2023 AIME I Problems/Problem 15

Problem 15

Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying

the real and imaginary parts of $z$ are integers;
$|z|=\sqrt{p}$ , and
there exists a triangle with side lengths $p$ , the real part of $z^{3}$ , and the imaginary part of $z^{3}$ .

Answer: 349

Suppose $z=a+bi$ ; notice that $\arg(z^{3})\approx 45^{\circ}$ , so by De Moivre’s theorem $\arg(z)\approx 15^{\circ}$ and $\tfrac{b}{a}\approx tan(15^{\circ})=2-\sqrt{3}$ . Now just try pairs $(a, b)=(t, \left(2-\sqrt{3}\right)t)$ going down from $t=\lfloor\sqrt{1000}\rfloor$ , writing down the value of $a^{2}+b^{2}$ on the right; and eventually we arrive at $(a, b)=(18, 5)$ the first time $a^{2}+b^{2}$ is prime. Therefore, $p=\boxed{349}$ .

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2023 AIME I Problems/Problem 15

Problem 15

Answer: 349