48-th Mathematical Olympiad in Poland First Round, September–December, 1996

1. Solve the system of equations:

\{ \begin{matrix}x · |x| + y · |y| & = & 1 \\ [x] + [y] & = & 1 . \end{matrix}

Note: [t] denotes the biggest integer not bigger than t.

2. Point P is an interior point of a parallelogram ABCD, such ^{}A̸BP = ^{}A̸DP. Prove that P̸ AB = ^{}P̸ CB.

3. Prove that for any real numbers a, b ≥ 1, c ≥ 0 and for any integer n ≥ 1 the inequality (ab + c)^{n} − c ≤ ((b + c)^{n} − c)a^{n} holds.

4. Prove that a natural number n ≥ 2 is composite if and only if there exist natural numbers

a, b, x, y ≥ 1 with a + b = n and \frac{x}{a} + \frac{y}{b} = 1 .

5. The angle bisectors of the internal angles A, B, C of triangle ABC intersect the opposite sides at points D, E, F, respectively and the circumcircle of ABC at K, L, M, respectively. Prove that

\frac{AD}{DK} + \frac{BE}{EL} + \frac{CF}{F M} ≥ 9 .

6. Polynomial P (x) of degree n satisfies the following condition:

P (k) = \frac{1}{k} for k = 1, 2, 4, 8, …, 2^{n} .

Find P (0).

7. Find the supremum of volumes of tetrahedra contained in a ball of given radius, whose one edge is a diameter of the ball.

8. Let a_{n} denote the number of all nonempty subsets of {1, 2, …, 6n}, whose sum of elements gives the remainder 5 upon division by 6; and let b_{n} be the number of all nonempty subsets of {1, 2, …, 7n}, whose product of elements gives the remainder 5 upon division by 7. Find the ratio a_{n}/b_{n}.

9. Determine all functions f: [1; ∞) → [1; ∞) satisfying the following two conditions:

(i) f(x + 1) = \frac{(f(x))^{2} − 1}{x} for x ≥ 1; (ii) the function g(x) = f(x)/x is bounded.

10. Points P, Q lie inside an acute triangle ABC and satisfy ^{}A̸CP = ^{}B̸CQ and ^{}C̸AP = ^{}B̸AQ. Points D, E, F are the orthogonal projections from P onto the lines BC, CA, AB, respectively. Prove that the angle DEF is right if and only if Q is the orthocenter of triangle BDF.

11. Given a natural number m ≥ 1 and a polynomial P (x) of positive degree with integer coefficients. Prove that if P (x) has at least three distinct integer roots, then P (x)+5^{m} has at most one integer root.

12. A group consisting of n people noticed that, for some period of time, three of them might be going for a dinner together, each pair meeting at exactly one dinner. Prove that n gives the remainder 1 or 3 upon division by 6.