https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Rep%27na&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T21:52:24ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=1999_AMC_8_Problems&diff=725551999 AMC 8 Problems2015-10-20T15:40:55Z<p>Rep'na: /* Problem 19 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>(6?3) + 4 - (2 - 1) = 5.</math> To make this statement true, the question mark between the 6 and the 3 should be replaced by<br />
<br />
<math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \qquad \text{(E)}\ \text{None of these}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?<br />
<br />
<asy><br />
draw(circle((0,0),2));<br />
dot((0,0));<br />
for(int i = 0; i < 12; ++i)<br />
{<br />
dot(2*dir(30*i));<br />
}<br />
<br />
label("$3$",2*dir(0),W);<br />
label("$2$",2*dir(30),WSW);<br />
label("$1$",2*dir(60),SSW);<br />
label("$12$",2*dir(90),S);<br />
label("$11$",2*dir(120),SSE);<br />
label("$10$",2*dir(150),ESE);<br />
label("$9$",2*dir(180),E);<br />
label("$8$",2*dir(210),ENE);<br />
label("$7$",2*dir(240),NNE);<br />
label("$6$",2*dir(270),N);<br />
label("$5$",2*dir(300),NNW);<br />
label("$4$",2*dir(330),WNW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90</math><br />
<br />
[[1999 AMC 8 Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which triplet of numbers has a sum NOT equal to 1?<br />
<br />
<math>\text{(A)}\ (1/2,1/3,1/6) \qquad \text{(B)}\ (2,-2,1) \qquad \text{(C)}\ (0.1,0.3,0.6) \qquad \text{(D)}\ (1.1,-2.1,1.0) \qquad \text{(E)}\ (-3/2,-5/2,5)</math><br />
<br />
[[1999 AMC 8 Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?<br />
<br />
<asy><br />
for (int a = 0; a < 6; ++a)<br />
{<br />
for (int b = 0; b < 6; ++b)<br />
{<br />
dot((4*a,3*b));<br />
}<br />
}<br />
draw((0,0)--(20,0)--(20,15)--(0,15)--cycle);<br />
draw((0,0)--(16,12));<br />
draw((0,0)--(16,9));<br />
<br />
label(rotate(30)*"Bjorn",(12,6.75),SE);<br />
label(rotate(37)*"Alberto",(11,8.25),NW);<br />
<br />
label("$0$",(0,0),S);<br />
label("$1$",(4,0),S);<br />
label("$2$",(8,0),S);<br />
label("$3$",(12,0),S);<br />
label("$4$",(16,0),S);<br />
label("$5$",(20,0),S);<br />
label("$0$",(0,0),W);<br />
label("$15$",(0,3),W);<br />
label("$30$",(0,6),W);<br />
label("$45$",(0,9),W);<br />
label("$60$",(0,12),W);<br />
label("$75$",(0,15),W);<br />
<br />
label("H",(6,-2),S);<br />
label("O",(8,-2),S);<br />
label("U",(10,-2),S);<br />
label("R",(12,-2),S);<br />
label("S",(14,-2),S);<br />
<br />
label("M",(-4,11),N);<br />
label("I",(-4,9),N);<br />
label("L",(-4,7),N);<br />
label("E",(-4,5),N);<br />
label("S",(-4,3),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 25 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35</math><br />
<br />
[[1999 AMC 8 Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?<br />
<br />
<math>\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500</math><br />
<br />
[[1999 AMC 8 Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Bo, Coe, Flo, Joe, and Moe have different amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Jo has more than Moe, but less than Bo. Who has the least amount of money?<br />
<br />
<math>\text{(A)}\ \text{Bo} \qquad \text{(B)}\ \text{Coe} \qquad \text{(C)}\ \text{Flo} \qquad \text{(D)}\ \text{Joe} \qquad \text{(E)}\ \text{Moe}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?<br />
<br />
<math>\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 110 \qquad \text{(D)}\ 120 \qquad \text{(E)}\ 130</math><br />
<br />
[[1999 AMC 8 Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is<br />
<br />
<asy><br />
draw((0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,2)--(2,2)--(2,3)--(0,3)--cycle);<br />
draw((1,3)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2));<br />
label("R",(.5,2.3),N);<br />
label("B",(1.5,2.3),N);<br />
label("G",(1.5,1.3),N);<br />
label("Y",(2.5,1.3),N);<br />
label("W",(2.5,.3),N);<br />
label("O",(3.5,1.3),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ \text{B} \qquad \text{(B)}\ \text{G} \qquad \text{(C)}\ \text{O} \qquad \text{(D)}\ \text{R} \qquad \text{(E)}\ \text{Y}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);<br />
