My try: WLOG a≥b≥c hence we have: am−n≥bm−n≥cm−n

⇒anbn+cn≥bncn+an≥cnan+bn

By Chebyshev's inequality we have: 

∑ambn+cn=∑am−n⋅anbn+cn

≥am−n+bm−n+cm−n3⋅∑anbn+cn

≥∑am−n+bm−n+cm−n2

We're done when a=b=c

P/s: This's discuss not A4 (Auto Ask Auto Answer)

My try: WLOG \(a\ge b\ge c\) hence we have: \(a^{m-n}\ge b^{m-n}\ge c^{m-n}\)

\(\Rightarrow\dfrac{a^n}{b^n+c^n}\ge\dfrac{b^n}{c^n+a^n}\ge\dfrac{c^n}{a^n+b^n}\)

By Chebyshev's inequality we have: 

\(\text{∑}\dfrac{a^m}{b^n+c^n}=\text{∑}a^{m-n}\cdot\dfrac{a^n}{b^n+c^n}\)

\(\ge\dfrac{a^{m-n}+b^{m-n}+c^{m-n}}{3}\cdot\text{∑}\dfrac{a^n}{b^n+c^n}\)

\(\ge\text{∑}\dfrac{a^{m-n}+b^{m-n}+c^{m-n}}{2}\)

We're done when \(a=b=c\)

P/s: This's discuss not A⁴ (Auto Ask Auto Answer)