Question

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 $m + n.$

答案

Solution:

Since the apex is equidistant from all three base vertices, its foot is the circumcenter of the base. The base is isosceles with sides $20, 20, 24:$ its altitude to the side of length $24$ is $\sqrt{20^2 - 12^2} = 16,$ so its area is $K = \frac{1}{2} \cdot 24 \cdot 16 = 192,$ and its circumradius is $R = \frac{abc}{4K} = \frac{20 \cdot 20 \cdot 24}{4 \cdot 192} = \frac{25}{2}.$

The height of the pyramid is $\sqrt{25^2 - \left(\frac{25}{2}\right)^2} = \frac{25\sqrt{3}}{2},$ so the volume is $\frac{1}{3} \cdot 192 \cdot \frac{25\sqrt{3}}{2} = 800\sqrt{3}.$ Then $m + n = 800 + 3 = 803.$

2017 AIME I Problem 4

Solution:

Problem 4 in Other Years