2005 Indonesia MO Problems/Problem 3
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 |