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Express $y$ in terms of $x_i$ and $a$, given that $1\geq x_i\geq0$, $a\geq1$, and

\begin{align} (1+x_1y)(1+x_2y)\cdots(1+x_ny)&=a,\\ x_1+x_2+\cdots+x_n&=1.\\ \end{align}

For $n=2,3$ I can find using quadratic and cubic formulas. But how to do it for general case?

Lee
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    I don't think that you can find $y$ explicitly if $n>4$ (there is no formula for solving polynomial equations of degree $n>5$). Instead, you can make some approximations. For example, you can prove that $y\geq\ln a$ (if $y>0$) by noting that $e^{x_iy}\geq 1+x_iy$. – richrow Jun 12 '20 at 13:58
  • @richrow There are formulas. Just not with radicals. – Phicar Jun 12 '20 at 14:00

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