47-th Mathematical Olympiad in Poland Third Round, March 29–30, 1996

First Day

1. Determine all pairs (n, r), where n is a positive integer, and r is real, for which the polynomial (x + 1)^{n} − r is divisible by 2x^{2} + 2x + 1.

2. Inside given triangle ABC there is chosen point P satisfying the conditions:

P̸ BC = ^{}P̸ CA < ^{}P̸ AB .

The line BP intersects the circumcircle of triangle ABC at points B and E. The circumcircle of triangle AP E intersects the line CE at points E and F. Prove that the points A, P, E, F are the consecutive vertices of a quadrilateral and that the ratio of the areas of the quadrilateral AP EF and the triangle ABP does not depend on the choice of the point P.

3. Given an integer n ≥ 2 and positive numbers a_{1}, a_{2}, …, a_{n} with the sum equal to 1.

(a) Prove that for any positive numbers x_{1}, x_{2}, …, x_{n} with the sum equal to 1, holds the following inequality:

2 ∑_{i<j} x_{i}x_{j} ≤ \frac{n − 2}{n − 1} + ∑_{i=1}^{n} \frac{a_{i}x_{i}^{2}}{1 − a_{i}} .

(b) Determine all numbers x_{1}, x_{2}, …, x_{n} for which the above inequality turns into the equality.

Second Day

4. In a tetrahedron ABCD hold the following equalities:

B̸AC = ^{}A̸CD and ^{}A̸BD = ^{}B̸DC .

Prove that the edges AB and CD have the same length.

5. For a natural number k ≥ 1 denote by p(k) the least prime number which is not a divisor of k. If p(k) > 2, then we define q(k) to be the product of all primes less than p(k); if p(k) = 2, we put q(k) = 1. Define the sequence (x_{n}) by the formulas

x_{0} = 1, x_{n+1} = \frac{x_{n}p(x_{n})}{q(x_{n})} for n = 0, 1, 2, ….

Determine all positive integers n with x_{n} = 111 111.

6. From a collection of all permutations f of the set {1, 2, …, n} satisfying the condition

f(i) ≥ i − 1 for i = 1, 2, …, n

we choose one at random. Let p_{n} be the probability that the chosen permutation satisfies

f(i) ≤ i + 1 for i = 1, 2, …, n .

Determine all positive integers n with p_{n} > 1/3.