2011 AMC 12A Problem 21

Below is the professionally curated solution for Problem 21 of the 2011 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 12A solutions, or check the answer key.

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Concepts:function radical recursion

Difficulty rating: 2270

21.

Let $f_1(x) = \sqrt{1 - x},$ and for integers $n \ge 2,$ let $f_n(x) = f_{n-1}\!\left(\sqrt{n^2 - x}\right).$ If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is $\{c\}.$ What is $N + c?$

$-226$

$-144$

$-20$

$20$

$144$

Solution:

Each step requires $\sqrt{n^2 - x}$ to lie in the domain of $f_{n-1}.$ Tracking the domains:

$f_1: (-\infty, 1].$ $f_2: \sqrt{4 - x} \in (-\infty, 1] \Rightarrow [3, 4].$ $f_3: \sqrt{9 - x} \in [3, 4] \Rightarrow [-7, 0].$ $f_4: \sqrt{16 - x} \in [-7, 0] \Rightarrow \{16\}$ (only the value $0$ is possible). $f_5: \sqrt{25 - x} = 16 \Rightarrow \{-231\}.$

For $f_6$ we would need $\sqrt{36 - x} = -231,$ impossible, so the domain is empty. Hence $N = 5,$ $c = -231,$ and $N + c = -226.$

Thus, the correct answer is A.

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