Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_N$ denote the probability that the product of these two integers has a units digit of $0$. The maximum possible value of $p_N$ over all possible choices of $N$ can be written as $\tfrac ab,$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

4. There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$? [i]Proposed by Tragic[/i]

Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M,$ or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order.

Consider a set of $16$ points arranged in $4 \times 4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define: $(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$; $(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities; $(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities. Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$ Proposed by David-Andrei Anghel, Romania.

Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{10}$ The chance of getting any given number is $\dfrac{1}{n}$ , so the probability of getting $7$ and heads is either $\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}$ or $\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}$. We get that either $n = 5$ or $n = 10$, but since rolling a $7$ is possible, only $n = \boxed{10}$ is a solution.[/hide]