HMMT 十一月 2010 · 冲刺赛 · 第 33 题

英文解析

33. [ 20 ] Convex quadrilateral BCDE lies in the plane. Lines EB and DC intersect at A , with AB = 2,
    AC = 5, AD = 200, AE = 500, and cos ∠ BAC = . What is the largest number of nonoverlapping 7
    circles that can lie in quadrilateral BCDE such that all of them are tangent to both lines BE and 9
    CD ?
    √
    √
    1+7
    7 θ 2 2 θ 1
    Answer: 5 Let θ = ∠ BAC , and cos θ = implies cos = = ; sin = ; BC =9
    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
    AB + BC + CA θrbe its radius; let O be tangent to line AB at point P . Then AP = and = tan =1
    1 1 1
    2 AP 2
    1 161
    √ √
    = ⇒ 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 nbe 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
    1 O P r rnn 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 + rn 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 then n − 1 n
    3 · 2 2
    1100
    500+200 −
    AX 5003
    √ √2
    incircle of 4 ADE be tangent to AD at X ; then the inradius of 4 ADE is = = .
    tanθ
    2 2 3 · 2 2
    500 n − 12
    √
    We want the highest n such that r ≤ ; thus 2 · 16 ≤ 500 = ⇒ n = 5.
    3 · 2 2 n