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, 2018

Geometry Difficulty 5.5 AIME, harder Prove it United States

Problem:

Circle $\omega_{1}$ of radius $1$ and circle $\omega_{2}$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_{1}$ and regular pentagon $MONEY$ in $\omega_{2}$ . It then writes down all $20$ (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

Solution

Solution:

We represent the vertices with complex numbers. Place the vertices of $CASH$ at $1$ , $i$ , $-1$ , $-i$ and the vertices of $MONEY$ at $2\alpha$ , $2\alpha\omega$ , $2\alpha\omega^{2}$ , $2\alpha\omega^{3}$ , $2\alpha\omega^{4}$ with $|\alpha|=1$ and $\omega=e^{\frac{2\pi i}{5}}$ . We have that the product of distances from a point $z$ to the vertices of $CASH$ is $|(z-1)(z-i)(z+1)(z+i)|=|z^{4}-1|$ , so we want to maximize

$| (16\alpha^{4}-1)(16\alpha^{4}\omega^{4}-1)(16\alpha^{4}\omega^{3}-1)(16\alpha^{4}\omega^{2}-1)(16\alpha^{4}\omega-1) |$

which just comes out to be $|2^{20}\alpha^{20}-1|$ . By the triangle inequality, this is at most $2^{20}+1$ , and it is clear that some $\alpha$ makes equality hold.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.