Difference between revisions of "1989 AIME Problems/Problem 8"

Revision as of 14:56, 13 March 2015

Problem

Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that $\begin{align*}x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7&=1\\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7&=12\\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7&=123\end{align*}.$

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

Solution

Solution 1

Notice that because we are given a system of $3$ equations with $7$ unknowns, the values $(x_1, x_2, \ldots, x_7)$ are not fixed; indeed one can take any four of the variables and assign them arbitrary values, which will in turn fix the last three.

Given this, we suspect there is a way to derive the last expression as a linear combination of the three given expressions. Let the coefficent of $x_i$ in the first equation be $y_i^2$ ; then its coefficients in the second equation is $(y_i+1)^{2}$ and the third as $(y_i+2)^2$ . We need to find a way to sum these to make $(y_i+3)^2$ [this is in fact a specific approach generalized by the next solution below].

Thus, we hope to find constants $a,b,c$ satisfying $ay_i^2 + b(y_i+1)^2 + c(y_i+2)^2 = (y_i + 3)^2$ . FOILing out all of the terms, we get

$[ay^2 + by^2 + cy^2] + [2by + 4cy] + b + 4c = y^2 + 6y + 9.$

Comparing coefficents gives us the three equation system:

$\begin{align*}a + b + c &= 1 \\ 2b + 4c &= 6 \\ b + 4c &= 9 \end{align*}$

Subtracting the second and third equations yields that $b = -3$ , so $c = 3$ and $a = 1$ . It follows that the desired expression is $a \cdot (1) + b \cdot (12) + c \cdot (123) = 1 - 36 + 369 = \boxed{334}$ .

Solution 2

Notice that we may rewrite the equations in the more compact form as:

$\sum_{i=1}^{7}i^2x_i=c_1,\ \ \sum_{i=1}^{7}(i+1)^2x_i=c_2,\ \ \sum_{i=1}^{7}(i+2)^2x_i=c_3,$ and $\sum_{i=1}^{7}(i+3)^2x_i=c_4,$

where $c_1=1, c_2=12, c_3=123,$ and $c_4$ is what we're trying to find.

Now consider the polynomial given by $f(z) := \sum_{i=1}^7 (z+i)^2x_i$ (we are only treating the $x_i$ as coefficients).

Notice that $f$ is in fact a quadratic. We are given $f(0), \ f(1), \ f(2)$ as $c_1, \ c_2, \ c_3$ and are asked to find $c_4$ . Using the concept of finite differences (a prototype of differentiation) we find that the second differences of consecutive values is constant, so that by arithmetic operations we find $c_4=334$ .

Alternatively, applying finite differences, one obtains $c_4 = {3 \choose 2}f(2) - {3 \choose 1}f(1) + {3 \choose 0}f(0) =334$ .

Solution 3

Notice that $3(n+2)^2-3(n+1)^2+n^2=(n+3)^2$

I'll number the equations for convenience

$\begin{align} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7&=1\\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7&=12\\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7&=123\\ 16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7&=\end{align}$

Let the coefficient of $x_i$ in $(1)$ be $n^2$ . Then the coefficient of $x_i$ in $(2)$ is $(n+1)^2$ etc.

Therefore, $3*(3)-3*(2)+(1)=(4)$

So $(4)=3*123-3*12+1=\boxed{334}$

Solution 4

Notice subracting the first equation from the second yields $3x_1 + 5x_2 + ... + 15x_7 = 11$ Then, repeating for the 2nd and 3rd equations, and then subtracting the result from the first obtained equation, we get $2x_1 + 2x_2 + ... +2x_7 = 100$ . Adding this twice to the first obtained equation gives difference of the desired equation and 3rd equation, which is 211. Adding to the 3rd equation, we get $\boxed{334}$

@@ Line 48: / Line 48: @@
 I'll number the equations for convenience
-<math>(1)  x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7&=1</math>
+<cmath>
+\begin{align}
-<math>(2)  4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7&=12</math>
+x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7&=1\\
+x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7&=12\\
-<math>(3)  9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7&=123</math>
+x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7&=123\\
+x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7&=\end{align}</cmath>
-<math>(4)  16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7</math>
 Let the coefficient of <math>x_i</math> in <math>(1)</math> be <math>n^2</math>. Then the coefficient of <math>x_i</math> in <math>(2)</math> is <math>(n+1)^2</math> etc.

1989 AIME (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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