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 \(16^{x^2+y}+16^{x+y^2}=1 \).

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