48-th Mathematical Olympiad in Poland Final Round, April 4–5, 1997

First Day

1. Positive integers x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} x_{7} satisfy the conditions:

x_{6} = 144 and x_{n+3} = x_{n+2}(x_{n+1} + x_{n}) for n = 1, 2, 3, 4.

Determine x_{7}.

2. Find all triples of real numbers x, y, z satisfying the system of equations

\{ \begin{matrix}3(x^{2} + y^{2} + z^{2}) = 1 \\ x^{2}y^{2} + y^{2}z^{2} + z^{2}x^{2} = xyz(x + y + z)^{3}. \end{matrix}

3. The medians of the lateral faces ABD, ACD, BCD of a pyramid ABCD taken from the vertex D make equal angles with the edges they were led to. Prove that the area of each lateral face is less than the sum of the areas of the remaining lateral faces.

Second Day

4. The sequence a_{1}, a_{2}, a_{3}, … is defined by

a_{1} = 0, a_{n} = a_{[n/2]} + (−1)^{n(n+1)/2} for n > 1.

For each integer k ≥ 0 determine the number of subscripts n satisfying the conditions

2^{k} ≤ n < 2^{k+1} , a_{n} = 0 .

Note: [n/2] denotes the biggest integer not bigger than n/2.

5. Given is a convex pentagon ABCDE with DC = DE and ^{}D̸CB = ^{}D̸EA = 90^{∘}. Let F be the interior point of segment AB determined by the condition AF : BF = AE: BC. Prove that ^{}F̸ CE = ^{}A̸DE and ^{}F̸ EC = ^{}B̸DC.

6. On a circle of radius 1, n distinct points are given. Let q be the number of the line segments whose length is greater than \sqrt{2} and whose endpoints lie in the given points. Prove that 3q ≤ n^{2}.