jasperE3 wrote: For any positive numbers $a,b,c$ and natural numbers $n,k$ prove the inequality $$\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.$$ Cyclic sum $(n+k)^2\frac{a^{n+k}}{b^n}+n(n+k)\frac{b^{n+k}}{c^n}+n^2\frac{c^{n+k}}{a^n}\ge (3n^2+3nk+k^2)a^k$

It's also true by rearrangement.