Question 155
Let $x_1, x_2, \dots, x_n$ (where $n \ge 2$) be real numbers with
$$x_1^2 + x_2^2 + \dots + x_n^2 = 1.$$
Prove that
$$
\sum_{k=1}^{n} \left( 1 - \frac{k}{\sum_{i=1}^{n} ix_i^2} \right)^2 \cdot \frac{x_k^2}{k} \le \left( \frac{n-1}{n+1} \right)^2 \sum_{k=1}^{n} \frac{x_k^2}{k}.
$$
Determine when the equality holds.
$$x_1^2 + x_2^2 + \dots + x_n^2 = 1.$$
Prove that
$$
\sum_{k=1}^{n} \left( 1 - \frac{k}{\sum_{i=1}^{n} ix_i^2} \right)^2 \cdot \frac{x_k^2}{k} \le \left( \frac{n-1}{n+1} \right)^2 \sum_{k=1}^{n} \frac{x_k^2}{k}.
$$
Determine when the equality holds.