How to prove $\frac{a}{a+bc}+\frac{b}{b+ac}+\frac{c}{c+ab}\geq \frac{3}{2}$ With $a + b + c = 3$ and $a, b, c>0$ prove these inequality:
1)$$\frac{a}{a+bc}+\frac{b}{b+ac}+\frac{c}{c+ab}\geq \frac{3}{2}$$
2)$$\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}+abc\geq 4$$
3)$$\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}+\frac{9}{4}abc\geq \frac{21}{4}$$
 A: (Note: The notation $\sum$ refers to a cyclic sum here.)
For the first one:
We can assume WLOG that $a\ge b\ge c$. We will prove a lemma:
Let $x, y$ be positive reals with $xy\ge 1$. Then: $$\frac{1}{1+x^2}+\frac{1}{1+y^2}\ge \frac{2}{1+xy}$$
Proof: Upon direct expanding, the inequality is equivalent to $(x-y)^2(xy-1)\ge 0$, which is indeed true.
Now applying the lemma with $x=\sqrt{\frac{ab}{c}}, y=\sqrt{\frac{ac}{b}}$, we have $xy=a\ge 1$ and thus:
$$\sum\frac{a}{a+bc}=\sum\frac{1}{1+\frac{bc}{a}}\ge \frac{2}{1+a}+\frac{a}{a+bc}$$
Note that $bc\le\frac{(b+c)^2}{4}=\frac{(3-a)^2}{4}=\frac{a^2-6a+9}{4}$. Thus:
$$\frac{2}{1+a}+\frac{a}{a+bc}\ge \frac{2}{1+a}+\frac{4a}{a^2-2a+9}$$ It suffices to show that the above is at least $\frac{3}{2}$. Indeed, direct calculation gives:
$$2(1+a)(a^2-2a+9)(\frac{2}{1+a}+\frac{4a}{a^2-2a+9}-\frac{3}{2})=3(3-a)(a-1)^2\ge 0$$
because $a\le 3$. 
A: For the second and the third question,a generalized version is presented here:

Let $a,b,c>0$ and $a+b+c=3$,then
$$f(a,b,c)=(\sum\frac{ab}{c})+\lambda abc\geq3+\lambda$$
for $1\leq \lambda\leq9/4$,where the constant $9/4$ is optimal.

It is easy to see that 9/4 is optimal(a simple comparation between $f(1,1,1)$ and $f(2,1/2,1/2)$).
Proof:
Without loss of generality,we assume that $a\geq b\geq c$.Then $1\leq a<3$.
$$f(a,b,c)=a(\frac{b}{c}+\frac{c}{b})+\frac{bc}{a}+\lambda abc=\frac{a(b+c)^2}{bc}+(\lambda a+\frac{1}{a})bc-2a$$
Recall that $a+b+c=3$,then
$$f(a,b,c)=\frac{a(3-a)^2}{bc}+(\lambda a+\frac{1}{a})bc-2a$$
Given $p,q>0$,it is obvious that function $u(x)=px+q/x$ is monotone decreasing on interval $(0,\sqrt{q/p}]$.We take $(\lambda a+\frac{1}{a},a(3-a)^2)$ as $(p,q)$,then $f(a,b,c)$ is a monotone decreasing function(for $bc$) on the interval $(0,\sqrt{q/p}]$ with $a$ fixed.
AM-GM inequality suggests that $$bc\leq\frac{(b+c)^2}{4}=\frac{(3-a)^2}{4},$$ and it is natural to test whether $(b+c)^2/4\leq\sqrt{q/p}$,i.e.,
$$\frac{(3-a)^2}{4}\leq\frac{a(3-a)}{\sqrt{\lambda a^2+1}}$$
It suffices to show that $$\frac{(3-a)^2}{16}\leq\frac{a^2}{\lambda a^2+1},$$i.e.,
$$\lambda\frac{(3-a)^2}{16}+\frac{1}{\lambda a^2+1}\leq 1$$.
Because $1\leq a<3$,$LHS\leq\lambda/4+1/(\lambda+1)\leq 1$ for every $1\leq \lambda\leq 3$
Hence $f(a,b,c)$ achieve its minimum(for fixed $a$) when $bc=(3-a)^2/4$.
$$\min_{a fixed}f(a,b,c)=(\lambda a+\frac{1}{a})\frac{(3-a)^2}{4}+2a$$.
We just need to show that
$$(\lambda a+\frac{1}{a})\frac{(3-a)^2}{4}+2a\geq 3+\lambda$$ 
for every $1\leq \lambda\leq 9/4$,which is equivalent to show that
$$\lambda(a-1)^2(a^2-4a+\frac{9}{\lambda})\geq 0$$
for every $1\leq \lambda\leq 9/4$.
It suffices to prove that $(a^2-4a+\frac{9}{\lambda})\geq 0$ for for every $1\leq \lambda\leq 9/4$,and it is quite obvious. 
Q.E.D.
A: We need to prove that:
$$\sum_{cyc}\frac{a}{a+bc}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\left(\frac{a}{a+bc}-\frac{1}{2}\right)\geq0$$ or
$$\sum_{cyc}\frac{a-bc}{a+bc}\geq0$$ or
$$\sum_{cyc}\frac{a(a+b+c)-3bc}{a+bc}\geq0$$ or
$$\sum_{cyc}\frac{(a-b)(a+3c)-(c-a)(a+3b)}{a+bc}\geq0$$ or
$$\sum_{cyc}(a-b)\left(\frac{a+3c}{a+bc}-\frac{b+3c}{b+ac}\right)\geq0$$ or
$$\sum_{cyc}\frac{(a-b)^2c(a+b+3c-3)}{(a+bc)(b+ac)}\geq0$$ or
$$\sum_{cyc}\frac{(a-b)^2c^2}{(a+bc)(b+ac)}\geq0.$$
Done!
A: For the first one:
Set $\left(\sqrt{\frac{bc}{a}},\sqrt{\frac{ca}{b}},\sqrt{\frac{ab}{c}}\right)\rightarrow (x,y,z)$
Thus $xy+yz+zx=3$ and we must prove $$\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\ge \frac{3}{2}$$
It's equivalent to $\sum x^2+3\ge 3x^2y^2z^2+\sum y^2z^2$
$$\Leftrightarrow (x+y+z-xyz)(x+y+z+3xyz)\ge 12$$
Set $p=x+y+z, \,\ q=xy+yz+zx=3, \,\ r=xyz$
We need to prove that $(p-r)(p+3r)\ge 12\Leftrightarrow (pq-3r)(pq+9r)\geq 4q^3$
$$\Leftrightarrow p^2q^2+6pqr\ge 4q^3+27r^2$$
Since $\sum x^3y+\sum xy^3\ge 2\sum x^2y^2\implies p^2q+3pr\ge 4q^2\implies p^2q^2+3pqr\ge 4q^3 \,\ (\star )$
Since $(x+y+z)(xy+yz+zx)\ge 9xyz\implies pq\ge 9r\implies 3pqr\ge 27r^2 \,\ (\star \star)$
$(\star) +(\star \star)\implies $ the desired result
Done
