2001_eng - k 1 , k 2 , , k n be given integers. For each of...

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42nd International Mathematical Olympiad Washington, DC, United States of America July 8–9, 2001 Problems Each problem is worth seven points. Problem 1 Let ABC be an acute-angled triangle with circumcentre O . Let P on BC be the foot of the altitude from A . Suppose that Ð BCA ³Ð ABC + 30 é . Prove that Ð CAB COP < 90 é . Problem 2 Prove that a ± ± ± ± ±!!! ! a 2 + 8 b c + b ± ± ± ± ±!!! ! b 2 + 8 c a + c ± ± ± ± ±!!! ! c 2 + 8 a b ³ 1 for all positive real numbers a , b and c . Problem 3 Twenty-one girls and twenty-one boys took part in a mathematical contest. Each contestant solved at most six problems. For each girl and each boy, at least one problem was solved by both of them. Prove that there was a problem that was solved by at least three girls and at least three boys. Problem 4 Let n be an odd integer greater than 1, and let
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Unformatted text preview: k 1 , k 2 , , k n be given integers. For each of the n ! permutations a = H a 1 , a 2 , , a n L of 1, 2, , n , let S H a L = i = 1 n k i a i . Prove that there are two permutations b and c , b c , such that n ! is a divisor of S H b L S H c L . http://imo.wolfram.com/ Problem 5 In a triangle ABC , let AP bisect BAC , with P on BC , and let BQ bisect ABC , with Q on CA . It is known that BAC = 60 and that AB + BP = AQ + QB . What are the possible angles of triangle ABC ? Problem 6 Let a , b , c , d be integers with a &gt; b &gt; c &gt; d &gt; 0. Suppose that a c + b d = H b + d + a c L H b + d a + c L . Prove that a b + c d is not prime. 2 IMO 2001 Competition Problems http://imo.wolfram.com/...
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2001_eng - k 1 , k 2 , , k n be given integers. For each of...

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