(1) Four circular chips are each centered on one of four 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 in square inches if the checkerboard's squares each measure two

inches on a side?

(2) Three faces of a right rectangular prism have areas of 48, 49 and 50 square

units. What is the volume of the prism, in cubic units? Express your answer to the nearest

whole number.

(3) The triangle with vertices A (6, 1), B (4, 1) and C (4, 4) is rotated 90

degrees counterclockwise about B. What are the coordinates of the image of C (the point

where C is located after the rotation)? Express your answer as an ordered pair.

(4) A square and an equilateral triangle have equal perimeters. The area of the

triangle is 2p3 square inches. What is the number of inches in the length of the diagonal

of the square?

(5) A jar of peanut butter which is 3 inches in diameter and 4 inches high sells

for $0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter

and 6 inches high?

(6) A cylindrical glass is half full of lemonade. The ratio of lemon juice to water

in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is

the volume of lemon juice in the glass? Express your answer as a decimal to the nearest

hundredth.

(7) Each point in the hexagonal lattice shown is one unit from its nearest

neighbor. How many equilateral triangles have all three vertices in the lattice?

(8) On a standard 12-hour clock, through how many positive degrees does the

hour hand move from 9:15 a.m. to 12 noon of the same day? Express your answer as a

decimal to the nearest tenth.

(9) How many di�erent triangles can be formed using three vertices of a

hexagon as vertices of a triangle?

(10) A rectangular garden has a length that is twice its width. The dimensions

are increased so that the perimeter is doubled and the new shape is a square with an area

of 3600 square feet. What was the area of the original garden, in square feet?

(11) Quadrilateral ABCD is a trapezoid with AB parallel to CD. We know

AB = 20 and CD = 12. What is the ratio of the area of triangle ACB to the area of the

trapezoid ABCD? Express your answer as a common fraction.

(12) What is the greatest number of whole 4-inch by 6-inch notecards that can

be cut from a 25-inch by 27-inch sheet of poster board if every cut must be made parallel

to a side of the poster board? No taping or gluing is allowed.

(13) How much greater, in square inches, is the area of a circle of radius 20

inches than a circle of diameter 20 inches? Express your answer in terms of �.

(14) Three of the four vertices of a rectangle are (5; 11), (16; 11) and (16;�2).What is the area of the intersection of this rectangular region and the region inside the

graph of the equation (x � 5)2 + (y + 2)2 = 9? Express your answer as a common fraction

in terms of �.

(15) A triangle has a side of length 6 cm, a side of length 8 cm and a right angle.

What is the shortest possible length of the remaining side of the triangle? Express your

answer in centimeters as a decimal to the nearest hundredth.

(16) In this quilt pattern, points E, F , G and H are midpoints of the sides of

square ABCD, and square EFGH is divided into nine congruent unit squares, as shown.

What percent of the total area of square ABCD does the total shaded area represent?

Express your answer to the nearest whole percent.

A B

CD

E

F

G

H

(17) A triangular region is enclosed by the lines with equations y = 12x + 3,

y = �2x + 6 and y = 1. What is the area of the triangular region? Express your answer as

a decimal to the nearest hundredth.

(18) An equilateral triangle has an area of 64p3 cm2. If each side of the triangle

is decreased by 4 cm, by how many square centimeters is the area decreased?

(19) Kelly drove north for 12 miles and then east for 9 miles at an average rate

of 42 miles per hour to arrive at the town of Prime. Brenda left from the same location, at

the same time, and drove along a straight road to Prime at an average rate of 45 miles per

hour. How many minutes earlier than Kelly did Brenda arrive?

(20) A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches.

The box is �lled with as many solid 3-inch cubes as possible, with all of the cubes entirely

inside the rectangular box. What percent of the volume of the box is taken up by the

cubes?

(21) A recipe for crispy rice treats results in a mixture that �lls a 9-inch by 13-inch

pan to a depth of one inch. If a crispy rice treats mixture resulting from 1.5 times the

original recipe is poured into a pan that is 10 inches by 15 inches, to what depth, in inches,

will the pan be �lled? Express your answer as a decimal to the nearest hundredth.

(22) A quadrilateral in the plane has vertices (1; 3), (1; 1), (2; 1) and

(2006; 2007). How many square units is the area of the quadrilateral?

(23) Points A(0, 0), B(6, 0), C(6, 10) and D(0, 10) are vertices of rectangle

ABCD, and E is on segment CD at (2, 10). What is the ratio of the area of triangle ADE

to the area of quadrilateral ABCE? Express your answer as a common fraction.

