MATHCOUNTS
2007 National Competition
Countdown Round 1-25

Eighty percent of adults drink
coffee and 70% drink tea.
What is the smallest possible
percent of adults who drink
both coffee and tea?
Answer: 50 (percent)
A pyramid with volume
40 cubic inches has a
rectangular base. If the length
of the base is doubled, the
width tripled and the height
increased by 50%, what is the
volume of the new pyramid, in
cubic inches?
Answer: 360 (cubic inches)
For positive integers x and y,
what is the least possible
value of x if x3 = y2 + 2?
Answer: 3
Cory has 3 apples, 2 oranges
and 2 bananas. If Cory eats
one piece of his fruit per day
for a week and the pieces of
fruit within each category are
indistinguishable, in how many
orders can Cory eat the fruit?
One such order is AAAOOBB.
Answer: 210 (orders)
If A:B:C = 2:1:4, what is the
value of (3A + 2B)  (4C – A)?
Express your answer as a
common fraction.
4
Answer: 7
Define the operation $ as
a $ b = (a)(b) – a. What is the
value of (3 $ 5) $ (5 $ 3)?
Answer: 108
What is the least possible
product obtained from
multiplying three distinct
members of the set
1
1
{- 2 , 9, 5, 12, 3 } ?
Answer: -54
A set of five integers has
unique mode 7, median 9, and
arithmetic mean 11. What is
the greatest possible value in
the set?
Answer: 22
Each of four students hands in a
homework paper. Later the
teacher hands back the graded
papers randomly, one to each of
the students. In how many ways
can the papers be handed back
such that every student receives
someone else’s paper? The
order in which the students
receive their papers is irrelevant.
Answer: 9 (ways)
1
,
x
1
,
y
Given that x, y,
z and
are all integers, how many
distinct values of
x + y + z are possible?
1
z
Answer: 4 (values)
A 1 x 1 x 1 wire frame cube is to
be made by gluing together
pieces of wire. The wire can be
bent to form corners of the
cube. If exactly 12 units of wire
is used to make the frame, what
is the fewest number of pieces
of wire that can be used to
make the frame?
Answer: 4 (pieces)
A family of five has Momma,
Poppa and three children.
Each child generates 2 loads
of laundry per week while
each parent generates
1.5 loads per week. How
many loads of laundry are
generated by this family in
52 weeks?
Answer: 468 (loads)
Jane and her brother each spin
this spinner once. The spinner
has five congruent sectors. If the
non-negative difference of their
numbers is less than 3, Jane
wins. Otherwise, her
brother wins. What
5
1
is the probability that
4
2
Jane wins? Express
3
your answer as a
common fraction.
19
Answer:
25
What is the ordered pair of
real numbers (x, y) which
satisfies the equation
x + y – 7 + 4x – y + 12 = 0 ?
Answer: (-1, 8)
In the figure, the visible gray area
within the larger circle is equal to
three times the area of the white
circular region. What is the ratio
of the radius
of the small circle to
the radius of the
large circle? Express
your answer as a common
fraction.
1
Answer: 2
How many integers are there
in the solution set of
x – 2  5.6?
Answer: 11 (integers)
The three-digit positive integer
N has a ones digit of 3. What
is the probability that N is
divisible by 3? Express your
answer as a common fraction.
1
Answer:
3
Pizzas are sized by diameter.
What percent increase in area
results if Chantel’s pizza
increases from a 10-inch pizza
to a 12-inch pizza?
Answer: 44 (percent)
Figure A is made from three unit
squares, and will be called “a trio.”
Figure B is a 4 by 4 grid of unit
squares. Without overlap, what is
the maximum number of trios that
can be placed on the grid such that
each trio covers
exactly three of
the unit squares
Figure A
in the grid?
Figure B
Answer: 5 (trios)
While standing in line to buy
concert tickets, Kit moved
60 feet closer to the ticket
window over a period of
30 minutes. At this rate, how
many minutes will it take her
to move the remaining
70 yards to the ticket window?
Answer: 105 (minutes)
On a number line, what is the
coordinate of the point that is
3
between point A at and
x
7
point B at 2x and is one-third
the distance from A to B?
Express your answer as a
common fraction in terms of x.
19
Answer:
6x
Sue has 122 crates of the same
size. Each crate contains at least
90 apples and at most
100 apples. Crates containing the
same number of apples are
stacked one on top of another in
their own stack. Not all of the
stacks have the same number of
crates. What is the least possible
number of crates in the tallest
stack of crates?
Answer: 12 (crates)
Here are two functions:
f(x) = 3x2 – 2x + 4
2
g(x) = x – kx – 6
If f(10) – g(10) = 10, what is
the value of k?
Answer: -18
All positive integers whose
digits add up to 11 are listed in
increasing order:
29, 38, 47, … . What is the
eleventh number in that list?
Answer: 137
How many subsets of {1, 2, 3,
4, 5} with two or more
elements have the property
that the sum of their elements
is a positive, even number?
Answer: 13 (subsets)