MATHCOUNTS
MATHCOUNTS
Team Round
2007
1. Four circular chips are each centered on one of four 1.
adjacent squares on a checkerboard such that the
centers of the chips are the four vertices of a square.
What is the area of this square if the checkerboard’s
squares each measure two inches on a side?
2. A sculpture of a clothespin is 20 feet high. A normal 2.
clothespin of this shape is five inches high. Using this
scale, how many feet tall would a sculpture of a
5-foot, 7-inch woman be?
3. All of the ID codes in Agueleo’s school are three-digit 3.
positive integers. For Agueleo’s ID code, the product
of the digits is 216, the sum of the digits is 19, and the
integer ID code is aslarge as possible.
What is Agueleo’s ID code?
4. A subtraction square is a 3 by 3 grid of integers where, 4.
in each row from left to right, and in each column from
top to bottom, the first integer minus the second
integer equals the third integer. The integer k is in the top k
left unit square of the subtraction square, as shown.
When the subtraction square is filled in completely,
what is the sum of all nine integers?
Express your answer in simplest form in terms of k.
5. Two right pyramids with congruent square bases and 5.
equilateral triangular faces can be joined at their bases to form
an octahedron with eight congruent, equilateral triangular
faces. The total surface area of a particular such octahedron
with edge length 2 units is x square units. The volume of
this octahedron is y cubic units. What is the value of x + y?
Express your answer as a decimal to the nearest tenth.
6. In a particular list of three-digit perfect squares, the 6.
first perfect square can be turned into each of the others
by rearranging its digits. What is the largest number of
distinct perfect squares that could be in the list?
7. Charlie plans to build a square pyramid-like figure 7.
using unit cubes. The top level will have one cube.
Given any level, the vertices of the largest bottom
square coincide with the centers
of the top faces of the four corner
cubes of the level below. The top
three levels are shown. When Charlie
finishes gluing together all of the unit
cubes of the first eight levels, what is
the total surface area of all of the faces
of the resulting solid?
8. If [pic] 8.
what is the largest possible value of r – s?
Express your answer as a mixed number.
9. For positive integers K and T, it is true that 27 × K = [pic] 9.
K > 100 and K ÷ T is an integer. What is the smallest
possible value of K ÷ T ?
10. In a bag of three marbles, there are exactly two 10.
blue marbles. If Kia randomly chooses two marbles
without replacement, the probability of choosing the two
blue marbles is one-third. However, before Kia chooses
her two marbles, additional marbles are added to the bag.
The probability of picking two blue marbles without
replacement is still one-third. What is the least number of
marbles that could be in the bag after the additional marbles
have been added?
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