Maths Olympiad Exam

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.)

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Maths Olympiad Other Question

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. Let ABC be a triangle and let BB1, CC1 be respectively the bisectors of angle B, C with B1 on AC and C1 on AB. Let E, F be the feet of perpendiculars drawn from A onto BB1, CC1 respectively. Suppose D is the point at which the incircle of ABC touches AB. Prove that AD = EF