# Challenge Problem #8: 3Σ(1/(√(a^3+1))≥2Σ(√(a+b))

• MHB
• Olinguito
In summary, the purpose of Challenge Problem #8 is to challenge scientists and mathematicians to solve a complex inequality equation using critical thinking and problem-solving skills. The symbols used in the equation represent summation, exponents, square roots, and greater than or equal to. The number 3 in the first part of the equation indicates the number of terms being summed. To solve this challenge problem, one can use algebraic manipulation, substitution, or mathematical software. Successfully solving this problem could have implications in various fields and contribute to the development of new techniques and algorithms.
Olinguito
Let $a,b,c$ be positive real numbers such that $a+b+c=2$. Prove that
$$3\left(\frac1{\sqrt{a^3+1}}+\frac1{\sqrt{b^3+1}}+\frac1{\sqrt{c^3+1}}\right)\ \geqslant\ 2\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right).$$

Solution:

We have
$$\frac3{a^3+1}\ =\ \frac{2-a}{a^2-a+1}+\frac1{a+1}\ \geqslant\ 2\sqrt{\dfrac{2-a}{a^3+1}}$$
putting into partial fractions and applying AM–GM (noting that all terms are positive).

Hence
$$\frac3{\sqrt{a^3+1}}\ \geqslant\ 2\sqrt{2-a}\ =\ 2\sqrt{b+c}.$$
Similarly
$$\frac3{\sqrt{b^3+1}}\ \geqslant\ 2\sqrt{c+a}$$
and
$$\frac3{\sqrt{c^3+1}}\ \geqslant\ 2\sqrt{a+b};$$
summing gives the required inequality.

Last edited:
Olinguito said:
Solution:

We have
$$\frac3{a^3+1}\ =\ \frac{2-a}{a^2-a+1}+\frac1{a+1}\ \leqslant\ 2\sqrt{\dfrac{2-a}{a^3+1}}$$
putting into partial fractions and applying AM–GM (noting that all terms are positive).

Hence
$$\frac3{\sqrt{a^3+1}}\ \leqslant\ 2\sqrt{2-a}\ =\ 2\sqrt{b+c}.$$
Similarly
$$\frac3{\sqrt{b^3+1}}\ \leqslant\ 2\sqrt{c+a}$$
and
$$\frac3{\sqrt{c^3+1}}\ \leqslant\ 2\sqrt{a+b};$$
summing gives the required inequality.

the question says it is $>=$ but answer says it is $<=$

I’ve fixed the typo in my solution.

## 1. What is the purpose of Challenge Problem #8?

The purpose of Challenge Problem #8 is to test your understanding of summation notation and inequalities in mathematics.

## 2. What does the symbol Σ represent in the challenge problem?

The symbol Σ represents summation, which is a mathematical operation that adds together a sequence of numbers.

## 3. How do I solve this challenge problem?

To solve this challenge problem, you will need to manipulate the given expressions using algebraic techniques and apply your knowledge of summation notation and inequalities. It may also be helpful to break the problem down into smaller steps and work through them systematically.

## 4. Is there a specific value of a and b that will satisfy the inequality in the challenge problem?

Yes, there are specific values of a and b that will satisfy the inequality. However, part of the challenge is to determine what those values are through the process of solving the problem.

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

Yes, you can use a calculator to help with calculations, but the solution should be presented in mathematical notation and clearly explained with steps and reasoning.

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