10.8. On a circle of length 2013, 2013 points are marked, dividing it into equal arcs. A chip is placed at each marked point. We define the distance between two points as the length of the shorter arc between them. For what largest $n$ can the chips be rearranged so that there is again one chip at each marked point, and the distance between any two chips that were initially no more than $n$ apart has increased?

( D. Khramov)
Answer. $n=670$.

Solution. Let's number the points and the chips placed on them in a clockwise direction with consecutive non-negative integers from 0 to 2012. Consider an arbitrary permutation and the chips with numbers 0, 671, and 1342, initially located at the vertices of an equilateral triangle. The pairwise distances between them are 671. After the permutation, the sum of the pairwise distances between these chips will not exceed the length of the circumference, which means that the distance between some two of them will not exceed $2013 / 3 = 671$; hence, the distance between these two chips will not increase. Therefore, for $n \geqslant 671$, the required permutation is impossible.

Now, let's provide an example of the desired permutation for $n=670$. Each chip with a number $i \leqslant 1006$ will be moved to the point with number $a_{i}=2i$, and each chip with a number $i \geqslant 1007$ will be moved to the point with number $a_{i}=2i-2013$. In other words, $a_{i}$ is the remainder of the division of $2i$ by 2013. It is easy to see that each point will have one chip. It remains to show that the distances between pairs of chips, initially separated by no more than 670, will increase.

Consider any two chips with numbers $i$ and $j$; let the distance between them be $d \leqslant 670$. Then one of the arcs between points $a_{i}$ and $a_{j}$ will have a length of $2d$, meaning the distance between these points is $d'=\min\{2d, 2013-2d\}$. Note that $2d > d$ and $2013-2d > d$ (the latter because $3d < 2013$). Therefore, $d' > d$, which is what we needed to prove.

Comment. Only the correct answer without justification - 0 points.

Proving that for $n \geqslant 671$ the required arrangement does not exist (making an estimate) - 3 points.

Providing an example showing that $n=670$ works, with justification that the example meets the condition - 4 points.

Providing a correct example showing that $n=670$ works, without sufficient justification that the example meets the condition - 3 points.

Points for progress in proving the estimate and in constructing the example are cumulative.