draw(circle((.3,-.1),.7));<br />
draw(circle((2.8,-.2),.8));<br />
label("A",(1.3,.5),N);<br />
label("B",(3.1,-.2),S);<br />
label("C",(.6,-.2),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 850 \qquad \text{(B)}\ 1000 \qquad \text{(C)}\ 1150 \qquad \text{(D)}\ 1300 \qquad \text{(E)}\ 1450</math><br />
<br />
[[1999 AMC 8 Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green?<br />
<br />
<math>\text{(A)}\ \frac{1}{4} \qquad \text{(B)}\ \frac{1}{3} \qquad \text{(C)}\ \frac{5}{12} \qquad \text{(D)}\ \frac{1}{2} \qquad \text{(E)}\ \frac{7}{12}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);<br />
draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 30</math><br />
<br />
[[1999 AMC 8 Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is <math>11/4</math>. To the nearest whole percent, what percent of its games did the team lose?<br />
<br />
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 45 \qquad \text{(E)}\ 73</math><br />
<br />
[[1999 AMC 8 Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?<br />
<br />
<math>\text{(A)}\ 26 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 29 \qquad \text{(E)}\ 30</math><br />
<br />
[[1999 AMC 8 Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
In trapezoid <math>ABCD</math>, the sides <math>AB</math> and <math>CD</math> are equal. The perimeter of <math>ABCD</math> is<br />
<br />
<asy><br />
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);<br />
draw((4,3)--(4,0),dashed);<br />
draw((3.2,0)--(3.2,.8)--(4,.8));<br />
<br />
label("$A$",(0,0),SW);<br />
label("$B$",(4,3),NW);<br />
label("$C$",(12,3),NE);<br />
label("$D$",(16,0),SE);<br />
label("$8$",(8,3),N);<br />
label("$16$",(8,0),S);<br />
label("$3$",(4,1.5),E);<br />
</asy><br />
<br />
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 34 \qquad \text{(E)}\ 48</math><br />
<br />
[[1999 AMC 8 Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <br />
<br />
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters?<br />
<br />
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 60</math><br />
<br />
[[1999 AMC 8 Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade?<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11</math><br />
<br />
[[1999 AMC 8 Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 15</math><br />
<br />
[[1999 AMC 8 Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cookies should they make for their smaller party?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11</math><br />
<br />
[[1999 AMC 8 Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.)<br />
<br />
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math><br />
<br />
[[1999 AMC 8 Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.<br />
<br />
Which of the following is the front view for the stack map in Fig. 4?<br />
<br />
<asy><br />
unitsize(24);<br />
<br />
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);<br />
draw((1,0)--(1,2));<br />
draw((0,1)--(2,1));<br />
<br />
draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle);<br />
draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3));<br />
draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3));<br />
draw((17/3,7/3)--(14/3,7/3));<br />
draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0));<br />
draw((5,1)--(6,1)--(6,0));<br />
draw((20/3,4/3)--(6,4/3));<br />
draw((17/3,13/3)--(16/3,14/3));<br />
draw((17/3,10/3)--(16/3,11/3));<br />
draw((14/3,10/3)--(13/3,11/3));<br />
draw((5,2)--(13/3,8/3));<br />
draw((5,1)--(13/3,5/3));<br />
draw((6,2)--(17/3,7/3));<br />
<br />
draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle);<br />
draw((11,3)--(10,3)--(10,0));<br />
draw((11,2)--(9,2));<br />
draw((11,1)--(9,1));<br />
<br />
draw((13,0)--(16,0)--(16,2)--(13,2)--cycle);<br />
draw((13,1)--(16,1));<br />
draw((14,0)--(14,2));<br />
draw((15,0)--(15,2));<br />
<br />
label("Figure 1",(1,0),S);<br />
label("Figure 2",(17/3,0),S);<br />
label("Figure 3",(10,0),S);<br />
