# 2006 Alabama ARML TST Problems/Problem 14

## Problem

Find the real solution $(x, y)$ to the system of equations $x^3-3xy^2=-610,$ $3x^2y-y^3=182.$

## Solution

Note that $(x+yi)^3 = x^3 + 3x^2yi + 3xy^2i^2 + y^3i^3 = (x^3-3xy^2) + (3x^2y-y^3)i = -610 + 182i$

So, we need to find $(x+yi)$ that satisfies $(x+yi)^3 = -610 + 182i$. Even though there are three solutions, for simplicity, lets assume that one exists in the first quadrant in the complex plane, and try to find that one.

Note that $|(x+yi)^3|^2 = (-610)^2 + (182)^2 = 372100 + 33124 = 405224 = 8 \cdot 50653 = 2^3 \cdot 37^3$.

Thus, $|x+yi|^2 = \sqrt{2^3 \cdot 37^3} = 2\cdot37 = 74$.

Therefore, $x^2 + y^2 = 74$. Note that $(x,y) = ( 5,7)$ and $(7,5)$ are the only positive integer solutions to $x^2 + y^2 = 74$. (Even though the solution may be non-integral, this is a good place to start.)

However, letting $r \cdot cis(\theta) = x+yi$, [ $(x+yi)^3 = r^3 \cdot cis(3\theta)$] and bounding $\theta$ yields: $-1 < \tan{(3\theta)} = -\dfrac{182}{610} < 0$. $135^\circ < 3\theta < 180^\circ$. (Since $3\theta$ is obviously in the 2nd quadrant). $45^\circ < \theta < 60^\circ$. $\tan{(\theta)} > 1$.

Thus, $y>x$. So, the only remaining positive integer solution is $(x,y) = (5,7)$.

As a quick check: $(5+7i)^3 = (5)^3 + 3(5)^2(7)i + 3(5)(7)^2i^2 + (7)^3i^3 = -610 + 182i$

Thus, the solution is $(x,y) = \boxed{(5,7)}$.

Note that $(5+7i)cis(120^\circ) = \left(\dfrac{-5-7\sqrt{3}}{2}\right) + \left(\dfrac{-7+5\sqrt{3}}{2}\right)i$ and $(5+7i)cis(240^\circ) = \left(\dfrac{-5+7\sqrt{3}}{2}\right) + \left(\dfrac{-7-5\sqrt{3}}{2}\right)i$

Thus, $(x,y) = \left(\dfrac{-5-7\sqrt{3}}{2},\dfrac{-7+5\sqrt{3}}{2} \right)$ and $\left(\dfrac{-5+7\sqrt{3}}{2},\dfrac{-7-5\sqrt{3}}{2} \right)$ are also solutions.