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 f(m, n) ≤ \frac{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.

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 = T B + T C.

3. Let x_{1}, x_{2}, …, x_{n} be real numbers satisfying the conditions:

|x_{1} + x_{2} + ⋯ + x_{n}| = 1

and

|x_{i}| ≤ \frac{n + 1}{2} 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

|y_{1} + 2y_{2} + ⋯ + ny_{n}| ≤ \frac{n + 1}{2} .

Each problem is worth 7 points. Time: 4\frac{1}{2} hours.