23. $a,b,c>0,abc=1$
$$1)\quad \frac{a-1}{b}+\frac{b-1}{c}+\frac{c-1}{a}\geq 0$$
$$2)\quad \frac{a-1}{b+c}+\frac{b-1}{c+a}+\frac{c-1}{a+b}\geq 0$$
27. $a,b,c>0$
$$\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\leq 3$$
52. $a,b,c\geq 0,a+b+c\geq 3$
$$\frac{1}{a^2+b+c}+\frac{1}{a+b^2+c}+\frac{1}{a+b+c^2}\leq 1$$
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