KZ-MO 2002

Let $O$ be the center of the inscribed circle of the triangle $ABC$ , tangent to the side of $BC$ . Let $M$ be the midpoint of $AC$ , and $P$ be the intersection point of $MO$ and $BC$ . Prove that $AB = BP$ if $\angle BAC = 2 \angle ACB$ .

Prove that there is a set $A$ consisting of $2002$ different natural numbers satisfying the condition: for each $a \in A$ , the product of all numbers from $A$ , except $a$ , when divided by $a$ gives the remainder $1$ .

On the plane is given the acute triangle $ABC$ . Let $A_1$ and $B_1$ be the feet of the altitudes of $A$ and $B$ drawn from those vertices, respectively. Tangents at points $A_1$ and $B_1$ drawn to the circumscribed circle of the triangle $CA_1B_1$ intersect at $M$ . Prove that the circles circumscribed around the triangles $AMB_1$ , $BMA_1$ and $CA_1B_1$ have a common point.

Find all polynomials $P (x)$ with real coefficients that satisfy the identity $P (x ^ 2) = P (x) P (x + 1)$ .

Prove that for any integers $n> m> 0$ the number $2 ^n-1$ has a prime divisor not dividing $2 ^m-1$ .

$N$ grasshoppers are lined up in a row. At any time, one grasshopper is allowed to jump over exactly two grasshoppers standing to the right or left of him. At what $n$ can grasshoppers rearrange themselves in reverse order?