45-th Mathematical Olympiad in Poland Final Round, Warszawa, April 10–11, 1994

First Day

1. Determine all triples (x, y, z) of positive rationals, such that the numbers x+y +z, \frac{1}{x} + \frac{1}{y} + \frac{1}{z},

xyz are all natural.

2. Given two parallel lines k and l and a circle not intersecting the line k. From the point A lying on k draw two tangents to the given circle that intersect the line l in two points B and C. Let m be the line passing through A and the midpoint of BC. Prove that all the lines m obtained in that way (corresponding to the various points A on k) have a common point.

3. Given an integer c ≥ 1. To each subset A of {1, 2, …, n} there is assigned an integer w(A) from

{1, 2, …, c}, such that the following condition is satisfied w(A ∩ B) = min(w(A), w(B)) for A, B ⊆ {1, 2, …, n} .

Let a(n) be the number of such assignments. Compute lim_{n→∞} ^{n} \sqrt{a(n)} .

Second Day

4. We have three vessels without the scale: the first one (empty) may contain at most m litres, the second one (empty) may contain at most n litres, and the third one (full of water) may contain at most m + n litres. m and n are relatively prime positive integers. Prove that for any k ∈ {1, 2, …, m+n−1}, pouring out of one vessel into another, one can obtain in the third vessel exactly k litres of water.

5. A_{1}, A_{2}, …, A_{8} are the vertices of the parallelepiped with the centre O. Prove that

4 · ∑_{i=1}^{8} |OA_{i}|^{2} ≤ (∑_{i=1}^{8} |OA_{i}|)^{2} .

6. Different real numbers x_{1}, x_{2}, …, x_{n} (n ≥ 4) satisfy the conditions:

∑_{i=1}^{n} x_{i} = 0 ,  ∑_{i=1}^{n} x_{i}^{2} = 1 .

Prove that among these numbers there exist four, say a, b, c, d, such that the inequalities

a + b + c + nabc ≤ ∑_{i=1}^{n} x_{i}^{3} ≤ a + b + d + nabd

are valid.