(24) A 1200-diameter pizza and a 1600-diameter pizza are each cut into eight

congruent slices. Jane ate three slices of the 1200 pizza. Mark ate three slices of the 1600

pizza. How many more square inches of pizza than Jane did Mark eat? Express your

answer as a common fraction in terms of �.

(25) Given that the diagonals of a rhombus are always perpendicular bisectors of

each other, what is the area of a rhombus with side lengthp89 units and diagonals that

di�er by 6 units?

Copyright MATHCOUNTS Inc. All rights reserved

Answer Sheet

Number Answer Problem ID

1 4 square inches A4A5

2 343 cubic units 1051

3 (1, 1) AB42

4 3 inches 41C1

5 3.60 $ 4BC02

6 .79 cu inches ADDB

7 8 05B3

8 82.5 degrees AC012

9 20 04B3

10 800 sq feet B4212

11 5/8 A0A1

12 26 notecards B3D5

13 300� 50A1

14 9�=4 sq units 4A01

15 5.29 cm B4C

16 22 percent 43D5

17 8.45 square units 55C

18 28p3 square centimeters 0CC51

19 10 minutes CC2C

20 54 percent 30312

21 1.17 inches 2A1

22 3008 sq units 42D

23 15

4B42

24 21�/2 52B1

25 80 sq units 05212

Copyright MATHCOUNTS Inc. All rights reserved

Solutions

(1) 4 square inches ID: [A4A5]

The distance from the center of one square to the center of an adjacent square is 2 inches.

Thus the area of the square resulting from connecting the centers of the chips is 22 = 4

square inches.

(2) 343 cubic units ID: [1051]

If the length, width, and height of the rectangular prism are a, b, and c , then we are given

ab = 48, bc = 49, and ac = 50. Since we are looking for abc , the volume of the

rectangular prism, we multiply these three equations to �nd

(ab)(bc)(ac) = 48 � 49 � 50 =)a2b2c2 = 48 � 49 � 50 =)(abc)2 = 48 � 49 � 50 =)

abc =p48 � 49 � 50

=√(16 � 3) � 72 � (2 � 52)

= 4 � 7 � 5p2 � 3

= 140p6;

which to the neatest whole number is 343 cubic units.

(3) (1, 1) ID: [AB42]

Draw point B and point C and rotate C 90 degrees counterclockwise about B, as shown.

Point C is 3 units above B, so its image is 3 units to the left of B at (4� 3; 1) = (1; 1) .

B

C

C ′

(4) 3 inches ID: [41C1]

Suppose that the triangle has side length s. Then the area of the triangle is s2p34= 2

p3,

so s = 2p2. We calculate the triangle's perimeter as 3s = 6

p2, so the square's side length

is 6p2

4= 3

p2

2. Finally, the square's diagonal is then 3

p2

2�p2 = 3 inches.

(5) 3.60 $ ID: [4BC02]

The �rst jar has a volume of V = �r 2h = �(32)24 = 9�. The second jar has a volume of

V = �r 2h = �(62)26 = 54�. Note that the volume of the second jar is 6 times greater than

that of the �rst jar. Because peanut butter is sold by volume, the second jar will be six

times more expensive than the �rst jar, for an answer of $0:60� 6 = $3:60 .

(6) .79 cu inches ID: [ADDB]

We can begin by calculating the volume of the liquid in the glass. Since the glass is half full,

the portion �lled with liquid has height 3 inches. The volume will be �r 2h = � � 12 � 3 = 3�.

Now, since the ratio of lemon juice to water is 1:11, the ratio of lemon juice to the liquid

will be 1:(1+11) which is 1:12. So, the volume of lemon juice in the glass is:

3� � 1

12=

�

4� :7854

So, the answer is :79 to the nearest hundredth.

(7) 8 ID: [05B3]

Number the points clockwise, beginning with the upper left as 1. Number the center point

7.

We can create six equilateral triangles with side length one: 176, 172, 273, 657, 574,

and 473.

We can also create two equilateral triangles with side lengthp3: 135 and 246.

Thus, there are 8 such equilateral triangles.

(8) 82.5 degrees ID: [AC012]

During a 12-hour period, the hour hand moves through 360 degrees. The 234hours from

9:15 to 12 make up 234=12 = 11=48 of a 12-hour period, so during that time the hour hand

moves through (11=48)(360�) = 82:5 degrees.

