NL-NWO 2007

Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$ . On the extension of side $BC$ , we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$ . Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$ , and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$ . Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$ . Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$ .

$A$ $A_1$ $A_2$ $B$ $B_1$ $B_2$ $C$ $C_1$ $C_2$

Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$ ? If so, give an integer having that form that is divisible by $13$ , if not, prove that such an integer cannot exist.

Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square. (And prove that your answer is correct.)

A triangle $ABC$ and a point $P$ inside this triangle are given. Define $D, E$ and $F$ as the midpoints of $AP, BP$ and $CP$ , respectively. Furthermore, let $R$ be the intersection of $AE$ and $BD, S$ the intersection of $BF$ and $CE$ , and $T$ the intersection of $CD$ and $AF$ . Prove that the area of hexagon $DRESFT$ is independent of the position of $P$ inside the triangle.

$A$ $B$ $C$ $D$ $E$ $F$ $P$ $R$ $S$ $T$