2003_eng - D to the lines AB,BC,CA are P,Q,R respectively....

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44th IMO 2003 Problem 1 . S is the set { 1 , 2 , 3 ,..., 1000000 } . Show that for any subset A of S with 101 elements we can ﬁnd 100 distinct elements x i of S , such that the sets { a + x i | a A } are all pairwise disjoint. Problem 2 . Find all pairs ( m,n ) of positive integers such that m 2 2 mn 2 - n 3 +1 is a positive integer. Problem 3 . A convex hexagon has the property that for any pair of opposite sides the distance between their midpoints is 3 / 2 times the sum of their lengths Show that all the hexagon’s angles are equal. Problem 4 . ABCD is cyclic. The feet of the perpendicular from
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Unformatted text preview: D to the lines AB,BC,CA are P,Q,R respectively. Show that the angle bisectors of ABC and CDA meet on the line AC i RP = RQ . Problem 5 . Given n > 2 and reals x 1 x 2 x n , show that ( i,j | x i-x j | ) 2 2 3 ( n 2-1) i,j ( x i-x j ) 2 . Show that we have equality i the sequence is an arithmetic progression. Problem 6 . Show that for each prime p , there exists a prime q such that n p-p is not divisible by q for any positive integer n . 1...
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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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