rogue wrote: Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$ Proof of Zhangyunhua:

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Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive numbers such that $ a_1a_2 \cdots a_n\geq 1.$ Prove that $$a_1^{n+1}+a_2^n+\cdots+a_{n-1}^3+a_n^2\geq a_1^{n}+a_2^{n-1}+\cdots+a_{n-1}^2+a_n.$$

sqing wrote: Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive numbers such that $ a_1a_2 \cdots a_n\geq 1.$ Prove that $$a_1^{n+1}+a_2^n+\cdots+a_{n-1}^3+a_n^2\geq a_1^{n}+a_2^{n-1}+\cdots+a_{n-1}^2+a_n.$$ Proof of ytChen: Since $x^{k+1}-x^k-x+1=\left(x^k-1\right)(x-1)=\left(x^{k-1}+x^{k-2}+\cdots+1\right)(x-1)^2\ge0$ for any $x>0$ and positive integer $k$, we have $x^{k+1}-x^k\ge x-1$ for any $x>0$ and positive integer $k$, which implies that \begin{align*}&a_1^{n+1}+a_2^n+\cdots+a_{n-1}^3+a_n^2-\left(a_1^{n}+a_2^{n-1}+\cdots+a_{n-1}^2+a_n\right)\\ \ge&(a_1-1)+(a_2-1)+\cdots+(a_n-1)\\ =&a_1+a_2+\cdots+a_n-n\ge n\sqrt[n]{a_1a_2 \cdots a_n}-n\ge 0. \end{align*}

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