label("Figure 4",(14.5,0),S);<br />
<br />
label("$1$",(1.5,.2),N);<br />
label("$2$",(.5,.2),N);<br />
label("$3$",(.5,1.2),N);<br />
label("$4$",(1.5,1.2),N);<br />
<br />
label("$1$",(13.5,.2),N);<br />
label("$3$",(14.5,.2),N);<br />
label("$1$",(15.5,.2),N);<br />
label("$2$",(13.5,1.2),N);<br />
label("$2$",(14.5,1.2),N);<br />
label("$4$",(15.5,1.2),N);<br />
</asy><br />
<br />
<br /> <br /><br />
<br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle);<br />
draw((0,3)--(1,3));<br />
draw((0,2)--(1,2)--(1,0));<br />
draw((0,1)--(3,1));<br />
draw((2,0)--(2,2));<br />
<br />
draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle);<br />
draw((8,3)--(7,3)--(7,0));<br />
draw((8,2)--(6,2)--(6,0));<br />
draw((8,1)--(5,1));<br />
<br />
draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle);<br />
draw((12,3)--(11,3)--(11,0));<br />
draw((12,2)--(10,2));<br />
draw((12,1)--(10,1));<br />
<br />
draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle);<br />
draw((17,3)--(16,3));<br />
draw((17,2)--(16,2)--(16,0));<br />
draw((17,1)--(14,1));<br />
draw((15,0)--(15,2));<br />
<br />
draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle);<br />
draw((22,3)--(20,3));<br />
draw((22,2)--(20,2));<br />
draw((22,1)--(20,1)--(20,0));<br />
draw((21,0)--(21,4));<br />
<br />
label("(A)",(1.5,0),S);<br />
label("(B)",(6.5,0),S);<br />
label("(C)",(11,0),S);<br />
label("(D)",(15.5,0),S);<br />
label("(E)",(20.5,0),S);<br />
</asy><br />
<br />
[[1999 AMC 8 Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The degree measure of angle <math>A</math> is<br />
<br />
<asy><br />
unitsize(12);<br />
draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle);<br />
draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW));<br />
draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW));<br />
draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW));<br />
label(rotate(30)*"$40^\circ$",(2,-8.9),ENE);<br />
label("$100^\circ$",(21/3,-2/3),SE);<br />
label("$110^\circ$",(900/83,-317/83),NNW);<br />
label("$A$",(0,0),NW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 45</math><br />
<br />
[[1999 AMC 8 Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?<br />
<br />
<math>\text{(A)}\ \frac{3}{8} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ 2\frac{2}{3} \qquad \text{(E)}\ 3\frac{1}{3}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Square <math>ABCD</math> has sides of length 3. Segments <math>CM</math> and <math>CN</math> divide the square's area into three equal parts. How long is segment <math>CM</math>?<br />
<br />
<asy><br />
pair A,B,C,D,M,N;<br />
A = (0,0);<br />
B = (0,3);<br />
C = (3,3);<br />
D = (3,0);<br />
M = (0,1);<br />
N = (1,0);<br />
draw(A--B--C--D--cycle);<br />
draw(M--C--N);<br />
label("$A$",A,SW);<br />
label("$M$",M,W);<br />
label("$B$",B,NW);<br />
label("$C$",C,NE);<br />
label("$D$",D,SE);<br />
label("$N$",N,S);<br />
</asy><br />
<br />
<math>\text{(A)}\ \sqrt{10} \qquad \text{(B)}\ \sqrt{12} \qquad \text{(C)}\ \sqrt{13} \qquad \text{(D)}\ \sqrt{14} \qquad \text{(E)}\ \sqrt{15}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
When <math>1999^{2000}</math> is divided by <math>5</math>, the remainder is <br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 0</math><br />
<br />
[[1999 AMC 8 Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
Points <math>B</math>, <math>D</math>, and <math>J</math> are midpoints of the sides of right triangle <math>ACG</math>. Points <math>K</math>, <math>E</math>, <math>I</math> are midpoints of the sides of triangle <math>JDG</math>, etc. If the dividing and shading process is done 100 times (the first three are shown) and <math>AC=CG=6</math>, then the total area of the shaded triangles is nearest<br />
<br />
<asy><br />
draw((0,0)--(6,0)--(6,6)--cycle);<br />
draw((3,0)--(3,3)--(6,3));<br />
draw((4.5,3)--(4.5,4.5)--(6,4.5));<br />
draw((5.25,4.5)--(5.25,5.25)--(6,5.25));<br />
fill((3,0)--(6,0)--(6,3)--cycle,black);<br />
fill((4.5,3)--(6,3)--(6,4.5)--cycle,black);<br />
fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black);<br />
<br />
label("$A$",(0,0),SW);<br />
label("$B$",(3,0),S);<br />
label("$C$",(6,0),SE);<br />
label("$D$",(6,3),E);<br />
label("$E$",(6,4.5),E);<br />
label("$F$",(6,5.25),E);<br />
label("$G$",(6,6),NE);<br />
label("$H$",(5.25,5.25),NW);<br />
label("$I$",(4.5,4.5),NW);<br />
label("$J$",(3,3),NW);<br />
label("$K$",(4.5,3),S);<br />
label("$L$",(5.25,4.5),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10</math><br />
<br />
[[1999 AMC 8 Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AMC8 box|year=1999|before=[[1998 AJHSME Problems|1998 AJHSME]]|after=[[2000 AMC 8 Problems|2000 AMC 8]]}}<br />
* [[AMC 8]]<br />
* [[AMC 8 Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1999_AMC_8_Problems&diff=725541999 AMC 8 Problems2015-10-20T15:36:38Z<p>Rep'na: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>(6?3) + 4 - (2 - 1) = 5.</math> To make this statement true, the question mark between the 6 and the 3 should be replaced by<br />
<br />
<math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \qquad \text{(E)}\ \text{None of these}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?<br />
<br />
<asy><br />
draw(circle((0,0),2));<br />
dot((0,0));<br />
for(int i = 0; i < 12; ++i)<br />
{<br />
dot(2*dir(30*i));<br />
}<br />
<br />
label("$3$",2*dir(0),W);<br />
label("$2$",2*dir(30),WSW);<br />
label("$1$",2*dir(60),SSW);<br />
label("$12$",2*dir(90),S);<br />
label("$11$",2*dir(120),SSE);<br />
label("$10$",2*dir(150),ESE);<br />
label("$9$",2*dir(180),E);<br />
label("$8$",2*dir(210),ENE);<br />
label("$7$",2*dir(240),NNE);<br />
label("$6$",2*dir(270),N);<br />
label("$5$",2*dir(300),NNW);<br />
label("$4$",2*dir(330),WNW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90</math><br />
<br />
[[1999 AMC 8 Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which triplet of numbers has a sum NOT equal to 1?<br />
<br />
<math>\text{(A)}\ (1/2,1/3,1/6) \qquad \text{(B)}\ (2,-2,1) \qquad \text{(C)}\ (0.1,0.3,0.6) \qquad \text{(D)}\ (1.1,-2.1,1.0) \qquad \text{(E)}\ (-3/2,-5/2,5)</math><br />
<br />
[[1999 AMC 8 Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?<br />
<br />
<asy><br />
for (int a = 0; a < 6; ++a)<br />
{<br />
for (int b = 0; b < 6; ++b)<br />
{<br />
dot((4*a,3*b));<br />
}<br />
}<br />
draw((0,0)--(20,0)--(20,15)--(0,15)--cycle);<br />
draw((0,0)--(16,12));<br />
draw((0,0)--(16,9));<br />
<br />
label(rotate(30)*"Bjorn",(12,6.75),SE);<br />
label(rotate(37)*"Alberto",(11,8.25),NW);<br />
<br />
label("$0$",(0,0),S);<br />
label("$1$",(4,0),S);<br />
label("$2$",(8,0),S);<br />
label("$3$",(12,0),S);<br />
label("$4$",(16,0),S);<br />
label("$5$",(20,0),S);<br />
label("$0$",(0,0),W);<br />
label("$15$",(0,3),W);<br />
label("$30$",(0,6),W);<br />
label("$45$",(0,9),W);<br />
label("$60$",(0,12),W);<br />
label("$75$",(0,15),W);<br />
<br />
label("H",(6,-2),S);<br />
label("O",(8,-2),S);<br />
label("U",(10,-2),S);<br />
label("R",(12,-2),S);<br />
label("S",(14,-2),S);<br />
<br />
label("M",(-4,11),N);<br />
label("I",(-4,9),N);<br />
label("L",(-4,7),N);<br />
label("E",(-4,5),N);<br />
label("S",(-4,3),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 25 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35</math><br />
<br />
[[1999 AMC 8 Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?<br />
<br />
<math>\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500</math><br />
<br />
[[1999 AMC 8 Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Bo, Coe, Flo, Joe, and Moe have different amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Jo has more than Moe, but less than Bo. Who has the least amount of money?<br />
<br />
<math>\text{(A)}\ \text{Bo} \qquad \text{(B)}\ \text{Coe} \qquad \text{(C)}\ \text{Flo} \qquad \text{(D)}\ \text{Joe} \qquad \text{(E)}\ \text{Moe}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?<br />
<br />
<math>\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 110 \qquad \text{(D)}\ 120 \qquad \text{(E)}\ 130</math><br />
<br />
[[1999 AMC 8 Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is<br />
