MathDB

1

Checkerboard with one inch squares

A checkerboard consists of one-inch squares. A square card,

1.5

inches on a side, is placed on the board so that it covers part or all of the area of each of

n

squares. The maximum possible value of

n

is

\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}

24

1

Number directly above 142

What number is directly above

142

in this array of numbers?

\begin{array}{cccccc} & & & 1 & & \\ & & 2 & 3 & 4 & \\ & 5 & 6 & 7 & 8 & 9 \\ 10 & 11 & 12 & \cdots & & \\ \end{array}

<span class='latex-bold'>(A)</span>\ 99 \qquad <span class='latex-bold'>(B)</span>\ 119 \qquad <span class='latex-bold'>(C)</span>\ 120 \qquad <span class='latex-bold'>(D)</span>\ 121 \qquad <span class='latex-bold'>(E)</span>\ 122

23

1

Not finishing third in a race

Five runners,

P

,

Q

,

R

,

S

,

T

, have a race, and

P

beats

Q

,

P

beats

R

,

Q

beats

S

, and

T

finishes after

P

and before

Q

. Who could NOT have finished third in the race?

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

22

1

Using the numbers 2 22 times in a sequence starting with 1

Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?

\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199

21

1

Area of new rectangle when length/width increased

If the length of a rectangle is increased by

20\%

and its width is increased by

50\%

, then the area is increased by

\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\%

20

1

Sum of digits of 10^(93)-93

When

10^{93}-93

is expressed as a single whole number, the sum of the digits is

\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833

19

1

Telescoping

(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) =

\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142

18

1

Area of quadrilateral

The rectangle shown has length

AC=32

, width

AE=20

, and

B

and

F

are midpoints of

\overline{AC}

and

\overline{AE}

, respectively. The area of quadrilateral

ABDF

is
[asy] pair A,B,C,D,EE,F; A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10); draw(A--C--D--EE--cycle); draw(B--D--F); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); label("

A

",A,NW); label("

B

",B,N); label("

C

",C,NE); label("

D

",D,SE); label("

E

",EE,SW); label("

F

",F,W); [/asy]

\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340

17

1

Surface area of box

Square corners,

5

units on a side, are removed from a

20

unit by

30

unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is
[asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy]

\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000

16

1

1/(1+1/(2+1/3))=

\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{3}}} =

\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{3}{10} \qquad \text{(C)}\ \dfrac{7}{10} \qquad \text{(D)}\ \dfrac{5}{6} \qquad \text{(E)}\ \dfrac{10}{3}

15

1

Mean of 3 numbers with one number and the mean of 4 numbers

The arithmetic mean (average) of four numbers is

85

. If the largest of these numbers is

97

, then the mean of the remaining three numbers is

\text{(A)}\ 81.0 \qquad \text{(B)}\ 82.7 \qquad \text{(C)}\ 83.0 \qquad \text{(D)}\ 84.0 \qquad \text{(E)}\ 84.3

14

1

Each row/column use the numbers 1,2,3 in 3*3 square

The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers

1,2,3

. Then

A+B=

\begin{tabular}{|c|c|c|} \hline 1 & & \\ \hline & 2 & A \\ \hline & & B \\ \hline \end{tabular}

\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6

13

1

Area of the white portion of help sign

The word "'''HELP'''" in block letters is painted in black with strokes

1

unit wide on a

5

by

15

rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is
[asy] unitsize(12); fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black); fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black); fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black); fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black); fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white); draw((0,0)--(15,0)--(15,5)--(0,5)--cycle); label("5\left\{ \begin{tabular}{c} \\ \\ \\ \\ \end{tabular}\right.",(1,2.5),W); label(rotate(90)*"

\{

",(0.5,0.1),S); label("

1

",(0.5,-0.6),S); label(rotate(90)*"

\{

",(3.5,0.1),S); label("

1

",(3.5,-0.6),S); label(rotate(90)*"

\{

",(7.5,0.1),S); label("

1

",(7.5,-0.6),S); label(rotate(90)*"

