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

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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

Rewrite the first condition as

b = a^{\left(c^{2005}\right)}

c \ge 2,

then

b

is astronomically larger than

2005;

only

c = 0

and

c = 1

can work

The condition

\log_a b = c^{2005}

means

b = a^{\left(c^{2005}\right)}.

c \ge 2,

then

b = a^{\left(c^{2005}\right)} \ge 2^{\left(2^{2005}\right)},

which vastly exceeds

2005,

a + b + c = 2005

is impossible.

For

c = 0:

b = a^0 = 1,

a + 1 + 0 = 2005

gives

(a, b, c) = (2004, 1, 0).

For

c = 1:

b = a^1 = a,

2a + 1 = 2005

gives

(a, b, c) = (1002, 1002, 1).

There are

2

such triples.

Thus, the correct answer is C.