Intereting Posts

I need the proving of $x\log(x)=(\frac{x-1}{x})+\frac{3}{2!}(\frac{x-1}{x})^2+\frac{11}{3!}(\frac{x-1}{x})^3+…\frac{S_{n}}{n!}(\frac{x-1}{x})^n$
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uniform convergence of few sequence of functions
Example of infinite groups such that all its elements are of finite order
When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?
Dot Product Intuition
Proof that ${2p\choose p}\equiv 2\pmod p$
Proving the area of a square and the required axioms
Do integrable functions vanish at infinity?
Computing $n$ such that $\phi(n) = m$
Defining a Free Module
Beginner material for mathematical logic
How to prove if this equation provides an integral solution divisible by $3$?
Are 14 and 21 the only “interesting” numbers?

Let $x,y,z\ge 0$, show that

$$\dfrac{y}{xy+2y+1}+\dfrac{z}{yz+2z+1}+\dfrac{x}{zx+2x+1}\le\dfrac{3}{4}$$

I had solve

$$\sum_{cyc}\dfrac{y}{xy+y+1}\le 1$$

becasuse After some simple computations, it is equivalent to

$$(1-xyz)^2\ge 0$$

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Note that by CS inequality,$$\frac{4y^2}{xy^2+2y^2+y} =\frac{(y+y)^2}{(xy^2+y^2)+(y^2+y)}\le \frac{y^2}{xy^2+y^2}+\frac{y^2}{y^2+y}=\frac{1}{x+1}+\frac{y}{y+1}$$

Summing three such terms we get

$$4\sum_{cyc}\frac{y}{xy+2y+1}\le \sum_{cyc}\frac1{x+1}+\sum_{cyc}\frac{x}{x+1}=3$$

P.S. You will need to handle the case $xyz=0$ separately, shouldnt be too difficult.

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