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2023 AIME I Problems/Problem 15

Revision as of 12:17, 8 February 2023 by R00tsofunity (talk | contribs) (Created page with "==Problem 15== Find the largest prime number <math>p<1000</math> for which there exists a complex number <math>z</math> satisfying * the real and imaginary parts of <math>z</m...")
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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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