MHB Challenge Problem #7: Σ(x/(y^3+2))≥1

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The discussion revolves around proving the inequality Σ(x/(y^3+2))≥1 for positive real numbers x, y, and z such that xyz=1. Participants are encouraged to explore various mathematical approaches to demonstrate the validity of the inequality. The conversation highlights the importance of manipulating the terms and applying known inequalities to achieve the proof. A hint is provided to guide the participants in their reasoning. Engaging with this problem can deepen understanding of inequalities in mathematical analysis.
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Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$
 
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A hint ...

is requested (Giggle)
 
lfdahl said:
A hint ...

is requested (Giggle)
Niiiiice. (Clapping)

-Dan
 

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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