PUMaC 2023 · 团队赛 · 第 2 题

PUMaC 2023 — Team Round — Problem 2

Let Γ and Γ be externally tangent circles with radii and , respectively. The line ℓ is a 1 2 2 8 common external tangent to Γ and Γ . For n ≥ 3, we define Γ as the smallest circle tangent 1 2 n a to Γ , Γ , and ℓ . The radius of Γ can be expressed as where a, b are relatively prime n − 1 n − 2 10 b positive integers. Find a + b .

解析

Let Γ and Γ be externally tangent circles with radii and , respectively. The line ℓ is a 1 2 2 8 common external tangent to Γ and Γ . For n ≥ 3, we define Γ as the smallest circle tangent 1 2 n a to Γ , Γ , and ℓ . The radius of Γ can be expressed as where a, b are relatively prime n − 1 n − 2 10 b positive integers. Find a + b . Proposed by Adam Huang Answer: 15843 1 1 1 1 √ √ √ √ Note that the radii r , r , r satisfy the recurrence + = . Let a := . n − 2 n − 1 n n r r r r n − 2 n − 1 n n √ √ Then a obeys the Fibonacci recurrence with initial conditions a = 2 and a = 2 2. It n 1 2 √ √ 1 1 2 √ follows that a = 89 2, so that = 89 2, r = and our answer is a + b = 1+89 · 2 = 10 10 2 r 89 · 2 10