OMCC 2000

Find all three-digit numbers $abc$ (with $a \neq 0$ ) such that $a^{2}+b^{2}+c^{2}$ is a divisor of 26.

Determine all positive integers $n$ such that it is possible to tile a $15 \times n$ board with pieces shaped like this:

Let $ABCDE$ be a convex pentagon. If $P$ , $Q$ , $R$ and $S$ are the respective centroids of the triangles $ABE$ , $BCE$ , $CDE$ and $DAE$ , show that $PQRS$ is a parallelogram and its area is $2/9$ of that of $ABCD$ .

Write an integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors.

Note: Two triangles are said to be neighbors if they have a common side.

Let $ABC$ be an acute-angled triangle. $C_{1}$ and $C_{2}$ are two circles of diameters $AB$ and $AC$ , respectively. $C_{2}$ and $AB$ intersect again at $F$ , and $C_{1}$ and $AC$ intersect again at $E$ . Also, $BE$ meets $C_{2}$ at $P$ and $CF$ meets $C_{1}$ at $Q$ . Prove that $AP=AQ$ .

Let's say we have a nice representation of the positive integer $n$ if we write it as a sum of powers of 2 in such a way that there are at most two equal powers in the sum (representations differing only in the order of their summands are considered to be the same).

a) Write down the 5 nice representations of 10.

b) Find all positive integers with an even number of nice representations.