AMC 10 Daily Practice Round 3

Since the floor function only outputs integers and is an integer, must be an integer. Let for integers (to prevent the number under the square root from being negative). If is incremented by will always increase. While or may not increase, they will never decrease. Thus, is a strictly increasing function along the nonnegative integers. Clearly, and . It should be checked if there is an integer such that (if not, the integer where is as closest to as possible). Clearly, and . Since is strictly increasing, if exists. In this range, can only be 44 and can only be .

Then, , which lies in the range. Hence, . By the strictly increasing behavior of , the values are distinct and the only values of that can be one of the first positive integers. Thus, exactly of those integers are covered by , so the number of integers not covered by is .