1990 AMC 8

Part of AMC 8

Subcontests

(25)

1

Patterns

How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
[asy] fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1)); draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1)); draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1)); fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray); draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1)); draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1)); draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1)); fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray); draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1)); draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1)); draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1)); fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray); draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1)); draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1)); draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1));[/asy]

\text{(A)}\ 3\qquad\text{(B)}\ 6\qquad\text{(C)}\ 8\qquad\text{(D)}\ 12\qquad\text{(E)}\ 18

1

A Lot of Symbols

\Delta

\diamondsuit

will balance nine

\bullet

\Delta

\diamondsuit

\bullet

. [asy] unitsize(5.5); fill((0,0)--(-4,-2)--(4,-2)--cycle,black); draw((-12,2)--(-12,0)--(12,0)--(12,2)); draw(ellipse((-12,5),8,3)); draw(ellipse((12,5),8,3)); label("

\Delta \hspace{2 mm}\Delta \hspace{2 mm}\Delta \hspace{2 mm}\diamondsuit

",(-12,6.5),S); label("

\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm} \bullet

",(12,5.2),N); label("

\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet

",(12,5.2),S); fill((44,0)--(40,-2)--(48,-2)--cycle,black); draw((34,2)--(34,0)--(54,0)--(54,2)); draw(ellipse((34,5),6,3)); draw(ellipse((54,5),6,3)); label("

\Delta

",(34,6.5),S); label("

\bullet \hspace{2 mm}\diamondsuit

",(54,6.5),S);[/asy]
How many

\bullet

's will balance the two

\diamondsuit

's in this balance? [asy] unitsize(5.5); fill((0,0)--(-4,-2)--(4,-2)--cycle,black); draw((-12,4)--(-12,2)--(12,-2)--(12,0)); draw(ellipse((-12,7),6.5,3)); draw(ellipse((12,3),6.5,3)); label("

?

",(-12,8.5),S); label("

\diamondsuit \hspace{2 mm}\diamondsuit

",(12,4.5),S);[/asy]

\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5

1

Trip Graph

The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest?
[asy] unitsize(12); for(int a=1; a<13; ++a) { draw((2a,-1)--(2a,1)); } draw((-1,4)--(1,4)); draw((-1,8)--(1,8)); draw((-1,12)--(1,12)); draw((-1,16)--(1,16)); draw((0,0)--(0,17)); draw((-5,0)--(33,0)); label("

0

",(0,-1),S); label("

1

",(2,-1),S); label("

2

",(4,-1),S); label("

3

",(6,-1),S); label("

4

",(8,-1),S); label("

5

",(10,-1),S); label("

6

",(12,-1),S); label("

7

",(14,-1),S); label("

8

",(16,-1),S); label("

9

",(18,-1),S); label("

10

",(20,-1),S); label("

11

",(22,-1),S); label("

12

",(24,-1),S); label("Time in hours",(11,-2),S); label("

500

",(-1,4),W); label("

1000

",(-1,8),W); label("

1500

",(-1,12),W); label("

2000

",(-1,16),W); label(rotate(90)*"Distance traveled in miles",(-4,10),W); draw((0,0)--(2,3)--(4,7.2)--(6,8.5)); draw((6,8.5)--(16,12.5)--(18,14)--(24,15));[/asy]

\text{(A)}\ \text{first (0-1)}\qquad\text{(B)}\ \text{second (1-2)}\qquad\text{(C)}\ \text{third (2-3)}\qquad\text{(D)}\ \text{ninth (8-9)}\qquad\text{(E)}\ \text{last (11-12)}

1

Taking Candy

Several students are seated at a large circular table. They pass around a bag containing

100

pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be

\text{(A)}\ 10\qquad\text{(B)}\ 11\qquad\text{(C)}\ 19\qquad\text{(D)}\ 20\qquad\text{(E)}\ 25

1

List Of Numbers

8

numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown:

\text{\underline{\hspace{3 mm}?\hspace{3 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}16\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}64\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{1 mm}1024\hspace{1 mm}}}

