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 == | ||
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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>. | |
− | 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 | ||
== See Also == | == See Also == |
Latest revision as of 13:04, 29 May 2021
- 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 .
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.