How many ordered triples of integers (a, b, c), with a ≥ 2, b ≥ 1, and c ≥ 0, satisfy both \log_a b = c^2005 and a + b + c = 2005?

2005 AMC 12A Problem 21 — Solution

2005 AMC 12A Problem 21

Below is the professionally curated solution for Problem 21 of the 2005 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.

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Concepts:logarithm Diophantine Equation bounding to limit cases

Difficulty rating: 2440

21.

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

$0$

$1$

$2$

$3$

$4$

Solution:

The condition $\log_a b = c^{2005}$ means $b = a^{\left(c^{2005}\right)}.$

If $c \ge 2,$ then $b = a^{\left(c^{2005}\right)} \ge 2^{\left(2^{2005}\right)},$ which vastly exceeds $2005,$ so $a + b + c = 2005$ is impossible.

For $c = 0:$ $b = a^0 = 1,$ so $a + 1 + 0 = 2005$ gives $(a, b, c) = (2004, 1, 0).$ For $c = 1:$ $b = a^1 = a,$ so $2a + 1 = 2005$ gives $(a, b, c) = (1002, 1002, 1).$

There are $2$ such triples.

Thus, the correct answer is C.

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