Mathematical Olympiad in Poland Second Round, February 26–27, 1999

First Day

1. Let f: ⟨0, 1⟩ → ℝ be a given function, such that f(\frac{1}{n}) = (−1)^{n} for n = 1, 2, . . .. Prove that there do not exist increasing functions g: ⟨0, 1⟩ → ℝ, h: ⟨0, 1⟩ → ℝ, such that f = g − h.

2. The cube S with edge length 2 consists of 8 unit cubes. We will call the cube S with one unit cube removed a piece. The cube T with edge length 2^{n} consists of (2^{n})^{3} unit cubes. Prove that if one unit cube is removed from T, then the remaining solid can be built with pieces.

3. The convex quadrilateral ABCD is inscribed in a circle. The points E and F lie on the sides AB and CD respectively and satisfy the condition AE: EB = CF : F D. The point P lies on the straight segment EF and satisfies EP : P F = AB: CD. Prove that the ratio of the areas of the triangles AP D and BP C does not depend on the choice of the points E and F.

Second Day

4. Point P lies inside the triangle ABC and satisfies < ) P AB = < ) P CA and < ) P AC = < ) P BA. The point O is the center of the circumcircle of the triangle ABC. Prove that < ) AP O is a right angle if O ≠ P.

5. Let S = {1, 2, 3, 4, 5}. Find out how many functions f: S → S exist with the following property: f^{50}(x) = x for all x ∈ S.

Note: f^{50}(x) = \underbrace{f ∘ f ∘ … ∘ f}_{50}(x).

6. Let the positive integer k ≥ 2 and the integers a_{1}, a_{2}, …, a_{n} be given and have

the following properties:

a_{1} + 2^{i}a_{2} + 3^{i}a_{3} + … + n^{i}a_{n} = 0 , for i = 1, 2, …, k−1. Prove that the number a_{1} + 2^{k}a_{2} + 3^{k}a_{3} + … + n^{k}a_{n} is divisible by k! .