Mathcounts

In the sum 4 + \(2\sqrt{2}\) + 2 + \(\sqrt{2}\) + … each term is obtained by dividing the previous term by \(\sqrt{2}\) . If the sum of the series, in simplest radical form, is m + \(n\sqrt{2}\) , what is the value of m + n?

In a certain state legislature, a proposed bill was defeated with 6 fewer votes for the bill than against it. After the bill was amended, 9 members who had previously voted against the bill were now for it. This resulted in 60% of the legislature now being in favor of the bill. If all those who previously voted for the bill remained in favor of it, how many members are in this state legislature, assuming every member voted each time?

During a game of paintball, ten friends were positioned in a field so that no two of them stood the same distance apart. Each person aimed at his or her closest opponent, and at the signal everyone fired. What is the maximum number of times one player could have been hit?

Kevin and Devin each make one hat per day for charity, but they started on different days. Today, Kevin made his 520th hat, and Devin made his 50th. A celebration is planned for the next day that Kevin’s hat count is evenly divisible by Devin’s hat count. In how many days from today will they celebrate?

Isosceles triangle ACB has a right angle at C and shares a leg with equilateral triangle BCD of side length 2 in. The triangles, otherwise, do not intersect. Segments BC and AD intersect at E. What is the value of BE/EC? Express your answer in simplest radical form.

The sum of the pairwise products of three consecutive natural numbers is 8111. What is the largest of the three numbers?

Points A(−1, 2), B(−2, −5) and C(7, −2) are on a circle. What is the area of the circle? Express your answer as a decimal to the nearest tenth.

On average, Josiah allows one hit for every five pitches. If he were to throw nine pitches, what is the probability that he would allow exactly three hits? Express your answer as a decimal to the nearest hundredth. 

Writing down the four-digit dollar amount of a deposit, a bank teller accidentally transposed two adjacent digits. What is the greatest possible value of the resulting difference between the correct amount and the incorrect amount?

Donald has nine one-day passes to Dizzyworld. He can go alone and use one, or he can take a friend and use two. If he visits every day until he uses all the passes, in how many different ways can he use them? Using two a day for four days and then going alone at the end (2, 2, 2, 2, 1) is different from reversing the order (1, 2, 2, 2, 2).