Difference between revisions of "2004 AMC 12B Problems/Problem 2"
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{{duplicate|[[2004 AMC 12B Problems|2004 AMC 12B #2]] and [[2004 AMC 10B Problems/Problem 5|2004 AMC 10B #5]]}} | {{duplicate|[[2004 AMC 12B Problems|2004 AMC 12B #2]] and [[2004 AMC 10B Problems/Problem 5|2004 AMC 10B #5]]}} | ||
== Problem 2 == | == Problem 2 == | ||
| − | In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result? | + | In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are <math>0</math>, <math>1</math>, <math>2</math>, and <math>3</math>, although not necessarily in that order. What is the maximum possible value of the result? |
| − | <math> | + | <math>\mathrm{(A)\ }5\qquad\mathrm{(B)\ }6\qquad\mathrm{(C)\ }8\qquad\mathrm{(D)\ }9\qquad\mathrm{(E)\ }10</math> |
| − | |||
== Solution == | == Solution == | ||
| Line 11: | Line 10: | ||
If <math>b=0</math>, the expression evaluates to <math>c-d\leq 2</math>. <br/> | If <math>b=0</math>, the expression evaluates to <math>c-d\leq 2</math>. <br/> | ||
Case <math>d=0</math> remains. | Case <math>d=0</math> remains. | ||
| + | In that case, we want to maximize <math>c\cdot a^b</math> where <math>\{a,b,c\}=\{1,2,3\}</math>. Trying out the six possibilities we get that the greatest is <math>(a,b,c)=(3,2,1)</math>, where <math>c\cdot a^b=1\cdot 3^2=\boxed{\mathrm{(D)}\ 9}</math>. | ||
| − | + | == Video Solution 1== | |
| + | https://youtu.be/QtH3vCiqLkU | ||
| + | |||
| + | ~Education, the Study of Everything | ||
== See Also == | == See Also == | ||
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{{AMC12 box|year=2004|ab=B|num-b=1|num-a=3}} | {{AMC12 box|year=2004|ab=B|num-b=1|num-a=3}} | ||
{{AMC10 box|year=2004|ab=B|num-b=4|num-a=6}} | {{AMC10 box|year=2004|ab=B|num-b=4|num-a=6}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 18:21, 22 October 2022
- The following problem is from both the 2004 AMC 12B #2 and 2004 AMC 10B #5, so both problems redirect to this page.
Problem 2
In the expression 
, the values of 
, 
, 
, and 
are 
, 
, 
, and 
, although not necessarily in that order. What is the maximum possible value of the result?
Solution
If 
or 
, the expression evaluates to 
.
If 
, the expression evaluates to 
.
Case 
remains.
In that case, we want to maximize 
where 
. Trying out the six possibilities we get that the greatest is 
, where 
.
Video Solution 1
~Education, the Study of Everything
See Also
| 2004 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 1 |
Followed by Problem 3 |
| 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 | |
| 2004 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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. 