\{

",(11.5,0.1),S); label("

1

",(11.5,-0.6),S); label(rotate(270)*"\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.",(1.5,4),N); label("

3

",(1.5,5.8),N); label(rotate(270)*"\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.",(5.5,4),N); label("

3

",(5.5,5.8),N); label(rotate(270)*"\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.",(9.5,4),N); label("

3

",(9.5,5.8),N); label(rotate(270)*"\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.",(13.5,4),N); label("

3

",(13.5,5.8),N); label("\left. \begin{tabular}{c} \\ \end{tabular}\right\} 2",(14,1),E); [/asy]

\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38

12

1

operations +, -, * are used once, find a possible result

If each of the three operation signs,

+

,

-

,

\times

, is used exactly ONCE in one of the blanks in the expression

5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3

then the value of the result could equal

\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 19

11

1

Evaluating the graph of 81 student's test scores

Consider this histogram of the scores for

81

students taking a test:
[asy] unitsize(12); draw((0,0)--(26,0)); draw((1,1)--(25,1)); draw((3,2)--(25,2)); draw((5,3)--(23,3)); draw((5,4)--(21,4)); draw((7,5)--(21,5)); draw((9,6)--(21,6)); draw((11,7)--(19,7)); draw((11,8)--(19,8)); draw((11,9)--(19,9)); draw((11,10)--(19,10)); draw((13,11)--(19,11)); draw((13,12)--(19,12)); draw((13,13)--(17,13)); draw((13,14)--(17,14)); draw((15,15)--(17,15)); draw((15,16)--(17,16));
draw((1,0)--(1,1)); draw((3,0)--(3,2)); draw((5,0)--(5,4)); draw((7,0)--(7,5)); draw((9,0)--(9,6)); draw((11,0)--(11,10)); draw((13,0)--(13,14)); draw((15,0)--(15,16)); draw((17,0)--(17,16)); draw((19,0)--(19,12)); draw((21,0)--(21,6)); draw((23,0)--(23,3)); draw((25,0)--(25,2));
for (int a = 1; a < 13; ++a) { draw((2*a,-.25)--(2*a,.25)); }
label("

40

",(2,-.25),S); label("

45

",(4,-.25),S); label("

50

",(6,-.25),S); label("

55

",(8,-.25),S); label("

60

",(10,-.25),S); label("

65

",(12,-.25),S); label("

70

",(14,-.25),S); label("

75

",(16,-.25),S); label("

80

",(18,-.25),S); label("

85

",(20,-.25),S); label("

90

",(22,-.25),S); label("

95

",(24,-.25),S);
label("

1

",(2,1),N); label("

2

",(4,2),N); label("

4

",(6,4),N); label("

5

",(8,5),N); label("

6

",(10,6),N); label("

10

",(12,10),N); label("

14

",(14,14),N); label("

16

",(16,16),N); label("

12

",(18,12),N); label("

6

",(20,6),N); label("

3

",(22,3),N); label("

2

",(24,2),N);
label("Number",(4,8),N); label("of Students",(4,7),N);
label("

<span class='latex-bold'>STUDENT TEST SCORES</span>

",(14,18),N); [/asy]
The median is in the interval labeled

\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80

10

1

Greatest month on line graph

This line graph represents the price of a trading card during the first

6

months of

1993

.
[asy] unitsize(18); for (int a = 0; a <= 6; ++a) { draw((4*a,0)--(4*a,10)); } for (int a = 0; a <= 5; ++a) { draw((0,2*a)--(24,2*a)); } draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5));
label("

Jan

",(2,0),S); label("

Feb

",(6,0),S); label("

Mar

",(10,0),S); label("

Apr

",(14,0),S); label("

May

",(18,0),S); label("

Jun

",(22,0),S); label("

<span class='latex-bold'>1993 PRICES FOR A TRADING CARD</span>

",(12,10),N);
label("\begin{tabular}{c}P \\ R \\ I \\ C \\ E \end{tabular}",(-2,5),W); label("

1

",(0,2),W); label("

2

",(0,4),W); label("

3

",(0,6),W); label("

4

",(0,8),W); label("

5

",(0,10),W); [/asy]
The greatest monthly drop in price occurred during

\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}

9

1

Evaluating a functional data graph

Consider the operation

*

defined by the following table:
\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{tabular}
For example,

3*2=1

. Then

(2*4)*(1*3)=

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

8

1

One supply of medicine lasts how long?

