Difference between revisions of "1987 USAMO Problems/Problem 1"
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==Problem== | ==Problem== | ||
− | Find all solutions to <math>(m^2+n)(m + n^2)= (m - n)^3</math>, where m and n are non-zero integers. | + | Find all solutions to <math>(m^2+n)(m + n^2)= (m - n)^3</math>, where m and n are non-zero integers. |
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+ | Do it | ||
==Solution== | ==Solution== |
Latest revision as of 02:27, 11 May 2024
Problem
Find all solutions to , where m and n are non-zero integers.
Do it
Solution
Expanding both sides,
Note that
can be canceled and as
,
can be factored out.
Writing this as a quadratic equation in
:
.
The discriminant
equals
, which we want to be a perfect square.
Miraculously, this factors as
. This is square iff (if and only if)
is square or
. It can be checked that the only nonzero
that work are
. Finally, plugging this in and discarding extraneous roots gives all possible ordered pairs
as
.
See Also
1987 USAMO (Problems • Resources) | ||
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.