HMMT-Feb 2024 geo, Problem 1

Inside an equilateral triangle of side length 6 , three congruent equilateral triangles of side length $x$ with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below. A smaller equilateral triangle formed between the three congruent equilateral triangles has side length 1. Compute $x$.

Solution: \section*{Solution:} Let $x$ be the side length of the shaded triangles. Note that the centers of the triangles with side lengths 1 and 6 coincide; call this common center $O$. The distance from $O$ to a side of the equilateral triangle with side length 1 is $\sqrt{3} / 6$. Similarly the distance from $O$ to a side of the equilateral triangle with side length 6 is $\sqrt{3}$. Notice the difference of these two distances is exactly the length of the altitude of one of shaded triangles. So \[ \sqrt{3}-\frac{\sqrt{3}}{6}=\frac{\sqrt{3}}{2} x \Longrightarrow x=\frac{5}{3} \] $\fbox{\frac{5}{3}}$.

Answer: \frac{5}{3}