To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains

60

pills, then the supply of medicine would last approximately

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

7

1

Powers of 3

3^3+3^3+3^3 =

\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}

6

1

Adults remaining food feed

A can of soup can feed

3

adults or

5

children. If there are

5

cans of soup and

15

children are fed, then how many adults would the remaining soup feed?

\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10

5

1

Bar graphs=circle graph

Which one of the following bar graphs could represent the data from the circle graph?
[asy] unitsize(36); draw(circle((0,0),1),gray); fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray); fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black); [/asy]
[asy] unitsize(4);
fill((1,0)--(1,15)--(5,15)--(5,0)--cycle,gray); fill((6,0)--(6,15)--(10,15)--(10,0)--cycle,black); draw((11,0)--(11,20)--(15,20)--(15,0));
fill((26,0)--(26,15)--(30,15)--(30,0)--cycle,gray); fill((31,0)--(31,15)--(35,15)--(35,0)--cycle,black); draw((36,0)--(36,15)--(40,15)--(40,0));
fill((51,0)--(51,10)--(55,10)--(55,0)--cycle,gray); fill((56,0)--(56,10)--(60,10)--(60,0)--cycle,black); draw((61,0)--(61,20)--(65,20)--(65,0));
fill((76,0)--(76,10)--(80,10)--(80,0)--cycle,gray); fill((81,0)--(81,15)--(85,15)--(85,0)--cycle,black); draw((86,0)--(86,20)--(90,20)--(90,0));
fill((101,0)--(101,15)--(105,15)--(105,0)--cycle,gray); fill((106,0)--(106,10)--(110,10)--(110,0)--cycle,black); draw((111,0)--(111,20)--(115,20)--(115,0));
for(int a = 0; a < 5; ++a) { draw((25*a,21)--(25*a,0)--(25*a+16,0)); }
label("(A)",(8,21),N); label("(B)",(33,21),N); label("(C)",(58,21),N); label("(D)",(83,21),N); label("(E)",(108,21),N); [/asy]

4

1

Scientific Notation

1000\times 1993 \times 0.1993 \times 10 =

\text{(A)}\ 1.993\times 10^3 \qquad \text{(B)}\ 1993.1993 \qquad \text{(C)}\ (199.3)^2 \qquad \text{(D)}\ 1,993,001.993 \qquad \text{(E)}\ (1993)^2

3

1

Largest Prime Factor

Which of the following numbers has the largest prime factor?

\text{(A)}\ 39 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 77 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 121

2

1

49/84 in simplest form

When the fraction

\dfrac{49}{84}

is expressed in simplest form, then the sum of the numerator and the denominator will be

\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133

1

Product not 36

Which pair of numbers does NOT have a product equal to

36

?

\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}

1993 AMC 8

Subcontests

Checkerboard with one inch squares

Number directly above 142

Not finishing third in a race

Using the numbers 2 22 times in a sequence starting with 1

Area of new rectangle when length/width increased

Sum of digits of 10^(93)-93

Telescoping

Area of quadrilateral

Surface area of box

1/(1+1/(2+1/3))=

Mean of 3 numbers with one number and the mean of 4 numbers

Each row/column use the numbers 1,2,3 in 3*3 square

Area of the white portion of help sign

operations +, -, * are used once, find a possible result

Evaluating the graph of 81 student's test scores

Greatest month on line graph

Evaluating a functional data graph

One supply of medicine lasts how long?

Powers of 3

Adults remaining food feed

Bar graphs=circle graph

Scientific Notation

Largest Prime Factor

49/84 in simplest form

Product not 36