CZ-SK-MO 2011

In a certain triangle $ABC$ , there are points $K$ and $M$ on sides $AB$ and $AC$ , respectively, such that if $L$ is the intersection of $MB$ and $KC$ , then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$ .

Find all triples $(p, q, r)$ of prime numbers for which $$(p+1)(q+2)(r+3)=4pqr.$$

Suppose that $x$ , $y$ , $z$ are real numbers satisfying $$x+y+z=12, \text{and} x^2+y^2+z^2=54.$$ Prove that:list Each of the numbers $xy$ , $yz$ , $zx$ is at least $9$ , but at most $25$ . (b) One of the numbers $x$ , $y$ , $z$ is at most $3$ , and another one is at least $5$ .[/list]

Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$ , $b\ge 2$ , $c\ge 2$ . Adam and Boris play the following game. They alternately take turns with Adam first. On Adam's turn, he can choose one of the polynomial's coefficients and replace it with the sum of the other two coefficients. On Boris's turn, he can choose one of the polynomial's coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$ , $b$ and $c$ , determine who has a winning strategy.

In acute triangle ABC, which is not equilateral, let $P$ denote the foot of the altitude from $C$ to side $AB$ ; let $H$ denote the orthocenter; let $O$ denote the circumcenter; let $D$ denote the intersection of line $CO$ with $AB$ ; and let $E$ denote the midpoint of $CD$ . Prove that line $EP$ passes through the midpoint of $OH$ .

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$ , we have $$f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.$$