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Unformatted text preview: July 25, 2007 Problem 1. Real numbers a 1 ,a 2 ,...,a n are given. For each i (1 ≤ i ≤ n ) define d i = max { a j : 1 ≤ j ≤ i }  min { a j : i ≤ j ≤ n } and let d = max { d i : 1 ≤ i ≤ n } . (a) Prove that, for any real numbers x 1 ≤ x 2 ≤ · · · ≤ x n , max { x i a i  : 1 ≤ i ≤ n } ≥ d 2 . ( * ) (b) Show that there are real numbers x 1 ≤ x 2 ≤ · · · ≤ x n such that equality holds in ( * ). Problem 2. Consider five points A,B,C,D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let ` be a line passing through A . Suppose that ` intersects the interior of the segment DC at F and intersects line BC at G . Suppose also that EF = EG = EC . Prove that ` is the bisector of angle DAB . Problem 3. In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of membersparticular, any group of fewer than two competitors is a clique....
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This note was uploaded on 09/07/2011 for the course RESEARCH R 101 taught by Professor T.s. during the Fall '11 term at Research College.
 Fall '11
 T.S.

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