Maths Olympiad Exam

Let (a1, a2, a3, . . . , a2011) be a permutation (that is a rearrangement) of the numbers
1, 2, 3, . . . , 2011. Show that there exist two numbers j, k such that 1 j < k 2011 and aj + j = ak + k.

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Maths Olympiad Other Question

Let ABC be a triangle. Let D, E, F be points respectively on the segments BC, CA, AB such that AD, BE, CF concur at the point K. Suppose BD/DC = BF/FA and ADB = AFC. Prove that ABE = CAD A natural number n is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting k from n and the larger one is obtained by adding l to n. Prove that n+kl is a perfect square.