HMMT 二月 2010 · 冲刺赛 · 第 21 题

HMMT February 2010 — Guts Round — Problem 21

[ 10 ] Let 4 ABC be a scalene triangle. Let h be the locus of points P such that | P B − P C | = | AB − AC | . a Let h be the locus of points P such that | P C − P A | = | BC − BA | . Let h be the locus of points P b c such that | P A − P B | = | CA − CB | . In how many points do all of h , h , and h concur? a b c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUND

解析

[ 10 ] Let 4 ABC be a scalene triangle. Let h be the locus of points P such that | P B − P C | = | AB − AC | . a Let h be the locus of points P such that | P C − P A | = | BC − BA | . Let h be the locus of points P b c such that | P A − P B | = | CA − CB | . In how many points do all of h , h , and h concur? a b c Answer: 2 The idea is similar to the proof that the angle bisectors concur or that the perpendicular bisectors concur. Assume WLOG that BC > AB > CA . Note that h and h are both hyperbolas. a b Therefore, h and h intersect in four points (each branch of h intersects exactly once with each branch a b a of h ). Note that the branches of h correspond to the cases when P B > P C and when P B < P C . b a Similarly, the branches of h correspond to the cases when P C > P A and P C < P A . b If either P A < P B < P C or P C < P B < P A (which each happens for exactly one point of intersection of h and h ), then | P C − P A | = | P C − P B | + | P B − P A | = | AB − AC | + | BC − BA | = | BC − AC | , a b and so P also lies on h . So, exactly two of the four points of intersection of h and h lie on h , c a b c meaning that h , h , and h concur in four points. a b c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUND