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2017 AIME I Problems/Problem 4

Revision as of 17:12, 8 March 2017 by A1b2 (talk | contribs) (→‎Solution)

Problem 4

A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Solution

Let the triangular base be $\triangle ABC$. Using Simplified Heron's formula for the area of an isosceles triangle gives $12\sqrt{32(8)}=192$.

$\vdots$

The volume of the pyramid is $800\sqrt{3}$, so the answer is $\boxed{803}$

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