2021 April MIMC 10 Problems/Problem 21
How many solutions are there for the equation . (Recall that
is the largest integer less than
, and
is the smallest integer larger than
.)
Solution
Let's split this question into different parts.
First case: . In this case, both
and
has to be equal to
. Therefore,
. Factoring this equation, we get that
which provides two solutions,
and
.
Second case: . In this case,
must be equal to
. Setting
, we can get that
. Calculating the discriminant which is
, we know that
is an irrational number. However, since
is a floor function of a number, it is always an integer. Therefore, there is no solution for this case.
In total, there are solutions for the equation.