MathDB

Prove that there are infinitely many natural numbers

n

such that we can divide

1,2,\ldots ,3n

into three sequences

(a_n),(b_n)

and

(c_n)

, with

n

terms in each, satisfying the following conditions: i)

a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n

and

a_1+b_1+c_1

is divisible by

6

; ii)

a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,

and

a_1+a_2+\ldots +a_n

is divisible by

6

Problem Statement