2006 AMC 12A Problem 10 — Solution | LIVE by Po-Shen Loh

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2006 AMC 12A Problem 10

Attempt Problem 10 of the 2006 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

10.

For how many real values of $x$ is $\sqrt{120 - \sqrt{x}}$ an integer?

$3$

$6$

$9$

$10$

$11$

Answer: E

Concepts:radical counting integers in a range

Difficulty rating: 1490

Solution:

Let $k = \sqrt{120 - \sqrt{x}}$ be an integer. Then $k \ge 0$ and $k^2 = 120 - \sqrt{x} \le 120,$ so $0 \le k \le 10.$

Each such $k$ gives $\sqrt{x} = 120 - k^2 \ge 0,$ hence a distinct value $x = (120 - k^2)^2.$ That is $11$ values.

Thus, the correct answer is E.

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Problem 10 in Other Years

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