Difference between revisions of "2002 AMC 12P Problems/Problem 18"

Revision as of 23:52, 29 December 2023

Problem

If $a,b,c$ are real numbers such that $a^2 + 2b =7$ , $b^2 + 4c= -7,$ and $c^2 + 6a= -14$ , find $a^2 + b^2 + c^2.$

$\text{(A) }14 \qquad \text{(B) }21 \qquad \text{(C) }28 \qquad \text{(D) }35 \qquad \text{(E) }49$

Solution

If $\log_{b} 729 = n$ , then $b^n = 729$ . Since $729 = 3^6$ , $b$ must be $3$ to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of $b \Longrightarrow \boxed{\mathrm{E}}$ .

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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

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

@@ Line 1: / Line 1: @@
 == Problem ==
-How many positive [[integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer?
+If <math>a,b,c</math> are real numbers such that <math>a^2 + 2b =7</math>, <math>b^2 + 4c= -7,</math> and <math>c^2 + 6a= -14</math>, find <math>a^2 + b^2 + c^2.</math>
-<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }  </math>
+<math>
+\text{(A) }14
+\qquad
+\text{(B) }21
+\qquad
+\text{(C) }28
+\qquad
+\text{(D) }35
+\qquad
+\text{(E) }49
+</math>
 == Solution ==

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Difference between revisions of "2002 AMC 12P Problems/Problem 18"

Revision as of 23:52, 29 December 2023

Problem

Solution

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