|Time allowed 4 hours 30 min.|
|Each problem is worth 7 points|
Determine all pairs of positive integers such that
is a prime,
not exceeded 2p, and
is divisible by .
Two circles and are contained inside the circle , and are tangent to at the distinct points and , respectively. passes through the center of . The line passing through the two points of intersection of and meets at and . The lines and meet at and , respectively.
Prove that is tangent to .
Determine all functions R --> R such that
for all real numbers .