2000 Turkey Junior National Olympiad

1

Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$. Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.

2

Find the least positive integer $n$ such that $15$ divides the product \[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\] , for every positive integers $a_1, a_2, \dots, a_{15}$.

3

$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation \[f(x)f(y)-af(xy)=x+y\] , for every real numbers $x,y$. Find all possible real values of $a$.