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Value of |PY|/|QY| such that |PQ| is maximized

Source: 1987 USAMO Problem 4

July 24, 2011
geometry unsolvedgeometry

Problem Statement

Three circles CiC_i are given in the plane: C1C_1 has diameter ABAB of length 11; C2C_2 is concentric and has diameter kk (1<k<31 < k < 3); C3C_3 has center AA and diameter 2k2k. We regard kk as fixed. Now consider all straight line segments XYXY which have one endpoint XX on C2C_2, one endpoint YY on C3C_3, and contain the point BB. For what ratio XB/BYXB/BY will the segment XYXY have minimal length?
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