
065.
Quasirounding is a substitution one of the two closest integers
instead of the given number.
Given n numbers. Prove that You can quasiround them in such a way,
that a sum of every subset of quasirounded numbers will deviate from
the sum of the same subset of initial numbers not greater than (n+1)/4.
066.
The tourist has come to the Moscow by train. Alldaylong he wandered
randomly through the streets. Than he had a supper in the cafe on the
square and decided to return to the station only through the streets
that he has passed an odd number of times.
Prove that he is always able to do that.
067.
a)
A certain committee has gathered 40 times. There were 10 members on
every meeting. Not a single couple has met on the meetings twice.
Prove that there were no less then 60 members in the committee.
b)
Prove that You can not construct more then 30 subcommittees
of 5 members from the committee of 25 members, with no
couple of subcommittees having more than one common member.
068.
Given two relatively prime numbers p>0 and q>0.
An integer n is called "good" if we can represent it as n = px + qy
with nonnegative integers x and y, and "bad" in the opposite case.
a)
Prove that there exist integer c such that in a pair
{n, cn} always one is "good" and one is "bad".
b)
How many there exist "bad" numbers.?
069.
An airplanespy flies on the circumference with the centre A and radius
10 km. Its speed is 1000 km/h. At the certain moment a rocket with the
same speed starts from the point A and moves remaining on one line with
the plain and A.
What time is necessary for it to hit the plane?
/* remember, no advanced math. were known to the participants VAP */
070.
Prove that the sum of the lengths of the polyhedron edges
exceeds its tripled diameter (distance between two farest vertices).
071.
On the surface of the planet lives one inhabitant, that can move with
the speed not greater than u. A spaceship approaches to the planet with
its speed v.
Prove that if v/u > 10 , the spaceship can find the
inhabitant, even it is trying to hide.
The sixth competition  Voronezh, 1966.
form
8 072 073a 074 075a 076
9 077 073b 075b 078 079
10,11 075b 080 081 082 083
072.
There is exactly one astronomer on every planet of a certain system. He
watches the closest planet. The number of the planets is odd and all of
the distances are different.
Prove that there one planet being not watched.
073.
a)
Points B and C are inside the segment [AD]. AB=CD.
Prove that for all of the points P on the plane holds inequality
PA+PD>PB+PC.
b)
Given four points A,B,C,D on the plane. For all of the points P
on the plane holds inequality PA+PD > PB+PC.
Prove that points B and C are inside the segment [AD] and AB=CD.
074.
Can both (x^{2}+y) and (y^{2}+x)
be exact squares for natural x and y?
075.
a)
Pupils of the 8th form are standing in a row. There is the pupil
of the 7th form in before each, and he is smaller (in height) than
the elder.
Prove that if You will sort the pupils in each of rows
with respect to their height, every 8former will still be taller
than the 7former standing before him.
b)
An infantry detachment soldiers stand in the rectangle, being
arranged in columns with respect to their height.
Prove that if You
rearrange them with respect to their height in every separate row,
they will still be staying with respect to their height in columns.
076.
A rectangle ABCD is drawn on the crosslined paper with its sides
laying on the lines, and AD is k times more than AB (k is an
integer). All the shortest paths from A to C coming along the lines are
considered.
Prove that the number of those with the first link on [AD]
is k times more then of those with the first link on [AB].
077.
Given the numbers a_{1}, a_{2}, ... , a_{n}; such that
0<=a_{1}<=a_{2}<=2a_{1} , a_{2}<=a_{3}<=2a_{2} , ... , a_{n1}<=a_{n}<=2a_{n1}.
Prove that in the sum s=+a_{1}+a_{2}+...+a_{n}
You can choose appropriate signs to make 0<=s<=a_{1}.
078.
Prove that You can always pose a circle of radius S/P
inside a convex polygon with the perimeter P and area S.
079.
For three arbitrary crossroads A,B,C in a certain city there exist a
way from A to B not coming through C.
Prove that for every couple of
the crossroads there exist at least two nonintersecting ways
connecting them. (there are at least two crossroads in the city)
080.
Given a triangle ABC. Consider all the tetrahedrons PABC
with PH  the smallest of all tetrahedron's heights.
Describe the set of all possible points H.
081.
Given 100 points on the plain.
Prove that You can cover them with a
family of circles with the sum of their diameters less than 100 and the
distance between any two of the circles more than one.
082.
The distance from A to B is d kilometres. A plain flying with the
constant speed in the constant direction along and over the line (AB)
is being watched from those points. Observers have reported that the
angle to the plain from the point A has changed by \alpha degrees and
from B  by \beta degrees within one second.
What can be the minimal speed of the plain?
083.
20 Numbers are written on the board: 1, 2, ... ,20. Two players are
putting signs before the numbers in turn (+ or ). The first wants to
obtain the minimal possible absolute value of the sum.
What is the maximal value of the absolute value of the sum that can be
achieved by the second player?
"Where is the beginning of that end, that ends the beginning?"
Kozma Prutkov.
"If You see a title 'buffalo' on the elephant's cage
 don't believe to Your eyes!"
Kozma Prutkov.
The name has been changed  the numeration was restarted.
The first competition  Tbilisi, 1967.
form
8 084a 085a 086a 087 088
9 087b 086a 085b 084b 088
10 090 086b 091 092 093
084.
a)
The maximal height AH of the acuteangled triangle ABC equals the
median BM.
Prove that the angle ABC isn't greater than 60 degrees.
b)
The height AH of the acuteangled triangle ABC equals the median
BM and bisectrix CD.
Prove that the angle ABC is equilateral.
085.
a)
The digits of a natural number were rearranged.
Prove that the sum
of given and obtained numbers can't equal 999...9 (1967 of nines).
b)
The digits of a natural number were rearranged.
Prove that if the
sum of the given and obtained numbers equals 10^{10}, than the
given number was divisible by 10.
086.
a)
A lamp of a lighthouse enlights an angle of 90 degrees.
Prove that You can turn the lamps of four arbitrary
posed lighthouses so, that all the plane will be enlightened.
b)
There are eight lamps in eight points of the space. Each can
enlighten an octant (threefaced space angle with three mutually
orthogonal edges).
Prove that You can turn them so, that all the space will be enlightened.
087.
a)
Can You pose the numbers 0,1,...,9 on the circumference in such a
way, that the difference between every two neighbours would be
either 3 or 4 or 5?
b)
The same question, but about the numbers 0,1,...,13.
088.
Prove that there exists a number divisible by 5^{1000}
not containing a single zero in its decimal notation.
089.
Find all the integers x,y satisfying equation
x^{2}+x=y^{4}+y^{3}+y^{2}+y.
090.
In the sequence of the natural (i.e. positive integers) numbers every
member from the third equals the absolute value of the difference of
the two previous.
What is the maximal possible length of such a
sequence, if every member is less or equal to 1967?
091.
"KINGTHE SUICIDER"
Given a chessboard 1000x1000, 499 white castles and a black king.
Prove that it does not matter neither the initial situation nor the way
white plays, but the king can always enter under the check in a finite
number of moves.
092.
Three vertices KLM of the rhombus (diamond) KLMN lays on
the sides [AB], [BC] and [CD] of the given unit square.
Find the area of the set of all the possible vertices N.
093.
Given natural number k with a property "if n is divisible by k, than
the number, obtained from n by reversing the order of its digits is
also divisible by k".
Prove that the k is a divisor of 99.
The second competition  Leningrad, 1968.
form first day second day
8 094 095 096 097 098  105a 106 107 108 109
9 099 100 101 097 102  110 111 105a 108 109
10 103 095 104 097 096  105b 112 113 114 109
094.
Given an octagon with the equal angles. The lengths of all the sides
are integers.
Prove that the opposite sides are equal in pairs.
095.
What is greater, 31^{11} or 17^{14}?
/* calculators were not available  VAP */
096.
The circumference with the radius 100cm is drawn on the crosslined
paper with the side of the squares 1cm. It neither comes through the
vertices of the squares, nor touches the lines.
How many squares can
it pass through?
097.
Some students on the faculty speak several languages and some  Russian
only. 50 of them know English, 50  French and 50  Spanish.
Prove that it is possible to divide them onto 5 groups, not necessary equal,
to get 10 of them knowing English, 10  French and 10  Spanish in
each group.
098.
Prove the equality
2/(x_{2}1)+4/(x_{2}4)+6/(x_{2}9)+...+20/(x_{2}100)=
=11(1/((x1)(x+10))+1((x2)(x+9))+...+1((x10)(x+1))
099.
The difference between the maximal and the minimal diagonals of the
right nangle equals to its side ( n > 5 ).
Find n.
100.
The sequence a_{1},a_{2},a_{3},..., is built according to the rule
a_{1}=1, a_{2}=a_{1}+1/a_{1}, ... , a_{n+1}=a_{n}+1/a_{n}, ...
Prove that a_{100} > 14.
101.
Given two acuteangled triangles ABC and A'B'C' with the points O and
O' inside. Three pairs of the perpendiculars are drawn:

[OA_{1}] to the side [BC], [O'A'_{1}] to the side [B'C'],

[OB_{1}] to the side [AC], [O'B'_{1}] to the side [A'C'],

[OC_{1}] to the side [AB], [O'C'_{1}] to the side [A'B'];
it is known that

[OA_{1}] is parallel to the [O'A'],

[OB_{1}] is parallel to the [O'B'],

[OC_{1}] is parallel to the [O'C']
and the following products are equal:
OA_{1}*O'A' = OB_{1}*O'B' = OC_{1}*O'C'.
Prove that

[O'A'_{1}] is parallel to the [OA],

[O'B'_{1}] is parallel to the [OB],

[O'C'_{1}] is parallel to the [OC]
and
O'A'_{1}*OA = O'B'_{1}*OB = O'C'_{1}*OC.
102.
Prove that You can represent an arbitrary number not exceeding n!
(nfactorial; n!= 1*2*3*...*n) as a sum of k different numbers (k<=n)
that are divisors of n!.
103.
Given a triangle ABC, point D on [AB], E on [AC]; AD = DE = AC ,
BD = AE , DE is parallel to BC.
Prove that the length BD equals
to the side of a right decagon (tenangle) inscribed in a circle with
the radius R=AC.
104.
Three spheres are built so that the edges [AB], [BC], [AD] of
the tetrahedron ABCD are their respective diameters.
Prove that the spheres cover all the tetrahedron.
105.
a)
+++++
 +    +  + 
+++++
 +  +  +  + 
+++++
 +  +  +  + 
+++++
 +  +  +  + 
+++++
The fields of the square table 4x4 are filled with the "+" or
"" signs. You are allowed to change the signs simultaneously in
the whole row, column, or diagonal to the opposite sign. That means, for
example, that You can change the sign in the corner square, because
it makes a diagonal itself.
Prove that You will never manage to obtain a table containing pluses only.
b)
The fields of the square table 8x8 are filled with the "+" or signs
except one noncorner field with "". You are allowed to change the
signs simultaneously in the whole row, column, or diagonal to the
opposite sign. That means, for example, that You can change the sign in
the corner field, because it makes a diagonal itself.
Prove that You
will never manage to obtain a table containing pluses only.
106.
Medians divide the triangle onto 6 smaller ones. 4 of the
circles inscribed in those small ones are equal.
Prove that the triangle is equilateral.
107.
Prove that the equation x^{2} + x + 1 = py has solution
(x,y) for the infinite number of simple p.
108.
Each of the 9 referees on the figure skating championship estimates the
program of 20 sportsmen by assigning him a place (from 1 to 20). The
winner is determined by adding those numbers. (The less is the sum 
the higher is the final place).
It was found, that for the each sportsman, the difference of the
places, received from the different referees was not greater than 3.
What can be the maximal sum for the winner?
109.
Two finite sequences a_{1},a_{2},...,a_{n};
b_{1},b_{2},...,b_{n}
are just rearranged sequence 1, 1/2, ... , 1/n.
a_{1}+b_{1}>=a_{2}+b_{2}>=...>=a_{n}+b_{n}.
Prove that for every m (1<=m<=n) a_{m}+a_{n}>=4/m.
110.
There is scales on the teacher's table. There is a set of weighs on the
scales, and there are some pupils' names (may be more than one) on the
every weigh. A pupil entering the classroom moves all the weight with
his name to another side of the scales.
Prove that You can let in such
a subset of the pupils, that the scales will change its position.
111.
The city is a rectangle divided onto squares by m streets coming from
the West to the East and n streets coming from the North to the South.
There are militioners (policemen) on the streets but not on the
crossroads. They watch the certain automobile, moving along the closed
route, marking the time and the direction of its movement. Its trace is
not known in advance, but they know, that it will not pass over the
same segment of the way twice.
What is the minimal number of the militioners providing the unique
determination of the route according to their reports?
112.
The circle inscribed in the triangle ABC touches the side [AC] in the
point K.
Prove that the line connecting the middle of the [AC] side
with the centre of the circle halves the [BK] segment.
113.
The sequence a_{1},a_{2},...,a_{n} satisfies the following conditions:
a_{1}=0, a_{2}=a_{1}+1, ..., a_{n}=a_{n1}+1.
Prove that (a_{1}+a_{2}+...+a_{n})/n>=1/2.
114.
Given a quadrangle ABCD. The lengths of all its sides and diagonals are
the rational numbers. Let O be the point of its diagonals intersection.
Prove that AO  the length of the [AO] segment is also rational.
The third competition  Kiev, 1969.
form first day second day
8 115 116 117  122 123 124a
9 118 119 115  124 125 126
10 119 120 121  125 126 128
115.
The point E lies on the base [AD] of the trapezoid ABCD. The triangles
ABE, BCE and CDE perimeters are equal.
Prove that BC = AD/2.
116.
There is a wolf in the centre of a square field, and four dogs in the
corners. The wolf can easily kill one dog, but two dogs can kill the
wolf. The wolf can run all over the field, and the dogs  along the
fence (border) only.
Prove that if the dog's speed is 1.5 times more
than the wolf's, than the dogs can prevent the wolf escaping.
117.
Given a finite sequence of 0's and 1's with two properties:

if you chose five sequential digits in one place and in
the second place, those will be two different binary numbers.
(Some last digits of the first number may be included as
the first digits in the second.)

