Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2024 SSMO Relay Round 2 Problems/Problem 1"

(Created page with "==Problem== In a regular hexagon <math>ABCDEF</math>, let <math>X</math> be a point inside the hexagon such that <math>XA=XB=3.</math> If the area of the hexagon is <math>6\s...")
 
 
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
 +
Let <math>s</math> be the sidelength of the hexagon. We have <math>\frac{6s^2\sqrt{3}}{4} = 6\sqrt{3}\implies s = 2.</math> Let <math>h</math> be the distance from <math>X</math> to <math>AB.</math> Thus, we have <math>\sqrt{h^2+\left(\frac{s}{2}\right)^2} = 3\implies h^2 = 8\implies h = 2\sqrt{2}.</math> Since the distance from <math>AB</math> to <math>DE</math> is <math>2\sqrt{3},</math> <cmath>XE = \sqrt{(2\sqrt{3}-2\sqrt{2})^2+\left(\frac{s}{2}\right)^2}\implies XE^2 = 21-8\sqrt{6}\implies 21-8+6 = \boxed{19}.</cmath>
 +
 +
~SMO_Team

Latest revision as of 14:43, 10 September 2025

Problem

In a regular hexagon $ABCDEF$, let $X$ be a point inside the hexagon such that $XA=XB=3.$ If the area of the hexagon is $6\sqrt{3}$, then $XE^2 = a+b\sqrt{c},$ for squarefree $c$. Find $a+b+c$.

Solution

Let $s$ be the sidelength of the hexagon. We have $\frac{6s^2\sqrt{3}}{4} = 6\sqrt{3}\implies s = 2.$ Let $h$ be the distance from $X$ to $AB.$ Thus, we have $\sqrt{h^2+\left(\frac{s}{2}\right)^2} = 3\implies h^2 = 8\implies h = 2\sqrt{2}.$ Since the distance from $AB$ to $DE$ is $2\sqrt{3},$ \[XE = \sqrt{(2\sqrt{3}-2\sqrt{2})^2+\left(\frac{s}{2}\right)^2}\implies XE^2 = 21-8\sqrt{6}\implies 21-8+6 = \boxed{19}.\]

~SMO_Team

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_SSMO_Relay_Round_2_Problems/Problem_1&oldid=260282"