HMIC 2022

Is

$$\prod_{k=0}^{\infty}\left(1-\frac{1}{2022^{k!}}\right)$$

rational?

Does there exist a regular pentagon whose vertices lie on edges of a cube?

For a nonnegative integer $n$, let $s(n)$ be the sum of digits of the binary representation of $n$. Prove that

$$\sum_{n=0}^{2^{2022}-1} \frac{(-1)^{s(n)}}{2022+n}>0$$

Call a simple graph $G$ quasi-colorable if we can color each edge blue, red, green, and white such that

• for each vertex $v$ of degree 3 in $G$, the three edges containing $v$ as an endpoint are either colored blue, red, and green, or all three edges are white,
• not all edges are white.

A connected graph $G$ has $a$ vertices of degree $4, b$ vertices of degree 3, and no other vertices, where $a$ and $b$ are positive integers. Find the smallest real number $c$ so that the following statement is true: "If $a / b>c$, then $G$ is quasi-colorable."

Let $p$ be a prime and let $\mathbb{F}_{p}$ be the set of integers modulo $p$. Call a function $f: \mathbb{F}_{p}^{2} \rightarrow \mathbb{F}_{p}$ quasiperiodic if there exist $a, b \in \mathbb{F}_{p}$, not both zero, so that $f(x+a, y+b)=f(x, y)$ for all $x, y \in \mathbb{F}_{p}$. Find, with proof, the number of functions $\mathbb{F}_{p}^{2} \rightarrow \mathbb{F}_{p}$ that can be written as the sum of some number of quasiperiodic functions. Note: The original version of this problem neglected to mention that p was a prime. We regret the error.