2013 AIME II Problems/Problem 12

Problem 12

Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$ , where $a$ , $b$ , and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$ .

Solution

Every cubic in real coefficients has to have either three real roots or one real and two nonreal roots which are conjugates. This follows from Vieta's formulas.

Case 1: $f(z)=(z-r)(z-\omega)(z-\omega^*)$ , where $r\in \mathbb{R}$ , $\omega$ is nonreal, and $\omega^*$ is the complex conjugate of omega (note that we may assume that $\Im(\omega)>0$ ).

The real root $r$ must be one of $-20$ , $20$ , $-13$ , or $13$ . By Viète's formulas, $a=-(r+\omega+\omega^*)$ , $b=|\omega|^2+r(\omega+\omega^*)$ , and $c=-r|\omega|^2$ . But $\omega+\omega^*=2\Re{(\omega)}$ (i.e., adding the conjugates cancels the imaginary part). Therefore, to make $a$ is an integer, $2\Re{(\omega)}$ must be an integer. Conversely, if $\omega+\omega^*=2\Re{(\omega)}$ is an integer, then $a,b,$ and $c$ are clearly integers. Therefore $2\Re{(\omega)}\in \mathbb{Z}$ is equivalent to the desired property. Let $\omega=\alpha+i\beta$ .

Subcase 1.1: $|\omega|=20$ .

In this case, $\omega$ lies on a circle of radius $20$ in the complex plane. As $\omega$ is nonreal, we see that $\beta\ne 0$ . Hence $-20<\Re{(\omega)}< 20$ , or rather $-40<2\Re{(\omega)}< 40$ . We count $79$ integers in this interval, each of which corresponds to a unique complex number on the circle of radius $20$ with positive imaginary part.

Subcase 1.2: $|\omega|=13$ .

In this case, $\omega$ lies on a circle of radius $13$ in the complex plane. As $\omega$ is nonreal, we see that $\beta\ne 0$ . Hence $-13<\Re{(\omega)}< 13$ , or rather $-26<2\Re{(\omega)}< 26$ . We count $51$ integers in this interval, each of which corresponds to a unique complex number on the circle of radius $13$ with positive imaginary part.

Therefore, there are $79+51=130$ choices for $\omega$ . We also have $4$ choices for $r$ , hence there are $4\cdot 130=520$ total polynomials in this case.

Case 2: $f(z)=(z-r_1)(z-r_2)(z-r_3)$ , where $r_1,r_2,r_3$ are all real.

In this case, there are four possible real roots, namely $\pm 13, \pm20$ . Let $p$ be the number of times that $13$ appears among $r_1,r_2,r_3$ , and define $q,r,s$ similarly for $-13,20$ , and $-20$ , respectively. Then $p+q+r+s=3$ because there are three roots. We wish to find the number of ways to choose nonnegative integers $p,q,r,s$ that satisfy that equation. By balls and urns, these can be chosen in $\binom{6}{3}=20$ ways.

Therefore, there are a total of $520+20=\boxed{540}$ polynomials with the desired property.

2013 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2013 AIME II Problems/Problem 12

Problem 12

Solution

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