1987 USAMO Problems/Problem 1

Problem

Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$ , where m and n are non-zero integers.

Do it

Solution

Expanding both sides, $m^3+mn+m^2n^2+n^3=m^3-3m^2n+3mn^2-n^3$ Note that $m^3$ can be canceled and as $n \neq 0$ , $n$ can be factored out. Writing this as a quadratic equation in $n$ : $2n^2+(m^2-3m)n+(3m^2+m)=0$ . The discriminant $b^2-4ac$ equals $(m^2-3m)^2-8(3m^2+m)$ $=m^4-6m^3-15m^2-8m$ , which we want to be a perfect square. Miraculously, this factors as $m(m-8)(m+1)^2$ . This is square iff (if and only if) $m^2-8m$ is square or $m+1=0$ . It can be checked that the only nonzero $m$ that work are $-1, 8, 9$ . Finally, plugging this in and discarding extraneous roots gives all possible ordered pairs $(m, n)$ as $\{(-1,-1),(8,-10),(9,-6),(9,-21)\}$ .

1987 USAMO (Problems • Resources)
Preceded by First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5
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1987 USAMO Problems/Problem 1

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