1987 USAMO Problems/Problem 1

Revision as of 00:12, 2 April 2017 by A1b2 (talk | contribs)

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. .......

See Also

1987 USAMO (ProblemsResources)
Preceded by
First
Problem
Followed by
Problem 2
1 2 3 4 5
All USAMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png