USEMO 2021

Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$ 's column and strictly less than the average of $c$ 's row.

Proposed by Holden Mui

Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors.

Proposed by Ankan Bhattacharya

Let $A_1C_2B_1A_2C_1B_2$ be an equilateral hexagon. Let $O_1$ and $H_1$ denote the circumcenter and orthocenter of $\triangle A_1B_1C_1$ , and let $O_2$ and $H_2$ denote the circumcenter and orthocenter of $\triangle A_2B_2C_2$ . Suppose that $O_1 \ne O_2$ and $H_1 \ne H_2$ . Prove that the lines $O_1O_2$ and $H_1H_2$ are either parallel or coincide.

Ankan Bhattacharya