1. In an equilateral triangle, 3 points were marked on each side so that they divide the side into 4 equal segments. These points and the vertices of the triangle were painted. How many isosceles triangles exist with vertices at the painted points?
# Answer. 20

Solution. Let the vertices of the triangle be denoted as A, B, C. Suppose points M, N (M closer to A) are taken on side AB, points P, Q (P closer to A) on side AC, and points K, L (K closer to B) on side BC. Then the following cases are possible for isosceles triangles with the vertex at point A: ABC, AKL, APM, AQN. For the vertex at point B: BNK, BPQ, BML. For the vertex at point C: CQL, CPK, CMN. Another 8 cases arise without using the vertices of the original triangle: MQK, PLM, and 6 triangles, two sides of which are equal to $1 / 3$ of the side of the original triangle: MNK, etc. In total, there are 20 cases.

Criteria. All cases found - 7 points. Number of cases of one type found - 1 point, two types - 3 points.