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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

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question

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

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

Official source

initial import