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 .)
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.