PAMO 2013

A positive integer $n$ is such that $n(n+2013)$ is a perfect square. a) Show that $n$ cannot be prime. b) Find a value of $n$ such that $n(n+2013)$ is a perfect square.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$ .

Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$ be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$ , $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if $$\frac{BC}{CD}=\frac{EF}{FA}.$$

Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$ . Let $P$ and $Q$ be the midpoints of $AC$ and $BD$ , respectively. Prove that if $\angle ABP=\angle CBD$ , then $\angle BCQ=\angle ACD$ .

The cells of an $n\times n$ board with $n\ge 5$ are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any $3\times 3$ square within the board, two of its corner squares are coloured black and two are coloured white.

Let $x$ , $y$ , and $z$ be real numbers such that $x<y<z<6$ . Solve the system of inequalities: $$\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \dfrac{1}{6-z}+2\le x \end{array}\right.$$