Version: English

First day

Mar del Plata, Argentina - July 24, 1997

1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard).

For any pair of positive integers
*m*
and
*n*,
consider a right-angled
triangle whose vertices have integer coordinates
and whose legs, of lengths *m*
and *n*, lie along edges of the squares.

Let
*S*_{1}
be the total area of the black part of the triangle and
*S*_{2}
be the total area of the white part. Let

*f*(*m*,*n*) = | *S*_{1} - *S*_{2} |.

(a) Calculate
*f*(*m*,*n*)
for all positive integers *m* and *n* which are either both
even or both odd.

(b) Prove that
for all *m* and *n*.

(c) Show that there is no constant *C* such that
*f*(*m*,*n*) < *C*
for all *m* and *n*.

2

Angle *A* is the smallest in the triangle *ABC*.

The points *B* and *C* divide the circumcircle of the triangle
into two arcs. Let *U* be an interior
point of the arc between *B* and *C* which does not
contain *A*.

The perpendicular bisectors of *AB* and *AC* meet the line *AU* at
*V* and *W*, respectively. The lines *BV* and
*CW* meet at *T*.

Show that

*AU = TB* + *TC*.

3

Let
*x*_{1}, *x*_{2}, ... , *x*_{n}
be real numbers satisfying the conditions:

|*x*_{1} + *x*_{2} + ... + *x*_{n} | = 1

and

for *i* = 1, 2, ... , *n*.

Show that there exists a permutation
*y*_{1}, *y*_{2}, ... , *y*_{n}
of
*x*_{1}, *x*_{2}, ... , *x*_{n}
such that

.

Each problem is worth 7 points

Time: 4 1/2 hours.