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IMO 2009 Germany – Day 1 Problems ខែកក្កដា 16, 2009

Posted by តារារស្មី in គណិតវិទ្យា.
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Problem 1. Let n be a positive integer and let a_1,a_2,a_3,...,a_k ( k\geq 2) be distinct integer in the set { 1,2,…,n} such that n divides a_i(a_{i+1}-1) for i = 1,2,...,k-1. Prove that n does not divide a_k(a_1-1).

Problem 2. Let ABC be a triangle with circumcenter 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 sub-sequences ss_{1},ss_{2},ss_{3}, … and ss_{1+1},ss_{2+1},ss_{3+1}, … are both arithmetic progressions. Prove that the sequence s1,s2,s3, … is itself an arithmetic progression.

មតិ»

1. តារារស្មី - ខែកក្កដា 17, 2009

I think problem 1 can be solved using the following lemma :
x,y are possitive integers, x and y+1 are distinct, x,y<a
if xy is divisble by a, then (x+1)(y+1) is also divisible by a.
I have proved it, let's try together…


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