45-th Mathematical Olympiad in Poland Second Round, February, 1994

First Day

1. Determine all polynomials P (x) of degree 5, with real coefficients, such that (x − 1)^{3} | P (x) + 1 and (x + 1)^{3} | P (x) − 1.

2. Let a_{1}, a_{2}, …, a_{n} be positive real numbers with

∑_{i=1}^{n} a_{i} = ∏_{i=1}^{n} a_{i} .

Let b_{1}, b_{2}, …, b_{n} be real numbers satisfying a_{i} ≤ b_{i} for i = 1, 2, …, n. Prove that

∑_{i=1}^{n} b_{i} ≤ ∏_{i=1}^{n} b_{i} .

3. A section of a cube, passing through a center of the cube, is a cyclic hexagon. Prove that this hexagon is regular.

Second Day

4. To each vertex of a cube there is assigned a number 1 or −1. To each face there is assigned a product of the four assigned numbers to the vertices of this face. Determine the set of all possible values that may be attained by the sum of all the 14 assigned numbers.

5. The circle o inscribed in a triangle ABC is tangent to the sides AB and BC of △ABC in the points P and Q. The line P Q intersects the angle bisector of ^{}B̸AC in the point S. Prove that this angle bisector is perpendicular to the line SC.

6. Given is a prime p. Prove that the following two sentences are equivalent:

(1) there exists an integer n with p | n^{2} − n + 3;

(2) there exists an integer m with p | m^{2} − m + 25.