15

Part of 2014 ASDAN Math Tournament

Problems(2)

2014 General #15

Source:

Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?

2014General Test

Source:

A point is "bouncing" inside a unit equilateral triangle with vertices

(0,0)

(1,0)

(1/2,\sqrt{3}/2)

. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the direction of

(1,1)

. After the ball has traveled a cumulative distance of

30\sqrt{2}

, compute its distance from the origin.