Let

\[f(x) = \sqrt{x - \sqrt{x}}.\]

Find the largest three-digit value of $x$ such that $f(x)$ is an integer.

sandwich Dec 14, 2023

#1**0 **

To find the largest three-digit value of x such that f(x) is an integer, we need to analyze the properties of f(x):

Domain: Since the inner square root cannot be negative, the expression under the outer square root must be non-negative, i.e., x−x≥0. This simplifies to x≥x.

Integer solutions: We want f(x) to be an integer. Since taking the square root can only produce rational numbers (except for perfect squares), x−x must be a perfect square.

Three-digit limit: We want the largest three-digit value of x. Knowing that x≥x, the smallest possible value of x for which f(x) is an integer is when x−x=1, resulting in x=2. Therefore, we need to investigate values of x from 100 down to 2.

Checking perfect squares: Starting from 99 (the largest three-digit perfect square), we check if x−x is a perfect square for any integers smaller than 99.

We find that:

For x=99, x−x=87.

For x=64, x−x=53.

For x=49, x−x=40.

For x=36, x−x=27.

For x=25, x−x=16.

Therefore, among three-digit values, the largest values of x for which f(x) is an integer are 99, 64, 49, 36, and 25.

Determining the largest value: Among these values, 99 is the largest three-digit integer. So, the largest three-digit value of x such that f(x) is an integer is 99.

BuiIderBoi Dec 17, 2023