NL-BXMO-TST 2010

Let $ABCD$ be a trapezoid with $AB // CD$ , $2|AB| = |CD|$ and $BD \perp BC$ . Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$ . Let $O$ be the intersection of $AM$ and $BD$ . Let $N$ be the intersection of $OE$ and $AB$ . (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$ .

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$ .

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$ . The common tangent that's on the same side as $P$ , intersects the circles at $A$ and $B$ ,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$ , and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$ . Let $E$ be the intersection of $AP$ and $BC$ , and let $F$ be the intersection of $BP$ and $AD$ . Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$ . Prove that $AMBEQF$ is a cyclic hexagon.

For any non-negative integer $n$ , we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\}$ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$ . Show that for any non-negative integer $n$ , there exists a quadratic permutation of $\{0,1,..., n\}$ .