Question 35
Let $x_1, \dots, x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$
1 \le \frac{\left(x_1 + \frac{x_2}{2} + \frac{x_3}{3} + \dots + \frac{x_n}{n}\right) \cdot \left(x_1 + 2x_2 + 3x_3 + \dots + nx_n\right)}{\left(x_1 + x_2 + x_3 + \dots + x_n\right)^2} \le \frac{(n+1)^2}{4n}.
$$
(ii) Show that, for each $n \ge 1$, both inequalities can hold as equalities.
(i) Prove that
$$
1 \le \frac{\left(x_1 + \frac{x_2}{2} + \frac{x_3}{3} + \dots + \frac{x_n}{n}\right) \cdot \left(x_1 + 2x_2 + 3x_3 + \dots + nx_n\right)}{\left(x_1 + x_2 + x_3 + \dots + x_n\right)^2} \le \frac{(n+1)^2}{4n}.
$$
(ii) Show that, for each $n \ge 1$, both inequalities can hold as equalities.