MathDB

1

1989 AJHSME Problem 25

Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even?

\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{3}{7} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{2}{3} \qquad \text{(E)}\ \frac{5}{7}

[asy] unitsize(36); draw(circle((-3,0),1)); draw(circle((0,0),1)); draw((0,0)--dir(30)); draw((0,0)--(0,-1)); draw((0,0)--dir(150)); draw((-2.293,.707)--(-3.707,-.707)); draw((-2.293,-.707)--(-3.707,.707)); fill((-2.9,1)--(-2.65,1.25)--(-2.65,1.6)--(-3.35,1.6)--(-3.35,1.25)--(-3.1,1)--cycle,black); fill((.1,1)--(.35,1.25)--(.35,1.6)--(-.35,1.6)--(-.35,1.25)--(-.1,1)--cycle,black); label("

5

",(-3,.2),N); label("

3

",(-3.2,0),W); label("

4

",(-3,-.2),S); label("

8

",(-2.8,0),E); label("

6

",(0,.2),N); label("

9

",(-.2,.1),SW); label("

7

",(.2,.1),SE); [/asy]

24

1

1989 AJHSME Problem 24

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}

[asy] draw((0,0)--(0,8)--(6,8)--(6,0)--cycle); draw((0,8)--(5,9)--(5,8)); draw((3,-1.5)--(3,10.3),dashed); draw((0,5.5)..(-.75,4.75)..(0,4)); draw((0,4)--(1.5,4),EndArrow); [/asy]

23

1

1989 AJHSME Problem 23

An artist has

14

cubes, each with an edge of

1

meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?

\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42

[asy] draw((0,0)--(2.35,-.15)--(2.44,.81)--(.09,.96)--cycle); draw((.783333333,-.05)--(.873333333,.91)--(1.135,1.135)); draw((1.566666667,-.1)--(1.656666667,.86)--(1.89,1.1)); draw((2.35,-.15)--(4.3,1.5)--(4.39,2.46)--(2.44,.81)); draw((3,.4)--(3.09,1.36)--(2.61,1.4)); draw((3.65,.95)--(3.74,1.91)--(3.29,1.94)); draw((.09,.96)--(.76,1.49)--(.71,1.17)--(2.2,1.1)--(3.6,2.2)--(3.62,2.52)--(4.39,2.46)); draw((.76,1.49)--(.82,1.96)--(2.28,1.89)--(2.2,1.1)); draw((2.28,1.89)--(3.68,2.99)--(3.62,2.52)); draw((1.455,1.135)--(1.55,1.925)--(1.89,2.26)); draw((2.5,2.48)--(2.98,2.44)--(2.9,1.65)); draw((.82,1.96)--(1.55,2.6)--(1.51,2.3)--(2.2,2.26)--(2.9,2.8)--(2.93,3.05)--(3.68,2.99)); draw((1.55,2.6)--(1.59,3.09)--(2.28,3.05)--(2.2,2.26)); draw((2.28,3.05)--(2.98,3.59)--(2.93,3.05)); draw((1.59,3.09)--(2.29,3.63)--(2.98,3.59)); [/asy]

22

1

1989 AJHSME Problem 22

The letters

\text{A}

,

\text{J}

,

\text{H}

,

\text{S}

,

\text{M}

,

\text{E}

and the digits

1

,

9

,

8

,

9

are "cycled" separately as follows and put together in a numbered list: \begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 9891 \\ 2. & & HSMEAJ & 8919 \\ 3. & & SMEAJH & 9198 \\ & & ........ & \end{tabular}
What is the number of the line on which

\text{AJHSME 1989}

will appear for the first time?

\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24

21

1

1989 AJHSME Problem 21

Jack had a bag of

128

apples. He sold

25\%

of them to Jill. Next he sold

25\%

of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?

\text{(A)}\ 7 \qquad \text{(B)}\ 63 \qquad \text{(C)}\ 65 \qquad \text{(D)}\ 71 \qquad \text{(E)}\ 111

20

1

1989 AJHSME Problem 20

The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner?
[asy] draw((0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,1)--(4,1)--(4,0)--(2,0)--(2,-1)--(1,-1)--(1,0)--cycle); draw((1,0)--(1,1)--(2,1)--(2,0)--cycle); draw((3,1)--(3,0)); label("

1

",(1.5,1.25),N); label("

2

",(1.5,.25),N); label("

3

",(1.5,-.75),N); label("

4

",(2.5,.25),N); label("

5

",(3.5,.25),N); label("

6

",(.5,.25),N); [/asy]

\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15

19

1

1989 AJHSME Problem 19

The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during

1988

. For example, about

.5

million had been spent by the beginning of February and approximately

2

million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August?

\text{(A)}\ 1.5 \qquad \text{(B)}\ 2.5 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4.5 \qquad \text{(E)}\ 5.5

[asy] unitsize(18); for (int a=1; a<13; ++a) { draw((a,0)--(a,.5)); } for (int b=1; b<6; ++b) { draw((-.5,2b)--(0,2b)); } draw((0,0)--(0,12)); draw((0,0)--(14,0)); draw((0,0)--(1,.9)--(2,1.9)--(3,2.6)--(4,4.3)--(5,4.5)--(6,5.7)--(7,8.2)--(8,9.4)--(9,9.8)--(10,10.1)--(11,10.2)--(12,10.5)); label("J",(.5,0),S); label("F",(1.5,0),S); label("M",(2.5,0),S); label("A",(3.5,0),S); label("M",(4.5,0),S); label("J",(5.5,0),S); label("J",(6.5,0),S); label("A",(7.5,0),S); label("S",(8.5,0),S); label("O",(9.5,0),S); label("N",(10.5,0),S); label("D",(11.5,0),S); label("month F=February",(16,0),S); label("

