[A permutation f of a set S is a one-to-one mapping of S onto itself. An element i of S is called a fixed point if f(i) = i.]
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A1. Let pn(k) be the number of permutations of the set {1, 2, 3, ... , n} which have exactly k fixed points. Prove that the sum from k = 0 to n of (k pn(k) ) is n!.
[A permutation f of a set S is a one-to-one mapping of S onto itself. An element i of S is called a fixed point if f(i) = i.] |
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| A2. In an acute-angled triangle ABC the interior bisector of angle A meets BC at L and meets the circumcircle of ABC again at N. From L perpendiculars are drawn to AB and AC, with feet K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas. |
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| A3. Let x1, x2, ... , xn be real numbers satisfying x12 + x22 + ... + xn2 = 1. Prove that for every integer k ≥ 2 there are integers a1, a2, ... , an, not all zero, such that |ai| ≤ k - 1 for all i, and |a1x1 + a2x2 + ... + anxn| ≤ (k - 1)√n/(kn - 1). |
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| B1. Prove that there is no function f from the set of non-negative integers into itself such that f(f(n)) = n + 1987 for all n. |
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| B2. Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of 3 points determines a non-degenerate triangle with rational area. |
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| B3. Let n be an integer greater than or equal to 2. Prove that if k2 + k + n is prime for all integers k such that 0 ≤ k ≤ √(n/3), then k2 + k + n is prime for all integers k such that 0 ≤ k ≤ n-2. |
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