English version

SECOND DAY

Bucharest, July 17, 1999

Problem 4.

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 $x,y$.