This is a fun problem that I saw somewhere on the internet a long time ago:
Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion which is determined to eat you. The lions start at a constant speed of $v_l$, and they are always running directly towards your current location. You start in the center, and can run at a constant speed $v_h$ (assume instantaneous acceleration for all parties). You are NOT enclosed in the triangle, you are free to try to run wherever you want. Which patch should you take in order to survive the longest time possible? How long can you survive?
At first, because everybody's speed is constant, I thought we can just work with functions of $x$ for the paths of the lions and the human, and try to maximize the arc length of the lions' paths. However, I think it's much easier to work with the functions in parametric form, because the tangent lines to the lions' path at $t=t_0$ should go trough your position at $t_0$. Also, I think it's reasonable to assume that at $t=0$ your direction is straight towards the midpoint of one of the sides, because any other direction would cause you to meet one of the lions faster.