PAMO 2006

Let $AB$ and $CD$ be two perpendicular diameters of a circle with centre $O$ . Consider a point $M$ on the diameter $AB$ , different from $A$ and $B$ . The line $CM$ cuts the circle again at $N$ . The tangent at $N$ to the circle and the perpendicular at $M$ to $AM$ intersect at $P$ . Show that $OP = CM$ .

Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.

For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$ . If $a, b, c$ are distinct real numbers, prove that $$\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}$$ is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.

For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$ . For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$ .

In how many ways can the integers from $1$ to $2006$ be divided into three non-empty disjoint sets so that none of these sets contains a pair of consecutive integers?

Let $ABC$ be a right angled triangle at $A$ . Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$ . The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$ . Show that $O_1, O_2, K$ and $L$ are on a line.