2004 Turkey Junior National Olympiad

1

Let $[AD]$ and $[CE]$ be internal angle bisectors of $\triangle ABC$ such that $D$ is on $[BC]$ and $E$ is on $[AB]$. Let $K$ and $M$ be the feet of perpendiculars from $B$ to the lines $AD$ and $CE$, respectively. If $|BK|=|BM|$, show that $\triangle ABC$ is isosceles.

2

The positive integer $n$ is the sum of two positive integers that divide $n+6$. Find all possible values of $n$

3

On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school?