if You add 0 or 1 either from the left or from the right
side, the previous property will not be held.
Prove that the first four digits of that sequence coincide with the
last four.
118.
Given positive numbers a,b,c,d.
Prove that the set of inequalities
a+b<c+d;
(a+b)(c+d)<ab+cd;
(a+b)cd<ab(c+d)
contain at least one wrong.
119.
For what minimal natural a the polynomial ax^{2} + bx + c with the
integer c and b has two different positive roots both less than one.
120.
Given natural n. Consider all the fractions 1/(pq), where p and q are
relatively prime;
0<p<q<=n ; p+q>n.
Prove that the sum of all such a fractions equals to 1/2.
121.
Given n points in the three dimensional space such, that the arbitrary
triangle with the vertices in three of those points contains an angle
greater than 120 degrees.
Prove that You can rearrange them to make a polyline (unclosed) with
all the angles between the sequent links greater than 120 degrees.
122.
Find four different threedigit decimal numbers starting with the same
digit, such that their sum is divisible by three of them.
123.
Every city in the certain state is connected by airlines with no more
than with three other ones, but one can get from every city to every
other city changing a plane once only or directly.
What is the maximal possible number of the cities?
124.
Given a pentagon with all equal sides.
a)
Prove that there exist such a point on the maximal diagonal,
that every side is seen from it inside a right angle.
/* I mean that the side AB is seen from the point C inside an
arbitrary angle that is greater or equal than angle ACB.  VAP */
b)
Prove that the circles build on its sides as on the diameters cannot
cover the pentagon entirely.
125.
Given an equation x^{3} + ?x^{2} + ?x + ? = 0. First player substitutes
an integer on the place of one of the interrogative marks, than the
same do the second with one of the two remained marks, and, finally,
the first puts the integer instead of the last mark.
Explain how can the first provide the existence of three integer roots
in the obtained equation. (The roots may coincide.)
126.
20 football teams participate in the championship. What minimal number
of the games should be played to provide the property: from the three
arbitrary teams we can find at least on pair that have already met in
the championship.
127.
Let h_{k} be an apothem of the right kangle inscribed into a circle with
radius R.
Prove that (n + 1)h_{n+1}  nh_{n} > R.
128.
Prove that for the arbitrary positive a_{1}, a_{2}, ... , a_{n}
the following inequality is held
a_{1}/(a_{2}+a_{3})+a_{2}/(a_{3}+a_{4})+....a_{n1}/(a_{n}+a_{1})+a_{n}/(a_{1}+a_{2})>n/4
The 4th competition  Simferopol, 1970.
form first day second day
8 129 130 131 132 133a
9 134 135 133b 136 137
10 138 139 133b 136 140  141 142 143
129.
Given a circle, its diameter [AB] and a point C on it.
Build (with the
help of compasses and ruler) two points X and Y, that are symmetric
with respect to (AB) line, such that (YC) is orthogonal to (XA).
130.
The product of three positive numbers equals to one, their
sum is strictly greater than the sum of the inverse numbers.
Prove that one and only one of them is greater than one.
131.
How many sides of the convex polygon can equal its longest diagonal?
132.
The digits of the 17digit number are rearranged in the reverse order.
Prove that at list one digit of the sum of the new and the initial
number is even.
133.
a)
A castle is equilateral triangle with the side of 100 metres. It is
divided onto 100 triangle rooms. Each wall between the rooms is 10
metres long and contain one door. You are inside and are allowed to
pass through every door not more than once.
Prove that You can visit not more than 91 room (not exiting the castle).
b)
Every side of the triangle is divided onto k parts by the lines
parallel to the sides. And the triangle is divided onto k^{2}
small triangles. Let us call the "chain" such a sequence of
triangles, that every triangle in it is included only once,
and the consecutive triangles have the common side.
What is the greatest possible number of the triangles in the chain?
134.
Given five segments. It is possible to build
a triangle of every subset of three of them.
Prove that at least one of those triangles is acuteangled.
135.
The bisector [AD], the median [BM] and the height [CH] of
the acuteangled triangle ABC intersect in one point.
Prove that the angle BAC is greater than 45 degrees.
136.
Given five ndigit binary numbers. For each two numbers
their digits coincide exactly on m places. There is no
place with the common digit for all the five numbers.
Prove that 2/5 <= m/n <= 3/5.
137.
Prove that from every set of 200 integers You can choose a
subset of 100 with the total sum divisible by 100.
138.
Given triangle ABC, middle M of the side [BC], the centre O of the
inscribed circle. The line (MO) crosses the height AH in the point E.
Prove that the distance AE equals the inscribed circle radius.
139.
Prove that for every natural number k there exists an infinite set of
such natural numbers t, that the decimal notation of t does not contain
zeroes and the sums of the digits of the numbers t and kt are equal.
140.
Two equal rectangles are intersecting in 8 points.
Prove that the
common part area is greater than the half of the rectangle's area.
141.
All the 5digit numbers from 11111 to 99999 are written on
the cards. Those cards lies in a line in an arbitrary order.
Prove that the resulting 444445digit number is not a power of two.
142.
All natural numbers containing not more than n digits are divided onto
two groups. The first contains the numbers with the even sum of the
digits, the second  with the odd sum.
Prove that if 0<k<n than
the sum of the kth powers of the numbers in the first group equals to
the sum of the kth powers of the numbers in the second group.
143.
The vertices of the right nangle are marked with some colours
(each vertex  with one colour) in such a way, that the vertices
of one colour represent the right polygon.
Prove that there are two equal ones among the smaller polygons.
The 5th competition  Riga, 1971.
form first day second day
8 144 145a 146a 147  152ab 153 154
9 144 145a 148 147 146b  156abc152c 155
10 149 145b 150 147 151b  156 157 158
144.
Prove that for every natural n there exists a number, containing only
digits "1" and "2" in its decimal notation, that is divisible by 2^{n}
( nth power of two ).
145.
a)
Given a triangle A_{1}A_{2}A_{3} and the points
B_{1} and D_{2} on the side [A_{1}A_{2}],
B_{2} and D_{3} on the side [A_{2}A_{3}],
B_{3} and D_{1} on the side [A_{3}A_{1}].
If You build parallelograms A_{1}B_{1}C_{1}D_{1},
A_{2}B_{2}C_{2}D_{2}
and A_{3}B_{3}C_{3}D_{3}, the lines
(A_{1}C_{1}), (A_{2}C_{2}) and (A_{3}C_{3}), will
cross in one point O.
Prove that if A_{1}B_{1} = A_{2}D_{2} and
A_{2}B_{2} = A_{3}D_{3}, than A_{3}B_{3} = A_{1}D_{1}.
b)
Given a convex polygon A_{1}A_{2} ... A_{n} and the points
B_{1} and D_{2} on the side [A_{1}A_{2}],
B_{2} and D_{3} on the side [A_{2}A_{3}],
........
B_{n} and D_{1} on the side [A_{n}A_{1}].
If You build parallelograms A_{1}B_{1}C_{1}D_{1},
A_{2}B_{2}C_{2}D_{2} ... ,
A_{n}B_{n}C_{n}D_{n}, the lines (A_{1}C_{1}), (A_{2}C_{2}), ..., (A_{n}C_{n}), will
cross in one point O.
Prove that
A_{1}B_{1}*A_{2}B_{2}*...*A_{n}B_{n} =
A_{1}D_{1}*A_{2}D_{2}*...*A_{n}D_{n}.
146.
a)
A game for two.
The first player writes two rows of ten numbers
each, the second under the first. He should provide the following
property: if number b is written under a, and d  under c, then a
+ d = b + c.
The second player has to determine all the numbers. He
is allowed to ask the questions like "What number is written in the
x place in the y row?"
What is the minimal number of the questions
asked by the second player before he founds out all the numbers?
b)
There was a table mxn on the blackboard with the property: if You
chose two rows and two columns, then the sum of the numbers in the
two opposite vertices of the rectangles formed by those lines equals
the sum of the numbers in two another vertices. Some of the numbers
are cleaned. but it is still possible to restore all the table.
What is the minimal possible number of the remaining numbers?
147.
Given an unite square and some circles inside. Radius of each circle is
less than 0.001, and there is no couple of points belonging to the
different circles with the distance between them 0.001 exactly.
Prove that the area, covered by the circles is not greater than 0.34.
148.
The volumes of the water containing in each of three big enough
containers are integers. You are allowed only to relocate some times
from one container to another the same volume of the water, that the
destination already contains.
Prove that You are able to discharge one of the containers.
149.
Prove that if the numbers p_{1}, p_{2}, q_{1}, q_{2} satisfy the condition
(q_{1}  q_{2})^{2} + (p_{1}  p_{2})(p_{1}q_{2}  p_{2}q_{1}) < 0,
then the square polynomials
x^{2} + p_{1}x + q_{1} and
x^{2} + p_{2}x + q_{2}
have real roots, and between the roots of each there is a root of another one.
150.
The projections of the body on two planes are circles.
Prove that they have the same radius.
151.
Some numbers are written along the ring. If inequality (ad)(bc) < 0
is held for the four arbitrary numbers in sequence a,b,c,d, You have to
change the numbers b and c places.
Prove that You will have to do this operation finite number of times.
152.
a)
Prove that the line dividing the triangle onto two polygons with
equal perimeters and equal areas passes through the centre of the
inscribed circle.
b)
Prove the same statement for the arbitrary polygon outscribed around
the circle.
c)
Prove that all the lines halving its perimeter and area
simultaneously, intersect in one point.
153.
Given 25 different positive numbers.
Prove that You can choose two of
them such, that none of the other numbers equals neither to the sum nor
to the difference between the chosen numbers.
154.
a)
The vertex A_{1} of the right 12angle (dodecagon) A_{1}A_{2}...A_{12} is
marked with "" and all the rest  with "+". You are allowed to
change the sign simultaneously in the 6 vertices in succession.
Prove that is impossible to obtain dodecagon with A_{2} marked with
"" and the rest of the vertices  with "+".
b)
Prove the same statement if it is allowed to change the signs not in
six, but in four vertices in succession.
c)
Prove the same statement if it is allowed to change the signs in
three vertices in succession.
155.
N unit squares on the infinite sheet of crosslined paper are painted
with black colour.
Prove that You can cut out the finite number of
square pieces and satisfy two conditions

all the black squares are contained in those pieces.

the area of black squares is not less than 1/5 and not greater than
4/5 of every piece area.
The task for the tenth form was as follows:
"You are given three serious problems. Try to investigate at least one,
but to obtain as many results, as You can. At the end of Your work
make a sort of resume, showing the main proved facts, challenged
examples and the hypotheses that seem to be true ..."
/* this form of competition was never repeated later  it had
required too much efforts from those who checked the works */
156.
A cube with the edge of length n is divided onto n^{3} unit ones.
Let us choose some of them and draw three lines parallel to the edges through
their centres.
What is the least possible number of the chosen small
cubes necessary to make those lines cross all the smaller cubes?
a)
Find the answer for the small n (n = 2,3,4).
b)
Try to find the answer for n = 10.
c)
If You can not solve the general problem, try to estimate that value
from the upper and lower side.
d)
Note, that You can reformulate the problem in such a way:
Consider all the triples (x_{1},x_{2},x_{3}), where x_{i} can be one of the
integers 1,2,...,n.
What is the minimal number of the triples
necessary to provide the property: for each of the triples there
exist the chosen one, that differs only in one coordinate.
Try to
find the answer for the situation with more than three coordinates,
for example, with four.
157.
a)
Consider the function f(x,y) = x^{2} + xy + y^{2}.
Prove that for the every point (x,y) there exist such
integers (m,n), that f((xm),(yn)) <= 1/2.
b)
Let us denote with g(x,y) the least possible value of the
f((xm),(yn)) for all the integers m,n. The statement a)
was equal to the fact g(x,y) <= 1/2.
Prove that in fact, g(x,y) <= 1/3.
Find all the points (x,y), where g(x,y)=1/3.
c)
Consider function f_{a}(x,y) = x^{2} + axy + y^{2} (0 <= a <= 2).
Find
any c such that g_{a}(x,y) <= c.
Try to obtain the closest estimation.
158.
1 2 1 2
    The switch with two inputs and two
v v v v outputs can be in one of two
++++ ++++ different positions. In the left
        part of the picture a) the first input
 \ /      is connected with the second output
 X      and we can denote this as
 / \      1 2
          and the position in the right
++++ ++++ V V part of the picture 1 2
    2 1 will be denoted as  
V V V V V V
1 2 1 2 1 2
fig.a)
1 2 3 The scheme on the picture b) is universal
   in that sense that changing the state of
V V  the element switches You can obtain all
++++  the six connections, i.e.
  
   1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
++++                   
   V V V V V V V V V V V V V V V V V V
 V V 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1
 ++++
   (Check it. Note, that the total number of the
   states is 2 * 2 * 2 = 8, because each element
 ++++ can be in two positions.)
  
