Problem (Geometry):
The six sides of convex hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings such that every triangle $A_{i} A_{j} A_{k}$ has at least one red side.

Solution:
Only two triangles have no sides that are sides of the original hexagon: $A_{1} A_{3} A_{5}$ and $A_{2} A_{4} A_{6}$. For each of these triangles, there are $2^{3}-1=7$ colorings in which at least one side is red, for a total of $7 \cdot 7=49$ colorings of those six diagonals. The colorings of the three central diagonals $\overline{A_{1} A_{4}}, \overline{A_{2} A_{5}}, \overline{A_{3} A_{6}}$ are irrelevant because the only triangles they can form include sides of the original hexagon, so they can be colored in $2^{3}=8$ ways, for a total of $8 \cdot 49=\mathbf{3 9 2}$ colorings.

Final answer: 392