2005 Indonesia MO Problems/Problem 3
Revision as of 11:33, 17 September 2019 by Rockmanex3 (talk | contribs) (Solution to Problem 3 -- squares)
Problem
Let and be positive integers such that is an integer.
(a) Prove that is rational.
(b) Prove that is a positive integer.
Solution
Let , where all variables are integers. Rearranging the expression results in .
Squaring both sides results in , and rearranging terms results in .
Squaring both sides results in . Solving for results in . Since the right side is rational, the left side must be rational. Therefore, is rational, and since is a positive integer, must be a positive integer.
See Also
2005 Indonesia MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 4 |
All Indonesia MO Problems and Solutions |