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ASU 281 All Soviet Union MO 1979 odd sum a_1a_{k+1}+ ... + a_{n-k}a_n

Source:

July 14, 2019

Sequencealgebra

Problem Statement

The finite sequence

a_1, a_2, ... , a_n

of ones and zeroes should satisfy a condition:
for every

k

from

0

(n-1)

the sum a_1a_{k+1} + a_2a_{k+2} + ... + a_{n-k}a_n should be odd.
a) Construct such a sequence for

n=25

.
b) Prove that there exists such a sequence for some

n > 1000