CMO 2007

What is the maximum number of non-overlapping $2\times 1$ dominoes that can be placed on a $8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.

Attachments:

You are given a pair of triangles for which two sides of one triangle are equal in length to two sides of the second triangle, and the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between $\frac {1}{2}(\sqrt {5} - 1)$ and $\frac {1}{2}(\sqrt {5} + 1)$ .

Suppose that $f$ is a real-valued function for which $$f(xy)+f(y-x)\geq f(y+x)$$ for all real numbers $x$ and $y$ .

a) Give a non-constant polynomial that satisfies the condition. b) Prove that $f(x)\geq 0$ for all real $x$ .

For two real numbers $a$ , $b$ , with $ab\neq 1$ , define the $\ast$ operation by $$a\ast b=\frac{a+b-2ab}{1-ab}.$$ Start with a list of $n\geq 2$ real numbers whose entries $x$ all satisfy $0<x<1$ . Select any two numbers $a$ and $b$ in the list; remove them and put the number $a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains.

$a.$ Prove that this single number is the same regardless of the choice of pair at each stage. $b.$ Suppose that the condition on the numbers $x$ is weakened to $0<x\leq 1$ . What happens if the list contains exactly one $1$ ?

Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at $D, E$ and $F,$ respectively. Let $\omega,\,\omega_{1},\,\omega_{2}$ and $\omega_{3}$ denote the circumcircles of triangle $ABC, AEF, BDF$ and $CDE$ respectively.

Let $\omega$ and $\omega_{1}$ intersect at $A$ and $P,\,\omega$ and $\omega_{2}$ intersect at $B$ and $Q,\,\omega$ and $\omega_{3}$ intersect at $C$ and $R.$

$a.$ Prove that $\omega_{1},\,\omega_{2}$ and $\omega_{3}$ intersect in a common point.

$b.$ Show that $PD, QE$ and $RF$ are concurrent.