MathDB
Heated Argument

Source: 2014 AIME I Problem 7

March 14, 2014
trigonometrycalculusderivativevectorratioanalytic geometrygraphing lines

Problem Statement

Let ww and zz be complex numbers such that ∣w∣=1|w| = 1 and ∣z∣=10|z| = 10. Let θ=arg⁡(w−zz)\theta = \arg\left(\tfrac{w-z}{z}\right). The maximum possible value of tan⁡2θ\tan^2 \theta can be written as pq\tfrac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q. (Note that arg⁡(w)\arg(w), for w≠0w \neq 0, denotes the measure of the angle that the ray from 00 to ww makes with the positive real axis in the complex plane.
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