OPM 2021

Joana divided $365$ by all integers from $1$ to $365$ and added all the remainders. Then she divided $366$ by all the integers from $1$ to $366$ and also added all the remainders. Which of the two sums is greater and what is the difference between them?

Let $ABC$ be a triangle such that $AB = AC$ . Let $D$ be a point in $[BC]$ and $E$ a point in $[AD]$ such that $\angle BE D = \angle BAC = 2 \angle DEC$ . Shows that $DB = 2CD$ .

All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$ . What is the sum of the smallest elements of all these sequences?