# 2004 Indonesia MO Problems/Problem 5

## Problem

Given a system of equations: $\left\{\begin{array}{l}x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 = 1\\4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 = 12\\9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 = 123\end{array}\right.$

Then determine the value of $S = 16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7$.

## Solution

Subtract the first equation from the second equation of the original to get $$3x_1 + 5x_2 + 7x_3 + 9x_4 + 11x_5 + 13x_6 + 15x_7 = 11$$ Subtract the second equation from the third equation of the original to get $$5x_1 + 7x_2 + 9x_3 + 11x_4 + 13x_5 + 15x_6 + 17x_7 = 111$$ The difference between the two above equations is $$2x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 + 2x_6 + 2x_7 = 100$$ That means $$7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7 = 211$$ Adding that to the third equation of the original results in $$16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7 = \boxed{334}$$

## Note

This problem is a copy of Problem 8 from the 1989 AIME.