NL-NWO 2004

Determine the number of pairs of positive integers $(a, b)$ , with $a \le b$ , for which lcm $(a, b) = 2004$ . lcm ( $a, b$ ) means the least common multiple of $a$ and $b$ . Example: lcm $(18, 24) = 72$ .

Two circles $A$ and $B$ , both with radius $1$ , touch each other externally. Four circles $P, Q, R$ and $S$ , all four with the same radius $r$ , lie such that $P$ externally touches on $A, B, Q$ and $S$ , $Q$ externally touches on $P, B$ and $R$ , $R$ externally touches on $A, B, Q$ and $S$ , $S$ externally touches on $P, A$ and $R$ . Calculate the length of $r.$

$A$ $B$ $P$ $Q$ $R$ $S$

Start with a stack of $100$ cards. Now repeat the following: choose a stack of at least $2$ cards and split them into two smaller piles (at least $1$ card of each). Continue this until there are finally $100$ stacks of $1$ card each. Every time you split a pile into two stacks you get a number of points that is equal to the product of the number of cards in the two new stacks. What is the maximum number of points that you can earn in total?

Two circles $C_1$ and $C_2$ touch each other externally in a point $P$ . At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$ . The line $QP$ still intersects $C_2$ at point $C$ . Prove that triangle $ABC$ is isosceles.

A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties: $a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer. Determine all possible values of $c$ and the associated values of $a$ and $b$ .