51-st Mathematical Olympiad in Poland Final Round, April 3–4, 2000

First Day

1. Let n ≥ 2 be a given integer. How many solutions (x_{1}, x_{2}, …, x_{n}) possesses the following system of equations

\{ \begin{matrix}x_{2} + x_{1}^{2} & = & 4x_{1} \\ x_{3} + x_{2}^{2} & = & 4x_{2} \\ x_{4} + x_{3}^{2} & = & 4x_{3} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ x_{n} + x_{n−1}^{2} & = & 4x_{n−1} \\ x_{1} + x_{n}^{2} & = & 4x_{n} \end{matrix}

in the set of nonnegative real numbers?

2. In the triangle ABC holds AC = BC. The point P lies inside the triangle ABC and is chosen such that ^{}P̸ AB = ^{}P̸ BC. The point M is the midpoint of the side AB. Prove that

A̸P M +^{}B̸P C = 180^{∘}.

3. The sequence (p_{n}) of natural numbers satisfies:

1^{∘} p_{1} and p_{2} are primes, 2^{∘} for n ≥ 3 the number p_{n} is the greatest divisor of the number

p_{n−1} + p_{n−2} + 2000 .

Prove that the sequence (p_{n}) is bounded.

Second Day

4. In the regular pyramid with the vertex S and the base A_{1}A_{2}...A_{n} each lateral edge makes the angle 60^{∘} with the base of the pyramid. For each natural number n ≥ 3 prove or disprove that: there exist points B_{2}, B_{3}, …, B_{n} lying on the edges A_{2}S, A_{3}S, …, A_{n}S, respectively such that the following inequality holds

A_{1}B_{2} + B_{2}B_{3} + B_{3}B_{4} + … + B_{n−1}B_{n} + B_{n}A_{1} < 2A_{1}S .

5. For the given natural number n ≥ 2 find the smallest number k with the following property: From each set of k unit squares of the n × n chessboard one can choose a subset such that the number of the unit squares contained in this subset and lying in a row or column of the chessboard is even.

6. Let P (x) be a real polynomial with an odd degree satisfying:

P (x^{2} − 1) = (P (x))^{2} − 1

for all x. Prove that for all x the following equality holds

P (x) = x .