MathDB

Prime Numbers, Triangle Inequality, Complex… Oh My!

Source: 2023 AIME I #15

February 8, 2023

AMCAIMEAIME I

Problem Statement

Find the largest prime number

p<1000

for which there exists a complex number

z

satisfying [*] the real and imaginary part of

z

are both integers; [*]

|z|=\sqrt{p}

, and [*] there exists a triangle whose three side lengths are

p

, the real part of

z^{3}

, and the imaginary part of

z^{3}