English version


Bucharest, July 17, 1999

Time allowed 4 hours 30 min.
Each problem is worth 7 points
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 G1 and G2 are contained inside the circle G, and are tangent to G at the distinct points M and N, respectively. G1 passes through the center of G2. The line passing through the two points of intersection of G1 and G2 meets G at A and B. The lines MA and MB meet G1 at C and D, respectively.

Prove that CD is tangent to G2.

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.