let $a,b,c$ are real numbers,and such $a+b+c=3$,show that $$\dfrac{1}{2a^2-6a+9}+\dfrac{1}{2b^2-6b+9}+\dfrac{1}{2c^2-6c+9}\le\dfrac{3} {5}\cdots (1)$$ I find sometimes,and I find this same problem: let $a,b,c$ are real numbers,and such $a+b+c=3$,show that $$ \frac{1}{5a^2-4a+11}+\frac{1}{5b^2-4b...

I'm looking for some ideas to deal with my inequality. P/s: I'd like to create simple look inequalities so we can call them as artificial inequalities. I agree that my inequality may be wrong but I checked it carefully before posting here. Problem. Let $a,b,c\ge 0: a+b+c=4.$ Prove that$$\color{b...

The Problem. Let $a,b,c$ be non negative real numbers such that $a+b+c=1$, show that the follwoing inequality holds true $$ \frac{2a}{-6a^2+5a+3}+\frac{2b}{-6b^2+5b+3}+\frac{2c}{-6c^2+5c+3}\geq \frac{1}{2}. $$ My Attempt(s) I first approached the problem simply by using Cauchy-Schwartz inequalit...

I’d like to bring to your attention a request from Michael Rozenberg, one of the most prominent members of the Mathematics Stack Exchange community and one of the best in proving inequalities. He has advocated for the creation of a tag for the bacteria method, a powerful technique widely used in ...

In this answer which I wrote a few days ago, I posit that for any polynomial $P \in \mathbb{N}[x]$, the asymptotic density of the set of natural numbers $n$ such that $P(n)$ is odious (that is; has an odd number of 1 bits in its binary expansion) is $\frac{1}{2}$, and likewise for evil numbers (t...

An evil number is a positive integer $n$ that has an even number of $1$s in its binary expansion. Many theorems exist about evil numbers, the most known ones are probably those that involve the Thue-Morse sequence. However, I find no information about prime numbers having an even number of $1$s...