About the problem

Suppose that a group of people are running around a track. They all start from the same point, and they each run with their own unique speed. A runner is lonely at time \(t\) if, at time \(t\), they are far away from every other runner. Is every runner sometimes lonely? The lonely runner conjecture says that the answer is yes: every runner is sometimes lonely.

Formally, the lonely runner conjecture says that for every set of positive integers \(v_1, \ldots, v_n\), there is a real number \(t\) such that \(||tv_i|| \geq \frac{1}{n+1}\) for every \(i \in \{1, \ldots, n\}\). This problem asks for a counterexample to the lonely runner conjecture.

The lonely runner conjecture is known to be true for \(n \leq 12\), and the proofs for \(7 \leq n \leq 12\) are all from the last year. This Quanta magazine article describes some of the ideas that went into proving \(n=7, 8\), and \(9\), which were the first new cases of the lonely runner conjecture to be proven since 2008. In addition to the new proofs for small \(n\), two conjectures which strengthened the lonely runner conjecture were recently disproven (see here and here), raising the possibility that the conjecture could be false.

A solution to this problem is simply a set \(v_1, \ldots, v_n\) of positive integers. The verifier checks that this set forms a counterexample to the lonely runner conjecture. The main risk to this problem is that the lonely runner conjecture might be true. Another risk is that the solution may not be as insightful as mathematicians hope. The ideal solution would have some arithmetic structure that sheds light on why the conjecture is false. But even a solution that seems random could lead to new math in the study of Diophantine approximation.

Prompt

# The Lonely Runner Conjecture

Consider n runners on a circular track of length 1, all starting at position 0 at time t=0, each running at a distinct positive integer speed. The conjecture states:

    For any such set of n runners, there exists a time t at which every runner is at distance at least 1/(n+1) from the starting position 0.

Here, distance on the circular track means the shorter arc length, i.e. the distance of runner k from 0 at time t is

    ||t * vk|| = min( {t * vk}, 1 - {t * vk} )

where {x} denotes the fractional part of x.

## Your task

Find a counterexample: a list of n >= 2 distinct positive integers v1, ..., vn such that for every time t, at least one runner is at distance strictly less than 1/(n+1) from position 0. In other words, there is no moment at which the starting position is "lonely".

## Submission format

Write your answer to a file containing the speeds as integers separated by spaces or newlines, for example:

    3 7 11 15

Submit your file path using the submit tool.