Question

A plane contains points $A$ and $B$ with $AB = 1.$ Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A'.$ Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A'$ to point $B'.$ Suppose $AB' = \frac{4}{3}.$ The value of $\cos\theta$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer

Solution:

Work in the complex plane with $B = 0$ and $A = 1.$ Rotating $z$ about $P$ through angle $\varphi$ counterclockwise gives $P + e^{i\varphi}(z - P).$ So $A' = e^{i\theta},$ and rotating $B$ clockwise through $\theta$ about $A'$ gives $B' = A' + e^{-i\theta}(0 - A') = e^{i\theta} - e^{-i\theta}e^{i\theta} = e^{i\theta} - 1.$

Then $AB'^2 = \left|e^{i\theta} - 2\right|^2 = (\cos\theta - 2)^2 + \sin^2\theta = 5 - 4\cos\theta.$ Setting this equal to $\left(\frac{4}{3}\right)^2 = \frac{16}{9}$ gives $4\cos\theta = 5 - \frac{16}{9} = \frac{29}{9},$ so $\cos\theta = \frac{29}{36}$ (indeed positive, consistent with $\theta$ acute). Thus $m + n = 29 + 36 = 65.$

2026 AIME I Problem 5

Solution:

Problem 5 in Other Years