

A1. Let m and n be positive integers such that:
m/n = 1  1/2 + 1/3  1/4 + ...  1/1318 + 1/1319.
Prove that m is divisible by 1979.


A2. A prism with pentagons A_{1}A_{2}A_{3}A_{4}A_{5} and B_{1}B_{2}B_{3}B_{4}B_{5} as the top and bottom faces is given. Each side of the two pentagons and each of the 25 segments A_{i}B_{j} is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Prove that all 10 sides of the top and bottom faces have the same color.


A3. Two circles in a plane intersect. A is one of the points of intersection. Starting simultaneously from A two points move with constant speed, each traveling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point P in the plane such that the two points are always equidistant from P.


B1. Given a plane k, a point P in the plane and a point Q not in the plane, find all points R in k such that the ratio (QP + PR)/QR is a maximum.


B2. Find all real numbers a for which there exist nonnegative real numbers x_{1}, x_{2}, x_{3}, x_{4}, x_{5} satisfying:
x_{1} + 2x_{2} + 3x_{3} + 4x_{4} + 5x_{5} = a,
x_{1} + 2^{3}x_{2} + 3^{3}x_{3} + 4^{3}x_{4} + 5^{3}x_{5} = a^{2},
x_{1} + 2^{5}x_{2} + 3^{5}x_{3} + 4^{5}x_{4} + 5^{5}x_{5} = a^{3}.


B3. Let A and E be opposite vertices of an octagon. A frog starts at vertex A. From any vertex except E it jumps to one of the two adjacent vertices. When it reaches E it stops. Let a_{n} be the number of distinct paths of exactly n jumps ending at E. Prove that:
a_{2n1} = 0
a_{2n} = (2 + √2)^{n1}/√2  (2  √2)^{n1}/√2.

