Find the ones digit of $1!+2!+3!+\cdots+99!$

$x,y$ and $z$ are positive integers where $xy=48,yz=72,zx=96$. Find the value of $x+y+z$.

The four corners of a square with sides of length 1 are cut to make a regular octagon. Find the exact length of the sides of the regular octagon.

Triangle $ABC$ has sides with lengths $AB=4,BC=3$ and $CA=2$. Two points $D$ and $E$ lie on side $AB$ such that $AD=1$ and $\angle ACD=\angle BCE$. Find the length of $BE$.

A person can climb one or two steps at a time. When a set of stairs has 4 steps, they can climb it in $5$ different ways (1-1-1-1, 2-1-1, 1-2-1, 1-1-2 and 2-2). How many different ways can they climb a set of stairs that has $10$ steps?

How many different sets of integers $\{a,b,c,d\}$ satisfy:
\begin{cases}
a+b=cd \\
c+d=ab
\end{cases}

A circle is inscribed inside a regular hexagon with sides of length $2$. $P$ is the point of intersection of side $DE$ and the circle. $Q,R$ are the points of intersection of sides $PA,PB$ and the circle, respectively. Find the exact area of triangle $PQR$.

$a_n$ is the value of $\sqrt{n}$ rounded to the nearest integer. The first few terms are:
$a_1=1,a_2=1,a_3=2,a_4=2,a_5=2,a_6=2,a_7=3$. Find the value of
$$\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots+\frac{1}{a_{1980}}$$

Four married couples (eight people in total) were at a party. At the end of the party, the host asked each person at the party how many people’s hands they shook and they all gave different answers. Assuming no person shakes their own partner’s hand, how many people’s hands did the host’s partner shake?

When real numbers $a,b,x,y$ satisfy the following simultaneous equations:
\begin{cases}
ax+by=3 \\
ax^2+by^2=7 \\
ax^3+by^3=16 \\
ax^4+by^4=42 \\
\end{cases}
find the value of $ax^5+by^5$.

$n$ is a positive four-digit number where the tens digit is non-zero. When the first two digits are multiplied by the last two digits, the product is a factor of $n$. Find all possible values of $n$.
*Separate your answers with commas