5/27 Theta Math Club Activity

A 1-by-2 rectangle is inscribed in a semicircle with its longer side on the diameter. What is the area of the semicircle?

Hint: the area of a triangle with an inscribed circle may also be represented as the product of its semi-perimeter (sum of the triangle's sides divided by 2) and the inradius (what we want) of the inscribed circle

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Hint: connect points B and D to form a right triangle

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Answer explanation

This question quizzes a fundamentally important concept in combinatorics. Start with the A. See how many ways there are to make the next letter, M. (There are 4.) Now, see how many ways there are to choose the next letter, C. (There are three.) Lastly, there are two ways to choose the number 8, leaving you with 24 ways in total to spell out AMC8.

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Answer explanation

List out the multiples of 9. Plug them into the equalities to see if they match.

Which of the following numbers is not a perfect square?

Answer explanation

Look for the values with odd exponents. Then, realize that 4²⁰¹⁹ can be rewritten as 2⁴⁰³⁸.

Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is 132.

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Dylan and Franklin are running laps around a 400-meter oval track. They started together, but Dylan has pulled ahead, because he runs 25% faster than Franklin. How many laps will Dylan have run when he first passes Franklin?

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2023 MATHCOUNTS Chapter Spring Round

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2023 MATHCOUNTS Chapter Sprint Round

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