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European Mathematical Cup 2016 senior division problem 1

Source:

December 31, 2016

number theory

Problem Statement

Is there a sequence

a_{1}, . . . , a_{2016}

of positive integers, such that every sum

a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}

(with

1 \le r \le s \le 2016

) is a composite number, but: a)

GCD(a_{i}, a_{i+1}) = 1

for all

i = 1, 2, . . . , 2015

; b)

GCD(a_{i}, a_{i+1}) = 1

for all

i = 1, 2, . . . , 2015

and

GCD(a_{i}, a_{i+2}) = 1

for all

y

?

GCD(x, y)

denotes the greatest common divisor of

x

,

y

.
Proposed by Matija Bucić

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