Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2002 AMC 10P Problems/Problem 12"

(→‎Solution 1)
(→‎Solution 1)
Line 57: Line 57:
 
=a^{2002}</math>
 
=a^{2002}</math>
  
−
Thus, our answer is <math>\box{\textbf{(C) }\text{ III and IV only}}.</math>
+
Thus, our answer is <math>\boxed{\textbf{(C) }\text{ III and IV only}}.</math>
  
 
== See also ==
 
== See also ==
 
{{AMC10 box|year=2002|ab=P|num-b=11|num-a=13}}
 
{{AMC10 box|year=2002|ab=P|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 04:53, 15 July 2024

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

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2002_AMC_10P_Problems/Problem_12&oldid=222526"