2014 UMO Problems/Problem 2
Problem
(a) Find all positive integers and
that satisfy
or prove that there are no solutions.
(b) Find all positive integers and
that satisfy
or prove that there are no
solutions.
Solution
(a) We see that we can rewrite as
. Since
and
are perfect squares, their modulo can only be
. Since none of those two combinations make
, there are no solutions to
such that
.
(b) Similarly, we can rewrite as
and therefore it also does not have integer solutions.
See Also
2014 UMO (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All UMO Problems and Solutions |