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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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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