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A1. Let En = (a1 - a2)(a1 - a3) ... (a1 - an) + (a2 - a1)(a2 - a3) ... (a2 - an) + ... + (an - a1)(an - a2) ... (an - an-1). Let Sn be the proposition that En ≥ 0 for all real ai.
Prove that Sn is true for n = 3 and 5, but for no other n > 2. |
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| A2. Let P1 be a convex polyhedron with vertices A1, A2, ... , A9. Let Pi be the polyhedron obtained from P1 by a translation that moves A1 to Ai. Prove that at least two of the polyhedra P1, P2, ... , P9 have an interior point in common. |
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| A3. Prove that we can find an infinite set of positive integers of the form 2n - 3 (where n is a positive integer) every pair of which are relatively prime. |
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B1. All faces of the tetrahedron ABCD are acute-angled. Take a point X in the interior of the segment AB, and similarly Y in BC, Z in CD and T in AD.
(a) If angle DAB + angle BCD ≠ angle CDA + angle ABC, then prove that none of the closed paths XYZTX has minimal length; (b) If angle DAB + angle BCD =angle CDA + angle ABC, then there are infinitely many shortest paths XYZTX, each with length 2 AC sin k, where 2k = angle BAC + angle CAD + angle DAB. |
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| B2. Prove that for every positive integer m we can find a finite set S of points in the plane, such that given any point A of S, there are exactly m points in S at unit distance from A. |
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| B3. Let A = (aij), where i, j = 1, 2, ... , n, be a square matrix with all aij non-negative integers. For each i, j such that aij = 0, the sum of the elements in the ith row and the jth column is at least n. Prove that the sum of all the elements in the matrix is at least n2/2. |
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