2007 Indonesia MO Problems/Problem 8
Problem
Let and be two positive integers. If there are infinitely many integers such that is a perfect square, prove that .
Solution (credit to crazyfehmy)
Note that we can complete the square to get , which equals .
Assume that . Since are positive, we know that . In order to prove that is not a perfect square, we can show that there are values of where .
Since , we know that . In the case where , we can expand and simplify to get
All steps are reversible, so there are values of where , so there are no values of where that results in infinite number of integers that satisfy the original conditions.
Now assume that . Since are positive, we know that . In order to prove that is not a perfect square, we can show that there are values of where .
Since , we know that . In the case where , we can expand and simplify to get
All steps are reversible, so there are values of where , so there are no values of where that results in infinite number of integers that satisfy the original conditions.
Now we need to prove that if , there are an infinite number of integers that satisfy the original conditions. By the Substitution Property, we find that . The expression can be factored into . Since the expression is a perfect square, for all integer values of , there are an infinite number of integers that satisfies the original conditions when .
See Also
2007 Indonesia MO (Problems) | ||
Preceded by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Last Problem |
All Indonesia MO Problems and Solutions |