V V 
++++ 
  
  
++++ 
  
V V V
1 2 3
fig.b)
a)
Try to build the universal scheme for 4 inputs and 4 outputs,
that can provide all of 24 possible connections.
b)
What is the minimal number of the element switches for such a
scheme?
Let us call a scheme with n inputs and n outputs nuniversal, if it can
provide all n! possible connections of n inputs with n outputs.
c)
Here is the scheme (picture c) with 8 inputs and 8 outputs, where A
and B are 4universal.
Prove that it is 8universal.
1 2 3 4 5 6 7 8
       
V V V V V V V V V V V V V V V V
++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
               
               
++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
 \  \  \  \ /  /  /  / 
 \  \  \  \ /  /  /  / 
 \  \  \  X  /  /  / 
......................................................................
/ / / / \ \ \ \
++++++ ++++++
   
 A   B 
   
++++++ ++++++
\ \ \ \ / / / /
......................................................................
 /  /  /  / \  \  \  \ 
 /  /  /  / \  \  \  \ 
V V V V V V V V V V V V V V V V
++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
               
               
++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
       
V V V V V V V V
1 2 3 4 5 6 7 8
fig.c)
d)
Estimate the upper and lower bound for the number of the element
switches in the nuniversal scheme.
The 6th competition  Chelyabinsk, 1972.
form first day second day
8 159 160 161  166 167 168
9 162a 163 161 164  169 170 171
10 162b 163 165 164  166 172 173
159.
Given a rectangle ABCD, points M  the middle of [AD] side, N  the
middle of [BC] side. Let us take a point P on the continuation of the
[DC] segment over the point D. Let us denote the point of intersection
of lines (PM) and (AC) as Q.
Prove that the angles QNM and MNP are equal.
160.
Given 50 segments on the line. Prove that one of the following
statements is valid:

1. Some 8 segments have the common point.

2. Some 8 segments do not intersect each other.
161.
Find the maximal x such that the expression 4^{27} + 4^{1000} + 4^{x}
is the exact square.
162.
a)
Let a,n,m be natural numbers, a > 1.
Prove that if (a^{m} + 1) is
divisible by (a^{n} + 1) than m is divisible by n.
b)
Let a,b,n,m be natural numbers, a>1, a and b are relatively prime.
Prove that if (a^{m}+b^{m}) is divisible by
(a^{n}+b^{n}) than m is
divisible by n.
163.
2
/ \
/ \
4 3
/ \ / \
16 5 9 4
/ \ / \ /\ / \
The triangle table is built according to the rule:
You put the natural number a>1 in the upper row, and
then You write under the number k from the left
side k^{2}, and from the right side  (k+1). For
example, if a = 2, You get the table on the picture.
Prove that all the numbers on
each particular line are different.
164.
Given several squares with the total area 1.
Prove that You can pose
them in the square of the area 2 without any intersections.
165.
Let O be the intersection point of the of the convex quadrangle ABCD
diagonals.
Prove that the line drawn through the points of
intersection of the medians of AOB and COD triangles is orthogonal to
the line drawn through the points of intersection of the heights of BOC
and AOD triangles.
166.
Each of the 9 straight lines divides the given square onto two
quadrangles with the areas related as 2:3.
Prove that there exist
three of them intersecting in one point.
167.
The 7angle A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}A_{7} is inscribed in a circle.
Prove that if the centre of the circle is inside the 7angle, than the
sum of A_{1},A_{2} and A_{3} angles is less than 450 degrees.
168.
A game for two. One gives a digit and the second substitutes
it instead of a star in the following difference:
****  **** =
Then the first gives the next digit, and so on 8 times.
The first wants to obtain the greatest possible difference,
the second  the least. Prove that:

1. The first can operate in such a way that the difference would be
not less than 4000, not depending on the second's behaviour.

