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 |