NL-NWO 2008

Suppose we have a square $ABCD$ and a point $S$ in the interior of this square. Under homothety with centre $S$ and ratio of magnification $k > 1$ , this square becomes another square $A'B'C'D'$ . Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$ .

$A$ $B$ $C$ $D$ $A'$ $B'$ $C'$ $D'$ $S$

Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$ .

Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ( $1$ and $2008$ included). Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$ .

Three circles $C_1,C_2,C_3$ , with radii $1, 2, 3$ respectively, are externally tangent. In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles. Find the radius of $C_4$ .

$C_1$ $C_2$ $C_3$

We're playing a game with a sequence of $2008$ non-negative integers. A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends. Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.