KZ-MO 2001

Prove that there are infinitely many natural numbers $n$ such that $2 ^ n + 3 ^ n$ is divisible by $n$ .

In the acute triangle $ABC$ , $L$ , $H$ and $M$ are the intersection points of bisectors, altitudes and medians, respectively, and $O$ is the center of the circumscribed circle. Denote by $X$ , $Y$ and $Z$ the intersection points of $AL$ , $BL$ and $CL$ with a circle, respectively. Let $N$ be a point on the line $OL$ such that the lines $MN$ and $HL$ are parallel. Prove that $N$ is the intersection point of the medians of $XYZ$ .

For positive numbers $x_1, x_2, \ldots, x_n$ $(n \geq 1)$ the following equality holds $$\frac {1} {{1 + x_1}} + \frac {1} {{1 + x_2}} + \ldots + \frac {1} {{1 + x_n}} = 1.$$ Prove that $x_1 \cdot x_2 \cdot \ldots \cdot x_n \geq (n-1) ^ n.$

Find all functions $f: \mathbb {R} \rightarrow \mathbb {R}$ satisfying the equality $f (x ^ 2-y ^ 2) = (x-y) (f (x) + f (y))$ for any $x, y \in \mathbb {R}$ .

Find all possible pairs of real numbers $(x, y)$ that satisfy the equalities $y ^ 2- [x] ^ 2 = 2001$ and $x ^ 2 + [y] ^ 2 = 2001$ .

Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $r$ . Prove that $r \geq \frac {{\sqrt 3}} {{10}}$ .

Two circles $w_1$ and $w_2$ intersect at two points $P$ and $Q$ . The common tangent to $w_1$ and $w_2$ , which is closer to the point $P$ than to $Q$ , touches these circles at $A$ and $B$ , respectively. The tangent to $w_1$ at the point $P$ intersects $w_2$ at the point $E$ (different from $P$ ), and the tangent to $w_2$ at the point $P$ intersects $w_1$ at $F$ (different from $P$ ). Let $H$ and $K$ be points on the rays $AF$ and $BE$ , respectively, such that $AH = AP$ and $BK = BP$ . Prove that the points $A$ , $H$ , $Q$ , $K$ and $B$ lie on the same circle.

There are $n \geq4$ points on the plane, the distance between any two of which is an integer. Prove that there are at least $\frac {1} {6}$ distances, each of which is divisible by $3$ .