Factorize $5(a^3+b^3+c^3)-3(a^2+b^2+c^2)(a+b+c)+12abc$.

Factorize $5(a^3+b^3+c^3)-3(a^2+b^2+c^2)(a+b+c)+12abc$.

My solution:

The given expression is written so neatly and beautifully at first glance, it's like it couldn't be factored. And we feel so reluctant to expand the second product, but we have to, if we want to factor the expression correctly, we have to get a clearer picture of what this expression is all about by expanding and then rearranging terms in decreasing order:

Analysis Quiz 18: Multiple-Choice Test (Develop Problem Solving Skill)

Please answer the following questions based on the proving of the inequality below:
Given $a,\,b,\,c$ and $d$ are all positive real numbers such that $a+b+c+d=4$. Prove that [MATH]\sum_{\text{cyclic}}^{} \frac{a}{a^3+8}\le \frac{4}{9}[/MATH].

Question 1: Do you think we can stick to the approach we employed in the previous quiz (17) for proving this inequality?
A. Yes.
B. No.

Quiz 18: Multiple-Choice Test (Develop Problem Solving Skill)

IMO Inequality Problem

Let $a,\,b,\,c$ be real numbers greater than $2$ such that $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1$.

Prove that $(a-2)(b-2)(c-2)\le 1$.

My solution:

Note that

$(a−2)(b−2)(c−2)$

A system of the form $a+b\sqrt{2}$.

Given $x$ and $y$ are of the form $a+b\sqrt{2}$ ($a$ and $b$ are both positive integers) that satisfy the equation

$2x+y-\sqrt{3x^2+3xy+y^2}=2+\sqrt{2}$.

Find such $x$ and $y$.

Find the minimum value of $\tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$

$x$, $y$ and $z$ are three acute angles from a triangle.

Find the minimum value of $\tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$.

The trick for solving this problem fast and effective depends on if you could use the implicit relation between $x,\,y$ and $z$ when they are the angles from a triangle:

What else could we generate from $A+B+C=\pi$, when $A,\, B,\,C$ are three angles from a triangle?

Following previous blog post, we have shown that if

If $A,\, B,\,C$ are three angles from a triangle, i.e. $A+B+C=\pi$, then we should have known the following equality holds.

[MATH]\tan A+\tan B+\tan C=\tan A\tan B\tan C[/MATH]

On this blog post, we now try to generate another relation between $A,\,B$ and $C$ for cotangent functions:

What could educators do to motivate students to learn well in mathematics?

Mathematics is one of the most powerful tools to shape the way we think and see the world. But it's no secret that success in learning math is very much depends on maintaining a high level of motivation. Without motivation and a sense of emotional involvement, it's hard if not difficult to have the stamina to keep learning.

Analysis Quiz 17: Multiple-Choice Test (Improve Logical Thinking and Develop Discerning Patterns Skills)

Question 1: Given $a,\,b,\,c$ and $d$ are all positive real numbers such that $a+b+c+d=4$. Prove that [MATH]\sum_{cyclic} \frac{a}{a^3+8}\lt \frac{12}{25}[/MATH].

Do you think you would need the given equality to prove for the inequality?

A. Yes.
B. No.

Answer:

Of course we need the given equality to prove for the target expression such that it must be less than [MATH]\frac{12}{25}[/MATH].

Quiz 17: Multiple-Choice Test (Improve Logical Thinking and Develop Discerning Patterns Skills)

Happy New Year 2016, everyone! :D