# RMO 2011

1. 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.
2. Let $$(a_1a_2a_3…..a_{2011})$$ be a permutation (that is arearrangement) of the numbers 1, 2, 3 . . . , 2011. Show that there exist two numbers j, k. such that $$1 \le j < k \le 2011$$ and $$|a_{j}-j|=|a_{k}-k|$$.
3. A natural number n is chosen strictly between two consecutive perfect square. 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.
4. Consider a 20-sided convex polygon K, with vertice $$A_1,A_2,….,A_{20}$$ 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. (For example $$A_1A_2, A_4A_5, A_{11}A_{12}$$ is an admissible triple while $$A_1A_2, A_4A_5, A_{19}A_{20}$$ is not).
5. Let ABC be a triangle and let $$BB_1,CC_1$$ be respectively the bisectors of ∠ B, ∠ C with $$B_1$$ on AC and $$C_1$$ on AB. Let E, F be the feet of perpendiculars drawn from A onto $$BB_1,CC_1$$ respectively. Suppose D is the point at which the incircle of ABC touches AB. Prove that AD = EF.
6. Find all pairs (x, y) of real numbers such that $$16^{x^2+y}+16^{x+y^2}=1$$.