46-th Mathematical Olympiad in Poland Final Round, Gdynia, March 31 – April 1, 1995

First Day

1. Determine the number of the subsets of {1, 2, …, 2n}, in which the equation x + y = 2n + 1 has no solutions.

2. The diagonals of a convex pentagon cut it into a pentagon and ten triangles. What is the largest number of the obtained triangles which may have an equal area?

3. Given is a prime p > 3; set q = p^{3}. Define the sequence (a_{n}) by:

a_{n} = \{ \begin{matrix}n & for & n = 0, 1, 2, …, p − 1, \\ a_{n−1} + a_{n−p} & for & n > p − 1 . \end{matrix}

Determine the remainder when a_{q} is divided by p.

Second Day

4. Numbers x_{1}, x_{2}, …, x_{n} are positive with the harmonic mean equal to 1. Determine the smallest

value of

x_{1} + \frac{x_{2}^{2}}{2} + \frac{x_{3}^{3}}{3} + … + \frac{x_{n}^{n}}{n} .

5. In the urn there are n sheets of paper labelled 1, 2, …, n. We draw the sheets one by one without putting them into the urn again. When we obtain a sheet with a number divisible by k, we stop the drawing. For a fixed n, determine all values of k for which the expected value of the number of the drawn sheets is equal to k.

6. Given three rays k, l, m in the space with a common beginning P and a point A, distinct from P, belonging to k. Prove that there exists exactly one pair of the points B and C belonging to l and m respectively, such that

P A + AB = P C + CB and P B + BC = P A + AC .