Ankoganit wrote: Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$ $uvw$ kills it because this inequality it's $f(v^2)\geq0$, where $f$ is a linear function.

Let $a=\frac{x}{y}$ $b=\frac{y}{z}$ $c=\frac{z}{x}$ So our inequality becomes $\sum_{cyc} \frac{x}{y}+ 3\ge 4\sum_{cyc} \frac{yz}{xz+y^2}$ Which is just A.M-G.M $4 \frac{yz}{xz+y^2}\le 2\sqrt{\frac{z}{x}}$ and $\frac{z}{x}+1\ge 2\sqrt{\frac{z}{x}}$ Write similar inequalities and sum up. Done!!!

Ankoganit wrote: Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$Dimitar Trenevski By AM-GM , $$\frac{a+b+c+3}{4}\geq \frac{1}{2}(\sqrt{a}+\sqrt{b}+\sqrt{c})= \frac{1}{2\sqrt{ab}}+\frac{1}{2\sqrt{bc}}+\frac{1}{2\sqrt{ca}}\geq\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$

Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+1}{3}\geqslant \frac{1}{a+b+c}+\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}.$$Anzhenping

sqing wrote: Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+1}{3}\geqslant \frac{1}{a+b+c}+\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}.$$Anzhenping Right<=4/3

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$. Prove that $$\frac{a+b+c+1}{3}\geqslant \frac{1}{a+b+c}+\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}.$$

sqing wrote: Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$. Prove that $$\frac{a+b+c+1}{3}\geqslant \frac{1}{a+b+c}+\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}.$$ Right<=4/3

sqing wrote: Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove that $$\frac{a+b+c+1}{3}\geqslant \frac{1}{a+b+c}+\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}.$$ Right>=4/3>=Left

we will prove (c+1)/4 ≥ 1/(a+b) or in fact ca + cb + a + b ≥ 4. ca = 1/b and cb = 1/a and we know a + b + 1/a + 1/b ≥ 2 + 2 = 4. we're Done.