2. The second can operate in such a way that the difference would be
not greater than 4000, not depending on the first's behaviour.
169.
Let x,y be positive numbers, s  the least of { x, (y+ 1/x), 1/y}.
What is the greatest possible value of s? To what x and y does it
correspond?
170.
The point O inside the convex polygon makes isosceles triangle with all
the pairs of its vertices.
Prove that O is the centre of the outscribed circle.
171.
Is it possible to put the numbers 0,1 or 2 in the unit squares of the
crosslined paper 100x100 in such a way, that every rectangle 3x4 (and
4x3) would contain three zeros, four ones and five twos?
172.
Let the sum of positive numbers x_{1,} x_{2}, ... , x_{n} be 1.
Let s be the greatest of the numbers
{x_{1}/(1+x_{1}), x_{2}/(1+x_{1}+x_{2}),
... x_{n}/(1+x_{1}+...+x_{n})}.
What is the minimal possible s? What x_{i} correspond it.
/* Derivatives were not included in the school plans in
Russia that time. They expected another solution. VAP */
173.
Oneround hockey tournament is finished (each plays with each one time,
the winner gets 2 points, looser  0, and 1 point for draw). For
arbitrary subgroup of teams there exists a team (may be from that
subgroup) that has got an odd number of points in the games with the
teams of the subgroup.
Prove that there was even number of the participants.
The 7th competition  Kishenew, 1973.
form first day second day
8 174a 175 176  182 183 184ab
9 174b 177 178  179 185 186 184c
10 180 177 181  187 188 184c
174.
Fourteen coins are submitted to the judge. An expert knows, that the
coins from number one to seven are false, and from 8 to 14  normal.
The judge is sure only that all the true coins have the same weight and
all the false coins weights equal each other, but are less then the
weight of the true coins. The expert has the scales without weights.
a)
The expert wants to prove, that the coins 17 are false.
How can he do it in three weighings?
b)
How can he prove, that the coins 17 are false and the coins 814
are true in three weighings?
175.
Prove that 9digit number, that contains all the decimal digits except
zero and does not ends with 5 can not be exact square.
176.
Given n points, n > 4. Prove that You can connect them with arrows, in
such a way, that You can reach every point from every other point,
having passed through one or two arrows. (You can connect every pair
with one arrow only, and move along the arrow in one direction only.)
177.
Given an angle with the vertex O and a circle touching its sides in the
points A and B. A ray is drawn from the point A parallel to [OB). It
intersects with the circumference in the point C. The segment [OC]
intersects the circumference in the point E. The straight lines (AE)
and (OB) intersect in the point K.
Prove that OK = KB.
178.
The real numbers a,b,c satisfy the condition: for all x, such
that for 1 <= x <= 1, the inequality
 ax^{2} + bx + c  <= 1 is held.
Prove that for the same x
 cx^{2} + bx + a  <= 2.
179.
The tennis federation has assigned numbers to 1024 sportsmen,
participating in the tournament, according to their skill. (The tennis
federation uses the olympic system of tournaments. The looser in the
pair leaves, the winner meets with the winner of another pair. Thus, in
the second tour remains 512 participants, in the third  256, et.c.
The winner is determined after the tenth tour.) It comes out, that in
the play between the sportsmen whose numbers differ more than on 2
always win that whose number is less.
What is the greatest possible number of the winner?
180.
The square polynomial f(x)=ax^{2}+bx+c is of such a sort,
that the equation f(x)=x does not have real roots.
Prove that the equation f(f(x)) does not have real roots also.
181.
n squares of the infinite crosslined sheet of paper are painted
with black colour (others are white). Every move all the squares
of the sheet change their colour simultaneously. The square gets
the colour, that had the majority of three ones: the square itself,
its neighbour from the right side and its neighbour from the upper side.
a)
Prove that after the finite number of the moves all the black
squares will disappear.
b)
Prove that it will happen not later than on the nth move.
182.
Three similar acuteangled triangles AC_{1}B, BA_{1}C and CB_{1}A are built
on the outer side of the acuteangled triangle ABC. (Equal triples of
the angles are AB_{1}C, ABC_{1}, A_{1}BC and BA_{1}C, BAC_{1}, B_{1}AC.)
a)
Prove that the circumferences outscribed around the outer
triangles intersect in one point.
b)
Prove that the straight lines AA_{1}, BB_{1} and CC_{1}
intersect in the same point.
183.
N men are not acquainted each other. You need to introduce some of them
to some of them in such a way, that all the men will have different
number of the acquaintances.
Prove that it is possible for all N.
184.
The king have revised the chessboard 8x8 having visited all the fields
once only and returned to the starting point. When his trajectory was
drawn (the centres of the squares were connected with the straight
lines), a closed broken line without selfintersections appeared.
a)
Give an example that the king could make 28 steps parallel
the sides of the board only.
b)
Prove that he could not make less than 28 such a steps.
c)
What is the maximal and minimal length of the broken line
if the side of a field is 1?
185.
Given a triangle with a,b,c sides and with the area 1. a >= b >= c.
Prove that b^{2} >= 2.
186.
Given a convex nangle with pairwise (mutually) nonparallel sides
/* who knows Russian  a letter "r" had been broken on the
organising committee's typewriter  and it became
an inexhaustible source of jokes for some years */
and a point inside it. Prove that there are not more than n straight
lines coming through that point and halving the area of the nangle.
187.
Prove that for every positive x_{1}, x_{2}, x_{3}, x_{4}, x_{5} holds inequality:
(x_{1} + x_{2} + x_{3} + x_{4} + x_{5})^{2} >=
>= 4(x_{1}x_{2} + x_{3}x_{4} + x_{5}x_{1} + x_{2}x_{3} + x_{4}x_{5}).
188.
Given 4 points in threedimensional space, not lying in one plane.
What is the number of such a parallelepipeds (bricks), that each point
is a vertex of each parallelepiped?
The 8th competition  Erevan, 1974.
form first day second day
8 189abc190 191  197 198 199 200a
9 190 192 189d 193  201 202 200b
10 194 195 196 193  203 204 200b
189.
a,b,c) Given some cards with either "1" or "+1" written on the opposite
side. You are allowed to choose a triple of cards and ask about the
product of the three numbers on the cards.
What is the minimal number
of questions allowing to determine all the numbers on the cards ...
a)
for 30 cards,
b)
for 31 cards,
c)
for 32 cards.
(You should prove, that You cannot manage with less questions.)
d)
Fifty abovementioned cards are lying along the circumference. You
are allowed to ask about the product of three consecutive numbers
only. You need to determine the product af all the 50 numbers.
What is the minimal number of questions allowing to determine it?
190.
Among all the numbers representable as 36^{k}  5^{l} (k and l are
natural numbers) find the smallest.
Prove that it is really the smallest.
191.
a)
Each of the side of the convex hexagon (6angle) is longer
than 1. Does it necessary have a diagonal longer than 2?
b)
Each of the main diagonals of the convex hexagon is longer
than 2. Does it necessary have a side longer than 1?
192.
Given two circles with the radiuses R and r, touching each other from
the outer side. Consider all the trapezoids, such that its lateral
sides touch both circles, and its bases touch different circles.
Find the shortest possible lateral side.
193.
Given n vectors of unit length in the plane. The length of
their total sum is less than one.
Prove that You can
rearrange them to provide the property: for every k, k<= n,
the length of the sum of the first k vectors is less than 2.
194.
Find all the real a,b,c such that the equality
ax+by+cz + bx+cy+az + cx+ay+bz = x+y+z
is valid for all the real x,y,z.
195.
Given a square ABCD. Points P and Q are in the sides [AB] and
[BC] respectively. BP=BQ. Let H be the base of the
perpendicular from the point B to the segment [PC].
Prove that the angle DHQ is a right one.
196.
Given some red and blue points. Some of them are connected by the
segments. Let us call "exclusive" the point, if its colour differs from
the colour of more than half of the connected points. Every move one
arbitrary "exclusive" point is repainted to the other colour.
Prove that after the finite number of moves there will remain no
"exclusive" points.
197.
Find all the natural n and k such that n^{n} has k digits and k^{k} has n
digits.
198.
Given points D and E on the legs [CA] and [CB], respectively, of the
isosceles right triangle. CD = CE. The extensions of the
perpendiculars from D and C to the line AE cross the hypotenuse AB in
the points K and L.
Prove that KL = LB.
199.
Two are playing the game "cats and rats" on the chessboard 8x8. The
first has one piece  a rat, the second  several pieces  cats. All
the pieces have four available moves  up, down, left, right  to the
neighbour field, but the rat can also escape from the board if it is on
the boarder of the chessboard. If they appear on the same field  the
rat is eaten. The players move in turn, but the second can move all the
cats in independent directions.
a)
Let there be two cats. The rat is on the interior field.
Is it possible to put the cats on such a fields on the border
that they will be able to catch the rat?
b)
Let there be three cats, but the rat moves twice during the
first turn. Prove that the rat can escape.
200.
a)
Prove that You can rearrange the numbers 1, 2, ... , 32 in
such a way, that for every couple of numbers none of the
numbers between them will equal their arithmetic mean.
b)
Can You rearrange the numbers 1, 2, ... , 100 in such a way,
that for every couple of numbers none of the numbers between
them will equal their arithmetic mean?
201.
Find all the threedigit numbers such that it equals to the arithmetic
mean of the six numbers obtained by rearranging its digits.
202.
Given a convex polygon. You can put no triangle with area 1 inside it.
Prove that You can put the polygon inside a triangle with the area 4.
203.
Given a function f(x) on the segment 0<=x<=1.
For all x,
f(x)>=0; f(1)=1.
For all the couples of {x_{1},x_{2}} such, that all
the arguments are in the segment f(x_{1}+x_{2})>=f(x_{1})+f(x_{2}).
a)
Prove that for all x holds f(x) <= 2x.
b)
Is the inequality f(x) <= 1.9x valid?
204.
Given a triangle ABC with the are 1. Let A',B' and C' are
the middles of the sides [BC], [CA] and [AB] respectively.
What is the minimal possible area of the common part of
two triangles A'B'C' and KLM, if the points K,L and M
are lying on the segments [AB'], [CA'] and [BC'] respectively?
The 9th competition  Saratov, 1975.
form first day second day
8 205a 206 207 208a  213 214 215
9 209 206 210 208b  216 215 217
10 211 212 205b 208  214 218 219
205.
a)
The triangle ABC was turned around the centre of the outscribed
circle by the angle less than 180 degrees and thus was obtained the
triangle A_{1}B_{1}C_{1}.
The corresponding segments [AB] and [A_{1}B_{1}]
intersect in the point C_{2};
[BC] and [B_{1}C_{1}]  A_{2}; [AC] and
[A_{1}C_{1}]  B_{2}.
Prove that the triangle A_{2}B_{2}C_{2} is similar to the triangle ABC.
b)
The quadrangle ABCD was turned around the centre of the outscribed
circle by the angle less than 180 degrees and thus was obtained
the quadrangle A_{1}B_{1}C_{1}D_{1}.
Prove that the points of intersection of the corresponding lines
( (AB) and (A_{1}B_{1}),
(BC) and (B_{1}C_{1}),
(CD) and (C_{1}D_{1}),
(DA) and (D_{1}A_{1}) )
are the vertices of the parallelogram.
206.
Given a triangle ABC with the unit area. The first player chooses a
point X on the side [AB], than the second  Y on [BC] side, and,
finally, the first chooses a point Z on [AC] side. The first tries to
obtain the greatest possible area of the XYZ triangle, the second 
the smallest.
What area can obtain the first for sure and how?
207.
What is the smallest perimeter of the convex 32angle, having all the
vertices in the nodes of crosslined paper with the sides of its
squares equal to 1?
208.
a)
Given a big square consisting of 7x7 squares. You should mark the
centres of k points in such a way, that no quadruple of the marked
points will be the vertices of a rectangle with the sides parallel
to the sides of the given squares.
What is the greatest k such that the problem has solution?
b)
The same problem for 13x13 square.
209.
Denote the middles of the convex hexagon
A_{1}A_{2}A_{3}A_{4}A_{5}A_{6} diagonals
A_{6}A_{2}, A_{1}A_{3}, A_{2}A_{4}, A_{3}A_{5}, A_{4}A_{6}, A_{5}A_{1} as B_{1}, B_{2}, B_{3}, B_{4},
B_{5}, B_{6} respectively.
Prove that if the hexagon B_{1}B_{2}B_{3}B_{4}B_{5}B_{6} is
convex, than its area equals to the quarter of the initial hexagon.
210.
Prove that it is possible to find 2^{n+1} of 2^{n} digit numbers
containing only "1" and "2" as digits, such that every two of them
distinguish at least in 2^{n1} digits.
211.
Given a finite set of polygons in the plane.
Every two of them have a common point.
Prove that there exists a straight line, that crosses all the polygons.
212.
Prove that for all the positive numbers a,b,c the following inequality
is valid:
a^{3}+b^{3}+c^{3}+3abc>ab(a+b)+bc(b+c)+ac(a+c).
213.
Three flies are crawling along the perimeter of the ABC triangle in
such a way, that the centre of their masses is a constant point. One
of the flies has already passed along all the perimeter.
Prove that the centre of the flies' masses coincides with the centre of
masses of the ABC triangle. (The centre of masses for the triangle is
the point of medians intersection.)
214.
Several zeros, ones and twos are written on the blackboard. An
anonymous clean in turn pairs of different numbers, writing, instead of
cleaned, the number not equal to each. (0 instead of pair {1,2}; 1
instead of {0,2}; 2 instead of {0,1}).
Prove that if there remains one
number only, it does not depend on the processing order.
215.
Given a horizontal strip on the plane (its sides are parallel lines)
and n lines intersecting the strip. Every two of them intersect inside
the strip, and not a triple has a common point. Consider all the paths
along the segments of those lines, starting on the lower side of the
strip and ending on the upper side with the properties: moving along
such a path we are constantly rising up, and, having reached the
intersection, we are obliged to turn to another line.
Prove that:
a)
there are not less than n/2 such a paths without common points;
b)
there is a path consisting of not less than of n segments;
c)
there is a path that goes along not more than along n/2+1 lines;
d)
there is a path that goes along all the n lines.
216.
For what k is it possible to construct a cube kxkxk of the black
and white cubes 1x1x1 in such a way that every small cube has
the same colour, that have exactly two his neighbours. (Two cubes
are neighbours, if they have the common face.)
217.
Given a polynomial P(x) with
a)
natural coefficients;
b)
integer coefficients;
Let us denote with a_{n} the sum of the digits of P(n) value.
Prove that there is a number encountered in the sequence
a_{1}, a_{2}, ... , a_{n}, ... infinite times.
218.
The world and the european champion are determined in the same
tournament carried in one round. There are 20 teams and k of them are
european. The european champion is determined according to the results
of the games only between those k teams.
What is the greatest k such
that the situation, when the single european champion is the single
world outsider, is possible if:
a)
it is hockey (draws allowed)?
b)
it is volleyball (no draws)?
219.
a)
Given real numbers a_{1},a_{2},b_{1},b_{2} and positive p_{1},p_{2},q_{1},q_{2}.
Prove that in the table 2x2
(a_{1} + b_{1})/(p_{1} + q_{1}) ,
(a_{1} + b_{2})/(p_{1} + q_{2})
(a_{2} + b_{1})/(p_{2} + q_{1}) ,
(a_{2} + b_{2})/(p_{2} + q_{2})
there is a number in the table, that is not less than another number
in the same row and is not greater than another number in the same
column (a saddle point).
b)
Given real numbers a_{1}, a_{2}, ... , a_{n}; b_{1}, b_{2}, ... , b_{n}
and positive p_{1}, p_{2}, ... , p_{n}; q_{1}, q_{2}, ... , q_{n}.
We build the table nxn, with the numbers (0 < i,j <= n)
(a_{i} + b_{j})/(p_{i} + q_{j})
in the intersection of the ith row and jth column.
Prove that there is a number in the table, that is not less than
arbitrary number in the same row and is not greater than arbitrary
number in the same column (a saddle point).
The 10th competition  Dushanbe, 1976.
form first day second day
8 220 221 222ab 223  229 230 231
9 222b 224 223 225  230 232 231
10 223 226 227 228 225  233 234 231
220.
There are 50 exact watches lying on a table.
Prove that there exist a
certain moment, when the sum of the distances from the centre of the
table to the ends of the minute hands is more than the sum of the
distances from the centre of the table to the centres of the watches.
221.
A row of 1000 numbers is written on the blackboard. We write a new
row, below the first according to the rule: We write under every number
a the natural number, indicating how many times the number a is
encountered in the first line. Then we write down the third line:
under every number b  the natural number, indicating how many times
the number b is encountered in the second line, and so on.
a)
Prove that there is a line that coincides with the preceding one.
b)
Prove that the eleventh line coincides with the twelfth.
c)
Give an example of the initial line such, that the tenth row differs
from the eleventh.
222.
Given three circumferences of the same radius in a plane.
a)
All three are crossing in one point K. Consider three arcs AK,CK,EK
: the A,C,E are the points of the circumferences intersection and
the arcs are taken in the clockwise direction. (Sorry, no picture.
Every arc is inside one circle, outside the second and on the border
of the third one)
Prove that the sum of the arcs is 180 degrees.
b)
Consider the case, when the three circles give a curvilinear
triangle BDF as there intersection (instead of one point K).
Prove that the sum of the AB, CD and EF arcs is 180 degrees.
(The arcs are taken in the clockwise direction. Every arc is inside
one circle, outside the second and on the border of the third one)
223.
The natural numbers x_{1} and x_{2} are less than 1000. We build a sequence:
x_{3} = x_{1}  x_{2};
x_{4} = min { x_{1}  x_{2}, x_{1}  x_{3}, x_{2}  x_{3}};
.......
x_{k} = min { x_{i}  x_{j}; 0 <i < j < k};
.......
Prove that x_{21} = 0.
224.
Can You mark the cube's vertices with the threedigit binary
numbers in such a way, that the numbers at all the possible
couples of neighbouring vertices differ in at least two digits?
225.
Given 4 vectors a,b,c,d in the plane, such that a+b+c+d=0.
Prove the following inequality:
a+b+c+d >= a+d+b+d+c+d.
226.
Given right 1976angle. The middles of all the sides and diagonals are
marked.
What is the greatest number of the marked points lying on one
circumference?
227.
There are n rectangles drawn on the rectangular sheet of paper with the
sides of the rectangles parallel to the sheet sides. The rectangles do
not have pairwise common interior points.
Prove that after cutting out
the rectangles the sheet will split into not more than n+1 part.
228.
There are three straight roads. Three pedestrians are moving
along those roads, and they are NOT on one line in the initial
moment.
Prove that they will be one line not more than twice.
229.
Given a chessboard 99x99 with a set F of fields marked on it (the set
is different in three tasks). There is a beetle sitting on every field
of the set F. Suddenly all the beetles have raised into the air and
flied to another fields of the same set. The beetles from the
neighbouring fields have landed either on the same field or on the
neighbouring ones (may be far from their starting point). (We consider
the fields to be neighbouring if they have at least one common vertex.)
Consider a statement:
"There is a beetle, that either stayed on the
same field or moved to the neighbouring one".
Is it always valid if
the figure F is:
a)
A central cross, i.e. the union of
the 50th row and the 50th column?
b)
A window frame, i.e. the union of
the 1st, 50th and 99th rows and
the 1st, 50th and 99th columns?
c)
All the chessboard?
230.
Let us call "big" a triangle with all sides longer than 1. Given a
equilateral triangle with all the sides equal to 5.
Prove that:
a)
You can cut 100 big triangles out of given one.
b)
You can divide the given triangle onto 100 big
nonintersecting ones fully covering the initial one.
c)
The same as b), but the triangles either do not have
common points, or have one common side, or one common vertex.
d)
The same as c), but the initial triangle has the side 3.
231.
Given natural n. We shall call "universal" such a sequence of natural
number a_{1}, a_{2}, ... , a_{k}; k>=n, if we can obtain every transposition
of the first n natural numbers (i.e such a sequence of n numbers, that
every one is encountered only once) by deleting some its members.
(Examples: (1,2,3,1,2,1,3) is universal for n=3, and (1,2,3,2,1,3,1) 
not, because You can't obtain (3,1,2) from it.)
The goal is to estimate
the length of the shortest universal sequence for given n.
a)
Give an example of the universal sequence of n^{2} members.
b)
Give an example of the universal sequence of (n^{2} n + 1) members.
c)
Prove that every universal sequence contains not less than
n(n + 1)/2 members
d)
Prove that the shortest universal sequence for n=4 contains
12 members
e)
Find as short universal sequence, as You can. The Organising
Committee knows the method for (n^{2}  2n +4) members.
232.
n numbers are written down along the circumference. Their
sum equals to zero, and one of them equals 1.
a)
Prove that there are two neighbours with their difference
not less than n/4.
b)
Prove that there is a number that differs from the arithmetic
mean of its two neighbours not less than on 8/{n^{2}}.
c)
Try to improve the previous estimation, i.e what number can be
used instead of 8?
d)
Prove that for n=30 there is a number that differs from the
arithmetic mean of its two neighbours not less than on 2/113;
give an example of such 30 numbers along the circumference,
that not a single number differs from the arithmetic mean of
its two neighbours more than on 2/113.
233.
Given right nangle wit the point O  its centre. All the vertices
are marked either with +1 or 1. We may change all the signs in the
vertices of right kangle (2 <= k <= n) with the same centre O.
(By 2angle we understand a segment, being halved by O.)
Prove that in a), b) and c) cases there exists such a set of
(+1)s and (1)s, that we can never obtain a set of (+1)s only.
a)
n = 15;
b)
n = 30;
c)
n > 2;
d)
Let us denote K(n) the maximal number of (+1) and (1) sets
such, that it is impossible to obtain one set from another.
Prove, for example, that K(200) = 2^{80}.
234.
Given a sphere of unit radius with the big circle (i.e of unit radius)
that will be called "equator". We shall use the words "pole",
"parallel","meridian" as selfexplanatory.
a)
Let g(x), where x is a point on the sphere, be the distance from
this point to the equator plane.
Prove that g(x) has the property
if x_{1}, x_{2}, x_{3} are the ends of the
pairwise orthogonal radiuses, than
g(x_{1})^{2} + g(x_{2})^{2} + g(x_{3})^{2} = 1. (*)
Let function f(x) be an arbitrary nonnegative
function on a sphere that satisfies (*) property.
b)
Let x_{1} and x_{2} points be on the same meridian between the
north pole and equator, and x_{1} is closer to the pole than x_{2}.
Prove that f(x_{1}) > f(x_{2}).
c)
Let y_{1} be closer to the pole than y_{2}.
Prove that f(y_{1}) > f(y_{2}).
d)
Let z_{1} and z_{2} be on the same parallel.
Prove that f(z_{1}) = f(z_{2}).
e)
Prove that for all x , f(x) = g(x).
The 11th competition  Tallinn, 1977.
form first day second day
8 235 236 237b 238  243 244ab 245 246
9 237a 239 235 240  247 248 249 250
