HMIC 2021

2021 people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up 1000 positions to the left of their original position.

Let $n$ be a positive integer. Alice writes $n$ real numbers $a_{1}, a_{2}, \ldots, a_{n}$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors $\left(a_{n}\right.$ is not a neighbor of $a_{1}$, nor vice versa). A number changes sign if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_{1}$ can change sign, across all possible values of $a_{1}, a_{2}, \ldots, a_{n}$ and all possible sequences of moves Alice may make.

Let $A$ be a set of $n \geq 2$ positive integers, and let $f(x)=\sum_{a \in A} x^{a}$. Prove that there exists a complex number $z$ with $|z|=1$ and $|f(z)|=\sqrt{n-2}$.

Let $A_{1} A_{2} A_{3} A_{4}, B_{1} B_{2} B_{3} B_{4}$, and $C_{1} C_{2} C_{3} C_{4}$ be three regular tetrahedra in 3-dimensional space, no two of which are congruent. Suppose that, for each $i \in\{1,2,3,4\}, C_{i}$ is the midpoint of the line segment $A_{i} B_{i}$. Determine whether the four lines $A_{1} B_{1}, A_{2} B_{2}, A_{3} B_{3}$, and $A_{4} B_{4}$ must concur.

In an $n \times n$ square grid, $n$ squares are marked so that every rectangle composed of exactly $n$ grid squares contains at least one marked square. Determine all possible values of $n$.