MathDB

1

1996 AHSME #8 -fixed-

A

and

B

are 10 units apart. Points

B

and

C

are 4 units apart. Points

C

and

D

are 3 units apart. If

A

and

D

are as close as possible, then the number of units between them is

\text{(A)}\ 0 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 17

25

1

1996 AJHSME Problem 25

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?

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

24

1

1996 AJHSME Problem 24

The measure of angle

ABC

is

50^\circ

,

\overline{AD}

bisects angle

BAC

, and

\overline{DC}

bisects angle

BCA

. The measure of angle

ADC

is
[asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C);
label("

A

",A,SW); label("

B

",B,N); label("

C

",C,SE); label("

D

",D,N); label("

50^\circ

",(9.4,8.8),SW); [/asy]

\text{(A)}\ 90^\circ \qquad \text{(B)}\ 100^\circ \qquad \text{(C)}\ 115^\circ \qquad \text{(D)}\ 122.5^\circ \qquad \text{(E)}\ 125^\circ

23

1

1996 AJHSME Problem 23

The manager of a company planned to distribute a

\$50

bonus to each employee from the company fund, but the fund contained

\$5

less than what was needed. Instead the manager gave each employee a

\$45

bonus and kept the remaining

\$95

in the company fund. The amount of money in the company fund before any bonuses were paid was

\text{(A)}\ 945\text{ dollars} \qquad \text{(B)}\ 950\text{ dollars} \qquad \text{(C)}\ 955\text{ dollars} \qquad \text{(D)}\ 990\text{ dollars} \qquad \text{(E)}\ 995\text{ dollars}

22

1

1996 AJHSME Problem 22

The horizontal and vertical distances between adjacent points equal

1

unit. The area of triangle

ABC

is
[asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 4; ++b) { dot((a,b)); } } draw((0,0)--(3,2)--(4,3)--cycle); label("

A

",(0,0),SW); label("

B

",(3,2),SE); label("

C

",(4,3),NE); [/asy]

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

21

1

1996 AJHSME Problem 21

How many subsets containing three different numbers can be selected from the set

\{ 89,95,99,132, 166,173 \}

so that the sum of the three numbers is even?

\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24

20

1

1996 AJHSME Problem 20

Suppose there is a special key on a calculator that replaces the number

x

currently displayed with the number given by the formula

\frac{1}{1-x}

. For example, if the calculator is displaying

2

and the special key is pressed, then the calculator will display

-1

since

\frac{1}{1-2}=-1

. Now suppose that the calculator is displaying

5

. After the special key is pressed 100 times in a row, the calculator will display

\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5

19

1

1996 AJHSME Problem 19

The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is

2000

and at West,

2500

. In the two schools combined, the percent of students who prefer tennis is
[asy] unitsize(18); draw(circle((0,0),4)); draw(circle((9,0),4)); draw((-4,0)--(0,0)--4*dir(352.8)); draw((0,0)--4*dir(100.8)); draw((5,0)--(9,0)--(4*dir(324)+(9,0))); draw((9,0)--(4*dir(50.4)+(9,0)));
label("

48\%

",(0,-1),S); label("bowling",(0,-2),S); label("

30\%

",(1.5,1.5),N); label("golf",(1.5,0.5),N); label("

22\%

",(-2,1.5),N); label("tennis",(-2,0.5),N);
label("

40\%

",(8.5,-1),S); label("tennis",(8.5,-2),S); label("

24\%

",(10.5,0.5),E); label("golf",(10.5,-0.5),E); label("

36\%

",(7.8,1.7),N); label("bowling",(7.8,0.7),N);
label("

<span class='latex-bold'>East JHS</span>

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

<span class='latex-bold'>2000 students</span>

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

<span class='latex-bold'>West MS</span>

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

<span class='latex-bold'>2500 students</span>

",(9,-5),S); [/asy]

\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%

18

1

1996 AJHSME Problem 18

Ana's monthly salary was

\$2000

in May. In June she received a

20 \%

raise. In July she received a

20 \%

pay cut. After the two changes in June and July, Ana's monthly salary was

\text{(A)}\ 1920\text{ dollars} \qquad \text{(B)}\ 1980\text{ dollars} \qquad \text{(C)}\ 2000\text{ dollars} \qquad \text{(D)}\ 2020\text{ dollars} \qquad \text{(E)}\ 2040\text{ dollars}

17

1

1996 AJHSME Problem 17

Figure

OPQR

is a square. Point

O

is the origin, and point

Q

has coordinates

(2,2)

. What are the coordinates for

T

so that the area of triangle

PQT

equals the area of square

OPQR

?
[asy] pair O,P,Q,R,T; O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0); draw((-5,0)--(3,0)); draw((0,-1)--(0,3)); draw(P--Q--R); draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8)); draw((-0.2,2.8)--(0,3)--(0.2,2.8)); draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2)); draw((2.8,-0.2)--(3,0)--(2.8,0.2)); draw(Q--T);
label("

O

",O,SW); label("

P

",P,S); label("

Q

",Q,NE); label("

R

",R,W); label("

T

",T,S); [/asy]
NOT TO SCALE

\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)

16

1

1996 AJHSME Problem 16

1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=

\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998

15

1

1996 AJHSME Problem 15

The remainder when the product

1492\cdot 1776\cdot 1812\cdot 1996

is divided by

5

is

\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4

14

1

1996 AJHSME Problem 14

Six different digits from the set

\{ 1,2,3,4,5,6,7,8,9\}

are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is
[asy] unitsize(18); draw((0,0)--(1,0)--(1,1)--(4,1)--(4,2)--(1,2)--(1,3)--(0,3)--cycle); draw((0,1)--(1,1)--(1,2)--(0,2)); draw((2,1)--(2,2)); draw((3,1)--(3,2)); label("

23

",(0.5,0),S); label("

12

",(4,1.5),E); [/asy]

\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 35

13

1

1996 AJHSME Problem 13

In the fall of

1996

, a total of

800

students participated in an annual school clean-up day. The organizers of the event expect that in each of the years

1997

,

1998

, and

1999

, participation will increase by

50 \%

over the previous year. The number of participants the organizers will expect in the fall of

1999

is

\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700

12

1

1996 AJHSME Problem 12

What number should be removed from the list

1,2,3,4,5,6,7,8,9,10,11

so that the average of the remaining numbers is

6.1

?

