Time allowed: 9 hours Version: English Each question is worth 7 points. Problem 1 Let n be a positive integer. Let T be the set of points (x, y) in the plane where x and y are non-negative integers and x + y < n. Each point of T is coloured red or blue. If a point (x, y) is red, then so are all points (x', y') of T with both x'≦x and y'≦y. Define an X-set to be a set of n blue points having distinct x-coordinates, and a Y-set to be a set of n blue points having distinct y-coordinates. Prove that the number of X-sets is equal to the number of Y-sets. Problem 2 Let BC be a diameter of the circle Γ with centre O. Let A be a point on Γ such that 0°<∠AOB < 120°. Let D be the midpoint of the arc AB not containing C. The line through O parallel to DA meets the line AC at J. The perpendicular bisector of OA meets Γ at E and at F. Prove that J is the incentre of the triangle CEF. Problem 3 Find all pairs of integers m,n≧3 such that there exist infinitely many positive integers a for which a^m + a - 1 / a^n + a^2 - 1 is an integer. Problem 4 Let n be an integer greater than 1. The positive divisors of n are d1, d2,…, dk where 1 = d1 < d2 < … < dk = n. Define D = d1d2 + d2d3 + ... + dk-1dk . (a) Prove that D < n^2 . (b) Determine all n for which D is a divisor of n^2 . Problem 5 Find all functions f from the set R of real numbers to itself such that ( f(x) + f(z) )( f(y) + f(t) ) = f( xy - zt ) + f( xt + yz ) for all x,y,z,t in R . Problem 6 Let Γ1 , Γ2 , … , Γn be circles of radius 1 in the plane, where n≧3. Denote their centres by O1,O2,…,On respectively. Suppose that no line meets more than two of the circles. Prove that Σ ( 1 / OiOj ) ≦ (n-1)π/4 1≦i＜j≦n --

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