\text{(A)}\ \frac{1}{64}\qquad\text{(B)}\ \frac{1}{4}\qquad\text{(C)}\ 1\qquad\text{(D)}\ 2\qquad\text{(E)}\ 4

1

Incomes Error

The annual incomes of

1000

families range from

8200

98000

dollars. In error, the largest income was entered on the computer as

980000

dollars. The difference between the mean of the incorrect data and the mean of the actual data is

\text{(A)}\ \text{882 dollars}\qquad\text{(B)}\ \text{980 dollars}\qquad\text{(C)}\ \text{1078 dollars}\qquad\text{(D)}\ \text{482,000 dollars}\qquad\text{(E)}\ \text{882,000 dollars}

1

Seats

120

seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?

\text{(A)}\ 30\qquad\text{(B)}\ 40\qquad\text{(C)}\ 41\qquad\text{(D)}\ 60\qquad\text{(E)}\ 119

1

Rectangular Prism

Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have? [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy]

\text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 42\qquad\text{(E)}\ 48

Assume that the planes cutting the prism do not intersect anywhere in or on the prism.

1

Concrete Sidewalk

A straight concrete sidewalk is to be

3

60

3

inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?

\text{(A)}\ 2\qquad\text{(B)}\ 5\qquad\text{(C)}\ 12\qquad\text{(D)}\ 20\qquad\text{(E)}\ \text{more than 20}

1

Sum and Subtract

1990-1980+1970-1960+\cdots-20+10 =

\text{(A)}\ -990\qquad\text{(B)}\ -10\qquad\text{(C)}\ 990\qquad\text{(D)}\ 1000\qquad\text{(E)}\ 1990

1

Perimeter Of Figure

The area of this figure is

100\text{ cm}^{2}

. Its perimeter is
[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed);[/asy]

\text{(A)}\ \text{20 cm}\qquad\text{(B)}\ \text{25 cm}\qquad\text{(C)}\ \text{30 cm}\qquad\text{(D)}\ \text{40 cm}\qquad\text{(E)}\ \text{50 cm}

1

Number Of Balls

A bag contains only blue balls and green balls. There are

6

blue balls. If the probability of drawing a blue ball at random from this bag is

\frac{1}{4}

, then the number of green balls in the bag is

\text{(A)}\ 12\qquad\text{(B)}\ 18\qquad\text{(C)}\ 24\qquad\text{(D)}\ 30\qquad\text{(E)}\ 36

1

Postage Rate

One proposal for new postage rates for a letter was

30

cents for the first ounce and

22

cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing

4.5

\text{(A)}\ \text{96 cents}\qquad\text{(B)}\ \text{1.07 dollars}\qquad\text{(C)}\ \text{1.18 dollars}\qquad\text{(D)}\ \text{1.20 dollars}\qquad\text{(E)}\ \text{1.40 dollars}

1

Sum of 4-Digit Numbers

There are twenty-four 4-digit numbers that use each of the four digits 2, 5, 7, and 4exactly once. Listed in numerical order from smallest to largest, the number in the

17th

position in the list is

\text{(A)}\ 4527\qquad\text{(B)}\ 5724\qquad\text{(C)}\ 5742\qquad\text{(D)}\ 7245\qquad\text{(E)}\ 7524

1

Faces Of Cube

The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); label("

15

",(1.5,1.2),N); label("

11

",(4,2.3),N); label("

14

",(2.5,3.7),N);[/asy]

\text{(A)}\ 75\qquad\text{(B)}\ 76\qquad\text{(C)}\ 78\qquad\text{(D)}\ 80\qquad\text{(E)}\ 81

1

Calender

On this monthly calendar, the date behind one of the letters is added to the date behind

C

. If this sum equals the sum of the dates behind

A

B

, then the letter is
[asy] unitsize(12); draw((1,1)--(23,1)); draw((0,5)--(23,5)); draw((0,9)--(23,9)); draw((0,13)--(23,13)); for(int a=0; a<6; ++a) { draw((4a+2,0)--(4a+2,14)); } label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N); label("Fri.",(16,14),N); label("Sat.",(20,14),N); label("C",(12,10.3),N); label("

