Difference between revisions of "2002 AMC 10P Problems/Problem 12"
(→Solution 2) |
(→See also) |
||
(One intermediate revision by one other user not shown) | |||
Line 61: | Line 61: | ||
== Solution 2 == | == Solution 2 == | ||
This is the much more realistic and less-time-consuming approach. | This is the much more realistic and less-time-consuming approach. | ||
− | Notice that all answer choices except <math>\text(C)</math> include <math>\text{II}.</math> in them. Therefore, it is sufficient to prove that <math>\text{II}.</math> is false. Similar to solution 1, a quick glance tells us: | + | Notice that all answer choices except <math>\text{(C)}</math> include <math>\text{II}.</math> in them. Therefore, it is sufficient to prove that <math>\text{II}.</math> is false. Similar to solution 1, a quick glance tells us: |
<math>\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)</math> | <math>\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)</math> | ||
Line 75: | Line 75: | ||
{{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}} | ||
+ | [[Category: Introductory Algebra Problems]] |
Latest revision as of 21:44, 15 October 2025
Contents
Problem 12
For and
consider
Which of these equal
Solution 1
We can solve this problem with a case by case check of and
Since
all cases must equal
Thus, our answer is
Solution 2
This is the much more realistic and less-time-consuming approach.
Notice that all answer choices except include
in them. Therefore, it is sufficient to prove that
is false. Similar to solution 1, a quick glance tells us:
Therefore, by process of elimination, our answer is
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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.