ZA-SAMO 2003

You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. Prove that you will never have 2003 pieces.

Given a parallelogram $ABCD$ , join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$ . $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$ . Prove that $M$ and $N$ divide $BD$ into three equal parts.

The first four digits of a certain positive integer $n$ are $1137$ . Prove that the digits of $n$ can be shuffled in such a way that the new number is divisible by 7.

In a given pentagon $ABCDE$ , triangles $ABC$ , $BCD$ , $CDE$ , $DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$ . Prove that $BM = EN$ .

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

In $\Delta ABC$ , the sum of the sides is $2s$ and the radius of the incircle is $r$ . Three semicircles with diameters $AB$ , $BC$ and $CA$ are drawn on the outside of $ABC$ . A circle with radius $t$ touches all three semicircles. Prove that $$\frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r.$$