In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
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Am I misunderstanding this? The two points $A,B$ connected by the smallest distance are connected to each other, so we have at most $n-1$ edges in our figure (because the edge going out from $A$ is the same as the edge going out from $B$). However, for a closed polygonal line with $n$ vertices we need $n$ edges.
grobber wrote:
Am I misunderstanding this? The two points $A,B$ connected by the smallest distance are connected to each other, so we have at most $n-1$ edges in our figure (because the edge going out from $A$ is the same as the edge going out from $B$). However, for a closed polygonal line with $n$ vertices we need $n$ edges.
Your solution is right. The problem is that trivial. Indeed the real shortlist problem had a stronger assertion, but unfortunately it was wrong .
Darij