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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_13 / A rectangular box P has distinct edge lengths a, b, and c. The sum of the lengths of all 12 edges of P is 13, the areas of all 6 faces of P is (11)/(2), and the volume of P is …

Problem

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

problem

A rectangular box

P

has distinct edge lengths

a

b

, and

c

. The sum of the lengths of all

12

edges of

P

13

, the areas of all

6

faces of

P

\frac{11}{2}

, and the volume of

P

\frac{1}{2}

. Find the length of the longest interior diagonal connecting two vertices of

P

. The final answer can be written in the form

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

Plain-text mathematical notation (without MathML)

A rectangular box P has distinct edge lengths a, b, and c. The sum of the lengths of all 12 edges of P is 13, the areas of all 6 faces of P is (11)/(2), and the volume of P is (1)/(2). Find the length of the longest interior diagonal connecting two vertices of P. The final answer can be written in the form (m)/(n), where m and n are relatively prime positive integers. What is m+n?

Original LaTeX notation

A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the areas of all $6$ faces of $P$ is $\frac{11}{2}$, and the volume of $P$ is $\frac{1}{2}$. Find the length of the longest interior diagonal connecting two vertices of $P$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_13
task_id: fbea87ee-d25a-5352-b4bc-01a0144a14a7
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2023~5fAMC~5f12B~5fProblems~2fProblem~5f13
task_revision_id: 2
config / split: default / train

Official source

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