1
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $$$\frac{m}{n},$$$ where $$$m$$$ and $$$n$$$ are relatively prime positive integers. Find $$$m+n.$$$
2
A plane contains $$$40$$$ lines, no $$$2$$$ of which are parallel. Suppose there are $$$3$$$ points where exactly $$$3$$$ lines intersect, $$$4$$$ points where exactly $$$4$$$ lines intersect, $$$5$$$ points where exactly $$$5$$$ lines intersect, $$$6$$$ points where exactly $$$6$$$ lines intersect, and no points where more than $$$6$$$ lines intersect. Find the number of points where exactly $$$2$$$ lines intersect.
3
The sum of all positive integers $$$m$$$ for which $$$\tfrac{13!}{m}$$$ is a perfect square can be written as $$$2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$$$, where $$$a, b, c, d, e,$$$ and $$$f$$$ are positive integers. Find $$$a+b+c+d+e+f$$$.
