PUMaC 2021 · 团队赛 · 第 5 题

PUMaC 2021 — Team Round — Problem 5

Given a real number t with 0 < t < 1, define the real-valued function f ( t, θ ) = t ω , n = −∞ iθ where ω = e = cos θ + i sin θ . For θ ∈ [0 , 2 π ), the polar curve r ( θ ) = f ( t, θ ) traces out an ellipse E with a horizontal major axis whose left focus is at the origin. Let A ( t ) be the area of t 1 aπ the ellipse E . Let A ( ) = , where a, b are relatively prime positive integers. Find 100 a + b . t 2 b

解析

Given a real number t with 0 < t < 1, define the real-valued function f ( t, θ ) = t ω , n = −∞ iθ where ω = e = cos θ + i sin θ . For θ ∈ [0 , 2 π ), the polar curve r ( θ ) = f ( t, θ ) traces out an ellipse E with a horizontal major axis whose left focus is at the origin. Let A ( t ) be the area of t 1 aπ the ellipse E . Let A ( ) = , where a, b are relatively prime positive integers. Find 100 a + b . t 2 b Proposed by: Sunay Joshi Answer: 503 Note that f ( t, θ ) can be written as the sum of two geometric series: X X n n n − n f ( t, θ ) = t ω + t ω − 1 . n ≥ 0 n ≥ 0 Using the formula for the sum of a geometric series, we find 2 1 − t 2 1 − t 2 1+ t f ( t, θ ) = = . 2 t 2 1 + t − 2 t cos θ 1 − cos θ 2 1+ t The polar formula for an ellipse whose left focus is at the origin and whose major axis is along √ 2 2 2 a (1 − e ) a − b the x -axis is given by r ( θ ) = , where e = , where a and b are the semimajor and 1 − e cos θ a 2 t semiminor axes, respectively. Matching parameters to r ( θ ) = f ( t, θ ), we find e = and 2 1+ t 2 2 2 1 − t 1+ t a (1 − e ) = . Solving for a, b , we find a = and b = 1. 2 2 1+ t 1 − t 2 5 / 4 1+ t 1 1 5 π Thus the area is given by A ( t ) = πab = π . At t = , we have A ( ) = π · = . Thus 2 1 − t 2 2 3 / 4 3 our answer is 500 + 3 = 503.