10 237a 239 241 242 235  251 244 246
235.
Given a closed broken line without selfintersections in a plane. Not
a triple of its vertices belongs to one straight line. Let us call
"special" a couple of line's segments if the one's continuation
intersects another.
Prove that there is even number of special pairs.
236.
Given several points, not all lying on one straight line. Some number
is assigned to every point. It is known, that if a straight line
contains two or more points, than the sum of the assigned to those
points equals zero.
Prove that all the numbers equal to zero.
237.
a)
Given a circle with two inscribed triangles T_{1} and T_{2}. The
vertices of T_{1} are the middles of the arcs with the ends in the
vertices of T_{2}. Consider a hexagon  the intersection of T_{1} and
T_{2}.
Prove that its main diagonals are parallel to T_{1} sides and
are intersecting in one point.
b)
The segment, that connects the middles of the arcs AB and AC of the
circle outscribed around the ABC triangle, intersects [AB] and [AC]
sides in D and K points.
Prove that the points A,D,K and O  the
centre of the circle  are the vertices of a diamond.
238.
Several black an white checkers (tokens?) are standing along the
circumference. Two men remove checkers in turn. The first removes
all the black ones that had at least one white neighbour, and the
second  all the white ones that had at least one black neighbour.
They stop when all the checkers are of the same colour.
a)
Let there be 40 checkers initially. Is it possible that after
two moves of each man there will remain only one (checker)?
b)
Let there be 1000 checkers initially. What is the minimal possible
number of moves to reach the position when there will remain only
one (checker)?
239.
Given infinite sequence a_{n}. It is known that the limit of
b_{n}=a_{n+1}a_{n}/2 equals zero.
Prove that the limit of a_{n} equals zero.
240.
There are direct routes from every city of a certain country to every
other city. The prices are known in advance. Two tourists (they do not
necessary start from one city) have decided to visit all the cities,
using only direct travel lines. The first always chooses the cheapest
ticket to the city, he has never been before (if there are several 
he chooses arbitrary destination among the cheapests). The second 
the most expensive (they do not return to the first city).
Prove that
the first will spend not more money for the tickets, than the second.
/* The fact seems to be evident, but the proof is not easy  VAP */
241.
Every vertex of a convex polyhedron belongs to three edges.
It is possible to outscribe a circle around all its faces.
Prove that the polyhedron can be inscribed in a sphere.
242.
The polynomial x^{10} + ?x^{9} + ?x^{8} + ... + ?x + 1 is written on the
blackboard. Two players substitute (real) numbers instead of one of the
question marks in turn. (9 turns total.) The first wins if the
polynomial will have no real roots.
Who wins?
243.
Seven elves are sitting at a round table. Each elf has a cup. Some
cups are filled with some milk. Each elf in turn and clockwise divides
all his milk between six other cups. After the seventh has done this,
every cup was containing the initial amount of milk.
How much milk did
every cup contain, if there was three litres of milk total?
244.
Let us call "fine" the 2ndigit number if it is exact square
itself and the two numbers represented by its first n digits
(first digit may not be zero) and last n digits (first digit may
be zero, but it may not be zero itself) are exact squares also.
a)
Find all two and fourdigit fine numbers.
b)
Is there any sixdigit fine number?
c)
Prove that there exists 20digit fine number.
d)
Prove that there exist at least ten 100digit fine numbers.
e)
Prove that there exists 30digit fine number.
245.
Given a set of n positive numbers. For each its nonempty subset
consider the sum of all the subset's numbers.
Prove that You can divide those sums onto n groups in such a way, that
the least sum in every group is not less than a half of the greatest
sum in the same group.
246.
There are 1000 tickets with the numbers 000, 001, ... , 999; and 100
boxes with the numbers 00, 01, ... , 99. You may put a ticket in a box,
if You can obtain the box number from the ticket number by deleting one
digit.
Prove that:
a)
You can put all the tickets in 50 boxes;
b)
40 boxes is not enough for that;
c)
it is impossible to use less than 50 boxes.
d)
Consider 10000 4digit tickets, and You are allowed to delete two
digits. Prove that 34 boxes is enough for storing all the tickets.
e)
What is the minimal used boxes set in the case of kdigit tickets?
247.
Given a square 100x100 on the sheet of crosslined paper. There are
several broken lines drawn inside the square. Their links consist of
the small squares sides. They are neither pairwise nor
selfintersecting (have no common points). Their ends are on the big
square boarder, and all the other vertices are in the big square
interior.
Prove that there exists (in addition to four big square
angles) a node (corresponding to the crosslining family, inside the
big square or on its side) that does not belong to any broken line.
248.
Given natural numbers x_{1},x_{2},...,x_{n};y_{1},y_{2},...,y_{m}.
The following condition is valid:
(x_{1}+x_{2}+...+x_{n})=(y_{1}+y_{2}+...+y_{m})<mn. (*)
Prove that it is possible to delete some terms from (*) (not
all and at least one) and to obtain another valid condition.
249.
Given 1000 squares on the plane with their sides parallel to the
coordinate axes. Let M be the set of those squares centres.
Prove that You can mark some squares in such a way, that every point of
M will be contained not less than in one and not more than in four
marked squares.
250.
Given scales and a set of n different weights. We take weights in turn
and add them on one of the scales sides. Let us denote "L" the scales
state with the left side down, and "R"  with the right side down.
a)
Prove that You can arrange the weights in such an order,
that we shall obtain the sequence LRLRLRLR... of the scales
states. (That means that the state of the scales will be
changed after putting every new weight.)
b)
Prove that for every nletter word containing R's and L's
only You can arrange the weights in such an order, that the
sequence of the scales states will be described by that word.
251.
Let us consider one variable polynomials with the senior
coefficient equal to one. We shall say that two polynomials
P(x) and Q(x) commute, if P(Q(x))=Q(P(x)) (i.e. we obtain
the same polynomial, having collected the similar terms).
a)
For every a find all Q such that the Q degree is not
greater than three, and Q commutes with (x^{2}  a).
b)
Let P be a square polynomial, and k is a natural number. Prove that
there is not more than one commuting with P kdegree polynomial.
c)
Find the 4degree and 8degree polynomials
commuting with the given square polynomial P.
d)
R and Q commute with the same square polynomial P.
Prove that Q and R commute.
e)
Prove that there exists a sequence P_{2}, P_{3}, ... , P_{n}, ... (P_{k} is
kdegree polynomial), such that P_{2}(x) = x^{2}  2, and all the
polynomials in this infinite sequence pairwise commute.
The 12th competition  Tashkent, 1978.
form first day second day
8 252 253 254 255ab  260 261 262 263
9 252 253 256 257  260 261 264 265
10 258 259 255cde257  260 266 267 268
252.
Let a_{n} be the closest to sqrt(n) integer.
Find the sum 1/a_{1} + 1/a_{2} + ... + 1/a_{1980}.
253.
Given a quadrangle ABCD and a point M inside it such that ABMD is a
parallelogram. the angle CBM equals to CDM.
Prove that the angle ACD equals to BCM.
254.
Prove that there is no m such that (1978^{m}  1) is divisible
by (1000^{m}  1).
255.
Given a finite set K_{0} of points (in the plane or space). The sequence
of sets K_{1}, K_{2}, ... , K_{n}, ... is build according to the rule: we
take all the points of K_{i}, add all the symmetric points with respect
to all its points, and, thus obtain K_{i+1}.
a)
Let K_{0} consist of two points A and B with the distance 1 unit
between them. For what n the set K_{n} contains the point that is
1000 units far from A?
b)
Let K_{0} consist of three points that are the vertices of the
equilateral triangle with the unit square. Find the area of minimal
convex polygon containing K_{n}.
K_{0} below is the set of the unit volume tetrahedron vertices.
c)
How many faces contain the minimal convex polyhedron containing K_{1}?
d)
What is the volume of the abovementioned polyhedron?
e)
What is the volume of the minimal convex polyhedron containing K_{n}?
256.
Given two heaps of checkers. the bigger contains m checkers, the
smaller  n (m>n). Two players are taking checkers in turn from the
arbitrary heap. The players are allowed to take from the heap a number
of checkers (not zero) divisible by the number of checkers in another
heap. The player that takes the last checker in any heap wins.
a)
Prove that if m > 2n, than the first can always win.
b)
Find all x such that if m > xn, than the first can always win.
257.
Prove that there exists such an infinite sequence {x_{i}}, that for
all m and all k (m<>k) holds the inequality x_{m}x_{k}>1/mk.
258.
Let f(x) = x^{2} + x + 1.
Prove that for every natural m>1 the numbers
m, f(m), f(f(m)), ... are relatively prime.
259.
Prove that there exists such a number A that You can inscribe 1978
different size squares in the plot of the function y = A sin(x). (The
square is inscribed if all its vertices belong to the plot.)
260.
Given three automates that deal with the cards with the pairs of
natural numbers. The first, having got the card with (a,b), produces
new card with (a+1,b+1); the second, having got the card with (a,b),
produces new card with (a/2,b/2), if both a and b are even and nothing
in the opposite case; the third, having got the pair of cards with
(a,b) and (b,c) produces new card with (a,c). All the automates return
the initial cards also. Suppose there was (5,19) card initially. Is it
possible to obtain
a)
(1,50)?
b)
(1,100)?
c)
Suppose there was (a,b) card initially (a<b). We want to obtain
(1,n) card. For what n is it possible?
261.
Given a circle with radius R and inscribed nangle with area S.
We mark one point on every side of the given polygon.
Prove that the perimeter of the polygon with the vertices in the
marked points is not less than 2S/R.
262.
The checker is standing on the corner field of a nxn chessboard.
Each of two players moves it in turn to the neighbour (i.e. that
has the common side) field. It is forbidden to move to the field,
the checker has already visited. That who cannot make a move losts.
a)
Prove that for even n the first can always win, and if n is odd,
than the second can always win.
b)
Who wins if the checker stands initially on the neighbour to the
corner field?
263.
Given n nonintersecting segments in the plane. Not a pair of those
belong to the same straight line. We want to add several segments,
connecting the ends of given ones, to obtain one nonselfintersecting
broken line. Is it always possible?
264.
Given 0 < a <= x_{1} <= x_{2} <= ... <= x_{n} <= b.
Prove that
(x_{1}+x_{2}+...+x_{n})(1/x_{1}+1/x_{2}+...+1/x_{n})<=((a+b)^{2}/4ab)n^{2}.
265.
Given a simple number p>3. Consider the set M of the pairs (x,y) with
the integer coordinates in the plane such that 0 <= x < p; 0 <= y < p.
Prove that it is possible to mark p points of M such that not a triple
of marked points will belong to one line and there will be no
parallelogram with the vertices in the marked points.
266.
Prove that for every tetrahedron there exist two planes such that the
projection areas on those planes relation is not less than sqrt(2).
267.
Given a_{1}, a_{2}, ... , a_{n}. Define b_{k} = (a_{1} + a_{2} + ... + a_{k})/k
for 1 <= k <= n. Let
C = (a_{1}  b_{1})^{2} + (a_{2}  b_{2})^{2} + ... + (a_{n}  b_{n})^{2};
D = (a_{1}  b_{n})^{2} + (a_{2}  b_{n})^{2} + ... + (a_{n}  b_{n})^{2}.
Prove that C <= <D <= 2C.
268.
Consider a sequence x_{n}=(1+sqrt(2)+sqrt(3))^{n}.
Each member can be represented as
x_{n}=q_{n}+r_{n}sqrt(2)+s_{n}sqrt(3)+t_{n}sqrt(6),
where q_{n}, r_{n}, s_{n}, t_{n} are integers.
Find the limits of the fractions r_{n}/q_{n},
s_{n}/q_{n}, t_{n}/q_{n}.
The 13th competition  Tbilisi, 1979.
form first day second day
8 269 270 271  274 275 276 277
9 269 272 271  278 279 280 281
10 273 272 271  276 275 282 283
269.
What is the least possible relation of two isosceles triangles areas,
if three vertices of the first one belong to three different sides of
the second one?
270.
A grasshopper is hopping in the angle x>=0, y>=0 of the coordinate
plane (that means that it cannot land in the point with negative
coordinate). If it is in the point (x,y), it can either jump to the
point (x+1,y1), or to the point (x5,y+7).
Draw a set of such an initial points (x,y), that having started from
there, a grasshopper cannot reach any point farther than 1000 from
the point (0,0). Find its area.
271.
Every member of a certain parliament has not more than 3 enemies.
Prove that it is possible to divide it onto two subparliaments so, that
everyone will have not more than one enemy in his subparliament. (A is
the enemy of B if and only if B is the enemy of A.)
272.
Some numbers are written in the notebook. We can add to that list the
arithmetic mean of some of them, if it doesn't equal to the number,
already having been included in it. Let us start with two numbers, 0
and 1.
Prove that it is possible to obtain
a)
1/5;
b)
an arbitrary rational number between 0 and 1.
273.
For every n the decreasing sequence {x_{k}} satisfies a condition
x_{1}+x_{4}/2+x_{9}/3+...+x_{n2}/n<=1.
Prove that for every n it also satisfies
x_{1}+x_{2}/2+x_{3}/3+...+x_{n}/n<=3.
274.
Given some points in the plane. For some pairs A,B the vector AB is
chosen. For every point the number of the chosen vectors starting in
that point equal to the number of the chosen vectors ending in that
point.
Prove that the sum of the chosen vectors equals to zero vector.
275.
What is the least possible number of the checkers being required
a)
for the 8x8 chessboard;
b)
for the nxn chessboard;
to provide the property: Every line (of the chessboard fields)
parallel to the side or diagonal is occupied by at least one checker?
276.
Find x and y (a and b parameters):
(xy*sqrt(x^{2}y^{2}))/(sqrt(1x^{2}+y^{2})) = a;
(yx*sqrt(x^{2}y^{2}))/(sqrt(1x^{2}+y^{2})) = b.
277.
Given some square carpets with the total area 4.
Prove that they can fully cover the unit square.
278.
Prove that for the arbitrary numbers
x_{1}, x_{2}, ... , x_{n} from the [0,1] segment
(x_{1} + x_{2} + ... + x_{n} + 1)^{2} >=
4(x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2}).
279.
Natural p and q are relatively prime. The [0,1] is divided onto
(p+q) equal segments.
Prove that every segment except two marginal contain exactly one from
the (p+q2) numbers {1/p, 2/p, ... , (p1)/p, 1/q, 2/q, ... , (q1q)}.
280.
Given the point O in the space and 1979 straight lines l_{1}, l_{2}, ... ,
l_{1979} containing it. Not a pair of lines is orthogonal.
Given a point A_{1} on l_{1} that doesn't coincide with O.
Prove that it is possible to choose the points A_{i} on l_{i} (i = 2, 3,
... , 1979) in so that 1979 pairs will be orthogonal:
A_{1}A_{3} and l_{2};
A_{2}A_{4} and l_{3};
..........
A_{i1}A_{i+1} and l_{i};
..........
A_{1977}A_{1979} and l_{1978};
A_{1978}A_{1} and l_{1979};
A_{1979}A_{2} and l_{1}
281.
The finite sequence a_{1}, a_{2}, ... , a_{n} of ones and zeroes should
satisfy a condition:
for every k from 0 to (n1) the sum
a_{1}a_{k+1} + a_{2}a_{k+2} + ... + a_{nk}a_{n}
should be odd.
a)
Build such a sequence for n=25.
b)
Prove that there exists such a sequence for some n > 1000.
282.
The convex quadrangle is divided by its diagonals onto four
triangles. The circles inscribed in those triangles are equal.
Prove that the given quadrangle is a diamond.
283.
Given n points (in sequence) A_{1}, A_{2}, ... , A_{n} on a line. All the
segments A_{1}A_{2}, A_{2},A_{3}, ... A_{n1}A_{n} are shorter than 1. We need to
mark (k1) points so that the difference of every two segments, with
the ends in the marked points, is shorter than 1.
Prove that it is possible
a)
for k=3;
b)
for every k less than (n1).
The 14th competition  Saratov, 1980.
form first day second day
8 284 285 286 287  293 294 295 296
9 288 289 286 290  295 297 298 299
10 291 289 292 290  300 301 302 303
284.
All the twodigit numbers from 19 to 80 are written in a line without spaces.
Is the obtained number (192021....7980) divisible by 1980?
285.
The vertical side of a square is divided onto n segments. The sum of
the segments with even numbers lengths equals to the sum of the
segments with odd numbers lengths. (n1) lines parallel to the
horizontal sides are drawn from the segments ends, and, thus, n strips
are obtained. The diagonal is drawn from the lower left corner to the
upper right one. This diagonal divides every strip onto left and right
parts.
Prove that the sum of the left parts of odd strips areas equals
to the sum of the right parts of even strips areas.
286.
The load for the space station "Salute" is packed in containers. There
are more than 35 containers, and the total weight is 18 metric tons.
There are 7 oneway transport spaceships "Progress", each able to bring
3 metric tons to the station. It is known that they are able to take an
arbitrary subset of 35 containers.
Prove that they are able to take all the load.
287.
The points M and P are the middles of [BC] and [CD] sides of a convex
quadrangle ABCD. It is known that AM + AP = a.
Prove that the ABCD area is less than {a^{2}}/2.
288.
Are there three simple numbers x,y,z, such that x^{2} + y^{3} = z^{4}?
289.
Given a point E on the diameter AC of the certain circle.
Draw a chord BD to maximise the area of the quadrangle ABCD.
290.
There are several settlements on the bank of the Big Round Lake. Some
of them are connected with the regular direct ship lines. Two
settlements are connected if and only if two next (counterclockwise) to
each ones are not connected.
Prove that You can move from the arbitrary settlement to another
arbitrary settlement, having used not more than three ships.
291.
The sixdigit decimal number contains six different nonzero digits and
is divisible by 37.
Prove that having transposed its digits You can
obtain at least 23 more numbers divisible by 37.
292.
Find real solutions of the system
sin x + 2 sin(x+y+z) = 0,
sin y + 3 sin(x+y+z) = 0,
sin z + 4 sin(x+y+z) = 0.
293.
Given 1980 vectors in the plane, and there are some noncollinear among
them. The sum of every 1979 vectors is collinear to the vector not
included in that sum.
Prove that the sum of all vectors equals to the zero vector.
294.
Let us denote with S(n) the sum of all the digits of n.
a)
Is there such an n that n+S(n)=1980?
b)
Prove that at least one of two arbitrary successive natural
numbers is representable as n + S(n) for some third number n.
295.
Some squares of the infinite sheet of crosslined paper are red.
Each 2x3 rectangle (of 6 squares) contains exactly two red squares.
How many red squares can be in the 9x11 rectangle of 99 squares?
296.
An epidemic influenza broke out in the elves city. First day some of
them were infected by the external source of infection and nobody later
was infected by the external source. The elf is infected when visiting
his ill friend. In spite of the situation every healthy elf visits all
his ill friends every day. The elf is ill one day exactly, and has the
immunity at least on the next day. There is no graftings in the city.
Prove that
a)
If there were some elves immunised by the external source on
the first day, the epidemic influenza can continue arbitrary
long time.
b)
If nobody had the immunity on the first day, the epidemic influenza
will stop some day.
297.
Let us denote with P(n) the product of all the digits of n. Consider
the sequence n_{k+1} = n_{k} + P(n_{k}).
Can it be unbounded for some n_{1}?
298.
Given equilateral triangle ABC. Some line, parallel to [AC] crosses
[AB] and [BC] in M and P points respectively. Let D be the centre of
PMB triangle, E  the middle of the [AP] segment.
Find the angles of DEC triangle.
299.
Let the edges of rectangular parallelepiped be x,y and z (x<y<z).
Let p=4(x+y+z), s=2(xy+yz+zx) and
d=sqrt(x^{2}+y^{2}+z^{2}) be its perimeter,
surface area and diagonal length, respectively.
Prove that
x < 1/3(p/4  sqrt(d^{2}  s/2)),
z > 1/3(p/4 + sqrt(d^{2}  s/2)).
300.
The A set consists of integers only. Its minimal element is 1
and its maximal element is 100. Every element of A except 1 equals
to the sum of two (may be equal) numbers being contained in A.
What is the least possible number of elements in A?
301.
Prove that there is an infinite number of such numbers B that the
equation [x^{3/2}] + [y^{3/2}] = B has at least 1980 integer
solutions (x,y). ([z] denotes the greatest integer not exceeding z.)
302.
The edge [AC] of the tetrahedron ABCD is orthogonal to [BC], and [AD]
is orthogonal to [BD].
Prove that the cosine of the angle between (AC)
and (BD) lines is less than CD/AB.
303.
The number x from [0,1] is written as an infinite decimal fraction.
Having rearranged its first five digits after the point we can
obtain another fraction that corresponds to the number x_{1}.
Having rearranged five digits of x_{k} from (k+1)th till (k+5)th
after the point we obtain the number x_{k+1}.
a)
Prove that the sequence x_{i} has limit.
b)
Can this limit be irrational if we have started with the
rational number?
c)
Invent such a number, that always produces irrational numbers,
no matter what digits were transposed.
The 15th competition  AlmaAta, 1981.
form first day second day
8 304 305 306 307  315 316 317 318
9 308 309 310 311  319 320 321 322
10 311 312 312 314  323 324 325 326
304.
Two equal chessboards (8x8) have the same centre, but one is rotated
by 45 degrees with respect to another.
Find the total area of black
fields intersection, if the fields have unit length sides.
305.
Given points A,B,M,N on the circumference. Two chords [MA_{1}] and [MA_{2}]
are orthogonal to (NA) and (NB) lines respectively.
Prove that (AA_{1}) and (BB_{1}) lines are parallel.
306.
Let us say, that a natural number has the property P(k) if it can be
represented as a product of k succeeding natural numbers greater than 1.
a)
Find k such that there exists n which has properties
P(k) and P(k+2) simultaneously.
b)
Prove that there is no number having properties
P(2) and P(4) simultaneously.
307.
The rectangular table has four rows. The first one contains arbitrary
natural numbers (some of them may be equal). The consecutive lines are
filled according to the rule: we look through the previous row from
left to the certain number n and write the number k if n was met k
times.
Prove that the second row coincides with the fourth one.
308.
Given real a. Find the least possible area of the rectangle
with the sides parallel to the coordinate axes and containing
the figure determined by the system of inequalities
y <= x^{2},
y >= x^{2}  2x + a.
/* as there is no derivatives in the school program of the 9th
form, the participants had to use parabola properties VAP */
309.
Three equilateral triangles ABC, CDE, EHK (the vertices are mentioned
counterclockwise) are lying in the plane so, that the vectors [AD[ and
[DK[ are equal.
Prove that the triangle BHD is also equilateral.
310.
There are 1000 inhabitants in a settlement. Every evening every
inhabitant tells all his friends all the news he had heard the previous
day. Every news becomes finally known to every inhabitant.
Prove that
it is possible to choose 90 of inhabitants so, that if You tell them a
news simultaneously, it will be known to everybody in 10 days.
311.
It is known about real a and b that the inequality
a cos(x) + b cos(3x) > 1 has no real solutions.
Prove that b <=1.
312.
The points K and M are the centres of the AB and CD sides of the convex
quadrangle ABCD. The points L and M belong to two other sides and KLMN
is a rectangle.
Prove that KLMN area is a half of ABCD area.
313.
Find all the sequences of natural k_{n} with two properties:

