n - 3 < 2d/p < [n/2] [(n+1)/2] - 2, where [x] denotes the greatest integer not exceeding x.
| A1. Prove that 0 ≤ yz + zx + xy - 2xyz ≤ 7/27, where x, y and z are non-negative real numbers satisfying x + y + z = 1. |
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| A2. Find one pair of positive integers a, b such that ab(a+b) is not divisible by 7, but (a+b)7 - a7 - b7 is divisible by 77. |
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| A3. Given points O and A in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point X in the plane, the circle C(X) has center O and radius OX + ∠AOX/OX, where ∠AOX is measured in radians in the range [0, 2π). Prove that we can find a point X, not on OA, such that its color appears on the circumference of the circle C(X). |
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| B1. Let ABCD be a convex quadrilateral with the line CD tangent to the circle on diameter AB. Prove that the line AB is tangent to the circle on diameter CD if and only if BC and AD are parallel. |
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B2. Let d be the sum of the lengths of all the diagonals of a plane convex polygon with n > 3 vertices. Let p be its perimeter. Prove that:
n - 3 < 2d/p < [n/2] [(n+1)/2] - 2, where [x] denotes the greatest integer not exceeding x. |
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| B3. Let a, b, c, d be odd integers such that 0 < a < b < c < d and ad = bc. Prove that if a + d = 2k and b + c = 2m for some integers k and m, then a = 1. |
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