\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8

11

1

1996 AJHSME Problem 11

Let

x

be the number

0.\underbrace{0000...0000}_{1996\text{ zeros}}1,

where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?

\text{(A)}\ 3+x \qquad \text{(B)}\ 3-x \qquad \text{(C)}\ 3\cdot x \qquad \text{(D)}\ 3/x \qquad \text{(E)}\ x/3

10

1

1996 AJHSME Problem 10

When Walter drove up to the gasoline pump, he noticed that his gasoline tank was

\frac{1}{8}

full. He purchased

7.5

gallons of gasoline for

\$10

. With this additional gasoline, his gasoline tank was then

\frac{5}{8}

full. The number of gallons of gasoline his tank holds when it is full is

\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5

9

1

1996 AJHSME Problem 9

If

5

times a number is

2

, then

100

times the reciprocal of the number is

\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500

7

1

1996 AJHSME Problem 7

Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has

4

goldfish at the same time that Gretel has

128

goldfish, then in how many months from that time will they have the same number of goldfish?

\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8

6

1

1996 AJHSME Problem 6

What is the smallest result that can be obtained from the following process?
*Choose three different numbers from the set

\{3,5,7,11,13,17\}

. *Add two of these numbers. *Multiply their sum by the third number.

\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56

4

1

1996 AJHSME Problem 4

\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=

\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{2}{3} \qquad \text{(C)}\ \dfrac{3}{2} \qquad \text{(D)}\ \dfrac{17}{3} \qquad \text{(E)}\ \dfrac{34}{3}

5

1

1996 AJHSME Problem 5

The letters

P

,

Q

,

R

,

S

, and

T

represent numbers located on the number line as shown.
[asy] unitsize(36); draw((-4,0)--(4,0)); draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1)); draw((3.9,0.1)--(4,0)--(3.9,-0.1));
for (int i = -3; i <= 3; ++i) { draw((i,-0.1)--(i,0)); } label("

-3

",(-3,-0.1),S); label("

-2

",(-2,-0.1),S); label("

-1

",(-1,-0.1),S); label("

0

",(0,-0.1),S); label("

1

",(1,-0.1),S); label("

2

",(2,-0.1),S); label("

3

",(3,-0.1),S);
draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1)); draw((-3.6,0)--(-3.6,0.25)); label("

P

",(-3.6,0.25),N); draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1)); draw((-1.2,0)--(-1.2,0.25)); label("

Q

",(-1.2,0.25),N); draw((0.1,0.1)--(0.2,0)--(0.3,0.1)); draw((0.2,0)--(0.2,0.25)); label("

R

",(0.2,0.25),N); draw((0.8,0.1)--(0.9,0)--(1,0.1)); draw((0.9,0)--(0.9,0.25)); label("

S

",(0.9,0.25),N); draw((1.4,0.1)--(1.5,0)--(1.6,0.1)); draw((1.5,0)--(1.5,0.25)); label("

T

",(1.5,0.25),N); [/asy]
Which of the following expressions represents a negative number?

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

3

1

1996 AJHSME Problem 3

The

64

whole numbers from

1

through

64

are written, one per square, on a checkerboard (an

8

by

8

array of

64

squares). The first

8

numbers are written in order across the first row, the next

8

across the second row, and so on. After all

64

numbers are written, the sum of the numbers in the four corners will be

\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134

2

1

1996 AJHSME Problem 2

Jose, Thuy, and Kareem each start with the number

10

. Jose subtracts

1

from the number

10

, doubles his answer, and then adds

2

. Thuy doubles the number

10

, subtracts

1

from her answer, and then adds

2

. Kareem subtracts

1

from the number

10

, adds

2

to his number, and then doubles the result. Who gets the largest final answer?

\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kareem} \qquad \text{(D)}\ \text{Jose and Thuy} \qquad \text{(E)}\ \text{Thuy and Kareem}

1

1996 AJHSME Problem 1

How many positive factors of

36

are also multiples of

4

?

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

1996 AMC 8

Subcontests

1996 AHSME #8 -fixed-

1996 AJHSME Problem 25

1996 AJHSME Problem 24

1996 AJHSME Problem 23

1996 AJHSME Problem 22

1996 AJHSME Problem 21

1996 AJHSME Problem 20

1996 AJHSME Problem 19

1996 AJHSME Problem 18

1996 AJHSME Problem 17

1996 AJHSME Problem 16

1996 AJHSME Problem 15

1996 AJHSME Problem 14

1996 AJHSME Problem 13

1996 AJHSME Problem 12

1996 AJHSME Problem 11

1996 AJHSME Problem 10

1996 AJHSME Problem 9

1996 AJHSME Problem 7

1996 AJHSME Problem 6

1996 AJHSME Problem 4

1996 AJHSME Problem 5

1996 AJHSME Problem 3

1996 AJHSME Problem 2

1996 AJHSME Problem 1