for all n k_{n} <= n sqrt(n);

for all m>n (k_{n}  k_{m}) is divisible by (mn).
314.
Is it possible to fill a rectangular table with black and white
squares (only) so, that the number of black squares will equal
to the number of white squares, and each row and each column
will have more than 75% squares of the same colour?
315.
The quadrangles AMBE, AHBT, BKXM, and CKXP are parallelograms.
Prove that the quadrangle ABTE is also parallelogram.
(the vertices are mentioned counterclockwise)
316.
Find the natural solutions of the equation x^{3}  y^{3} = xy + 61.
317.
Eighteen soccer teams have played 8 tours of a oneround tournament.
Prove that there is a triple of teams, having not met each other yet.
318.
The points C_{1}, A_{1}, B_{1} belong to [AB], [BC], [CA] sides, respectively,
of the ABC triangle.
AC_{1}/C_{1}B =
BA_{1}/A_{1}C =
CB_{1}/B_{1}A = 1/3.
Prove that the perimeter P of the ABC triangle and the perimeter p
of the A_{1}B_{1}C_{1} triangle satisfy inequality
P/2 < p < 3P/4.
319.
Positive numbers x,y satisfy equality x^{3}+y^{3}=xy.
Prove that x^{2}+y^{2}<1.
320.
A pupil has tried to make a copy of a convex polygon, drawn inside
the unit circle. He draw one side, from its end  another, and so on.
Having finished, he has noticed that the first and the last vertices
do not coincide, but are situated d units of length far from each other.
The pupil draw angles precisely, but made relative error less than p
in the lengths of sides.
Prove that d < 4p.
321.
A number is written in the each vertex of a cube. It is allowed to add
one to two numbers written in the ends of one edge.
Is it possible to obtain the cube with all equal numbers if the numbers
were initially as on the pictures:
00 74 01
/ / / / / /
00  65  10 
           
 01  23  00
/ / / / / /
00 14 00
fig.a fig.b fig.c
322.
Find n such that each of the numbers n,(n+1),...,(n+20) has the common
divider greater than one with the number 30030 = 2*3*5*7*11*13.
323.
The natural numbers from 100 to 999 are written on separate cards. They
are gathered in one pile with their numbers down in arbitrary order.
Let us open them in sequence and divide into 10 piles according to the
least significant digit. The first pile will contain cards with 0 at
the end, ... , the tenth  with 9. Then we shall gather 10 piles in
one pile, the first  down, then the second, ... and the tenth  up.
Let us repeat the procedure twice more, but the next time we shall
divide cards according to the second digit, and the last time  to the
most significant one.
What will be the order of the cards in the obtained pile?
324.
Six points are marked inside the 3x4 rectangle.
Prove that there is a pair of marked points with the
distance between them not greater than sqrt(5).
325.
a)
Find the minimal value of the polynomial
P(x,y) = 4 + (x^{2})(y^{4}) + (x^{4})(y^{2})  3(x^{2})(y^{2})
/* the usage of derivatives is forbidden, as they were
not included in the school programs  VAP */
b)
Prove that it cannot be represented as a sum of the squares
of some polynomials of x,y.
326.
The segments [AD], [BE] and [CF] are the side edges of the _right_
triangle prism. (I mean, that the equilateral triangle is a base  VAP)
Find all the points in its base ABC, situated on the equal
distances from the (AE), (BF) and (CD) lines.
The 16th competition  Odessa, 1982.
form first day second day
8 327 328 329a 330  337 338 339 340
9 331 332 333 334  341 342 343 344
10 335 332 329b 336  345 346 347 348
327.
Given two points M and K on the circumference with radius r_{1} and
centre O_{1}. The circumference with radius r_{2} and centre O_{2} is
inscribed in MO_{1}K angle.
Find the MO_{1}KO_{2} quadrangle area.
328.
Every member, starting from the third one, of two sequences {a_{n}} and
{b_{n}} equals to the sum of two preceding ones. First members are: a_{1}
= 1, a_{2} = 2, b_{1} = 2, b_{2} = 1.
How many natural numbers are
encountered in both sequences (may be on the different places)?
329.
a)
Let m and n be natural numbers. For some nonnegative integers k_{1},
k_{2}, ... , k_{n} the number 2^{k1}+2^{k2}+...+2^{kn} is
divisible by (2^{m}1).
Prove that n >= m.
b)
Can You find a number, divisible by 111...1 (m times "1"),
that has the sum of its digits less than m?
330.
A nonnegative real number is written at every cube's vertex. The sum
of those numbers equals to 1. Two players choose in turn faces of the
cube, but they cannot choose the face parallel to already chosen one
(the first moves twice, the second  once).
Prove that the first player can provide the number, at the common for
three chosen faces vertex, to be not greater than 1/6.
331.
Once upon a time, three boys visited a library for the first time.
The first decided to visit the library every second day.
The second decided to visit the library every third day.
The third decided to visit the library every fourth day.
The librarian noticed, that the library doesn't work on Wednesdays.
The boys decided to visit library on Thursdays, if they have to
do it on Wednesdays, but to restart the day counting in these cases.
They strictly obeyed these rules.
Some Monday later I met them
all in that library.
What day of week was when they visited a library for the first time?
332.
The parallelogram ABCD isn't a diamond. The relation of the diagonal
lengths AC/BD equals to k. The [AM) ray is symmetric to the [AD)
ray with respect to the (AC) line. The [BM) ray is symmetric to the
[BC) ray with respect to the (BD) line. (M point is those rays
intersection.)
Find the AM/BM relation.
333.
3k points are marked on the circumference. They divide it onto 3k
arcs. Some k of them have length 1, other k of them have length 2,
the rest k of them have length 3.
Prove that some two of the marked points are the ends of one diameter.
334.
Given a point M inside a right tetrahedron.
Prove that at least one
tetrahedron edge is seen from the M in an angle, that has a cosine not
greater than 1/3. /* I mean that if A and B are the vertices,
corresponding to that edge, cos(AMB) <= 1/3 */
335.
Three numbers a,b,c belong to ]0,pi/2[ interval.
cos(a) = a; sin(cos(b)) = b; cos(sin(c)) = c.
Sort those numbers in increasing order.
336.
The closed broken line M has odd number of vertices  A_{1},A_{2},...,
A_{2n+1} in sequence. Let us denote with S(M) a new closed broken line
with vertices B_{1},B_{2},...,B_{2n+1}  the middles of the first line
links: B_{1} is the middle of [A_{1}A_{2}], ... , B_{2n+1}  of
[A_{2n+1}A_{1}].
Prove that in a sequence M_{1}=S(M), ... , M_{k} =
S(M_{k1}), ... there is a broken line, homothetic to the M.
337.
All the natural numbers from 1 to 1982 are gathered in an array in an
arbitrary order in computer's memory. The program looks through all the
sequent pairs (first and second, second and third,...) and exchanges
numbers in the pair, if the number on the lower place is greater than
another. Then the program repeats the process, but moves from another
end of the array. The number, that stand initially on the 100th place
reserved its place. Find that number.
338.
Cucumber river in the Flower city /* authors mean the N.Nosov's tale
about the sort of elves country */ has parallel banks with the distance
between them 1 metre. It has some islands with the total perimeter 8
metres. Mr. KnowAll claims that it is possible to cross the river in a
boat from the arbitrary point, and the trajectory will not exceed 3
metres. Is he right?
339.
There is a parabola y = x^{2} drawn on the coordinate plane. The axes
are deleted. Can You restore them with the help of compass and ruler?
340.
The square table nxn is filled by integers. If the fields have
common side, the difference of numbers in them doesn't exceed 1.
Prove that some number is encountered not less than
a)
not less than [n/2] times ([] mean the whole part);
b)
not less than n times.
341.
Prove that the following inequality is valid for the positive x:
2^{(x1/12)} + 2^{(x1/4)} >= 2^{(1 + x1/6)} .
342.
What minimal number of numbers from the set {1,2,...,1982}
should be deleted to provide the property: none of the remained
numbers equals to the product of two other remained numbers?
/* This particular problem sounds bad due to the translator,
but I had read the Moscow competition organising committee
announcement with the word "prize" 7 times repeated in
one sentence  VAP */
343.
Every square on the infinite sheet of crosslined paper contains
some real number.
Prove that some square contains a number that
does not exceed at least four of eight neighbouring numbers.
344.
Given a sequence of real numbers a_{1}, a_{2}, ... , a_{n}. Prove that
it is possible to choose some of the numbers providing 3 conditions:

a)
not a triple of successive members is chosen;

b)
at least one of every triple of successive members is chosen;

c)
the absolute value of chosen numbers sum is not less that one
sixth part of the initial numbers' absolute values sum.
345.
Given the square table nxn with (n1) marked fields.
Prove that it is possible to move all the marked fields below
the diagonal by moving rows and columns.
346.
Prove that the following inequality holds for all real a and natural n:
a*a1*a2*...*an >= F(a)*n!/(2^{n}),
F(a) is the distance from a to the closest integer.
347.
Can You find three polynomials P,Q,R of three variables x,y,z,
providing the condition:
a)
P(xy+z)^{3} + Q(yz1)^{3} +R(z2x+1)^{3} = 1;
b)
P(xy+z)^{3} + Q(yz1)^{3} +R(zx+1)^{3} = 1;
for all x,y,z?
348.
The KLMN tetrahedron (triangle pyramid) vertices are situated inside or
on the faces or on the edges of the ABCD tetrahedron.
Prove that KLMN
perimeter is less than 4/3 ABCD perimeter.
The 17th competition  Kishenew, 1983.
form first day second day
8 349 350 351 352  360 361 362 363
9 353 354 355 356  364 365 366 367
10 357 354 358 359  360 368 369 370
349.
+++++
    
