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.

Let be a permutation (that is arearrangement) of the numbers 1, 2, 3 . . . , 2011. Show that there exist two numbers j, k. such that and .

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 is a perfect square.

Consider a 20-sided convex polygon K, with vertice 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 is an admissible triple while is not).

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