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?

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?