Difference between revisions of "2014 AMC 12A Problems/Problem 24"

Revision as of 12:27, 13 February 2014

Let $f_0(x)=x+|x-100|-|x+100|$ , and for $n\geq 1$ , let $f_n(x)=|f_{n-1}(x)|-1$ . For how many values of $x$ is $f_{100}(x)=0$ ?

$\textbf{(A) }299\qquad \textbf{(B) }300\qquad \textbf{(C) }301\qquad \textbf{(D) }302\qquad \textbf{(E) }303\qquad$

1. Draw the graph of $f_0(x)$ by dividing the domain into three parts.

2. Look at the recursive rule. Take absolute of the previous function and down by 1 to get the next function.

3. Count the x intercepts of the each function and find the pattern.

The pattern turns out to be $3n+1$ solutions, and the answer is thus $\textbf{(C) }301\qquad$ . (Revised by Flamedragon)

2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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 ==Solution==
-. Draw the graph of f_0(x) by dividing the domain into three parts.
+. Draw the graph of <math>f_0(x)</math> by dividing the domain into three parts.
 . Look at the recursive rule. Take absolute of the previous  function and down by 1 to get the next function.
-. Count the x intercepts of the each function and find the pattern
+. Count the x intercepts of the each function and find the pattern.
+The pattern turns out to be <math>3n+1</math> solutions, and the answer is thus <math>\textbf{(C) }301\qquad</math>.
 (Revised by Flamedragon)