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AIME 2024 / AIME 2024 train b3f10a3a-4864-5944-898c-5892a42d404b

Problem

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

Problem

Let

ABCD

be a tetrahedron such that

AB=CD= \sqrt{41}

AC=BD= \sqrt{80}

, and

BC=AD= \sqrt{89}

. There exists a point

I

inside the tetrahedron such that the distances from

I

to each of the faces of the tetrahedron are all equal. This distance can be written in the form

\frac{m \sqrt n}{p}

, where

m

n

, and

p

are positive integers,

m

and

p

are relatively prime, and

n

is not divisible by the square of any prime. Find

m+n+p

Plain-text mathematical notation (without MathML)

Let ABCD be a tetrahedron such that AB=CD=√(41), AC=BD=√(80), and BC=AD=√(89). There exists a point I inside the tetrahedron such that the distances from I to each of the faces of the tetrahedron are all equal. This distance can be written in the form (m√(n))/(p), where m, n, and p are positive integers, m and p are relatively prime, and n is not divisible by the square of any prime. Find m+n+p.

Original LaTeX notation

Let $ABCD$ be a tetrahedron such that $AB=CD= \sqrt{41}$, $AC=BD= \sqrt{80}$, and $BC=AD= \sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt n}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$.

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Source and history

task_id: b3f10a3a-4864-5944-898c-5892a42d404b
task_key: default--train--b3f10a3a-4864-5944-898c-5892a42d404b
task_revision_id: 2
config / split: default / train

Official source

initial import