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 (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|.
3. 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.
4. 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. (For example (A1A2,A4A5,A11A12) is an admissible triple while (A1A2,A4A5,A19A20) is not.)
5. Let ABC be a triangle and let BB1, CC1 be respectively the bisectors of ∠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.
6. Find all pairs (x, y) of real numbers such that 16x2+y + 16x+y2= 1.
Back