2023 AMC 10B Problems/Problem 22

Problem

How many distinct values of 𝑥 satisfy where denotes the largest integer less than or equal to 𝑥?

Solution (Quick)

A quadratic equation can have up to 2 real solutions. With the , it could also help generate another pair. We have to verify that the solutions are real and distinct.

First, we get the trivial solution by ignoring the floor. , we get as our first pair of solutions.

Up to this point, we can rule out A,E.

Next, we see that This implies that must be an integer. We can guess and check as which yields

So we got 4 in total

~Technodoggo