$x_1,x_2...x_m$ are integers where $-2\leq x_j\leq1$, for all j=1,2,...m and $S_r=\sum_{j=1}^mx_j^r,S_1=22,S_4=184.$ Then $\max(S_3)-\min(S_3)$ is?

Question

$x_1,x_2...x_m$ are integers where $-2\leq x_j\leq1$, for all j=1,2,...m and $S_r=\sum_{j=1}^mx_j^r,S_1=22,S_4=184.$ Then $\max(S_3)-\min(S_3)$ is?

My attempt:

$$ \begin{align*} S_1&=x_1+x_2+x_3+\ldots =22\\ S_4&=x^4_1+x^4_2+x^4_3+\ldots=184 \end{align*}$$

$x_j$ can be $-2,-1,0,1$

I think $x_3$ can take multiple values because the answer is not zero, however this is not mentioned anywhere in the question. I don't know how to proceed further

Answer 1

First of all, we are not interested in all $i$ with $x_i=0$, because these $x_i$ do not effect any sum. Let $a$ be the number of $x_i$, that are 1, and let $b$ be the number of $x_i$, that are $-1$ and let $c$ be the number of $x_i$, that are $-2$. We have

$$ \begin{align*} 22 &= a-b-2c\\ 184 &= a+b+16c\end{align*} $$ Adding both equations respectively substracting both equations yields

$$\begin{align*} 103 &=a+7c\\ 81 &=b+9c \end{align*}$$ Since $a,b,c\geq 0$, there are only $10$ choices for $c$ namely $0\leq c\leq 9$. You have to do the rest on your own.