\[f(x) = \sqrt{x - \sqrt{x}}.\]
Find the largest three-digit value of $x$ such that $f(x)$ is an integer.

 Dec 14, 2023

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​.

 Dec 17, 2023

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