# 2009_eng - Language English Day 1 Wednesday Problem 1 Let n...

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Problem 1. Let n be a positive integer and let a 1 ,...,a k ( k 2 ) be distinct integers in the set { 1 ,...,n } such that n divides a i ( a i +1 - 1) for i = 1 ,...,k - 1 . Prove that n does not divide a k ( a 1 - 1) . Problem 2. Let ABC be a triangle with circumcentre O . The points P and Q are interior points of the sides CA and AB , respectively. Let K , L and M be the midpoints of the segments BP , CQ and PQ , respectively, and let Γ be the circle passing through K , L and M . Suppose that the line PQ is tangent to the circle Γ . Prove that OP = OQ . Problem 3. Suppose that s 1 ,s 2 ,s 3 ,... is a strictly increasing sequence of positive integers such that the subsequences s s 1 ,s s 2 ,s s 3 ,... and s s 1 +1 ,s s 2 +1 ,s s 3 +1 ,... are both arithmetic progressions. Prove that the sequence s 1 ,s 2 ,s 3 ,... is itself an arithmetic pro- gression. Language: English

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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.

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2009_eng - Language English Day 1 Wednesday Problem 1 Let n...

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