Problem 1

We call a

5

-tuple of integers arrangeable if its elements can be labeled

a,b,c,d,e

in some order so that

a-b+c-d+e=29

. Determine all

2017

-tuples of integers

n_1,n_2,\ldots,n_{2017}

such that if we place them in a circle in clockwise order, then any

5

-tuple of numbers in consecutive positions on the circle is arrangeable.

Step 3 of 4: The common parity is even

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m_1+m_2+m_3+m_4+m_5\equiv0\pmod2 \implies \text{common parity is even}

Detailed analysis

Any five consecutive $m_i$ 's sum to an even number modulo $2$ , in particular $m_1+m_2+m_3+m_4+m_5\equiv0\pmod2$ . Since all $m_i$ share one common parity, this sum of five terms of that parity is even only if the common parity itself is even. Hence every $m_i$ is even.