Transcript In the Village League, the team to win two of three

MATHCOUNTS
®
2000 National Competition
Countdown 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.
44
Answer:
125
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
A
center before the radii
are lined up again as
shown?
B
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: 2 2 (inches)
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
1- 2 + 3 - 4 + 5 - 6 +
... - 98 + 99 - 100?
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
2
2
2
when x + y + z is divided
by 4?
Answer: 3
In how many consecutive zeroes
does the product
115  116  117  L 201 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.
3
Answer:
4
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
2
the prism is 198 m , 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
1
1
1
triangle are in the ratio of 3 : 4 : 6 .
What is the number of degrees in
the measure of the largest angle?
Answer: 80 (degrees)
How many integers 1–9 are
divisors of the five-digit
number 24,516?
Answer: 6 (integers)