<s>[INST] If $ab+bc+ca+abc=4$, then $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq 3\leq a+b+c$ Let $a,b,c$ be positive reals such that $ab+bc+ca+abc=4$.
Then prove
$\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq 3\leq a+b+c $
So high guys im a high schooler trying to solve this inequality. I did a few things such as try to set a> b>c and then try to generalize it but i couldn't make much progress at all. I would appreciate a clear solution to help me understand the problem in depth.
I did manage to generalize that 
$\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq a+b+c $
By using the AM-GM inequality on each of a b and c however havent made much progress in proving that the LHS is less than 3 and the RHS is greater than 3.
 [/INST] Both inequalities we can prove by the Contradiction method.
Indeed, for the right inequality let $a+b+c<3$, $a=kx$, $b=ky$ and $c=kz$, 
where $k>0$ and $x+y+z=3$.
Thus, $k(x+y+z)<3$, which gives $k<1$.
Hence, $4=ab+ac+bc+abc=k^2(xy+xz+yz)+k^3xyz<xy+xz+yz+xyz$, 
which is contradiction because we'll prove now that
$$4\geq xy+xz+yz+xyz$$ or
$$\frac{4(x+y+z)^3}{27}\geq\frac{(x+y+z)(xy+xz+yz)}{3}+xyz$$ or
$$\sum_{cyc}(4x^3+3x^2y+3x^2z-10xyz)\geq0,$$
which is true by Muirhead.
Also, we can use AM-GM.
$$\sum_{cyc}x^3=x^3+y^3+z^3\geq3\sqrt[3]{x^3y^3z^3}=3xyz;$$
$$\sum_{cyc}(x^2y+x^2z)=x^2y+x^2z+y^2x+y^2z+z^2x+z^2y\geq6\sqrt[6]{x^6y^6z^6}=6xyz.$$
Thus, $$\sum_{cyc}(4x^3+3x^2y+3x^2z-10xyz)=$$
$$=4(x^3+y^3+z^3)+3(x^2y+x^2z+y^2x+y^2z+z^2x+z^2y)-30xyz\geq$$
$$\geq4\cdot3xyz+3\cdot6xyz-30xyz=0.$$
The left inequality.
Let $\sqrt{ab}+\sqrt{ac}+\sqrt{bc}>3$, $a=kx$, $b=ky$ and $c=kz$, 
where $k>0$ and $\sqrt{xy}+\sqrt{xz}+\sqrt{yz}=3$.
Thus, $k(\sqrt{xy}+\sqrt{xz}+\sqrt{yz})>3$, which gives $k>1$ and
$$4=ab+ac+bc+abc=k^2(xy+xz+yz)+k^3xyz>xy+xz+yz+xyz,$$
which is contradiction because we'll prove now that
$$4\leq xy+xz+yz+xyz.$$
Indeed, let $\sqrt{xy}=r$, $\sqrt{xz}=q$ and $\sqrt{yz}=p$.
Thus, $p+q+r=3$ and we need to prove that
$$p^2+q^2+r^2+pqr\geq4$$ or
$$\frac{(p+q+r)(p^2+q^2+r^2)}{3}+pqr\geq\frac{4(p+q+r)^3}{27}$$ or
$$\sum_{cyc}(5p^3-3p^2q-3p^2r+pqr)\geq0$$ or
$$\sum_{cyc}(p^3-p^2q-p^2r+pqr)+2\sum_{cyc}(2p^3-p^2q-p^2r)\geq0,$$
which is true by Schur and Muirhead.
The inequality $\sum\limits_{cyc}(2p^3-p^2q-p^2r)\geq0$ we can prove also without Muirhead:
$$\sum_{cyc}(2p^3-p^2q-p^2r)=\sum_{cyc}(p^3-p^2q-pq^2+q^3)=\sum_{cyc}(p-q)^2(p+q)\geq0.$$
Done!
Also, we can use the following trigonometric substitution.
Let $\sqrt{ab}=2\cos\gamma$, $\sqrt{ac}=2\cos\beta$ and $\sqrt{bc}=\cos\alpha$.
Thus, $$\cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma=1,$$
where $\alpha$, $\beta$ and $\gamma$ are measures of acute angles.
Hence, easy to see that $\alpha+\beta+\gamma=180^{\circ}$ and we need to prove that
$$\cos\alpha+\cos\beta+\cos\gamma\leq\frac{3}{2}$$ and
$$\sum_{cyc}\frac{\cos\alpha\cos\beta}{\cos\gamma}\geq\frac{3}{2},$$
which is obvious.
Done again!
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