Question

Assume that $x_1,$ $x_2,$ $\ldots,$ $x_7$ are real numbers such that

$\begin{aligned}x_1+4x_2+9x_3&\\+16x_4+25x_5&\\+36x_6+49x_7&=1,\\4x_1+9x_2+16x_3&\\+25x_4+36x_5&\\+49x_6+64x_7&=12,\\9x_1+16x_2+25x_3&\\+36x_4+49x_5&\\+64x_6+81x_7&=123.\end{aligned}$

Find the value of

$\begin{aligned}16x_1+25x_2+36x_3&\\+49x_4+64x_5&\\+81x_6+100x_7.&\end{aligned}$

Answer

Answer: 334

Concepts:pattern recognition polynomial system of equations

Difficulty rating: 2230

Small Hint:

Let $S(t)=\sum_{i=1}^7(i+t)^2x_i$

Big Hint:

Because $S(t)$ is quadratic in $t,$ its second finite differences are constant

Solution:

Define $S(t)=\sum_{i=1}^7(i+t)^2x_i.$ This is a quadratic polynomial in $t,$ and the equations say $S(0)=1,$ $S(1)=12,$ and $S(2)=123.$ Its first two differences are $11$ and $111,$ so the constant second difference is $100.$ The next first difference is therefore $211,$ giving $S(3)=123+211=334.$

1989 AIME Problem 8

Answer: 334

Difficulty rating: 2230

Small Hint:

Big Hint:

Solution:

Problem 8 in Other Years