IMOC 2021 A11

anhcanhsat97

给定 $ n \geq 2 $ 实数 $ x_1, x_2,..., x_n$ 证明
$$ \prod_{1 \leq i <j \leq n} (x_i-x_j) ^ 2 \leq \prod_ {i = 0}^{n-1}\left (\sum_{j = 1}^{n }x_j^{2i} \right) $$ 并找到所有 $(x_1, x_2,..., x_n)$ 在相同的位置

anhcanhsat97

回复 1# anhcanhsat97
A9 (ckliao914). For any given positive integer $n$, find
\[
\sum_{k=0}^n\left(\frac{\binom{n}{k} \cdot(-1)^k}{(n+1-k)^2}-\frac{(-1)^n}{(k+1)(n+1)}\right)
\]
A10 (Untro368). For any positive reals $x, y, z$ with $x y z+x y+y z+z x=4$, prove that
\[
\sqrt{\frac{x y+x+y}{z}}+\sqrt{\frac{y z+y+z}{x}}+\sqrt{\frac{z x+z+x}{y}} \geq 3 \sqrt{\frac{3(x+2)(y+2)(z+2)}{(2 x+1)(2 y+1)(2 z+1)}}
\]
A11 (Untro368). Given $n \geq 2$ reals $x_1, x_2, \ldots, x_n$. Show that
\[
\prod_{1 \leq i<j \leq n}\left(x_i-x_j\right)^2 \leq \prod_{i=0}^{n-1}\left(\sum_{j=1}^n x_j^{2 i}\right)
\]
and find all the $\left(x_1, \ldots, x_n\right)$ where the equality holds.

hbghlyj

“在相同的位置”是指什么？