Let <img alt="$a, b, c,$" height="18" src="https://latex.artofproblemsolving.com/2/d/0/2d02bb2275100261109b043261e702124460a83e.png" width="48" /> and <img alt="$d$" height="14" src="https://latex.artofproblemsolving.com/9/6/a/96ab646de7704969b91c76a214126b45f2b07b25.png" width="11" /> be real numbers that satisfy the system of equations   <img alt="\begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*}" height="87" src="https://latex.artofproblemsolving.com/d/c/2/dc245cfac4be4095f79b4f754d04e180a1181dca.png" style="float: left;" width="222" />     There exist relatively prime positive integers <img alt="$m$" height="10" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="18" /> and <img alt="$n$" height="10" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="13" /> such that   <img alt="\[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]" height="32" src="https://latex.artofproblemsolving.com/c/2/0/c20fc78cacf6996c3bfc20ed66cd4b0ac20f7bb3.png" width="178" /> Find <img alt="$m + n$" height="12" src="https://latex.artofproblemsolving.com/5/3/b/53bf1afcb274a3bb84338109fffec8a1f1c02008.png" width="48" />.

Question No: 445

Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations

$\begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*}$

There exist relatively prime positive integers $m$ and $n$ such that

$a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.$

Find $m + n$ .

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Tags: Algebra

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