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

• MHB
• Olinguito
In summary, the challenge problem #7 is a mathematical inequality involving the summation of a series, asking to find the minimum value of the series Σ(x/(y^3+2)) such that it is greater than or equal to 1. There are various approaches to solving this problem, such as using the concept of convergence and divergence of infinite series, simplifying the series, or using mathematical inequalities. The variables x and y in the problem are arbitrary real numbers, allowing for a wide range of possible solutions. There is no specific formula or algorithm to solve this problem, requiring a combination of mathematical knowledge and problem-solving skills. This challenge problem is important in developing critical thinking and problem-solving skills in mathematics, as well as exploring and
Olinguito
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$

A hint ...

is requested (Giggle)

lfdahl said:
A hint ...

is requested (Giggle)
Niiiiice. (Clapping)

-Dan

## 1. What is the purpose of Challenge Problem #7?

The purpose of Challenge Problem #7 is to test your understanding of summation notation and your ability to solve a complex mathematical problem.

## 2. How do I approach solving this 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.

## 3. Can I use a calculator to solve this problem?

No, calculators are not allowed for this challenge problem. You are expected to solve it using your mathematical skills and knowledge.

## 4. Is there a specific method or formula I should use to solve this problem?

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.

## 5. What are some tips for solving this problem efficiently?

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.

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