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Invariance Principle (Tomato Subjective 36)

Let $$a_1 , a_2 , … , a_n$$ be n numbers such that each $$a_i$$ is either 1 or -1. If $$a_1 a_2 a_3 a_4 + a_2 a_3 a_4 a_5 + … + a_n a_1 a_2 a_4 = 0$$ then prove that 4 divides n.

Solution: Let us denote each product of four numbers as $$b_i$$ . For example $$b_1 = a_1 a_2 a_3 a_4$$. Note that the last one begins with $$a_n$$ indicating there are n $$b_i$$’s.

We check the equation modulo 4 (that is in each step we change ‘something’ in the equation and check what happens to the remainder of the sum when divided by 4).In the first step we note that equation is 0 mod 4 (since 0 when divided by 4 gives 0 as the remainder).