1995 AMC 12 Problem 24

Attempt Problem 24 of the 1995 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1995 AMC 12 solutions, or check the answer key.

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24.

There exist positive integers

A,

B,

and

C,

with no common factor greater than

1,

such that

A\log_{200}5+B\log_{200}2=C.

What is

A+B+C?

6

7

8

9

10

Combine the left side into one logarithm

Use

200=2^3\cdot5^2

and compare prime exponents

Combining logarithms gives

\log_{200}(5^A2^B)=C,

5^A2^B=200^C=5^{2C}2^{3C}.

Hence

A=2C

and

B=3C.

The relatively prime positive triple is

(A,B,C)=(2,3,1),

whose sum is

6.

Thus the correct answer is A.

Problem 24 in other years