1. Given triangle ABC the point J is the centre of the excircle opposite the vertex A. This excircle is tangent to the side BC at M, and to the lines AB and AC at K and L, respectively. The lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of intersection of the lines AF and BC , and let T be the point of intersection of the lines AG and BC. Prove that M is the midpoint of ST. (The excircle of ABC opposite the vertex A is the circle that is tangent to the line segment BC, to the ray AB beyond B, and to the ray AC beyond C.) 2. If positive reals a_2, a_3,...,a_n satisfy a_2 * a_3 * ... * a_n = 1 and n > 2 prove that (a_2 + 1)^2 * (a_3 + 1)^3 * ... * (a_n + 1)^n > n^n 3. The "liar's" guessing game is a game played between two players A and B. The rules of the game depend on two positive integers k and n which are known to both players. At the start of the game A chooses integers x and N with 1≦x≦N. Player keeps x secret, and truthfully tells N to player B. Player B now tries to obtain information about x by asking player A questions as follows: each question consists of B specifying an arbitrary set S of positive integers (possibly one specified in some previous question), and asking A whether x belongs to S. Player B may ask as many questions as he wishes. After each question, player A must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is that, among any k+1 consecutive answers, at least one answer must be truthful. After B has asked as many questions as he wants, he must specify a set X of at most n positive integers. If x belongs to X, then wins; otherwise, he loses. Prove that: 1. If n≧2^k, then B can guarantee a win. 2. For all sufficiently large k , there exists an integer n ≧(1.99)^k such that B cannot guarantee a win. --

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