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Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$
$$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$
Niiiiice. (Clapping)lfdahl said:A hint ...
is requested (Giggle)
The purpose of Challenge Problem #7 is to test your understanding of summation notation and your ability to solve a complex mathematical problem.
First, rewrite the given expression in summation notation. Then, find the values of x and y that satisfy the given inequality. Finally, prove that the inequality holds true for those values.
No, calculators are not allowed for this challenge problem. You are expected to solve it using your mathematical skills and knowledge.
There is no specific method or formula that you must use. However, it may be helpful to use properties of summation and algebraic manipulation to simplify the expression and make it easier to solve.
One tip is to carefully read and understand the given expression before attempting to solve it. Another tip is to break the problem down into smaller, more manageable steps. It may also be helpful to check your work and make sure your solution satisfies the original inequality.