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.

Solution

Simply both sides completely $m^3+mn+m^2n^2+n^3=m^3-3m^2n+3mn^2-n^3$ Canceling out like terms gives us $mn+m^2n^2+n^3=-3m^2n+3mn^2-n^3$ Moving $2n^3$ to the RHS and factoring out the $3n$ gives us $n(m+m^2n-n^2)=3n(-m^2+mn-n^2)$ Only nonzero solutions are needed, so $n$ can be divided off. $m+m^2n-n^2=3(-m^2+mn-n^2)$ Move all terms with factors of $n$ to the RHS and simplifying $m-3m^2=3mn-2n^2-m^2n$ We should remove the cubic term. .......

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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