<p>Let <img alt="$x_1, x_2, \ldots , x_n$" height="13" src="https://latex.artofproblemsolving.com/a/2/3/a23db01b0ac9aec2cd9fefed107a6daad55092d3.png" width="105" /> be a sequence of integers such that (i) <img alt="$-1 \le x_i \le 2$" height="14" src="https://latex.artofproblemsolving.com/2/b/f/2bfacfb9e8c9ad3c29bd8c2a34b93437392b98d1.png" width="96" /> for <img alt="$i = 1,2, \ldots n$" height="16" src="https://latex.artofproblemsolving.com/c/1/1/c11f76a2169a08e7dfcf37e42dd9f1e70b389d62.png" width="99" /> (ii) <img alt="$x_1 + \cdots + x_n = 19$" height="15" src="https://latex.artofproblemsolving.com/0/b/4/0b48051152ac4cc100f778c860ce0e7bc0178744.png" width="144" />; and (iii) <img alt="$x_1^2 + x_2^2 + \cdots + x_n^2 = 99$" height="19" src="https://latex.artofproblemsolving.com/8/b/b/8bbb4f803f8a0b73eea3880bd231c15a03317045.png" width="184" />. Let <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$M$" height="12" src="https://latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png" width="19" /> be the minimal and maximal possible values of <img alt="$x_1^3 + \cdots + x_n^3$" height="19" src="https://latex.artofproblemsolving.com/d/e/b/deba43f2e5de9b17b2912893b5ca7baf68092384.png" width="102" />, respectively. Then <img alt="$\frac Mm =$" height="37" src="https://latex.artofproblemsolving.com/b/e/e/beeb17c0b637ef8d14c4f8834851274949a444de.png" width="42" /> ?</p>

Question No: 231

Let $x_1, x_2, \ldots , x_n$ be a sequence of integers such that (i) $-1 \le x_i \le 2$ for $i = 1,2, \ldots n$ (ii) $x_1 + \cdots + x_n = 19$ ; and (iii) $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$ . Let $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \cdots + x_n^3$ , respectively. Then $\frac Mm =$ ?

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Tags: Algebra Combinatorics

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