MATHCOUNTS® Problem of the Week Archive More State Problems – March 11, 2013 Problems & Solutions This month, students from 56 states and territories are competing for a chance to go to Washington, DC in May and participate in the 2013 Raytheon MATHCOUNTS National Competition. During the Countdown Round, students are asked to solve problems like these from the 2012 State Competition in 45 seconds or less. A set S contains 19 different integers with a mean of 101. The mean of the ten smallest integers in S is 50, and the mean of the ten largest integers in S is 150. What is the median of all the integers in S? (Countdown #30) The mean of the integers in S is 101, so the sum of the integers in S is 101 × 19 = 1919. The mean of the 10 smallest integers in S is 50, so the sum of these 10 integers is 50 × 10 = 500. The mean of the 10 largest integers in S is 150, so the sum of these 10 integers is 150 × 10 = 1500. Each of these sums includes the tenth integer in S, which is also the median. Therefore, the median of S is 500 + 1500 – 1919 = 2000 – 1919 = 81. Eight swimmers, numbered from 1 to 8, stand in ascending order clockwise around the edge of a circular swimming pool. Going around clockwise, every second swimmer still standing will jump into the pool. Swimmer #2 is the first to jump in the pool. What is the number of the swimmer who is the seventh to jump into the pool? (Countdown #31) Based on the information provided, we see that the swimmers will jump into the pool in the following order: swimmer #2, swimmer #4, swimmer #6, swimmer #8, swimmer #3, swimmer #7, swimmer #5, at which time only swimmer 1 remains. Therefore, the seventh swimmer to jump in the pool was swimmer #5. 7 8 1 6 5 2 4 3 Square ABCD has sides of length 22 cm. What is the area, in square centimeters, of the region shaded gray? (Countdown #34) Triangle AED has a base of length 22 cm and a height of 22cm. That means the area of the triangle is half the area of the square, and the region shaded gray accounts for the other half of the square’s area. Therefore, the area of the region shaded gray is ½ × 22 × 22 = 242 cm2. Katie writes down six different prime numbers, p, q, r, s, t and u, each less than 20. If p + q = r + s = t + u, what is the sum of all the prime numbers less than 20 that Katie did not write down? (Countdown #35) There are eight prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17 and 19. We get 19 + 5 = 17 + 7 = 13 + 11 = 24. So, it follows that the two numbers Katie did not write down are 2 and 3, and their sum is 2 + 3 = 5.