PUMaC 2023 · 团队赛 · 第 10 题

PUMaC 2023 — Team Round — Problem 10

The sum k cos can be written in the form 4041 k =1 pπ a cos( ) − b q , pπ 2 c sin ( ) q where a, b, c are relatively prime positive integers and p, q are relatively prime positive integers where p < q. Determine a + b + c + p + q. az + b 2 3

解析

The sum k cos can be written in the form 4041 k =1 pπ a cos( ) − b q , 2 pπ c sin ( ) q 4 where a, b, c are relatively prime positive integers and p, q are relatively prime positive integers where p < q. Determine a + b + c + p + q. Proposed by Frank Lu Answer: 4049 2020 P 4 ijπ 4041 We convert this into complex numbers, writing this as the real part of the sum je . j =1 2020 2020 P P 4 ikπ 4041 Using the formula for the sum of a geometric series, we instead write this as e = j =1 k = j 4 jiπ 2020 8084 iπ 8084 iπ 8084 iπ 4 iπ P 8082 iπ/ 4041 4041 4041 4041 4041 4041 e − e 2020 e e − e 1010 e = − . Now, we can rewrite this as + 4 iπ 4 iπ 4 iπ − 2 iπ 2 iπ 2 4041 4041 4041 e − 1 e − 1 ( e − 1) 4041 4041 i (( e − e ) / 2 i ) j =1 8080 iπ 8082 π 8080 π 1010 sin( ) cos( ) − 1 4041 e − 1 4041 4041 . Taking the real part of this yields the expression + . − 2 iπ 2 π 2 π 2 iπ 2 sin( ) 4 sin ( ) 2 4041 4041 4041 4041 (( e − e ) / 2 i ) 2 π cos( ) − 1 4041 But we know that sin(2 π ) = 0 , which means that we can rewrite this as . We thus 2 2 π 4 sin ( ) 4041 get the answer 1 + 1 + 4 + 2 + 4041 = 4049 . az + b 2 3