47-th Mathematical Olympiad in Poland Second Round, February 23–24, 1996

First Day

1. Can every polynomial with integer coefficients be expressed as a sum of cubes of polynomials with integer coefficients ?

2. A circle with center O inscribed in a convex quadrilateral ABCD is tangent to the lines AB, BC, CD, DA at points K, L, M, N respectively. Assume that the lines KL and MN are not parallel and intersect at the point S. Prove that BD ⊥ OS.

3. Prove that if a, b, c ≥ −3/4 and a + b + c = 1, then

\frac{a}{a^{2} + 1} + \frac{b}{b^{2} + 1} + \frac{c}{c^{2} + 1} ≤ \frac{9}{10} .

Second Day

4. Given a sequence a_{1}, a_{2}, …, a_{99} of one-digit numbers with the poperty that if for some n we have a_{n} = 1, then a_{n+1} ≠ 2; and if for some n we have a_{n} = 3, then a_{n+1} ≠ 4. Prove that there exist two numbers k, l ∈ {1, 2, …, 98} such that a_{k} = a_{l} and a_{k+1} = a_{l+1}.

5. Find all integer solutions to the equation

x^{2}(y − 1) + y^{2}(x − 1) = 1 .

6. Prove that each interior point of a parallelepiped with edges a, b, c is at most

\frac{1}{2}\sqrt{a^{2} + b^{2} + c^{2}}

distance off some vertex of the parallelepiped.