PUMaC 2023 · 团队赛 · 第 11 题

PUMaC 2023 — Team Round — Problem 11

Let f ( z ) = for a, b, c, d ∈ C . Suppose that f (1) = i , f (2) = i , and f (3) = i . If the real cz + d m 2 2 part of f (4) can be written as for relatively prime positive integers m, n , find m + n . n i

解析

Let f ( z ) = for a, b, c, d ∈ C . Suppose that f (1) = i , f (2) = i , and f (3) = i . If the real cz + d m 2 2 part of f (4) can be written as for relatively prime positive integers m, n , find m + n . n Proposed by Sunay Joshi and Aleksa Milojevic Answer: 34 Note that M¨ obius transformations (such as f ) preserve the cross ratio z − z z − z 2 1 3 ( z, z ; z , z ) = · 1 2 3 z − z z − z 3 1 2 2 3 In particular, if w = f ( z ), we must have ( z, 1; 2 , 3) = ( w, i ; i , i ). In other words, 2 3 z − 2 1 − 3 w − i i − i · = · 3 2 z − 3 1 − 2 w − i i − i Plugging in z = 4 and solving for w , we find 3 4 w = − i, 5 5 2 2 and so our answer is 3 + 5 = 34. i