Join the center to each of the corners and color each segment a different color and arbitrarily coloring each ambiguous point. Radially extend the colored sections to infinity.
To prove f(x) has a fixed point consider g(x) = f(x) - x which can take values outside of the triangle. To find a fixed point, we simply need to show that g(x) will map at least one point to the center. It is easy to prove that each corner will map to the color opposite and each edge can only map to points of a different color (unless it passes through the center, in which case we would have obtained out proof). At this point the problem reduces to 5 assuming that we construct a big enough circle.
Rough approach for qu 6:
Join the center to each of the corners and color each segment a different color and arbitrarily coloring each ambiguous point. Radially extend the colored sections to infinity.
To prove f(x) has a fixed point consider g(x) = f(x) - x which can take values outside of the triangle. To find a fixed point, we simply need to show that g(x) will map at least one point to the center. It is easy to prove that each corner will map to the color opposite and each edge can only map to points of a different color (unless it passes through the center, in which case we would have obtained out proof). At this point the problem reduces to 5 assuming that we construct a big enough circle.