HMMT 十一月 2015 · 冲刺赛 · 第 6 题

HMMT November 2015 — Guts Round — Problem 6

[ 6 ] Let AB be a segment of length 2 with midpoint M . Consider the circle with center O and radius r that is externally tangent to the circles with diameters AM and BM and internally tangent to the circle with diameter AB . Determine the value of r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2015, November 14, 2015 — GUTS ROUND Organization Team Team ID#

解析

[ 6 ] Let AB be a segment of length 2 with midpoint M . Consider the circle with center O and radius r that is externally tangent to the circles with diameters AM and BM and internally tangent to the circle with diameter AB . Determine the value of r . Proposed by: Sam Korsky 1 Answer: 3 1 1 Let X be the midpoint of segment AM . Note that OM ⊥ M X and that M X = and OX = + r 2 2 and OM = 1 − r . Therefore by the Pythagorean theorem, we have ( ) 2 1 1 2 2 2 2 OM + M X = OX = ⇒ (1 − r ) + = + r 2 2 2 1 which we can easily solve to find that r = . 3