CONO-SUR 1995 Problem 2

There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$ . Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?