Question 176
Let $f : [0, 1] \to \mathbb{R}$ be a continuous function and $n \ge 3$ be an integer. Show that one can find $n$ numbers $a_1, a_2, \dots, a_n$ in the interval $[0, 1]$, in arithmetic progression, such that
$$
\int_{0}^{1} f(x) \, dx = \frac{1}{n} \sum_{k=1}^{n} f(a_k).
$$
$$
\int_{0}^{1} f(x) \, dx = \frac{1}{n} \sum_{k=1}^{n} f(a_k).
$$