HMMT 二月 2016 · 冲刺赛 · 第 24 题

HMMT February 2016 — Guts Round — Problem 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 CB 1 be 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 to i 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2016, February 20, 2016 — GUTS ROUND Organization Team Team ID#

解析

[ 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 CB 1 be 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 to i 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 for k k +1 2 k 1 2016 1 4030 k any k ≥ 4031.