Escaping a stampede in Buffalo Country 
A person is standing at a random point in square ABCD (vertices labelled clockwise) of side length 10 units. The person is capable of instantly reaching and running at 1 unit/second.
A herd of buffaloes forming a straight front 10 units wide is charging in a straight line through the square ABCD, at uniform speed 2 units/second. They enter through AB exactly and exit through CD exactly.
Assume the person becomes aware of the stampede when the buffaloes first enter the square, and that the person takes the path which gives them the best chance of running out of the way of the stampede. What is the chance the person escapes the stampede?

Will the best escape path always be straight in this case?
What is the general strategy for solving these types of pursuit/escape problems?
 A: One way to see that the best escape path must be straight is to note that once the stampede of buffaloes overtakes you, you can never overtake the stampede of buffaloes. Therefore you can get to a point on the perimeter of the square iff you can reach that point before the buffaloes.
Now let's assume our square $ABCD$ is given by $[0,10]\times[0,10]$ and that we start at some location $(x,y)$, where without loss of generality $y \leq 5$ (if $y>5$ simply reflect across $y=5$). 
Note that if $x > 5$ we can just run to the line $x=10$ (so all such points are safe) and otherwise the buffalo will get to the line $x=10$ first. Hence assume $x\leq 5$. We'll now find the best choice of point on the $x$-axis to run to.
Note that a point $(x',0)$ on the $x$-axis is safe if $x'/2 \geq \sqrt{(x-x')^2 + y^2}$. Simplifying, this reduces to
$$3x'^2 - 8xx'+4x^2 + 4y^2 \leq 0$$
This is possible only if the discriminant of this quadratic (in $x'$) is positive, i.e.,
$$64x^2 - 12(4x^2 +4 y^2) \geq 0$$
or
$$x^2 \geq 3y^2$$
Since $x$ and $y$ are positive, this happens when $y\leq x/\sqrt{3}$. Therefore the safe points in the region $[0,10]\times[0,5]$ are given by the union of the triangle $\{0 \leq y \leq x/\sqrt{3} \mid y \in [0,5]\}$ and the right half rectangle $[5,10]\times[0,5]$. Together, these have area $25(6+\sqrt{3})/6$, so the probability the person can escape equals
$$\frac{6+\sqrt{3}}{12}$$
