2002_eng - k for which k n k 2-1 divides k m k-1 Problem 4...

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43rd IMO 2002 Problem 1 . S is the set of all ( h,k ) with h,k non-negative integers such that h + k < n . Each element of S is colored red or blue, so that if ( h,k ) is red and h 0 h,k 0 k , then ( h 0 ,k 0 ) is also red. A type 1 subset of S has n blue elements with different first member and a type 2 subset of S has n blue elements with different second member. Show that there are the same number of type 1 and type 2 subsets. Problem 2 . BC is a diameter of a circle center O . A is any point on the circle with 6 AOC > 60 o . EF is the chord which is the perpendicular bisector of AO. D is the midpoint of the minor arc AB . The line through O parallel to AD meets AC at J . Show that J is the incenter of triangle CEF . Problem 3 . Find all pairs of integers m > 2 ,n > 2 such that there are infinitely many positive integers
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Unformatted text preview: k for which k n + k 2-1 divides k m + k-1. Problem 4 . The positive divisors of the integer n > 1 are d 1 < d 2 < ... < d k , so that d 1 = 1 ,d k = n . Let d = d 1 d 2 + d 2 d 3 + ··· + d k-1 d k . Show that d < n 2 and find all n for which d divides n 2 . Problem 5 . Find all real-valued functions on the reals such that ( f ( x ) + f ( y ))(( f ( u ) + f ( v )) = f ( xu-yv ) + f ( xv + yu ) for all x,y,u,v . Problem 6 . n > 2 circles of radius 1 are drawn in the plane so that no line meets more than two of the circles. Their centers are O 1 ,O 2 , ··· ,O n . Show that ∑ i<j 1 /O i O j ≤ ( n-1) π/ 4. 1...
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