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AIME 2025 I / Let ABCDE be a convex pentagon with AB=14,BC=7,CD=24,DE=13,EA=26, and…

Problem

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

question

Let

ABCDE

be a convex pentagon with

AB=14, BC=7, CD=24, DE=13, EA=26,

and

\angle B=\angle E=60^\circ

. For each point

X

in the plane, define

f(X)=AX+BX+CX+DX+EX

. The least possible value of

f(X)

can be expressed as

m+n\sqrt{p}

, where

m

and

n

are positive integers and

p

is not divisible by the square of any prime. Find

m+n+p

Plain-text mathematical notation (without MathML)

Let ABCDE be a convex pentagon with AB=14,BC=7,CD=24,DE=13,EA=26, and ∠B=∠E=60^(∘). For each point X in the plane, define f(X)=AX+BX+CX+DX+EX. The least possible value of f(X) can be expressed as m+n√(p), where m and n are positive integers and p is not divisible by the square of any prime. Find m+n+p.

Original LaTeX notation

Let $ABCDE$ be a convex pentagon with $AB=14, BC=7, CD=24, DE=13, EA=26,$ and $\angle B=\angle E=60^\circ$. For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX$. The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p}$, where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.

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task_id: 3b06c8ec-d855-5dbb-927a-fb648086fd14
task_key: AIME2025~2dI--test--3b06c8ec-d855-5dbb-927a-fb648086fd14
task_revision_id: 2
config / split: AIME2025-I / test

Official source

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