HMMT 二月 2016 · 冲刺赛 · 第 24 题

英文解析

24. [ 12 ] Let ∆ A B C be a triangle with ∠ A B C = 90 and = 5 + 2. For any i ≥ 2, define A to
    1 1 1 1 i
    1CBbe the point on the line A C such that A B ⊥ A C and define B to be the point on the line B C
    1 i i − 1 1 i 1
    such that A B ⊥ B C . Let Γ be the incircle of ∆ A B C and for i ≥ 2, Γ be the circle tangent toi i 1 1 1 1 i
    Γ , A C, B C which is smaller than Γ .
    i − 1 1 1 i − 1
    How many integers k are there such that the line A B intersects Γ ?
    1 2016 k
    Proposed by:
    Answer: 4030
    We claim that Γ is the incircle of 4 B A C . This is because 4 B A C is similar to A B C with
    2 1 2 1 2 1 1
    √
    dilation factor 5 − 2 , and by simple trigonometry, one can prove that Γ is similar to Γ with the
    2 1
    same dilation factor. By similarities, we can see that for every k , the incircle of 4 A B C is Γ ,
    k k 2 k − 1
    and the incircle of 4 B A C is Γ . Therefore, A B intersects all Γ , . . . , Γ but not Γ fork k +1 2 k 1 2016 1 4030 kany k ≥ 4031.