Susie thinks of a positive integer $n$. She notices that, when she divides $2023$ by $n$, she is left with a remainder of $43$. Find how many possible values of $n$ there are.

The two positive integers $a, b$ with $a > b$ are such that $a\%$ of $b\%$ of $a$ and $b\%$ of $a\%$ of $b$ differ by $0.003$. Find all possible pairs $(a, b)$.

The $n$th term of a sequence is the first non-zero digit of the decimal expansion of $\frac{1}{\sqrt{n}}$. How many of the first one million terms of the sequence are equal to $1$?

In the parallelogram $ABCD$, a line through $A$ meets $BD$ at $P$, $CD$ at $Q$ and $BC$ extended at $R$. Prove that $\frac{PQ}{PR} = \left(\frac{PD}{PB}\right)^2$.

Mickey writes down on a board $n$ consecutive whole numbers, the smallest of which is $2023$. He repeatedly replaces the largest two numbers with their difference until only one number remains. For which values of $n$ is the last remaining number $0$?

Find all triples $(m, n, p)$ which satisfy $p^n + 3600 = m^2$, where $p$ is prime and $m, n$ are positive integers.