AMC 10 · 2005 · #21

Grade 11 algebranumber-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

logarithm-propertiesexponentsexponential-function caseworkbound-inequality-then-enumerateconvert-to-algebra ↑ Prerequisites: logarithm-propertiesexponents

📏 Medium solution 💡 3 insights

Problem

Integer triples must satisfy a logarithm condition with a huge exponent and also add up to 2005. Count the ordered triples that satisfy both.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Make a Systematic List

Rewriting the logarithm as a power makes b a function of a and c, which leaves c as the only real branching variable. Splitting on c gives a list that is guaranteed complete once the large values are killed, and they are killed by a size bound rather than by a feeling that the exponent is big. A complete case list is what turns a count into a proof.

1STEP 1

Rewrite the logarithm as a power

Rewriting the logarithm as a power removes it entirely.

log_a b = c²⁰⁰⁵ ⇔ b = a^{ c²⁰⁰⁵}

2STEP 2

Cap every part with the sum

The sum condition puts a cap on every part.

b = 2005 - a - c ≤ 2005 - 2 - 0 = 2003

3STEP 3

Case c equals 0

A zero exponent forces one value, giving one triple.

b = a⁰ = 1 → a = 2004 → (a,b,c) = (2004, 1, 0)

4STEP 4

Case c equals 1

An exponent of one gives exactly one more.

2a + 1 = 2005 → a = b = 1002 → (a,b,c) = (1002, 1002, 1)

5STEP 5

Case c at least 2 is too big

Anything larger blows past the cap, so that case is empty.

c ≥ 2 → c²⁰⁰⁵ ≥ 2²⁰⁰⁵ ＞ 12 → b ≥ 2¹² = 4096 ＞ 2003 ≥ b

6STEP 6

Collect the complete case list

The cases cover everything, so the count is 2, choice (C).

(2004, 1, 0) and (1002, 1002, 1)

Answer

Both surviving triples were written out and substituted back into both original equations, so the count is at least 2. Every other value of c was ruled out by the explicit inequality b ≥ 4096 ＞ 2003, so the count is at most 2. Both directions are needed: exhibiting examples alone would not rule out a third triple, and the size bound alone would not confirm that the two candidates actually work.

💡Key takeaway

A logarithm is an exponent, and exponents grow so fast that only the smallest ones can squeeze inside a fixed total like 2005.

Rewrite the logarithm as a power
Cap every part with the sum
Case c equals 0
Case c equals 1
Case c at least 2 is too big
Collect the complete case list