Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\]Determine the value of $f(2023)/f(2022)$.

Find all ordered pairs of positive integers $(r, s)$ for which there are exactly $35$ ordered pairs of positive integers $(a, b)$ such that the least common multiple of $a$ and $b$ is $2^r \cdot 3^s$.

Let $ABCD \dots KLMN$ be a regular polygon with $14$ sides. Show that the diagonals $AE$, $BG$, and $CK$ are concurrent.

A teacher wants her $N$ students to know each other, so she creates various clubs of three people, so that each student can participate in several clubs. The clubs are formed in such a way that if $A$ and $B$ are two people, then there is a single club such that $A$ and $B$ are two of its three members. (1) Show that there is no way for the teacher to form the clubs if $N = 11$. (2) Show that the teacher can do it if $N = 9$.

Let $t$ be a positive real number such that $t^4 + t^{-4} = 2023$. Determine the value of $t^3 + t^{-3}$ in the form of $a\sqrt b$, where $a$ and $b$ are positive integers.

Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$. (1) Find a $6$-digit positive integer $N$ such that \[\frac{u(N)}{N} = \frac{23}{35}.\](2) Prove that there is no positive integer $N$ with less than $6$ digits such that \[\frac{u(N)}{N} = \frac{23}{35}.\]