TUYMAADA 1999

Find all polynomials $P(x)$ such that $$P(x^3+1)=P(x^3)+P(x^2).$$

Proposed by A. Golovanov

What maximum number of elements can be selected from the set $\{1, 2, 3, \dots, 100\}$ so that no sum of any three selected numbers is equal to a selected number?

Proposed by A. Golovanov

Prove the inequality $${x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,$$ where $2 < x, y, z < 4.$ Proposed by A. Golovanov

In the triangle $ABC$ we have $\angle ABC=100^\circ$ , $\angle ACB=65^\circ$ , $M\in AB$ , $N\in AC$ , and $\angle MCB=55^\circ$ , $\angle NBC=80^\circ$ . Find $\angle NMC$ .

St.Petersburg folklore

Can the graphs of a polynomial of degree 20 and the function $y={1\over x^{40}}$ have exactly 30 points of intersection?

Proposed by K. Kokhas

A sequence of integers $a_0,\ a_1,\dots a_n \dots$ is defined by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each $n\in \mathbb{N}$ , and $a_{n+1}$ is the minimum number such that no three numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression. Prove that $a_{2^n}=3^n$ for each $n \in \mathbb{N}.$

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares.

Proposed by A. Golovanov