IMO 2021 Problem 5

Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts for the winter. Jumpy numbers the walnuts from 1 through $2021$, and digs $2021$ little holes in a circular pattern in the ground around their favorite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of $2021$ moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.

Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

===============================

Sketch of proof:

Equivalently, we want to show the following is impossible to achieve. In a circle of $2021$ cells which are initially all unmarked, we mark them one by one sequentially, such that each cell, when marked, is adjacent to either two marked or two unmarked cells.

IMO 2021 Problem 1

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

====================================

Proof:

There must be an integer $k$ such that $2k^2-4k$, $2k^2+1$, and $2k^2+4k$ are all within $[n, 2n]$. Any two of them sum up to a perfect square.

Q.E.D.