2001 AMC 12 Problem 9 — Solution | LIVE by Po-Shen Loh

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2001 AMC 12 Problem 9

Below is the professionally curated solution for Problem 9 of the 2001 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 12 solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:functional equation substitution

Difficulty rating: 1440

9.

Let $f$ be a function satisfying $f(xy) = \dfrac{f(x)}{y}$ for all positive real numbers $x$ and $y.$ If $f(500) = 3,$ what is the value of $f(600)?$

$1$

$2$

$\dfrac{5}{2}$

$3$

$\dfrac{18}{5}$

Solution:

Choose $x = 500$ and $y = \dfrac{6}{5}$ so that $xy = 600.$ Then $f(600) = \dfrac{f(500)}{6/5} = \dfrac{3}{6/5} = \dfrac{5}{2}.$

Thus, the correct answer is C.

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