ABC

\sqrt{\frac{2(p-a)}{c}}+\sqrt{\frac{2(p-b)}{a}}+\sqrt{\frac{2(p-c)}{b}}\geqslant\frac{p^{2}}{a^{2}+b^{2}+c^{2}-p^{2}},

BC=a

CA=b

AB=c

p

u

v

t

uv+ut+vt\leqslant u^{2}+v^{2}+t^{2}.

uv+ut+vt\leqslant u^{2}+v^{2}+t^{2}~\Leftrightarrow~2uv+2ut+2vt\leqslant2u^{2}+2v^{2}+2t^{2}~\Leftrightarrow

\Leftrightarrow~u^{2}+v^{2}-2uv+u^{2}+t^{2}-2ut+v^{2}+t^{2}-2vt\geqslant0~\Leftrightarrow

\Leftrightarrow~(u-v)^{2}+(u-t)^{2}+(v-t)^{2}\geqslant0.

k

l

m

K

L

M

\frac{k^{2}}{K}+\frac{l^{2}}{L}+\frac{m^{2}}{M}\geqslant\frac{(k+l+m)^{2}}{K+L+M}.

k^{2}LM(K+L+M)+l^{2}KM(K+L+M)+m^{2}KL(K+L+M)-

-KLM(k+l+m)^{2}=

=k^{2}LM(K+L+M)+l^{2}KM(K+L+M)+m^{2}KL(K+L+M)-

-KLM(k^{2}+l^{2}+m^{2}+2kl+2km+2lm)^{2}=

=k^{2}LM(L+M)+L^{2}KM(K+M)+m^{2}KL(K+L)-

-(2kl+2lm+km)KLM=

=M(k^{2}L^{2}-2klKL+l^{2}K^{2})+K(l^{2}M^{2}-2lmLM+m^{2}L^{2})+

+L(m^{2}K^{2}-2mkMK+k^{2}M^{2})=

=M(kL-lK)^{2}+K(lM-mL)^{2}+L(mK-kM)^{2}\geqslant0.

a=y+z

b=z+x

c=x+y

x,y,z\geqslant0

p=x+y+z,~a^{2}+b^{2}+c^{2}-p^{2}=

=(x+y)^{2}+(y+z)^{2}+(z+x)^{2}+(x+y+z)^{2}=x^{2}+y^{2}+z^{2}.

\sqrt{\frac{2(p-a)}{c}}+\sqrt{\frac{2(p-b)}{a}}+\sqrt{\frac{2(p-c)}{b}}=

=\sqrt{\frac{2x}{x+y}}+\sqrt{\frac{2y}{y+z}}+\sqrt{\frac{2z}{z+x}}=

=\frac{2x}{\sqrt{2x(x+y)}}+\frac{2y}{\sqrt{2y(y+z)}}+\frac{2z}{\sqrt{2z(z+x)}}\geqslant

\geqslant\frac{4x}{3x+y}+\frac{4y}{3y+z}+\frac{4z}{3z+x}=

=\frac{4x^{2}}{3x^{2}+xy}+\frac{4y^{2}}{3y^{2}+yz}+\frac{4z^{2}}{3z^{2}+xz}\geqslant\frac{4(x+y+z)^{2}}{3(x^{2}+y^{2}+z^{2})+xy+yz+zx}\geqslant

\geqslant\frac{4(x+y+z)^{2}}{3(x^{2}+y^{2}+z^{2})+x^{2}+y^{2}+z^{2}}\geqslant\frac{(x+y+z)^{2}}{x^{2}+y^{2}+z^{2}}=\frac{p^{2}}{a^{2}+b^{2}+c^{2}-p^{2}}.

a=y+z

b=z+x

c=x+y

p=x+y+z,~a^{2}+b^{2}+c^{2}-p^{2}=

=(y+z)^{2}+(z+x)^{2}+(x+y)^{2}-(x+y+z)^{2}=x^{2}+y^{2}+z^{2},

0\lt a+b-c=(y+z)+(z+x)-(x+y)=2z~\Rightarrow~z\gt0.

x\gt0

y\gt0

\sqrt{\frac{2x}{x+y}}+\sqrt{\frac{2y}{y+z}}+\sqrt{\frac{2z}{z+x}}\geqslant\frac{(x+y+z)^{2}}{x^{2}+y^{2}+z^{2}},

\sqrt{2x(x+y)}\leqslant\frac{3x+y}{2},~\sqrt{2y(y+z)}\leqslant\frac{3y+z}{2},~\sqrt{2z(z+x)}\leqslant\frac{3z+x}{2}.

(\sqrt{3x+y},\sqrt{3y+z},\sqrt{3z+x})~\mbox{и}~\left(\frac{2x}{\sqrt{3x^{2}+xy}},\frac{2y}{\sqrt{3y^{2}+yx}},\frac{2z}{\sqrt{3z^{2}+zx}}\right)

(2x+2y+2z)^{2}\leqslant\left(\frac{4x^{2}}{3x^{2}+xy}+\frac{4y^{2}}{3y^{2}+zy}+\frac{4z^{2}}{3z^{2}+zx}\right)(3(x^{2}+y^{2}+z^{2})+xy+yz+zx).

(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\geqslant0~\Rightarrow~xy+yz+zx\leqslant x^{2}+y^{2}+z^{2}.

\sqrt{\frac{2x}{x+y}}+\sqrt{\frac{2y}{y+z}}+\sqrt{\frac{2z}{z+x}}=\frac{2x}{\sqrt{2x(x+y)}}+\frac{2y}{\sqrt{2y(y+z)}}+\frac{2z}{\sqrt{2z(z+x)}}\geqslant

\geqslant\frac{4x}{3x+y}+\frac{4y}{3y+z}+\frac{4z}{3z+x}=\frac{4x^{2}}{3x^{2}+xy}+\frac{4y^{2}}{3y^{2}+yz}+\frac{4z^{2}}{3z^{2}+zx}\geqslant

\geqslant\frac{(2x+2y+2z)^{2}}{3(x^{2}+y^{2}+z^{2})+xy+yz+zx}\geqslant\frac{4(x+y+z)^{2}}{3(x^{2}+y^{2}+z^{2})+x^{2}+y^{2}+z^{2}}=

=\frac{(x+y+z)^{2}}{x^{2}+y^{2}+z^{2}}=\frac{p^{2}}{a^{2}+b^{2}+c^{2}-p^{2}}.