mathcounts ® 2000 national competition countdown round
TRANSCRIPT
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MATHCOUNTS®
2000 National CompetitionCountdown Round
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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.
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Answer: 44
125
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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?
A
B
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Answer: 2 (times)
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A line that passes through (-5,-8) and (-3,-4) will cross the y-axis at what y-coordinate?
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Answer: 2
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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.
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Answer: (inches) 2 2
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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.
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Answer: 16 (revolutions)
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What is the value of
1 2 3 4 5 698 99 100
... ?
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Answer: –50
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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?
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Answer: 16 (feet)
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x, y and z are positive odd
integers. What is the remainder
when is divided
by 4?
x y z2 2 2
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Answer: 3
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In how many consecutive zeroes does the product end?115 116 117 201
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Answer: 23 (zeroes)
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What is the total number of different committees that can be formed by selecting one or more persons from a group of six people?
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Answer: 63 (committees)
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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?
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Answer: 4 (centimeters)
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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?
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Answer: 60
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The average of 11 consecutive even integers is 24. What is the greatest of these integers?
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Answer: 34
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What is the fewest possible number of units in the perimeter of a triangle with side lengths that are relatively prime integers?
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Answer: 12 (units)
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Four distinct digits are used to make 2 two-digit numbers. What is the greatest possible product of the two numbers formed?
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Answer: 8352
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How many whole numbers from 10 to 99 have a units digit greater than the tens digit?
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Answer: 36 (numbers)
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A 200% increase is the same as a 50% increase followed by what other percent increase?
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Answer: 100 (percent)
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What percent of the first 200 prime numbers have reciprocals less than 0.05?
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Answer: 96 (percent)
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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.
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Answer: 3
4
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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?
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Answer: 6 (centimeters)
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How many miles per hour is 1298 feet per second?
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Answer: 885 (mph)
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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?
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Answer: 162 (cubic meters)
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In 1984, July 4 fell on a Wednesday. On what day of the week did July 4, 1990, fall?
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Answer: Wednesday
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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?
13
14
16: :
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Answer: 80 (degrees)
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How many integers 1–9 are divisors of the five-digit number 24,516?
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Answer: 6 (integers)