Task 1. Answer: $k=16$ collisions.

A wire in the shape of a circle with radius $R$ has $m$ beads strung on it, equally spaced from each other. At a certain moment, $(m-1)$ beads are made to move with the same speed $v(1/$ sec) in the counterclockwise direction, while the remaining bead is made to move in the opposite direction at the same speed. It is assumed that after the collision of two beads, their speed remains unchanged, and the direction instantly reverses. How many collisions will occur between the beads in $T$ seconds.