A Magic Proof about an Inequality

Problem

Let ${a}$ , ${b}$ , ${c}$ be positive real number such that ${21ab+2bc+8ca\leq 12}$ . Prove the inequality

$\displaystyle \frac{1}{a}+\frac{2}{b}+\frac{3}{c}\geq \frac{15}{2}$

Continue reading →

A Short Summary of Describing Polynomials

Let’s see some problems about describing polynomials first.

Problems

Let $n=2^m,\ m\in\mathbb{N}^{*}$ , describe all polynomials $f(x)$ with a degree of $n$ and coefficients in $\mathbb{R}$ that satisfy equality $f(x^2+1)=f(x)^2+1$ .
Describe all polynomials $f:\mathbb{N}^*\to\mathbb{N}^*$ that satisfy condition $f(y)+x^2\mid f(x)^2+y$ for all $x,y\in\mathbb{N}^*$ .
Describe all polynomials $f(x)$ that satisfy $(x-1)\cdot f(x+1)-(x+2)\cdot f(x)=0$ .

Continue reading →

Why the Area of a Circle is Equal to πr^2?

I have been told that the area of a circle is equal to $\pi r^2$ since I was in primary school by my math teacher, and from then on I use this formula to solve many problems about circles. And these days, when I used this formula again, I suddenly realized that I didn’t know why the formula will look like this in fact.