+++++
    
+++++
    
+++++
    
+++++
Every cell of the net, drawn on the
picture has 1x1 size. Is it possible
to represent this net as a union of
the following sets:
a)
Eight broken lines of length five each?
b)
Five broken lines of length eight each?
350.
Three numbers were written with a chalk on the blackboard. The
following operation was repeated several times: One of the numbers was
cleared and the sum of two other numbers, decreased by 1, was written
instead of it. The final set of numbers is {17, 1967, 1983} /* In some
organisers opinion "17" is the most distinguishing number, and those
three numbers were met on the picturesque jubilee slogans. In fact, in
1967 the competition was just renamed VAP */. Is it possible to admit
that the initial numbers were
a)
{2, 2, 2}?
b)
{3, 3, 3}?
351.
Three disks touch pairwise from outside in the points X,Y,Z. Then the
radiuses of the disks were expanded by 2/sqrt(3) times, and the centres
were reserved.
Prove that the XYZ triangle is completely covered by
the expanded disks.
353.
Find all the solutions of the system
y^{2} = x^{3}  3x^{2} + 2x,
x^{2} = y^{3}  3y^{2} + 2y.
354.
Natural number k has n digits in its decimal notation. It was rounded
up to tens, then the obtained number was rounded up to hundreds, and so
on (n1) times.
Prove that The obtained number m satisfies inequality
m < 18*k/13.
(Examples of rounding: 191>190>200; 135>140>100.)
355.
The point D is the middle of the [AB] side of the ABC triangle. The
points E and F belong to [AC] and [BC] respectively.
Prove that the DEF triangle area does not exceed the sum of the ADE and
BDF triangles areas.
356.
The sequences a_{n} and b_{n} members are the last digits of
[(sqrt(10))^{n}] and [(sqrt(2))^{n}] respectively (here [] denotes
the whole part of a number). Are those sequences periodical?
357.
Two acute angles a and b satisfy condition
(sin(a))^{2} + (sin(b))^{2} = sin(a+b).
Prove that a + b = pi/2.
358.
The points A_{1},B_{1},C_{1},D_{1} and A_{2},B_{2},C_{2},D_{2} are orthogonal
projections of the ABCD tetrahedron vertices on two planes.
Prove that it is possible to move one of the planes to provide
the parallelness of (A_{1}A_{2}), (B_{1}B_{2}), (C_{1}C_{2}) and (D_{1}D_{2}) lines.
359.
The pupil is training in the square equation solution. Having the
recurrent equation solved, he stops, if it doesn't have two roots, or
solves the next equation, with the free coefficient equal to the
greatest root, the coefficient at x equal to the least root, and the
coefficient at x^{2} equal to 1.
Prove that the process cannot be
infinite. What maximal number of the equations he will have to solve?
360.
Given natural n,m,k. It is known that m^{n} is divisible by n^{m}; and n^{k}
is divisible by k^{n}.
Prove that m^{k} is divisible by k^{m}.
361.
The Abba tribe language alphabet contains two letters only.
Not a word of this language is a beginning of another word.
Can this tribe vocabulary contain 3 fourletter, 10 fiveletter,
30 sixletter and 5 sevenletter words?
362.
Can You fill the squares of the infinite crosslined paper with
integers so, that the sum of the numbers in every 4x6 fields rectangle
would be
a)
10?
b)
1?
363.
The points A_{1},B_{1},C_{1} belong to [BC],[CA],[AB] sides of the ABC
triangle respectively. The [AA_{1}], [BB_{1}], [CC_{1}] segments
split the ABC onto 4 smaller triangles and 3 quadrangles.
It is known, that the smaller triangles have the same area.
Prove that the quadrangles have equal areas. What is the
quadrangle area, it the small triangle has the unit area?
364.
The kindergarten group is standing in the column of pairs. The number
of boys equals the number of girls in each of the two columns. The
number of mixed (boy and girl) pairs equals to the number of the rest
pairs.
Prove that the total number of children in the group is
divisible by eight.
365.
One side of the rectangle is 1cm. It is known that the rectangle can be
divided by two orthogonal lines onto four rectangles, and each of the
smaller rectangles has the area not less than 1 square centimetre, and
one of them is not less than 2 square centimetres.
What is the least possible length of another side of big rectangle?
366.
Given a point O inside ABC triangle. Prove that
S_{A}*[OA) + S_{B}*[OB) + S_{C}*[OC) = [OO),
where S_{A}, S_{B}, S_{C} denote BOC, COA, AOB triangles areas respectively
( [AB) denotes vector from A to B , [OO)  zero vector).
367.
Given (2m+1) different integers, each absolute value is not greater
than (2m1).
Prove that it is possible to choose three numbers among
them, with their sum equal to zero.
368.
The points D,E,F belong to the sides ]AB[, ]BC[ and ]CA[ of the ABC
triangle respectively (but they are not vertices). Let us denote with
d_{0}, d_{1}, d_{2}, and d_{3} the maximal side length of the DEF, DEA, DBF,
CEF, triangles respectively.
Prove that
d_{0} >= (sqrt(3)/2) min{d_{1}, d_{2}, d_{3}}. When the equality takes place?
369.
The M set consists of k nonintersecting segments on the line. It is
possible to put an arbitrary segment shorter than 1cm on the line in
such a way, that his ends will belong to M.
Prove that the total sum
of the segment lengths is not less than 1/k cm.
370.
The infinite decimal notation of the real number x contains all the
digits. Let v_{n} be the number of different ndigit segments encountered
in x notation.
Prove that if for some n v_{n} <= (n+8), than x is a
rational number.
The 18th competition  Ashkhabad, 1984.
form first day second day
8 371 372 373 374  383 384 385 386
9 375 376 377 378  387 388 389 390
10 379 380 381 382  391 392 393 394
371.
a)
The product of n integers equals n, and their sum is zero.
Prove that n is divisible by 4.
b)
Let n is divisible by 4.
Prove that there exist n integers such, that their product
equals n, and their sum is zero.
372.
Prove that every positive a and b satisfy inequality
((a+b)^{2})/2 + (a+b)/4 >= a*sqrt(b) + b*sqrt(a).
373.
Given two equilateral triangles A_{1}B_{1}C_{1} and A_{2}B_{2}C_{2} in the plane.
(The vertices are mentioned counterclockwise.) We draw vectors [OA[,
[OB[, [OC[, from the arbitrary point O, equal to [A_{1}A_{2}[, [B_{1}B_{2}[,
[C_{1}C_{2}[ respectively.
Prove that the triangle ABC is equilateral.
374.
Given four colours and enough square plates 1x1. We have to paint four
edges of every plate with four different colours and combine plates,
putting them with the edges of the same colour together.
Describe all the pairs m,n, such that we can combine those plates in a
nxm rectangle, that has every edge of one colour, and its four edges
have different colours.
375.
Prove that every positive x,y and real a satisfy inequality
x^{((sin a)2)} * y^{((cos a)2)} < x + y.
376.
Given a cube and two colours. Two players paint in turn a triple of
arbitrary unpainted edges with his colour. (Everyone makes two moves.)
The first wins if he has painted all the edges of some face
with his colour. Can he always win?
377.
n natural numbers (n>3) are written on the circumference. The relation
of the two neighbours sum to the number itself is a whole number.
Prove that the sum of those relations is
a)
not less than 2n;
b)
less than 3n.
378.
The circle with the centre O is inscribed in the ABC triangle. The
circumference touches its sides [BC], [CA], [AB] in A_{1}, B_{1}, C_{1}
points respectively. The [AO], [BO], [CO] segments cross the
circumference in A_{2}, B_{2}, C_{2} points respectively.
Prove that (A_{1}A_{2}),(B_{1}B_{2}) and (C_{1}C_{2}) lines intersect in one point.
379.
Find integers m and n such that (5 + 3 sqrt(2))^{m} = (3 + 5 sqrt(2))^{n}.
380.
n real numbers are written in increasing order in a line. The same
numbers are written in the second line below in unknown order. The
third line contains the sums of the pairs of numbers above from two
previous lines. It comes out, that the third line is arranged in
increasing order.
Prove that the second line coincides with the first one.
381.
Given ABC triangle. From the P point three lines (PA),(PB),(PC) are
drawn. They cross the outscribed circumference in A_{1}, B_{1},C_{1} points
respectively. It comes out that the A_{1}B_{1}C_{1} triangle equals to the
initial one.
Prove that there are not more than eight such a points P in a plane.
382.
Positive x,y,z satisfy a system:
x^{2} + xy + (y^{2})/3 = 25;
(y^{2})/3 + z^{2} = 9;
z^{2} + zx + x^{2} = 16.
Find the value of (xy + 2yz + 3zx) expression.
383.
The teacher wrote on a blackboard: " x^{2} + 10x + 20 " Then all the
pupils in the class came up in turn and either decreased or increased
by 1 either the free coefficient or the coefficient at x, but not both.
Finally they have obtained: " x^{2} + 20x + 10 ".
Is it true that some time during the process there was written the
square polynomial with the integer roots?
384.
The centre of the coin with radius r is moved along some polygon with
the perimeter P, that is outscribed around the circle with radius R
(R>r).
Find the coin trace area (a sort of polygon ring).
385.
There are scales and (n+1) weights with the total weight 2n. Each
weight weight is an integer. We put all the weights in turn on the
lighter side of the scales, starting from the heaviest one, and if the
scales is in equilibrium  on the left side.
Prove that when all the
weights will be put on the scales, they will be in equilibrium.
386.
Let us call "absolutely prime" the prime number, if having transposed
its digits in an arbitrary order, we obtain prime number again.
Prove
that its notation cannot contain more than three different digits.
387.
The x and y figures satisfy a condition: for every n>=1 the number
xx...x6yy...y4 (n times x and n times y) is an exact square.
Find all possible x and y.
388.
The A,B,C and D points (from left to right) belong to the straight line.
Prove that every point E, that doesn't belong to the line satisfy:
AE + ED +  AB  CD  > BE + CE.
389.
Given a sequence {x_{n}}. x_{1} = x_{2} = 1. x_{n+2} = (x_{n+1})^{2}  {x_{n}}/2.
Prove that the sequence has limit and find it.
390.
The white fields of 1983x1984 chessboard are filled with either +1
or 1. For every black field, the product of neighbouring numbers is +1.
Prove that all the numbers are +1.
391.
The white fields of 3x3 chessboard are filled with either +1 or 1.
For every field, let us calculate the product of neighbouring numbers.
Then let us change all the numbers by the respective products.
Prove that we shall obtain only +1's, having repeated this operation
finite number of times.
392.
What is more 2/201 or ln(101/100)? (No differential calculus allowed).
393.
Given three circles c_{1},c_{2},c_{3} with r_{1},r_{2},r_{3} radiuses, r_{1} > r_{2};
r_{1} > r_{3}. Each lies outside of two others. The A point  an
intersection of the outer common tangents to c_{1} and c_{2}  is outside
c_{3}. The B point  an intersection of the outer common tangents to c_{1}
and c_{3}  is outside c_{2}. Two pairs of tangents  from A to c_{3} and
from B to c_{2}  are drawn.
Prove that the quadrangle, they make, is
outscribed around some circle and find its radius.
394.
Prove that every cube's crosssection, containing its centre,
has the area not less then its face's area.
The 19th competition  Mogilev, 1985.
form first day second day
8 395 396 397 398  407 408 409 410
9 399 400 401 402  411 410 412 413
10 403 404 405 406  414 415 416 417
395.
Two perpendiculars are drawn from the middles of each side of the
acuteangle triangle to two other sides. Those six segments make
hexagon.
Prove that the hexagon area is a half of the triangle area.
396.
Is there any number n, such that the sum of its digits in the decimal
notation is 1000, and the sum of its square digits in the decimal
notation is 1000000?
397.
What maximal number of the dames (? Am I right? The checker
turns into a dame, when it reaches the last row? ) can be
put on the chessboard 8x8 so, that every dame can be taken
by at least one other dame?
398.
You should paint all the sides and diagonals of the right nangle so,
that every pair of segments, having the common point, would be painted
with different colours.
How many colours will You require?
399.
Given a straight line and the point O out of the line.
Prove that it
is possible to move an arbitrary point A in the same plane to the O
point, using only rotations around O and symmetry with respect to the l.
400.
The senior coefficient a in the square polynomial
P(x) = ax^{2} + bx + c is more than 100.
What is the maximal
number of integer values of x, such that P(x)<50?
401.
Different natural numbers a,b,...,k are written in the diagram.
k