<br />
<asy><br />
draw((0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,2)--(2,2)--(2,3)--(0,3)--cycle);<br />
draw((1,3)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2));<br />
label("R",(.5,2.3),N);<br />
label("B",(1.5,2.3),N);<br />
label("G",(1.5,1.3),N);<br />
label("Y",(2.5,1.3),N);<br />
label("W",(2.5,.3),N);<br />
label("O",(3.5,1.3),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ \text{B} \qquad \text{(B)}\ \text{G} \qquad \text{(C)}\ \text{O} \qquad \text{(D)}\ \text{R} \qquad \text{(E)}\ \text{Y}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);<br />
draw(circle((.3,-.1),.7));<br />
draw(circle((2.8,-.2),.8));<br />
label("A",(1.3,.5),N);<br />
label("B",(3.1,-.2),S);<br />
label("C",(.6,-.2),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 850 \qquad \text{(B)}\ 1000 \qquad \text{(C)}\ 1150 \qquad \text{(D)}\ 1300 \qquad \text{(E)}\ 1450</math><br />
<br />
[[1999 AMC 8 Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green?<br />
<br />
<math>\text{(A)}\ \frac{1}{4} \qquad \text{(B)}\ \frac{1}{3} \qquad \text{(C)}\ \frac{5}{12} \qquad \text{(D)}\ \frac{1}{2} \qquad \text{(E)}\ \frac{7}{12}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);<br />
draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 30</math><br />
<br />
[[1999 AMC 8 Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is <math>11/4</math>. To the nearest whole percent, what percent of its games did the team lose?<br />
<br />
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 45 \qquad \text{(E)}\ 73</math><br />
<br />
[[1999 AMC 8 Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?<br />
<br />
<math>\text{(A)}\ 26 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 29 \qquad \text{(E)}\ 30</math><br />
<br />
[[1999 AMC 8 Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
In trapezoid <math>ABCD</math>, the sides <math>AB</math> and <math>CD</math> are equal. The perimeter of <math>ABCD</math> is<br />
<br />
<asy><br />
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);<br />
draw((4,3)--(4,0),dashed);<br />
draw((3.2,0)--(3.2,.8)--(4,.8));<br />
<br />
label("$A$",(0,0),SW);<br />
label("$B$",(4,3),NW);<br />
label("$C$",(12,3),NE);<br />
label("$D$",(16,0),SE);<br />
label("$8$",(8,3),N);<br />
label("$16$",(8,0),S);<br />
label("$3$",(4,1.5),E);<br />
</asy><br />
<br />
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 34 \qquad \text{(E)}\ 48</math><br />
<br />
[[1999 AMC 8 Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <br />
<br />
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters?<br />
<br />
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 60</math><br />
<br />
[[1999 AMC 8 Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade?<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11</math><br />
<br />
[[1999 AMC 8 Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 15</math><br />
<br />
[[1999 AMC 8 Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cookies should they make for their smaller party?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11</math><br />
<br />
[[1999 AMC 8 Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC<math>\to</math>8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.)<br />
<br />
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math><br />
<br />
[[1999 AMC 8 Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.<br />
<br />
Which of the following is the front view for the stack map in Fig. 4?<br />
<br />
<asy><br />
unitsize(24);<br />
<br />
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);<br />
draw((1,0)--(1,2));<br />
draw((0,1)--(2,1));<br />
<br />
draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle);<br />
draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3));<br />
draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3));<br />
draw((17/3,7/3)--(14/3,7/3));<br />
draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0));<br />
draw((5,1)--(6,1)--(6,0));<br />
draw((20/3,4/3)--(6,4/3));<br />
draw((17/3,13/3)--(16/3,14/3));<br />
