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permutation of $\{1, 2, ... , 2n\}$

Source: 39-th Vietnamese Mathematical Olympiad 2001

March 19, 2007

combinatorics unsolvedcombinatorics

Problem Statement

(a_{1}, a_{2}, ... , a_{2n})

is a permutation of

\{1, 2, ... , 2n\}

such that

|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}|

for

i \neq j

. Show that

a_{1}= a_{2n}+n

iff

1 \leq a_{2i}\leq n

for

i = 1, 2, ... n.