Competition problem — Geometry — Maths Olympiad Prep

Library / /27 of 129

Geometry Difficulty 4.5 AIME Find the answer Slovenia

The equilateral triangles $ABC$ and $DEF$ with the sides of length $1$ are divided into congruent equilateral triangles as shown in the figure. A circle is inscribed into each smaller triangle. What is the ratio of the shaded area of the triangle $ABC$ to the shaded area of the triangle $DEF$ ?

Pick one

Solution

The proportion of the area of the triangle covered by the inscribed circle is $\frac{\pi\sqrt{3}}{9}$ . For each of the nine congruent triangles in the triangle $ABC$ the shaded area represents $\frac{\pi\sqrt{3}}{9}$ of the total area, so this is also the proportion of the shaded area in the triangle $ABC$ . The same is true for the triangle $DEF$ , so both ratios are equal.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.