draw((17/3,10/3)--(16/3,11/3));<br />
draw((14/3,10/3)--(13/3,11/3));<br />
draw((5,2)--(13/3,8/3));<br />
draw((5,1)--(13/3,5/3));<br />
draw((6,2)--(17/3,7/3));<br />
<br />
draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle);<br />
draw((11,3)--(10,3)--(10,0));<br />
draw((11,2)--(9,2));<br />
draw((11,1)--(9,1));<br />
<br />
draw((13,0)--(16,0)--(16,2)--(13,2)--cycle);<br />
draw((13,1)--(16,1));<br />
draw((14,0)--(14,2));<br />
draw((15,0)--(15,2));<br />
<br />
label("Figure 1",(1,0),S);<br />
label("Figure 2",(17/3,0),S);<br />
label("Figure 3",(10,0),S);<br />
label("Figure 4",(14.5,0),S);<br />
<br />
label("$1$",(1.5,.2),N);<br />
label("$2$",(.5,.2),N);<br />
label("$3$",(.5,1.2),N);<br />
label("$4$",(1.5,1.2),N);<br />
<br />
label("$1$",(13.5,.2),N);<br />
label("$3$",(14.5,.2),N);<br />
label("$1$",(15.5,.2),N);<br />
label("$2$",(13.5,1.2),N);<br />
label("$2$",(14.5,1.2),N);<br />
label("$4$",(15.5,1.2),N);<br />
</asy><br />
<br />
<br /> <br /><br />
<br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle);<br />
draw((0,3)--(1,3));<br />
draw((0,2)--(1,2)--(1,0));<br />
draw((0,1)--(3,1));<br />
draw((2,0)--(2,2));<br />
<br />
draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle);<br />
draw((8,3)--(7,3)--(7,0));<br />
draw((8,2)--(6,2)--(6,0));<br />
draw((8,1)--(5,1));<br />
<br />
draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle);<br />
draw((12,3)--(11,3)--(11,0));<br />
draw((12,2)--(10,2));<br />
draw((12,1)--(10,1));<br />
<br />
draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle);<br />
draw((17,3)--(16,3));<br />
draw((17,2)--(16,2)--(16,0));<br />
draw((17,1)--(14,1));<br />
draw((15,0)--(15,2));<br />
<br />
draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle);<br />
draw((22,3)--(20,3));<br />
draw((22,2)--(20,2));<br />
draw((22,1)--(20,1)--(20,0));<br />
draw((21,0)--(21,4));<br />
<br />
label("(A)",(1.5,0),S);<br />
label("(B)",(6.5,0),S);<br />
label("(C)",(11,0),S);<br />
label("(D)",(15.5,0),S);<br />
label("(E)",(20.5,0),S);<br />
</asy><br />
<br />
[[1999 AMC 8 Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The degree measure of angle <math>A</math> is<br />
<br />
<asy><br />
unitsize(12);<br />
draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle);<br />
draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW));<br />
draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW));<br />
draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW));<br />
label(rotate(30)*"$40^\circ$",(2,-8.9),ENE);<br />
label("$100^\circ$",(21/3,-2/3),SE);<br />
label("$110^\circ$",(900/83,-317/83),NNW);<br />
label("$A$",(0,0),NW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 45</math><br />
<br />
[[1999 AMC 8 Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?<br />
<br />
<math>\text{(A)}\ \frac{3}{8} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ 2\frac{2}{3} \qquad \text{(E)}\ 3\frac{1}{3}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Square <math>ABCD</math> has sides of length 3. Segments <math>CM</math> and <math>CN</math> divide the square's area into three equal parts. How long is segment <math>CM</math>?<br />
<br />
<asy><br />
pair A,B,C,D,M,N;<br />
A = (0,0);<br />
B = (0,3);<br />
C = (3,3);<br />
D = (3,0);<br />
M = (0,1);<br />
N = (1,0);<br />
draw(A--B--C--D--cycle);<br />
draw(M--C--N);<br />
label("$A$",A,SW);<br />
label("$M$",M,W);<br />
label("$B$",B,NW);<br />
label("$C$",C,NE);<br />
label("$D$",D,SE);<br />
label("$N$",N,S);<br />
</asy><br />
<br />
<math>\text{(A)}\ \sqrt{10} \qquad \text{(B)}\ \sqrt{12} \qquad \text{(C)}\ \sqrt{13} \qquad \text{(D)}\ \sqrt{14} \qquad \text{(E)}\ \sqrt{15}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
When <math>1999^{2000}</math> is divided by <math>5</math>, the remainder is <br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 0</math><br />
<br />
[[1999 AMC 8 Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