(9) 20 ID: [04B3]

We can make a triangle out of any three vertices, so the problem is really asking how many

ways there are to choose three vertices from six. There are six choices for the �rst vertex,

�ve for the second, and four for the third. However, we've overcounted, so we have to

determine how many di�erent orders there are in which we could choose those same three

vertices. That is, if we choose x for the �rst vertex, y for the second, and z for the third,

it will be the same triangle as if we had chosen y for the �rst vertex, z for the second, and

x for the third. We can pick any three of the vertices �rst, any two second, and then the

last is determined, so we've overcounted by a factor of six. Thus, our �nal answer is6�5�46

= 20 triangles.

(10) 800 sq feet ID: [B4212]

Let w be the width of the original rectangular garden. The perimeter of the rectangle is

2(w + 2w) = 6w , so the perimeter of the square is 12w . The dimensions of the square are

3w � 3w and its area is (3w)(3w) = 9w 2, so set 9w 2 = 3600 ft.2 to �nd w 2 = 400 square

feet. The area of the original rectangle is (2w)(w) = 2w 2 = 2 � 400 ft.2 = 800 square

feet.

(11) 5/8 ID: [A0A1]

Let the length of the height of trapezoid ABCD be h; note that this is also the length of

the height of triangle ACB to base AB. Then the area of ABCD is 20+122

� h = 16h. On the

other hand, the area of triangle ACB is 12� 20 � h = 10h. Thus the desired ratio is 10

16=

5

8.

(12) 26 notecards ID: [B3D5]

Because the dimensions of our notecards are both even, the amount of notecards that can

be cut from a 25-inch by 27-inch sheet is the same as the amount of notecards that can be

cut from a 24-inch by 26-inch sheet of poster board. We can cut this new poster board

into two pieces: a 24-inch by 20-inch board, and a 24-inch by 6-inch board. From the

24-inch by 20-inch board, we can cut a total of (24=6) � (20=4) = 20 6-inch by 4-inch

notecards. From the 24-inch by 6-inch board, we can cut a total of (24=4) � (6=6) = 6

4-inch by 6-inch notecards. This is a total of 20 + 6 = 26 notecards.

(13) 300� ID: [50A1]

A circle of diameter 20 inches has radius 10 inches. Thus the di�erence in the areas of

these two circles is 202� � 102� = 300� square inches.

(14) 9�=4 sq units ID: [4A01]

The sides of the rectangle are parallel to the axes, so the fourth point must make a vertical

line with (5,11) and a horizontal one with (16,-2); this means that the fourth point is

(5,-2). The graph of the region inside the equation is a circle with radius 3 and center

(5,-2):

Since each angle of a rectangle is 90� and the corner coincides with the center of the

circle, the rectangle covers exactly a quarter of the circle. The area of the intersection is

thus 14r 2� = 1

4� 32� =

9

4� .

(15) 5.29 cm ID: [B4C]

The remaining side is minimized if it is a leg of the triangle rather than the hypotenuse.

Then its length isp82 � 62 = 2

p7 � 5:29 cm.

(16) 22 percent ID: [43D5]

The shaded region is 4=9 of the area of square EFGH. Because E; F; G; and H are

midpoints of the sides of square ABCD, the area of EFGH is 1=2 of the area of ABCD. It

follows that the shaded region is (4=9)(1=2) = 2=9 of the area of ABCD. 2=9 = :2, so our

answer is 22 .

(17) 8.45 square units ID: [55C]

The vertices of the triangle are the points where two of the lines intersect.

y = 12x + 3 intersects y = 1 when

1

2x + 3 = 1) x = �4

y = �2x + 6 intersects y = 1 when

�2x + 6 = 1) x =5

2y = 1

2x + 3 intersects y = �2x + 6 when

1

2x + 3 = �2x + 6) x =

6

5and

y = �2(6

5

)+ 6 =

18

5

Thus the vertices of the triangle are (�4; 1),(52; 1

), and

(65; 185

). We can let the base of

the triangle lie along the line y = 1. It will have length

4 +5

2=

13

2The altitude from

(65; 185

)to this line will have length

18

5� 1 =

13

5Thus the area of the triangle is

1

2� 13

2� 13

5=

169

20= 8:45

(18) 28p3 square centimeters ID: [0CC51]

We �rst consider an equilateral triangle with side length s. If we construct an altitude, it

will divide the equilateral triangle into two congruent 30� 60� 90 triangles with the

longest side having length s and the altitude opposite the 60� angle. Since the side lengths

of a 30� 60� 90 triangle are in a 1 :p3 : 2 ratio, the altitude will have length s

p3

2. Since

the base of this equilateral triangle is s, its area will be 12bh = 1

2s(

s

p3

2

)= s

2p3

4.

