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### Transcript of MATHCOUNTS ® 2000 National Competition Countdown Round

• 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.

• 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

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

• 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.

• 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.

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

• 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?

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

• In how many consecutive zeroes does the product end?

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

• 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?

• 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?

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

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

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

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

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

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

• 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.

• 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?

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

• 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?