My solution to https://artofproblemsolving.com/community/c4t48f4h2595840_equal_angles_in_equilateral_grid__201617_savin_competition_79_p18

 In the attached grid made by unit regular triangles there are points A(0,1), B(4,0), C(2,1), D(3,5). Prove that <ABC=<CAD.

My proof:
Take the equilateral triangle XYZ, X(0,0), Y(5,0), Z(0,5) and the point E(1,4) onto YZ. Clearly triangle ABE is equilateral, since triangles XBA and ZAE are congruent (s.a.s.). Take now F(4,4) and notice trat triangles BFD and AZE are congruent, thus BD||=AE, ABDE is a rhombus with a 60 degs angle, therefore <BAD=30 degs=<CBX. With <ABX=<BAC we get the required <ABC=<CAD.