English version
Time allowed 4 hours 30 min. |
Each problem is worth 7 points |
Determine all pairs $(n,p)$ of positive integers such that
$p$ is a prime,
$n$ not exceeded 2p, and
$(p-1)n+\; 1$ is divisible by $np-1$.
Problem 5.
Two circles $G$_{1} and $G$_{2} are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G$_{1} passes through the center of $G$_{2}. The line passing through the two points of intersection of $G$_{1} and $G$_{2} meets $G$ at $A$ and $B$. The lines $MA$ and $MB$ meet $G$_{1} at $C$ and $D$, respectively.
Prove that $CD$ is tangent to $G$_{2}.
Problem 6.
Determine all functions $f:$ R --> R such that
$f(x-f(y))=f(f(y))\; +\; x\; f(y)+f(x)-1$
for all real numbers $x,y$.