MHB Absolute value of real numbers

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The discussion focuses on maximizing the expression $\dfrac{1}{1+|x|}+\dfrac{1}{1+|y|}+\dfrac{1}{1+|z|}$ under the constraint $3x+2y+z=1$, where $x$, $y$, and $z$ are real numbers. Participants explore the relationship between the absolute values of $x$, $y$, and $z$ and their contributions to the overall sum. The goal is to express the maximum value as a fraction $\dfrac{q}{p}$, where $p$ and $q$ are relatively prime positive integers. The final task is to calculate the sum $p+q$. The discussion emphasizes the mathematical approach to finding the optimal values of $x$, $y$, and $z$.
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Reals $x,\,y$ and $z$ satisfies $3x+2y+z=1$. For relatively prime positive integers $p$ and $q$, let the maximum of $\dfrac{1}{1+|x|}+\dfrac{1}{1+|y|}+\dfrac{1}{1+|z|}$ be $\dfrac{q}{p}$. Find $p+q$.
 
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