2005 AMC 12A Problems/Problem 21

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Problem

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 $a + b + c = 2005$?

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

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 $\mathrm{(C)}$.

See also

2005 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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

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