Library / Estonia — Open Contests /11 of 63

Estonia — Open Contests

Geometry Difficulty 4.8 AIME Prove it Estonia

In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36°$ . Find all possibilities, how big the angles of this triangle can be.

Solution

Based on the initial conditions the angles of the triangle are $\alpha$ , $2\alpha$ and $180° - 3\alpha$ and they all have to be different. It remains to perform calculations for three cases: $\alpha = 36°$ , $2\alpha = 36°$ and $180° - 3\alpha = 36°$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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