<span class='latex-bold'>A</span>

",(16,10.3),N); label("Q",(12,6.3),N); label("S",(4,2.3),N); label("

<span class='latex-bold'>B</span>

",(8,2.3),N); label("P",(12,2.3),N); label("T",(16,2.3),N); label("R",(20,2.3),N);[/asy]

\text{(A)}\ \text{P}\qquad\text{(B)}\ \text{Q}\qquad\text{(C)}\ \text{R}\qquad\text{(D)}\ \text{S}\qquad\text{(E)}\ \text{T}

1

Grades

The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were: \begin{tabular}[t]{lllllllll}89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75,\\ 63, & 84, & 78, & 71, & 80, & 90. & & &\\ \end{tabular} In Mr. Freeman's class, what percent of the students received a grade of C?
\boxed{\begin{tabular}[t]{cc}A: & 93-100\\ B: & 85-92\\ C: & 75-84\\ D: & 70-74\\ F: & 0-69\end{tabular}}

\text{(A)}\ 20\%\qquad\text{(B)}\ 25\%\qquad\text{(C)}\ 30\%\qquad\text{(D)}\ 33\frac{1}{3}\%\qquad\text{(E)}\ 40\%

1

Discounts

A dress originally priced at 80 dollars was put on sale for

25\%

10\%

tax was added to the sale price, then the total selling price (in dollars) of the dress was

\text{(A)}\ \text{45 dollars}\qquad\text{(B)}\ \text{52 dollars}\qquad\text{(C)}\ \text{54 dollars}\qquad\text{(D)}\ \text{66 dollars}\qquad\text{(E)}\ \text{68 dollars}

1

Largest Possible Product

When three different numbers from the set

\{-3,-2,-1, 4, 5\}

are multiplied, the largest possible product is

\text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60

1

Largest?

Which of these five numbers is the largest?

\text{(A)}\ 13579+\frac{1}{2468}\qquad\text{(B)}\ 13579-\frac{1}{2468}\qquad\text{(C)}\ 13579\times\frac{1}{2468}

\text{(D)}\ 13579\div\frac{1}{2468}\qquad\text{(E)}\ 13579.2468

1

Closest Product

Which of the following is closest to the product

(.48017)(.48017)(.48017)

?

\text{(A)}\ 0.011\qquad\text{(B)}\ 0.110\qquad\text{(C)}\ 1.10\qquad\text{(D)}\ 11.0\qquad\text{(E)}\ 110

1

Units Digit

Which of the following could not be the unit's digit [one's digit] of the square of a whole number?

\text{(A)}\ 1\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 8

1

Fraction Shaded

What fraction of the square is shaded?
[asy] draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3)); fill((0,0)--(0,1)--(1,1)--cycle,grey); fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey); fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey);[/asy]

\text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{2}{5}\qquad\text{(C)}\ \frac{5}{12}\qquad\text{(D)}\ \frac{3}{7}\qquad\text{(E)}\ \frac{1}{2}

1

Changing Digits

0.12345

, when changed to

9

, gives the largest number?

\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5

1

Find The Sum

What is the smallest sum of two

3

-digit numbers that can be obtained by placing each of the six digits

4,5,6,7,8,9

in one of the six boxes in this addition problem?
[asy] unitsize(12); draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle); draw((4,1)--(6,1)--(6,3)--(4,3)--cycle); draw((4,4)--(6,4)--(6,6)--(4,6)--cycle); draw((7,1)--(9,1)--(9,3)--(7,3)--cycle); draw((7,4)--(9,4)--(9,6)--(7,6)--cycle);[/asy]

\text{(A)}\ 947\qquad\text{(B)}\ 1037\qquad\text{(C)}\ 1047\qquad\text{(D)}\ 1056\qquad\text{(E)}\ 1245