parmenides51 wrote: We will call the sequence of positive real numbers. $a_1,a_2,\dots ,a_n$ of length $n$ when $$a_1\geq \frac{a_1+a_2}{2}\geq \dots \geq \frac{a_1+a_2+\cdots +a_n}{n}.$$Let $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ be sequences of length $n.$ Prove that $$\sum_{i = 1}^{n}x_iy_i\geq\frac{1}{n}\left(\sum_{i = 1}^{n}x_i\right)\left(\sum_{i = 1}^{n}y_i\right).$$ (tatari/nightmare) Nice. "length $n$" is equivalent to "increasing sequence". Thus, the required inequality is just Cebishev