Version: English

First day
Mar del Plata, Argentina - July 24, 1997


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 S1 be the total area of the black part of the triangle and S2 be the total area of the white part. Let

f(m,n) = | S1 - S2 |.

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

(b) Prove that f(m,n) =< 1/2 max{m,n} for all m and n.

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


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.


Let x1, x2, ... , xn be real numbers satisfying the conditions:

|x1 + x2 + ... + xn | = 1


|x_1| <= (n+1)/2 for i = 1, 2, ... , n.

Show that there exists a permutation   y1, y2, ... , yn  of   x1, x2, ... , xn  such that

| y_1 + 2 y_2 + ... + n y_n | <= (n+1)/2.

Each problem is worth 7 points
Time: 4 1/2 hours.