Maths Olympiad Exam

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.

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Maths Olympiad Other Question

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. Consider a 20-sided convex polygon K, with vertices A1,A2, . . . ,A20 in that order. Find the number of ways in which three sides of K can be chosen so that every pair among them has at least two sides of K between them. (Forexample (A1A2,A4A5,A11A12) is an admissible triple while (A1A2,A4A5,A19A20)is not.)