6

Part of 2000 239 Open Mathematical Olympiad

Problems(2)

cockroaches moving with constant speed in plane

Source: 239 2000 S6

n

cockroaches are sitting on the plane at the vertices of the regular

n

-gon. They simultaneously begin to move at a speed of

v

on the sides of the polygon in the direction of the clockwise adjacent cockroach, then they continue moving in the initial direction with the initial speed. Vasya a entomologist moves on a straight line in the plane at a speed of

u

. After some time, it turned out that Vasya has crushed three cockroaches. Prove that

v = u

physicscombinatorics

Not very easy geometry [quadrilateral and tangent circles]

Source: 239MO 2000, classes 8-9, problem 6

Let ABCD be a convex quadrilateral, and let M and N be the midpoints of its sides AD and BC, respectively. Assume that the points A, B, M, N are concyclic, and the circumcircle of triangle BMC touches the line AB. Show that the circumcircle of triangle AND touches the line AB, too. Darij

geometrycircumcirclegeometry proposed