Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the start and finished training. At what maximum $S$ can we knowingly say that at some point the sum of the pairwise distances between athletes was at least $S$? Proposed by A. Golovanov, I. Rubanov
Problem
Source: Tuymaada 2024 Junior P3
Tags: number theory, estimates
sami1618
11.07.2024 19:01
Answer: $1.6$ Upper bound:
To do this draw a track with $4$ ticks and look at the runners placements after each minute, in some scenarios you have to verify that $S$ is never more than $1.6$ during the minute. You only need to simulate $5$ minutes as the runners will all meet on the other side of the track.
. Lower bound: Clearly the ratios of their speeds must be rational so let them be able to complete $1$, $\frac{p}{q}$, and $\frac{r}{s}$ laps in a minute for $p<q$, $r<s$, $(p,q)=1$, and $(r,s)=1$ for some positive integers $p,q,r$ and $s$. If one of $p$ and $q$ is even then after $q$ minutes the first two runners are at different ends of the track thus we must have that $p,q,r$ and $s$ are odd. Clearly $q$ and $s$ can not both be $3$ so assume that $5\leq q$. Let $r$ be an odd integer such that $pr\equiv 1 \pmod{q}$ then after $r$ minutes $S$ is at least $\tfrac{2(q-1)}{q}$, as desired.