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Unformatted text preview: Problem 4 . Let n ≥ 3 be an integer. Let t 1 ,t 2 ,...,t n be positive real numbers such that n 2 + 1 > ( t 1 + t 2 + ... + t n ) ± 1 t 1 + 1 t 2 + ... + 1 t n ² . Show that t i ,t j ,t k are side lengths of a triangle for all i , j , k with 1 ≤ i < j < k ≤ n . Problem 5 . In a convex quadrilateral ABCD the diagonal BD does not bisect the angles ABC and CDA . The point P lies inside ABCD and satisﬁes 6 PBC = 6 DBA and 6 PDC = 6 BDA. Prove that ABCD is a cyclic quadrilateral if and only if AP = CP . 1 Problem 6 . We call a positive integer alternating if every two consecutive digits in its decimal representation are of diﬀerent parity. Find all positive integers n such that n has a multiple which is alternating. 2...
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 Fall '11
 T.S.
 convex quadrilateral ABCD

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