2022 HMIC

Part of Harvard-MIT Mathematics Tournament

Subcontests

(5)

1

HMIC loves "quasi"

\mathbb{F}_p

be the set of integers modulo

p

. Call a function

f : \mathbb{F}_p^2 \to \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)

x,y \in \mathbb{F}_p

. Find the number of functions

\mathbb{F}_p^2 \to \mathbb{F}_p

that can be written as the sum of some number of quasiperiodic functions.

1

Quasicolorable coloring of a graph

Call a simple graph

G

quasicolorable if we can color each edge blue, red, green, or white such that
[*] for each vertex v of degree 3 in G, the three edges incident to v are either (1) red, green, and blue, or (2) all white, [*] not all edges are white.
A simple connected graph

G

a

vertices of degree

4

b

vertices of degree

3

, and no other vertices, where

a

b

are positive integers. Find the smallest real number

c

so that the following statement is true: “If

a/b > c

G

must be quasicolorable.”

1

Trivial by direct computation

For a nonnegative integer

n

s(n)

be the sum of the digits of the binary representation of

n

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

1

An extraordinary regular pentagon

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

1

Rational Infinite Product

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