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3

Part of 2009 Costa Rica - Final Round

Problems(1)

Costa Rican Math Olympiad Problem 3 2009

Source:

11/29/2009
Let triangle ABC ABC acutangle, with m∠ACB≤ m∠ABC m \angle ACB\leq\ m \angle ABC. M M the midpoint of side BC BC and P P a point over the side MC MC. Let C1 C_{1} the circunference with center C C. Let C2 C_{2} the circunference with center B B. P P is a point of C1 C_{1} and C2 C_{2}. Let X X a point on the opposite semiplane than B B respecting with the straight line AP AP; Let Y Y the intersection of side XB XB with C2 C_{2} and Z Z the intersection of side XC XC with C1 C_{1}. Let m\angle PAX \equal{} \alpha and m\angle ABC \equal{} \beta. Find the geometric place of X X if it satisfies the following conditions: (a) \frac {XY}{XZ} \equal{} \frac {XC \plus{} CP}{XB \plus{} BP} (b) \cos(\alpha) \equal{} AB\cdot \frac {\sin(\beta )}{AP}
trigonometrygeometry unsolvedgeometry