JBMO 2006

If $n>4$ is a composite number, then $2n$ divides $(n-1)!$ .

The triangle $ABC$ is isosceles with $AB=AC$ , and $\angle{BAC}<60^{\circ}$ . The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$ , and $\angle{ABD}\equiv\angle{CBE}$ . Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$ . Compute $\angle{COD}$ .

We call a number perfect if the sum of its positive integer divisors(including $1$ and $n$ ) equals $2n$ . Determine all perfect numbers $n$ for which $n-1$ and $n+1$ are prime numbers.

Consider a $2n \times 2n$ board. From the $i$ th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board?