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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. |
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| A2. A prism with pentagons A1A2A3A4A5 and B1B2B3B4B5 as the top and bottom faces is given. Each side of the two pentagons and each of the 25 segments AiBj 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. |
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| 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. |
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| 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. |
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B2. Find all real numbers a for which there exist non-negative real numbers x1, x2, x3, x4, x5 satisfying:
x1 + 2x2 + 3x3 + 4x4 + 5x5 = a,
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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 an be the number of distinct paths of exactly n jumps ending at E. Prove that:
a2n-1 = 0
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