49th Mathematical Olympiad in Poland

Problems of the first round, September – December 1997

1. Solve the system of equations:

\{ \begin{matrix}|x − y| − \frac{|x|}{x} = −1 \\ |2x − y| + |x + y − 1| + |x − y| + y − 1 = 0. \end{matrix}

2. The lines containing the sides of triangle ABC, inscribed in a circle of center O, intersect at H. It is known that AO = AH. Find the measure of the angle CAB.

3. The sequences (a_{n}), (b_{n}), (c_{n}) are given by the conditions: a_{1} = 4, a_{n+1} = a_{n}(a_{n} − 1), 2^{b_{n}} = a_{n}, 2^{n−c_{n}} = b_{n} for n = 1, 2, 3, …. Prove that the sequence (c_{n}) is bounded.

4. Given a positive number a. Determine all real numbers c with the following property: for

any pair of positive numbers x, y the following inequality holds (c − 1)x^{a+1} ≤ (cy − x)y^{a}.

5. Given is an integer n ≥ 1. Solve the equation:

| tan^{n} x − cot^{n} x| = 2n| cot 2x|.

6. In triangle ABC with AB > AC, D is the midpoint of the side BC; E lies on the side AC. Points P and Q are the feet of the perpendiculars from B and E, respectively to the line AD. Prove that BE = AE + AC if and only if AD = P Q.

7. Given positive integers m, n. Set A = {1, 2, 3, …, n}. Determine the number of functions

f: A → A attaining exactly m values and satisfying the condition if k, ℓ ∈ A, k ≤ ℓ, then f(f(k)) = f(k) ≤ f(ℓ).

8. Determine if there exists a convex polyhedron having exactly k edges and a plane not passing through any of its vertices and cutting r edges with 3r > 2k.

9. Let a_{0} = 0,91 and a_{k} = 0, \underbrace{99 … 9}_{2^{k}} \underbrace{00 … 0}_{2^{k}−1} 1 for k = 1, 2, 3, …. Compute

lim_{n→∞}(a_{0}a_{1} … a_{n}).

10. The medians AD, BE, CF of triangle ABC intersect at G. The quadrilaterals AF GE and BDGF are cyclic. Prove that the triangle ABC is equilateral.

11. In a tennis tournament n players take part. Everyone has played against everyone else, there were no draws. Prove that there exists a player A who won with every player B directly or indirectly, i.e. either A won with B or A won with some player C, who had won with B.

12. Let g(k) be the biggest prime divisor of an integer k if |k| ≥ 2, and let g(−1) = g(0) = g(1) = 1. Determine if there exists a polynomial W of positive degree with integer coefficients for which the set of the numbers of the form g(W (x)) (x — an integer) is finite.