AoPS - Problem

Source: Lusophon 2012 CPLP P1

Tags: combinatorics, combinatorial geometry, collinear, geometry, incenter

parmenides51

30.08.2018 01:17

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

Easily we can see that Arnaldo does $\frac{1}{2}$ lap per minute and Bernaldo does $\frac{1}{3}$ lap per minute, as we want to discover when they are in opposite sides of the circle we need de Bernaldo position to be $\frac{1}{2} + ($Arnaldo position$)(mod 1)$, as de Arnaldo possition is $\frac{1}{2} t$, we need to find all t's where $\frac{1}{3} t \equiv \frac{1}{2} + \frac{1}{2} t (mod 1)$ so $t \equiv 3(mod 6)$ which implies that $t = 3+6k$ therefore there are $10$ times between minute $1$ and $61$ where arnaldo bernaldo and the mast are colinear.