MATHCOUNTS ® 2000 National Competition Countdown Round

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  • MATHCOUNTS2000 National CompetitionCountdown Round

  • In the Village League, the team to win two of three softball games becomes the champion. If the probability of Team Alpha beating Team Beta is 60% for every game, what is the probability that Beta wins the championship? Express your answer as a common fraction.

  • Answer:

  • Circles A and B are congruent. Circle A is rolled around circle B, remaining tangent at all times. Circle A rolls around circle B exactly once. How many times will circle A revolve around its own center before the radii are lined up again as shown?AB

  • Answer: 2 (times)

  • A line that passes through (-5,-8) and (-3,-4) will cross the y-axis at what y-coordinate?

  • Answer: 2

  • What is the number of inches in the radius of a circle whose area is one-half the area of a circle with radius 4 inches? Express your answer in simplest radical form.

  • Answer: (inches) 22

  • A unicycle has a wheel with a radius of 1 foot. How many complete revolutions will the wheel make when the unicycle rolls 100 feet? Express your answer to the nearest whole number.

  • Answer: 16 (revolutions)

  • What is the value of 1234569899100-+-+-+-+-...?

  • Answer: 50

  • A 3-foot high tree was planted and grows by an equal number of feet each year. At the end of the seventh year, it is 1/9 taller than at the end of the sixth year. How many feet tall will it be at the end of the 13th year?

  • Answer: 16 (feet)

  • x, y and z are positive odd integers. What is the remainder when is divided by 4?

  • Answer: 3

  • In how many consecutive zeroes does the product end?

  • Answer: 23 (zeroes)

  • What is the total number of different committees that can be formed by selecting one or more persons from a group of six people?

  • Answer: 63 (committees)

  • In ABC, AB = 5 cm, BC = 10 cm, and the altitude drawn to AB is 8 cm. What is the number of centimeters in the length of the altitude to BC?

  • Answer: 4 (centimeters)

  • When four numbers are added three at a time, the four sums are 42, 43, 47 and 48. What is the sum of the four numbers?

  • Answer: 60

  • The average of 11 consecutive even integers is 24. What is the greatest of these integers?

  • Answer: 34

  • What is the fewest possible number of units in the perimeter of a triangle with side lengths that are relatively prime integers?

  • Answer: 12 (units)

  • Four distinct digits are used to make 2 two-digit numbers. What is the greatest possible product of the two numbers formed?

  • Answer: 8352

  • How many whole numbers from 10 to 99 have a units digit greater than the tens digit?

  • Answer: 36 (numbers)

  • A 200% increase is the same as a 50% increase followed by what other percent increase?

  • Answer: 100 (percent)

  • What percent of the first 200 prime numbers have reciprocals less than 0.05?

  • Answer: 96 (percent)

  • What is the probability that the product of two numbers chosen randomly from the set of all positive integers is divisible by 2? Express your answer as a common fraction.

  • Answer: 34

  • A circle has a radius of 10 centimeters and a chord of the circle is 16 centimeters long. How many centimeters is the midpoint of the chord from the center of the circle?

  • Answer: 6 (centimeters)

  • How many miles per hour is 1298 feet per second?

  • Answer: 885 (mph)

  • The ratio of length to width to height of a rectangular prism is 3:2:1. If the surface area of the prism is 198 m2, how many cubic meters are in its volume?

  • Answer: 162 (cubic meters)

  • In 1984, July 4 fell on a Wednesday. On what day of the week did July 4, 1990, fall?

  • Answer: Wednesday

  • The measures of the angles of a triangle are in the ratio of . What is the number of degrees in the measure of the largest angle?

  • Answer: 80 (degrees)

  • How many integers 19 are divisors of the five-digit number 24,516?

  • Answer: 6 (integers)