Now we can set this expression equal to 64p3 and solve for s to �nd the side length of

our original triangle. Doing this, we get that s2p3

4= 64

p3. We can then multiply both

sides of the equation by 4p3to get that s2 = 256. Taking the square root of both sides, we

�nd that s = 16, so the original triangle had a side length of 16 cm. If we decrease this by

4 cm, we get that the new triangle has side length 12 cm and therefore has an area of144

p3

4= 36

p3 cm. Therefore, the area is decreased by 64

p3� 36

p3 = 28

p3 cm.

(19) 10 minutes ID: [CC2C]

We can start by drawing a diagram of the routes that each of them took:

9

12

15

The two routes form a 9-12-15 right triangle. Kelly drove for 9 + 12 = 21 miles at a rate

of 42 miles per hour, so she drove for 21=42 = :5 hours which equals 30 minutes. Brenda

drove for 15 miles at an average rate of 45 miles per hour, so she drove for 15=45 � 60 = 20

minutes. Thus, Brenda arrived 30� 20 = 10 minutes earlier than Kelly.

(20) 54 percent ID: [30312]

Three-inch cubes can �ll a rectangular box only if the edge lengths of the box are all

integer multiples of 3 inches. The largest such box whose dimensions are less than or equal

to those of the 600 � 500 � 1000 box is a 600 � 300 � 900 box. The ratio of the volumes of these

two boxes is6 � 3 � 96 � 5 � 10 =

3 � 95 � 10 =

27

50;

which is 54 percent.

(21) 1.17 inches ID: [2A1]

The volume of crispy rice treats resulting from the original recipe is 9 � 13 � 1 = 117 cubic

inches. Thus the volume obtained by making 1.5 times the original recipe is

1:5 � 117 = 175:5 cubic inches. So the depth to which the pan is �lled is 175:510�15 = 1:17

inches.

(22) 3008 sq units ID: [42D]

The quadrilateral is shown below:

AB

C

D

Divide the quadrilateral into two triangles with the dashed line. We will �nd the area of

these two triangles separately. Since AB is horizontal, the area of triangle ABC is half the

product of the length AB multiplied by the length of the vertical altitude from C to line

AB, or 1�20062

= 1003. Since AD is vertical, the area of triangle ACD is half the product of

the length AD multiplied by the length of the horizontal altitude from C to line AD, or2�2005

2= 2005. The area of the entire quadrilateral is 1003 + 2005 = 3008 square units.

(23) 15

ID: [4B42]

The area of triangle ADE is 12(10)(2) = 10 square units, and the area of rectangle ABCD

is (6)(10) = 60 square units. Subtracting, we �nd that the area of ABCE is 50 square

units. Therefore, the ratio of the area of triangle ADE to the area of quadrilateral ABCE

is 10=50 = 1=5 .

(24) 21�/2 ID: [52B1]

They both ate 38of a pizza. Therefore, the quantity that Mark ate in excess of Jane is

simply 38� the di�erence in total area of the pizzas. The 16" pizza has 64� area, and the

12" pizza has 36� area, making for a di�erence of 28�. 38� 28� =

21

2�

(25) 80 sq units ID: [05212]

Because the diagonals of a rhombus are perpendicular bisectors of each other, they divide

the rhombus into four congruent right triangles. Let x be half of the length of the shorter

diagonal of the rhombus. Then x + 3 is half of the length of the longer diagonal. Also, x

and x + 3 are the lengths of the legs of each of the right triangles. By the Pythagorean

theorem,

x2 + (x + 3)2 =(p

89)2

:

Expanding (x + 3)2 as x2 + 6x + 9 and moving every term to the left-hand side, the

equation simpli�es to 2x2 + 6x � 80 = 0. The expression 2x2 + 6x � 80 factors as

2(x � 5)(x + 8), so we �nd x = 5 and x = �8. Discarding the negative solution, we

calculate the area of the rhombus by multiplying the area of one of the right triangles by 4.

The area of the rhombus is 4 �(12� 5(5 + 3)

)= 80 square units.

x

x + 3

√

89

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