44-th Mathematical Olympiad in Poland Second Round, February, 1993

First Day

1. Prove that for any positive numbers x, y, u, v the inequality

\frac{xy + xv + uy + uv}{x + y + u + v} ≥ \frac{xy}{x + y} + \frac{uv}{u + v}

holds.

2. Given is a circle with center O and a point P lying outside of this circle. Let l be a line passing through the point P and cutting the given circle at points A and B. Let C be the symmetric point to A with respect to the line OP and let m be the line connecting the points B and C. Prove that all the lines m, determined by the various lines l, have a common point.

3. On the edge OA_{1} of a tetrahedron OA_{1}B_{1}C_{1} there are chosen points A_{2}, A_{3}, such that OA_{1} > OA_{2} > OA_{3} > 0. Let B_{2}, B_{3} be the points of the edge OB_{1}, and C_{2}, C_{3} – the points of the edge OC_{1} such that the planes A_{1}B_{1}C_{1}, A_{2}B_{2}C_{2}, A_{3}B_{3}C_{3} are parallel. Let V_{i} (i = 1, 2, 3) be the volume of the tetrahedron OA_{i}B_{i}C_{i} and V be the volume of the tetrahedron OA_{1}B_{2}C_{3}. Prove that

V_{1} + V_{2} + V_{3} ≥ 3V.

Second Day

4. Let (x_{n}) be the sequence of natural numbers such that:

x_{1} = 1 and x_{n} < x_{n+1} ≤ 2n for n = 1, 2, 3 ….

Prove that for every natural number k, there exist the subscripts r and s, such that x_{r} − x_{s} = k.

5. On the sides BC, CA, AB of a triangle ABC there are chosen points D, E, F (respectively), such that inradii of the triangles AEF, BF D, CDE are all equal to r_{1}. Inradii of the triangles DEF and ABC are equal to r_{2} and r respectively. Prove that r_{1} + r_{2} = r.

6. Given is a continuous function f: R ⟶ R satisfying the condition:

f(1000) = 999 ,  f(x) · f(f(x)) = 1  for x ∈ R.

Find f(500).