Given \(m\ge n>0\) and \(a,b,c\) are positive real numbers. Prove this inequality \(\dfrac{a^m}{b^n+c^n}+\dfrac{b^m}{c^n+a^n}+\dfrac{c^m}{a^n+b^n}\ge\dfrac{a^{m-n}+b^{m-n}+c^{m-n}}{2}\)

 it's a general formula of Nesbitt's inequality, i have a method but i need more :)), Help me, thanks 

For \(a,x,y,z\) are positive real numbers satisfy \(xyz=1\) and \(a\ge 1\). Prove that \(\dfrac{x^a}{y+z}+\dfrac{y^a}{x+z}+\dfrac{z^a}{x+y}\ge\dfrac{3}{2}\)

-Like Nesbitt's inequality, i have a problem it's a general formula of Nesbitt's inequality,  if necessary please inbox to me :)

Prove that with \(n>3\) and \(x_1;x_2;...;x_n>0\) satisfy \(\Pi^n_{i=1}x_i=1\), so \(\dfrac{1}{1+x_1+x_1x_2}+\dfrac{1}{1+x_2+x_2x_3}+...+\dfrac{1}{1+x_n+x_nx_1}>1\)

- Source: The problem from Russia,2004

P/s: Spammer go away, Contraindication for young buffalo !!