HMMT 二月 2022 · 冲刺赛 · 第 29 题

HMMT February 2022 — Guts Round — Problem 29

[16] Let a ̸ = b be positive real numbers and m, n be positive integers. An m + n -gon P has the property that m sides have length a and n sides have length b . Further suppose that P can be inscribed in a circle of radius a + b . Compute the number of ordered pairs ( m, n ), with m, n ≤ 100, for which such a polygon P exists for some distinct values of a and b .

解析

[16] Let a ̸ = b be positive real numbers and m, n be positive integers. An m + n -gon P has the property that m sides have length a and n sides have length b . Further suppose that P can be inscribed in a circle of radius a + b . Compute the number of ordered pairs ( m, n ), with m, n ≤ 100, for which such a polygon P exists for some distinct values of a and b . Proposed by: Daniel Zhu Answer: 940 a Solution: Letting x = , we have to solve a + b x 1 − x m arcsin + n arcsin = π. 2 2 This is convex in x , so if it is to have a solution, we must find that the LHS exceeds π at one of the endpoints. Thus max( m, n ) ≥ 7. If min( m, n ) ≤ 5 we can find a solution by by the intermediate value theorem. Also if min( m, n ) ≥ 7 then x 1 − x m arcsin + n arcsin ≥ 14 arcsin(1 / 4) > π. 2 2 π The inequality arcsin(1 / 4) > can be verified by noting that 14 π π 3 . 5 1 sin < < = . 14 14 14 4 The final case is when min( m, n ) = 6. We claim that this doesn’t actually work. If we assume that n = 6, we may compute the derivative at 0 to be √ m 1 m − 48 − 6 · √ = > 0 , 2 2 3 so no solution exists.