1

",(-.6,2),W); label("

2

",(-.6,4),W); label("

3

",(-.6,6),W); label("

4

",(-.6,8),W); label("

5

",(-.6,10),W); label("dollars in millions",(0,11.9),N); [/asy]

18

1

1989 AJHSME Problem 18

Many calculators have a reciprocal key

\boxed{\frac{1}{x}}

that replaces the current number displayed with its reciprocal. For example, if the display is

\boxed{00004}

and the

\boxed{\frac{1}{x}}

key is depressed, then the display becomes

\boxed{000.25}

. If

\boxed{00032}

is currently displayed, what is the fewest number of times you must depress the

\boxed{\frac{1}{x}}

key so the display again reads

\boxed{00032}

?

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

17

1

1989 AJHSME Problem 17

The number

\text{N}

is between

9

and

17

. The average of

6

,

10

, and

\text{N}

could be

\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16

16

1

1989 AJHSME Problem 15

In how many ways can

47

be written as the sum of two primes?

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

15

1

1989 AJHSME Problem 15

The area of the shaded region

\text{BEDC}

in parallelogram

\text{ABCD}

is
[asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S); label("

10

",(9,8),N); label("

6

",(7,0),S); label("

8

",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy]

\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80

14

1

1989 AJHSME Problem 14

When placing each of the digits

2,4,5,6,9

in exactly one of the boxes of this subtraction problem, what is the smallest difference that is possible?

\text{(A)}\ 58 \qquad \text{(B)}\ 123 \qquad \text{(C)}\ 149 \qquad \text{(D)}\ 171 \qquad \text{(E)}\ 176

\begin{tabular}[t]{cccc} & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{tabular}

13

1

1989 AJHSME Problem 13

\frac{9}{7\times 53} =

\text{(A)}\ \frac{.9}{.7\times 53} \qquad \text{(B)}\ \frac{.9}{.7\times .53} \qquad \text{(C)}\ \frac{.9}{.7\times 5.3} \qquad \text{(D)}\ \frac{.9}{7\times .53} \qquad \text{(E)}\ \frac{.09}{.07\times .53}

12

1

1989 AJHSME Problem 12

\frac{1-\frac{1}{3}}{1-\frac{1}{2}} =

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

11

1

1989 AJHSME Problem 11

10

1

1989 AJHSME Problem 10

What is the number of degrees in the smaller angle between the hour hand and the minute hand on a clock that reads seven o'clock?

\text{(A)}\ 50^\circ \qquad \text{(B)}\ 120^\circ \qquad \text{(C)}\ 135^\circ \qquad \text{(D)}\ 150^\circ \qquad \text{(E)}\ 165^\circ

9

1

1989 AJHSME Problem 9

There are

2

boys for every

3

girls in Ms. Johnson's math class. If there are

30

students in her class, what percent of them are boys?

\text{(A)}\ 12\% \qquad \text{(B)}\ 20\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 60\% \qquad \text{(E)}\ 66\frac{2}{3}\%

8

1

1989 AJHSME Problem 8

(2\times 3\times 4)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right) =

\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 26

7

1

1989 AJHSME Problem 7

If the value of

20

quarters and

10

dimes equals the value of

10

quarters and

n

dimes, then

n=

\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 45

6

1

1989 AJHSME Problem 6

If the markings on the number line are equally spaced, what is the number

\text{y}

?
[asy] draw((-4,0)--(26,0),Arrows); for(int a=0; a<6; ++a) { draw((4a,-1)--(4a,1)); } label("0",(0,-1),S); label("20",(20,-1),S); label("y",(12,-1),S); [/asy]

\text{(A)}\ 3 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 16

5

1

1989 AJHSME Problem 5

-15+9\times (6\div 3) =

\text{(A)}\ -48 \qquad \text{(B)}\ -12 \qquad \text{(C)}\ -3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 12

4

1

1989 AJHSME Problem 4

Estimate to determine which of the following numbers is closest to

\frac{401}{.205}

.

\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000

3

1

1989 AJHSME Problem 3

Which of the following numbers is the largest?

\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009

2

1

1989 AJHSME Problem 2

\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =

\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246

1

2010 AMC 8 Problem 1-Group of 50 sums

(1+11+21+31+41)+(9+19+29+39+49)=

\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 \qquad \text{(E)}\ 250

1989 AMC 8

Subcontests

1989 AJHSME Problem 25

1989 AJHSME Problem 24

1989 AJHSME Problem 23

1989 AJHSME Problem 22

1989 AJHSME Problem 21

1989 AJHSME Problem 20

1989 AJHSME Problem 19

1989 AJHSME Problem 18

1989 AJHSME Problem 17

1989 AJHSME Problem 15

1989 AJHSME Problem 15

1989 AJHSME Problem 14

1989 AJHSME Problem 13

1989 AJHSME Problem 12

1989 AJHSME Problem 11

1989 AJHSME Problem 10

1989 AJHSME Problem 9

1989 AJHSME Problem 8

1989 AJHSME Problem 7

1989 AJHSME Problem 6

1989 AJHSME Problem 5

1989 AJHSME Problem 4

1989 AJHSME Problem 3

1989 AJHSME Problem 2

2010 AMC 8 Problem 1-Group of 50 sums