Version : English
39th International Mathematical Olympiad
First Day - Taipei - July 15, 1998
Problem 1
In the convex quadrilateral ABCD, the diagonals AC and BD
are perpendicular and the opposite sides AB and DC are not
parallel. Suppose that the point P, where the perpendicular bisectors
of AB and DC meet, is inside ABCD. Prove that ABCD
is a cyclic quadrilateral if and only if the triangles ABP and CDP
have equal areas.
Problem 2
In a competition, there are a contestants and b judges, where
b ³ 3 is an odd integer. Each judge
rates each contestant as either ``pass'' or ``fail''. Suppose k
is a number such that, for any two judges, their ratings coincide for at
most k contestants. Prove that
Problem 3
For any positive integer n, let d(n) denote the number
of positive divisors of n (including 1 and
n itself).
Determine all positive integers k such that
for some n.
Time Allowed : 4[1/2] hours.
Each problem is worth 7 points.
Version : English
39th International Mathematical Olympiad
Second Day - Taipei - July 16, 1998
Problem 4
Determine all pairs (a,b) of positive integers such that
ab2+b+7 divides a2b+a+b.
Problem 5
Let I be the incentre of triangle ABC. Let the incircle of
ABC touch the sides BC, CA and AB at K,
L and M, respectively. The line through B parallel
to MK meets the lines LM and LK at R and S,
respectively. Prove that ÐRIS is
acute.
Problem 6
Consider all functions f from the set N of all positive
integers into itself satisfying
| f( t2f( s)
) = s( f( t) )2, |
|
for all s and t in N. Determine the least possible
value of f(1998).
Time Allowed : 4[1/2] hours.
Each problem is worth 7 points.