The Problems of Ukrainian Authors

Question

Andriy has counted the sum of the squares of positive integers from

1

to

2016

, Vitaliy has counted the sum of the squares of natural numbers from

1

to

2016

. Yuriy has added Andriy's and Vitaliy's numbers, multiplied by

3

and added

2016

. What number has Yuriy got?

(Andriy Anikushin)

Accepted Answer

Let's write down the number that Yuriy had got: aligned & 3 (1^2 + 2^2 + + 2016^2) + 3 (1 + 2 + + 2016) + 2016 = & = (3 1^2 + 3 1 + 1) + (3 2^2 + 3 2 + 1) + + (3 2016^2 + 3 2016 + 1) = & = (1^3 + 3 1^2 + 3 1 + 1) + (2^3 + 3 2^2 + 3 2 + 1) + + (2016^3 + 3 2016^2 + 3 2016 + 1) - & - (1^3 + 2^3 + + 2016^3) = ((1+1)^3 + (2+1)^3 + + (2016+1)^3) & - (1^3 + 2^3 + + 2016^3) = & = (2^3 + 3^3 + + 2017^3) - (1^3 + 2^3 + + 2016^3) = 2017^3 - 1. aligned