Points <math>B</math>, <math>D</math>, and <math>J</math> are midpoints of the sides of right triangle <math>ACG</math>. Points <math>K</math>, <math>E</math>, <math>I</math> are midpoints of the sides of triangle <math>JDG</math>, etc. If the dividing and shading process is done 100 times (the first three are shown) and <math>AC=CG=6</math>, then the total area of the shaded triangles is nearest<br />
<br />
<asy><br />
draw((0,0)--(6,0)--(6,6)--cycle);<br />
draw((3,0)--(3,3)--(6,3));<br />
draw((4.5,3)--(4.5,4.5)--(6,4.5));<br />
draw((5.25,4.5)--(5.25,5.25)--(6,5.25));<br />
fill((3,0)--(6,0)--(6,3)--cycle,black);<br />
fill((4.5,3)--(6,3)--(6,4.5)--cycle,black);<br />
fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black);<br />
<br />
label("$A$",(0,0),SW);<br />
label("$B$",(3,0),S);<br />
label("$C$",(6,0),SE);<br />
label("$D$",(6,3),E);<br />
label("$E$",(6,4.5),E);<br />
label("$F$",(6,5.25),E);<br />
label("$G$",(6,6),NE);<br />
label("$H$",(5.25,5.25),NW);<br />
label("$I$",(4.5,4.5),NW);<br />
label("$J$",(3,3),NW);<br />
label("$K$",(4.5,3),S);<br />
label("$L$",(5.25,4.5),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10</math><br />
<br />
[[1999 AMC 8 Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AMC8 box|year=1999|before=[[1998 AJHSME Problems|1998 AJHSME]]|after=[[2000 AMC 8 Problems|2000 AMC 8]]}}<br />
* [[AMC 8]]<br />
* [[AMC 8 Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1999_AMC_8_Problems&diff=725531999 AMC 8 Problems2015-10-20T15:33:36Z<p>Rep'na: /* Problem 17 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>(6?3) + 4 - (2 - 1) = 5.</math> To make this statement true, the question mark between the 6 and the 3 should be replaced by<br />
<br />
<math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \qquad \text{(E)}\ \text{None of these}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?<br />
<br />
<asy><br />
draw(circle((0,0),2));<br />
dot((0,0));<br />
for(int i = 0; i < 12; ++i)<br />
{<br />
dot(2*dir(30*i));<br />
}<br />
<br />
label("$3$",2*dir(0),W);<br />
label("$2$",2*dir(30),WSW);<br />
label("$1$",2*dir(60),SSW);<br />
label("$12$",2*dir(90),S);<br />
label("$11$",2*dir(120),SSE);<br />
label("$10$",2*dir(150),ESE);<br />
label("$9$",2*dir(180),E);<br />
label("$8$",2*dir(210),ENE);<br />
label("$7$",2*dir(240),NNE);<br />
label("$6$",2*dir(270),N);<br />
label("$5$",2*dir(300),NNW);<br />
label("$4$",2*dir(330),WNW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90</math><br />
<br />
[[1999 AMC 8 Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which triplet of numbers has a sum NOT equal to 1?<br />
<br />
<math>\text{(A)}\ (1/2,1/3,1/6) \qquad \text{(B)}\ (2,-2,1) \qquad \text{(C)}\ (0.1,0.3,0.6) \qquad \text{(D)}\ (1.1,-2.1,1.0) \qquad \text{(E)}\ (-3/2,-5/2,5)</math><br />
<br />
[[1999 AMC 8 Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?<br />
<br />
<asy><br />
for (int a = 0; a < 6; ++a)<br />
{<br />
for (int b = 0; b < 6; ++b)<br />
{<br />
dot((4*a,3*b));<br />
}<br />
}<br />
draw((0,0)--(20,0)--(20,15)--(0,15)--cycle);<br />
draw((0,0)--(16,12));<br />
draw((0,0)--(16,9));<br />
<br />
label(rotate(30)*"Bjorn",(12,6.75),SE);<br />
label(rotate(37)*"Alberto",(11,8.25),NW);<br />
<br />
label("$0$",(0,0),S);<br />
label("$1$",(4,0),S);<br />
label("$2$",(8,0),S);<br />
label("$3$",(12,0),S);<br />
label("$4$",(16,0),S);<br />
label("$5$",(20,0),S);<br />
label("$0$",(0,0),W);<br />
label("$15$",(0,3),W);<br />
label("$30$",(0,6),W);<br />
label("$45$",(0,9),W);<br />
label("$60$",(0,12),W);<br />
label("$75$",(0,15),W);<br />
<br />
label("H",(6,-2),S);<br />
label("O",(8,-2),S);<br />
label("U",(10,-2),S);<br />
label("R",(12,-2),S);<br />
label("S",(14,-2),S);<br />
<br />
label("M",(-4,11),N);<br />
label("I",(-4,9),N);<br />
label("L",(-4,7),N);<br />
label("E",(-4,5),N);<br />
label("S",(-4,3),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 25 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35</math><br />
<br />
[[1999 AMC 8 Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?<br />
<br />
<math>\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500</math><br />
<br />
