At the same time, car X leaves city A for B and car Y leaves B for A. Their individual speeds are constant, and Y is 20 km/h faster than X. They first meet at C after 30 minutes. On reaching B and A respectively, they immediately turn back and meet for the second time at D. Given $AB=2CD$, find AB and both speeds. Answer as (AB in km, X speed, Y speed in km/h).
Some students stand in a circle. A teacher distributes sweets clockwise from any starting student: the first receives 1 sweet, the second 2, the third 3, and each subsequent distribution has one more sweet than the preceding one. The total distributed in the second round exceeds that in the first by 100. How many students are there?
Triangle $ABC$ is isosceles at A with $\angle BAC=40^\circ$. Choose D on side BC, different from B and C. Let M be the midpoint of BD, and draw $DH\perp AB$, with $H\in AB$. The line through M perpendicular to AM meets DH at N. Find $\angle ANM$ in degrees.
Construct equilateral triangles $ABD$ and $ACE$ externally to triangle $ABC$. Let F be outside the triangle with $FB=FC$ and $\angle BFC=120^\circ$. Prove that $AF\perp DE$.
Two motorcyclists start simultaneously from A and B towards each other. The first reaches B and turns back; the second reaches A and turns back immediately. Their first meeting is 15 km from A and their second is 9 km from B, with opposite directions at the second meeting. Find AB.