2005 AMC 12A Problems/Problem 21: Difference between revisions

Latest revision as of 14:03, 1 July 2025

Problem

How many ordered triples of integers $(a,b,c)$ , with $a \geq 2$ , $b \geq 1$ , and $c \geq 0$ , satisfy both $\log_{a}b = c^{2005}$ and $a + b + c = 2005$ ?

$\mathrm{(A)} \ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4$

Solution

$a^{c^{2005}} = b$

Casework upon $c$ :

$c = 0$ : Then $a^0 = b \Longrightarrow b = 1$ . Thus we get $(2004,1,0)$ .
$c = 1$ : Then $a^1 = b \Longrightarrow a = b$ . Thus we get $(1002,1002,1)$ .
$c \ge 2$ : Then the exponent of $a$ becomes huge, and since $a \ge 2$ there is no way we can satisfy the second condition. Hence we have two ordered triples $\mathrm{(C)}$ .

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

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 ==Problem==
-How many ordered triples of [[integer]]s <math>(a,b,c)</math>, with <math>a \ge 2</math>, <math>b\ge 1</math>, and <math>c \ge 0</math>, satisfy both <math>\log_a b = c^{2005}</math> and <math>a + b + c = 2005</math>?
+How many ordered triples of integers <math>(a,b,c)</math>, with <math>a \geq 2</math>, <math>b \geq 1</math>, and <math>c \geq 0</math>, satisfy both <math>\log_{a}b = c^{2005}</math> and <math>a + b + c = 2005</math>?
 <math>\mathrm{(A)} \ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4</math>

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2005 AMC 12A Problems/Problem 21: Difference between revisions

Latest revision as of 14:03, 1 July 2025

Problem

Solution

See also