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Let

n

be a positive integer, and let

\left( a_{1},\ a_{2} ,\ ...,\ a_{n}\right)

\left( b_{1},\ b_{2},\ ...,\ b_{n}\right)

and

\left( c_{1},\ c_{2},\ ...,\ c_{n}\right)

be three sequences of integers such that for any two distinct numbers

i

and

j

from the set

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

, none of the seven integers a_{i}\minus{}a_{j}; \left( b_{i}\plus{}c_{i}\right) \minus{}\left( b_{j}\plus{}c_{j}\right); b_{i}\minus{}b_{j}; \left( c_{i}\plus{}a_{i}\right) \minus{}\left( c_{j}\plus{}a_{j}\right); c_{i}\minus{}c_{j}; \left( a_{i}\plus{}b_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\right); \left( a_{i}\plus{}b_{i}\plus{}c_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\plus{}c_{j}\right) is divisible by

n

. Prove that: a) The number

n

is odd. b) The number

n

is not divisible by

3

. [hide="Source of the problem"]Source of the problem: This question is a generalization of one direction of Theorem 2.1 in: Dean Alvis, Michael Kinyon, Birkhoff's Theorem for Panstochastic Matrices, American Mathematical Monthly, 1/2001 (Vol. 108), pp. 28-37. The original Theorem 2.1 is obtained if you require b_{i}\equal{}i and c_{i}\equal{}\minus{}i for all

i

, and add in a converse stating that such sequences

\left( a_{1},\ a_{2},\ ...,\ a_{n}\right)

\left( b_{1},\ b_{2},\ ...,\ b_{n}\right)

and

\left( c_{1} ,\ c_{2},\ ...,\ c_{n}\right)

indeed exist if

n

is odd and not divisible by

3

Problem Statement