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2002 AMC 10P Problems/Problem 12

Revision as of 04:56, 15 July 2024 by Wes (talk | contribs) (→‎Solution 1)

Problem 12

For $f_n(x)=x^n$ and $a \neq 1$ consider

$\text{I. } (f_{11}(a)f_{13}(a))^{14}$

$\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)$

$\text{III. } (f_{11}(f_{13}(a)))^{14}$

$\text{IV. } f_{11}(f_{13}(f_{14}(a)))$

Which of these equal $f_{2002}(a)?$

$\text{(A) I and II only} \qquad \text{(B) II and III only} \qquad \text{(C) III and IV only} \qquad \text{(D) II, III, and IV only} \qquad \text{(E) all of them}$

Solution 1

We can solve this problem with a case by case check of $\text{I., II., III.,}$ and $\text{IV.}$ Since $f_n=x^n,$ $f_{2002}(a)=a^{2002},$ all cases must equal $a^{2002}.$

$\text{I. } (f_{11}(a)f_{13}(a))^{14}$

$(f_{11}(a)f_{13}(a))^{14} =(a^{11}a^{13})^{14} = (a^{24})^14 = a^{336} \neq a^{2002}$

$\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)$

$f_{11}(a)f_{13}(a)f_{14}(a) =a^{11}a^{13}a^{14} =a^{38} \neq a^{2002}$

$\text{III. } (f_{11}(f_{13}(a)))^{14}$

$(f_{11}(f_{13}(a)))^{14} =((a^{13})^{11})^{14} =a^{13 \cdot 11 \cdot 14} =a^{2002}$

$\text{IV. } f_{11}(f_{13}(f_{14}(a)))$

$f_{11}(f_{13}(f_{14}(a))) =((a^{14})^{13})^{11} =a^{14 \cdot 13 \cdot 11} =a^{2002}$

Thus, our answer is $\boxed{\textbf{(C) }\text{ III and IV only}}.$

Solution 2

This is the much more realistic and less-time-consuming approach. Notice that all answer choices except $\text(C)$ include $\text{II}.$ in them. Therefore, it is sufficient to prove that $\text{II}.$ is false. Similar to solution 1, a quick glance tells us:

$\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)$

$f_{11}(a)f_{13}(a)f_{14}(a) =a^{11}a^{13}a^{14} =a^{38} \neq a^{2002}$

Therefore, $\boxed{\textbf{(C) }\text{ III and IV only}}.$

See also

2002 AMC 10P (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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 10 Problems and Solutions

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