AMC 8 Daily Practice Round 1
Complete problem set with solutions and individual problem pages
Connect the diagonals of square , and color each of the four vertices either red or yellow. A triangle whose vertices are all the same color is called a monochromatic triangle. How many coloring methods in which there is at least one monochromatic triangle?
- A.
- B.
- C.
- D.
- E.
Each vertex can be colored in two ways, so there are coloring methods in total.
For there to be a monochromatic triangle, the case of “two vertices red and two vertices yellow” must be excluded. This case has methods. Therefore, the number of coloring methods that yield at least one monochromatic triangle is
Equivalently, to have a monochromatic triangle, we must exclude the case where “the two diagonals are colored differently.” This case has methods. Hence, the number of coloring methods with at least one monochromatic triangle is also
