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AIME 2024 / AIME 2024 train c93df5c3-615b-53b2-a818-7428b52651df

Problem

Answer published by the source. Consult the official source to check your work against its answer.

Problem

Let

x,y

and

z

be positive real numbers that satisfy the following system of equations: \[\log_2\left({x \over yz}\right) = {1 \over 2}\] \[\log_2\left({y \over xz}\right) = {1 \over 3}\] \[\log_2\left({z \over xy}\right) = {1 \over 4}\] Then the value of

\left|\log_2(x^4y^3z^2)\right|

\tfrac{m}{n}

where

m

and

n

are relatively prime positive integers. Find

m+n

Plain-text mathematical notation (without MathML)

Let x,y and z be positive real numbers that satisfy the following system of equations: 
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of |log₂(x⁴y³z²)| is (m)/(n) where m and n are relatively prime positive integers. Find m+n.

Original LaTeX notation

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: 
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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Source and history

task_id: c93df5c3-615b-53b2-a818-7428b52651df
task_key: default--train--c93df5c3-615b-53b2-a818-7428b52651df
task_revision_id: 2
config / split: default / train

Official source

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