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?

Defeat the AMC: lather rinse repeat

I track the scores of my students who are prepping for various math contests. Here is an example of student who is hoping to qualify for AIME next year. My method is to assign an old AMC 10 as homework, and the student takes it under timed exam conditions. The student grades her exam at home, and then spends as much time as she likes to solve any remaining problems. We meet to discuss the problems that could not be solved, and I explain the solutions and offer quick proofs and explanations of underlying theorems.

There isn’t a lot of strategy or overthinking here. Old AMCs are freely available online, with 2 per year, there is no end of practice tests. Take an exam, study what you could not solve, take another exam, lather, rinse, repeat.

In just a few months this student has raised her score to very close to the AIME floor most years. Notice she’s more than doubled the number of correctly solved problems.

There’s no magic or secret way to prepare. Time spent researching books and classes is better spent taking as many exams as possible and most importantly, studying the problems you couldn’t solve. Good luck!