V
h>i
 
V V
e>f>g
  
V V V
a>b>c>d
Every number in the diagram equals to the sum of two numbers at the
beginning of two arrows, leading to the number.
What is the minimal possible value of d?
402.
Given unbounded strictly increasing sequence a_{1}, a_{2}, ... , a_{n}, ...
of positive numbers. Prove that
a)
there exists a number k_{0} such that for all k>k_{0}
the following inequality is valid:
a_{1}/a_{2} + a_{2}/a_{3} + ... + a_{k}/a_{k1} < k  1
b)
there exists a number k_{0} such that for all k>k_{0}
the following inequality is valid:
a_{1}/a_{2} + a_{2}/a_{3} + ... + a_{k}/a_{k1} < k  1985
403.
Find all the pairs (x,y) such that sin(x)sin(y) + sin(x)sin(y) <= 0.
404.
The convex pentagon ABCDE was drawn in the plane.
A_{1} was symmetric to A with respect to B.
B_{1} was symmetric to B with respect to C.
C_{1} was symmetric to C with respect to D.
D_{1} was symmetric to D with respect to E.
E_{1} was symmetric to E with respect to A.
How is it possible to restore the initial pentagon with the
compasses and ruler, knowing A_{1},B_{1},C_{1},D_{1},E_{1} points?
405.
The sequence a_{1}, a_{2}, ... , a_{k}, ... is built according to the rules:
a_{2n} = a_{n}; a_{4n+1} = 1; a_{4n+3} = 0.
Prove that it is nonperiodical sequence.
406.
n straight lines are drawn in a plane. They divide the plane onto
several parts. Some of the parts are painted. Not a pair of painted
parts has nonzero length common bound.
Prove that the number of painted parts is not more than (n^{2} + n)/3.
407.
Given a cube, a cubic box, that exactly suits for the cube, and six
colours. First man paints each side of the cube with its (side's)
unique colour. Another man does the same with the box.
Prove that the third man can put the cube in the box in such a way,
that every cube side will touch the box side of different colour.
408.
The [A_{0}A_{5}] diameter divides a circumference with the O centre onto
two hemicircumferences. One of them is divided onto five equal arcs
A_{0}A_{1}, A_{1}A_{2}, A_{2}A_{3}, A_{3}A_{4}, A_{4}A_{5}. The (A_{1}A_{4}) line crosses
(OA_{2}) and (OA_{3}) lines in M and N points.
Prove that (A_{2}A_{3} + MN) equals to the circumference radius.
409.
If there are four numbers (a,b,c,d) in four registers of the
calculating machine, they turn into (ab,bc,cd,da) numbers whenever
You press the button.
Prove that if not all the initial numbers are
equal, machine will obtain at least one number more than 1985 after
some number of the operations.
410.
Numbers 1,2,3,...,2n are divided onto two equal groups. Let
a_{1},a_{2},...,a_{n} be the first group numbers in the increasing order, and
b_{1,}b_{2},...,b_{n}  the second group numbers in the decreasing order.
Prove that a_{1}  b_{1} + a_{2}  b_{2} + ... + a_{n}  b_{n} = n^{2}.
411.
The parallelepiped is constructed of the equal cubes. Three
parallelepiped faces, having the common vertex are painted.
Exactly half of all the cubes have at least one face painted.
What is the total number of the cubes?
412.
One of two circumferences of radius R comes through A and B
vertices of the ABCD parallelogram. Another comes through
B and D. Let M be another point of circumferences intersection.
Prove that the circle outscribed around AMD triangle has radius R.
413.
Given right hexagon. The lines parallel to all the sides are drawn from
all the vertices and middles of the sides (consider only the interior,
with respect to the hexagon, parts of those lines). Thus the hexagon
is divided onto 24 triangles, and the figure has 19 nodes. 19 different
numbers are written in those nodes.
Prove that at least 7 of 24
triangles have the property: the numbers in its vertices increase
(from the least to the greatest) counterclockwise.
414.
Solve the equation ("2" encounters 1985 times):
x
 = 1
x
2 + 
2 + ........................
x
2 + 
1 + sqrt(1 + x)
415.
All the points situated more close than 1cm to ALL the vertices of
the right pentagon with 1cm side, are deleted from that pentagon.
Find the area of the remained figure.
416.
Given big enough sheet of crosslined paper with the side of the squares
equal to 1. We are allowed to cut it along the lines only.
Prove that
for every m>12 we can cut out a rectangle of the greater than m area
such, that it is impossible to cut out a rectangle of m area from it.
417.
The ABCDA_{1}B_{1}C_{1}D_{1} cube has unit length edges.
Find the distance between two circumferences, one of those is inscribed
into the ABCD base, and another comes through A,C and B_{1} points.
The 20th competition  Ulyanovsk, 1986.
form first day second day
8 418 419 420 421  430 431 432 433
9 422 423 424 425  434 433b 435 436
10 426 427 428 429  437 438 439 440
418.
The square polynomial x^{2}+ax+b+1 has natural roots.
Prove that (a^{2}+b^{2}) is a composite number.
419.
Two equal squares, one with red sides, another with blue ones, give an
octagon in intersection.
Prove that the sum of red octagon sides
lengths is equal to the sum of blue octagon sides lengths.
420.
The point M belongs to the [AC] side of the acuteangle triangle ABC.
Two circles are outscribed around ABM and BCM triangles.
What M
position corresponds to the minimal area of those circles intersection?
421.
Certain king of a certain state wants to build n cities and n1 roads,
connecting them to provide a possibility to move from every city to
every city. (Each road connects two cities, the roads do not intersect,
and don't come through another city.) He wants also, to make the
shortests distances between the cities, along the roads, to be
1,2,3,...,n(n1)/2 kilometres.
Is it possible for
a)
n=6;
b)
n=1986?
422.
Prove that it is impossible to draw a convex quadrangle, with one
diagonal equal to doubled another, the angle between them 45 degrees,
on the coordinate plane, so, that all the vertices' coordinates would
be integers.
423.
Prove that the rectangle mxn table can be filled with exact squares so,
that the sums in the rows and the sums in the columns will be exact
squares also.
424.
Two circumferences, with the distance d between centres, intersect in
P and Q points. Two lines are drawn through the A point on the
first circumference (Q<>A<>P) and P and Q points. They intersect the
second circumference in the B and C points.
a)
Prove that the radius of the circle, outscribed around the ABC
triangle, equals d.
b)
Describe the set of the new circle's centres, if the A point moves
along all the first circumference.
425.
Given right hexagon. Each side is divided onto 1000 equal segments.
All the points of division are connected with the segments, parallel to
sides. Let us paint in turn the triples of unpainted nodes of obtained
net, if they are vertices of the unilateral triangle, doesn't matter of
what size an orientation. Suppose, we have managed to paint all the
vertices except one.
Prove that the unpainted node is not a hexagon vertex.
426.
Find all the natural numbers equal to the square of its divisors number.
427.
Prove that the following inequality holds for all positive {a_{i}}.
1/a_{1} + 2/(a_{1}+a_{2}) + ... + n/(a_{1}+...+a_{n}) < 4(1/a_{1} + ... + 1/a_{n})
428.
A line is drawn through the A vertex of ABC triangle with AB<>AC.
Prove that the line can not contain more than one point M such, that
M is not a triangle vertex, and the angles ABM abd ACM are equal.
What lines do not contain such a point M at all?
429.
A cube with edge of length n (n>=3) consists of n^{3} unit cubes.
Prove that it is possible to write different n^{3} integers on all the
unit cubes to provide the zero sum of all integers in the every row
parallel to some edge.
430.
The decimal notation of three natural numbers consists of equal digits:
n digits x for a, n digits y for b and 2n digits z for c.
For every n > 1 find all the possible triples of digits x,y,z such,
that a^{2} + b = c.
431.
Given two points inside a convex dodecagon (twelve sides) situated
10cm far from each other.
Prove that the difference between the sum of distances, from the
point to all the vertices, is less than 1m for those points.
432.
Given 30 equal cups with milk. An elf tries to make the amount of milk
equal in all the cups. He takes a pair of cups and aligns the milk
level in two cups.
Can there be such an initial distribution of milk
in the cups, that the elf will not be able to achieve his goal in a
finite number of operations?
433.
Find the relation of the black part length and the white part length
for the main diagonal of the
a)
100x99 chessboard;
b)
101x99 chessboard.
434.
Given right nangle A_{1}A_{2}...A_{n}. Prove that if
a)
n is even number, than for the arbitrary point M in the plane,
it is possible to choose signs in an expression
+[MA_{1}[ +[MA_{2}[ + ... +[MA_{n}[
to make it equal to the zero vector (here [MA[ denotes vector).
b)
n is odd, than the abovementioned expression equals to the zero
vector for the finite set of M points only.
435.
All the fields of a square nxn (n>2) table are filled with +1 or 1
according to the rules:
At the beginning 1 are put in all the boundary fields.
The number put in the field in turn (the field is chosen arbitraryly)
equals to the product of the closest, from the different sides, numbers
in its row or in its column.
a)
What is the minimal
b)
What is the maximal
possible number of +1 in the obtained table?
436.
Prove that for every natural n the following inequality is valid:
sin 1 + sin 2 + sin(3n1) + sin(3n) > 8n/5.
437.
Prove that the sum of all numbers representable as 1/(mn), where
m,n  natural numbers, 1 <= m < n <= 1986; is not an integer.
438.
A triangle and a square are outscribed around the unit circle.
Prove that the intersection area is more than 3.4.
Is it possible to assert that it is more than 3.5?
439.
Let us call a polynomial "admissible" if all it's coefficients are 0,
1, 2 or 3.
For given n find the number of all the admissible
polynomials P such, that P(2) = n.
440.
Consider all the tetrahedrons AXBY, outscribed around the sphere. Let
A and B points be fixed.
Prove that the sum of angles in the nonplane
quadrangle AXBY doesn't depend on X and Y points.
The 21st competition  Frunze, 1987.
form first day second day
8 441 442 443 444  452 453 454 455
9 445 446a 447 448  456 457 458 459
10 449 450 451 446b  455 460 461 462
441.
Ten sportsmen have taken part in a tabletennis tournament (each pair
has met once only, no draws). Let x_{i} be the number of ith player
victories, y_{i}  losses.
Prove that x_{1}^{2} + ... + x_{10}^{2} = y_{1}^{2} + ... + y_{10}^{2}
442.
It is known that, having 6 weighs, it is possible to balance the scales
with loads, which weights are successing natural numbers from 1 to 63.
Find all such sets of weighs.
443.
Given right heptagon (7angle) A_{1}...A_{7}.
Prove that 1/A_{1}A_{5} + 1/A_{1}A_{3} = 1/A_{1}A_{7}.
444.
The "Sea battle" game.
a)
You are trying to find the 4field ship  a rectangle 1x4, situated
on the 7x7 playing board. You are allowed to ask a question, whether
it occupies the particular field or not. How many questions is it
necessary to ask to find that ship surely?
b)
The same question, but the ship is a connected (i.e. its fields have
common sides) set of 4 fields.
445.
Prove that (1^{1987} + 2^{1987} + ... + n^{1987}) is divisible by (n+2).
446.
++
 
+++
  
+++
a)
How many such figures can You put on a 8x8 board without
intersections?
b)
An arbitrary field is cut out of a 1987x1987 board.
Prove that the reminding part of the board can be always
cut on such parts.
447.
Three lines are drawn parallel to the sides of the triangles in the
opposite to the vertex, not belonging to the side, part of the plane.
The distance from each side to the corresponding line equals the length
of the side.
Prove that six intersection points of those lines with
the continuations of the sides are situated on one circumference.
448.
Given two closed broken lines in the plane with odd numbers of edges.
All the lines, containing those edges are different, and not a triple
of them intersects in one point.
Prove that it is possible to chose one edge from each line such, that
the chosen edges will be the opposite sides of a convex quadrangle.
449.
Find a set of five different relatively prime natural numbers such,
that the sum of an arbitrary subset is a composite number.
450.
Given a convex pentagon. The angles ABC and ADE are equal. The angles
AEC and ADB are equal too.
Prove that the angles BAC and DAE are equal also.
451.
Prove such a, that all the numbers cos a, cos 2a, cos 4a, ... ,
cos((2^{n})a) are negative.
452.
The positive numbers a,b,c,A,B,C satisfy a condition
a + A = b + B = c + C = k.
Prove that aB + bC + cA <= k^{2}.
453.
Each field of the 1987x1987 board is filled with numbers, which
absolute value is not greater than one. The sum of all the numbers
in every 2x2 square equals 0.
Prove that the sum of all the numbers is not greater than 1987.
454.
The B vertex of the ABC angle lies out the circle, and the [BA) and
[BC) beams intersect it. The K point belongs to the intersection of the
[BA) beam and the circumference. The KP chord is orthogonal to the
angle ABC bisector. The (KP) line intersects the BC beam in the M
point.
Prove that the [PM] segment is twice as long as the distance
from the circle centre to the angle ABC bisector.
455.
Two players are writting in turn natural numbers not exceeding p.
The rules forbid to write the divisors of the numbers already having
been written. Those who cannot make his move looses.
a)
Who, and how, can win if p=10?
b)
Who wins if p=1000?
456.
Every evening uncle Chernomor (see Pushkin's tales) appoints either
9 or 10 of his 33 "knights" /* let's call them so */ in the "night
guard" /* let's call it so */.
When it can happen, for the first time,
that every knight has been on duty the same number of times?
457.
Some points with the integer coordinates are marked on the coordinate
plane. Given a set of nonzero vectors. It is known, that if You apply
the beginnings of those vectors to the arbitrary marked point, than
there will be more marked ends of the vectors, than not marked.
Prove that there is infinite number of marked points.
458.
The convex nangle (n>=5) is cut along all its diagonals.
Prove that there are at least a pair of parts with the different areas.
459.
The T_{0} set consists of all the numbers, representable as (2^{k})!,
k = 0, 1, 2, ... , n, ... The T_{m} set is obtained from T_{m1} by
adding all the finite sums of different numbers, that belong to T_{m1}.
Prove that there is a natural number, that doesn't belong to T_{1987}.
460.
The plot of the y=f(x) function, being rotated by the (right) angle
around the (0,0) point is not changed.
a)
Prove that the equation f(x)=x has the unique solution.
b)
Give an example of such a function.
461.
All the faces of a convex polyhedron are the triangles.
Prove that it is possible to paint all its edges in red and blue colour
in such a way, that it is possible to move from the arbitrary vertex
to every vertex along the blue edges only and along the red edges only.
462.
Prove that for every natural n the following inequality is held:
(2n + 1)^{n} >= (2n)^{n} + (2n  1)^{n}.
From the Holy Land, with respect
/\ /\ Vladimir A. Pertsel S/W engineer
((ovo)) Email: \/ Sagantec Israel
():::() voldemar@sagantec.co.il (o o) tel.97248572781
PVAooO(_)Ooo
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