HMMT 十一月 2010 · 冲刺赛 · 第 33 题

HMMT November 2010 — Guts Round — Problem 33

[ 20 ] Convex quadrilateral BCDE lies in the plane. Lines EB and DC intersect at A , with AB = 2, 7 AC = 5, AD = 200, AE = 500, and cos ∠ BAC = . What is the largest number of nonoverlapping 9 circles that can lie in quadrilateral BCDE such that all of them are tangent to both lines BE and CD ?

解析

[ 20 ] Convex quadrilateral BCDE lies in the plane. Lines EB and DC intersect at A , with AB = 2, 7 AC = 5, AD = 200, AE = 500, and cos ∠ BAC = . What is the largest number of nonoverlapping 9 circles that can lie in quadrilateral BCDE such that all of them are tangent to both lines BE and CD ? √ √ 7 1+ 7 θ 2 2 θ 1 9 Answer: 5 Let θ = ∠ BAC , and cos θ = implies cos = = ; sin = ; BC = 9 2 2 3 2 3 √ 7 11 4 + 25 − 2(2)(5) = . Let O be the excircle of 4 ABC tangent to lines AB and AC , and let r 1 1 9 3 r AB + BC + CA θ 1 be its radius; let O be tangent to line AB at point P . Then AP = and = tan = 1 1 1 2 AP 2 1 1 16 √ √ = ⇒ r = . Let O be a circle tangent to O and the lines AB and AC , and let r 1 n n − 1 n 2 2 3 · 2 2 O P θ 1 n n be its radius; let O be tangent to line AB at point P . Then = sin = ; since 4 AP O ∼ n n n n AO 2 3 n 1 O P r r n n n n 4 AP O and O O = r + r , we have = = = = ⇒ n − 1 n − 1 n n − 1 n n − 1 3 AO AO + O O 3 r + r + r n n − 1 n − 1 n n − 1 n n − 1 n − 1 16 √ r = 2 r = 2 . We want the highest n such that O is contained inside 4 ADE . Let the n n − 1 n 3 · 2 2 1100 500+200 − 3 AX 500 2 √ √ incircle of 4 ADE be tangent to AD at X ; then the inradius of 4 ADE is = = . θ tan 2 2 3 · 2 2 2 500 n − 1 √ We want the highest n such that r ≤ ; thus 2 · 16 ≤ 500 = ⇒ n = 5. n 3 · 2 2