Three students each flip three fair coins. What is the probability that all three students get the same number of tails? Express your answer as a common fraction. Twins Taylor and Tyler were born on 05/02/07. This date is referred to as a sum date because the sum of the month and day is equal to the two-digit year: 05 + 02 = 07. How many years in the 21st century will have a sum date in each month during that year? The area of rectangle WXYZ is 90 cm 2 . P and Q are points on diagonal WY such that 3(WP + QY) = 2PQ. What is the area of triangle PQZ? There are 50 equally spaced points marked on a circle. Sara numbers them clockwise from 1 to 50. Starting at 1, she then draws congruent, connected segments between points that are 8 spaces apart, moving clockwise from the end of the previous segment. For example, the first segment is drawn from 1 to 9, and the second from 9 to 17. What is the sum of the numbers on the endpoints of the 23rd segment drawn? The five squares of the diagram shown are to be colored orange, yellow, green, blue and indigo, with exactly one color per square. Two colorings are the same if one is just a rotation of the other (but not if the diagram must be flipped over). How many distinct colorings are there? What is the greatest four-digit palindrome that is divisible by 7 and 8? Tryouts were held for three positions on the school basketball team: center, guard and forward. There were 4 players who tried out for center, 10 for guard and 10 for forward. These numbers include 1 player who tried out for center and guard, 3 who tried out for center and forward, and 4 who tried out for guard and forward. All six of these counts include 1 player who tried out for all three positions. If 17 players, in all, tried out for the team, how many players tried out only for guard? When ice melts and becomes water, its volume decreases by 8%. A cylindrical block of ice completely fills a container with a height of 10 cm and a radius of 4 cm. When all of the ice melts, what will be the height of the water in the cylinder? Express your answer as a decimal to the nearest tenth. A triangle exists with side lengths 2 x , 3 x + 7 and 6 x – 5 for how many integer values of x? If x and y are integers, such that x > y , ( x + y ) 2 = 9 and x 2 + y 2 = 29, what is the smallest possible value for x ? years W P Q X Z Y cm 2 colorings players cm values
MATHCOUNTS 2016-2017 30 A circular pizza is cut along four diameters into eight identical sectors. If the total perimeter of each sector is 10 inches, what is the area of the whole pizza? Express your answer as a decimal to the nearest tenth. Because of a traffic jam, Alana’s 18-mile commute to work took 4 minutes longer than usual, and her average speed was decreased by 9 mi/h. How many minutes did it take her to get to work that day? The standard gravitational acceleration of an object near Earth’s surface is g n ≈ 32 ft/s 2 .
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