# 2022 MathCounts State Team Round #9

A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up three times in a row. What is the probability that the coin will be tossed more than $10$ times? Express your answer as a common fraction.

# 2022 MathCounts State Team Round #10

Consider the equilateral triangle $ABC$ with sides of length $8 \sqrt{3}$ cm. A point in the interior of $ABC$ is said to be “special” if it is a distance of $3$ cm from one side of the triangle and a distance of $7$ cm from another side. Consider the convex polygon whose vertices consist of the special points. What is the area of this polygon? Express your answer as a decimal to the nearest tenth.

# 2021 MathCounts National Sprint #28 (60 second solve!)

The three coin denominations used in Coinistan have values 7 cents, 12 cents and 23 cents. Fareed and Krzysztof notice that their two Coinistan coin collections have the same number of coins and the same total value, but not the same number of 7-cent coins. What is the smallest possible value of Krzysztof’s collection?

# 2021 National MathCounts Sprint #21 (video)

Four lines are drawn through the figure shown (see problem statement in video). What is the maximum number of non-overlapping regions created inside the figure?

# 2021 National MathCounts Team Round #9 (video solution)

Gretchen labels each of the six faces of a cube with a distinct positive integer so that for each vertex of the cube, the product of the three numbers on the faces touching the vertex is a perfect square. What is the least possible value of the sum of the numbers on this cube?

# 2021 MathCounts National Target Round #7 (video solution)

This one uses a rarely used formula for the area of a triangle.

What is the largest possible perimeter of a triangle whose sides have integer side lengths and that can fit inside a circle of radius 20 cm?

# 2013 MathCounts Chapter Target #3

This problem is a good example of finding a bijection or a 1:1 correspondence between a set that is difficult to count and another that is easier to count, in this case using binary numbers.

A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner by colored?