Question 29
Let $n \ge 3$ and $u_1, ..., u_n$ be real numbers satisfying
$$ \sum_{i=1}^{n} u_i^{2018} = 1, \quad \sum_{i=1}^{n} u_i^{2019} = 0. $$
Show that there exist $1 \le k_1 \lt k_2 \le n$ such that
$$
\sum_{i=1}^{n} u_i^{2020} \le |u_{k_1} u_{k_2}|.
$$
$$ \sum_{i=1}^{n} u_i^{2018} = 1, \quad \sum_{i=1}^{n} u_i^{2019} = 0. $$
Show that there exist $1 \le k_1 \lt k_2 \le n$ such that
$$
\sum_{i=1}^{n} u_i^{2020} \le |u_{k_1} u_{k_2}|.
$$