[[1999 AMC 8 Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Bo, Coe, Flo, Joe, and Moe have different amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Jo has more than Moe, but less than Bo. Who has the least amount of money?<br />
<br />
<math>\text{(A)}\ \text{Bo} \qquad \text{(B)}\ \text{Coe} \qquad \text{(C)}\ \text{Flo} \qquad \text{(D)}\ \text{Joe} \qquad \text{(E)}\ \text{Moe}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?<br />
<br />
<math>\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 110 \qquad \text{(D)}\ 120 \qquad \text{(E)}\ 130</math><br />
<br />
[[1999 AMC 8 Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is<br />
<br />
<asy><br />
draw((0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,2)--(2,2)--(2,3)--(0,3)--cycle);<br />
draw((1,3)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2));<br />
label("R",(.5,2.3),N);<br />
label("B",(1.5,2.3),N);<br />
label("G",(1.5,1.3),N);<br />
label("Y",(2.5,1.3),N);<br />
label("W",(2.5,.3),N);<br />
label("O",(3.5,1.3),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ \text{B} \qquad \text{(B)}\ \text{G} \qquad \text{(C)}\ \text{O} \qquad \text{(D)}\ \text{R} \qquad \text{(E)}\ \text{Y}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);<br />
draw(circle((.3,-.1),.7));<br />
draw(circle((2.8,-.2),.8));<br />
label("A",(1.3,.5),N);<br />
label("B",(3.1,-.2),S);<br />
label("C",(.6,-.2),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 850 \qquad \text{(B)}\ 1000 \qquad \text{(C)}\ 1150 \qquad \text{(D)}\ 1300 \qquad \text{(E)}\ 1450</math><br />
<br />
[[1999 AMC 8 Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green?<br />
<br />
<math>\text{(A)}\ \frac{1}{4} \qquad \text{(B)}\ \frac{1}{3} \qquad \text{(C)}\ \frac{5}{12} \qquad \text{(D)}\ \frac{1}{2} \qquad \text{(E)}\ \frac{7}{12}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);<br />
draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 30</math><br />
<br />
[[1999 AMC 8 Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is <math>11/4</math>. To the nearest whole percent, what percent of its games did the team lose?<br />
<br />
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 45 \qquad \text{(E)}\ 73</math><br />
<br />
[[1999 AMC 8 Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?<br />
<br />
<math>\text{(A)}\ 26 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 29 \qquad \text{(E)}\ 30</math><br />
<br />
[[1999 AMC 8 Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
In trapezoid <math>ABCD</math>, the sides <math>AB</math> and <math>CD</math> are equal. The perimeter of <math>ABCD</math> is<br />
<br />
<asy><br />
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);<br />
draw((4,3)--(4,0),dashed);<br />
draw((3.2,0)--(3.2,.8)--(4,.8));<br />
<br />
label("$A$",(0,0),SW);<br />
label("$B$",(4,3),NW);<br />
label("$C$",(12,3),NE);<br />
label("$D$",(16,0),SE);<br />
label("$8$",(8,3),N);<br />
label("$16$",(8,0),S);<br />
label("$3$",(4,1.5),E);<br />
</asy><br />
<br />
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 34 \qquad \text{(E)}\ 48</math><br />
<br />
[[1999 AMC 8 Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <br />
<br />
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters?<br />
<br />
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 60</math><br />
<br />
[[1999 AMC 8 Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade?<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11</math><br />
<br />
[[1999 AMC 8 Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 15</math><br />
<br />
[[1999 AMC 8 Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC<math>\to</math>8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cookies should they make for their smaller party?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11</math><br />
<br />
[[1999 AMC 8 Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Problems 17, 18, and 19 refer to the following:<br />
<br />
At Central Middle School the 108 students who take the AMC<math>\to</math>8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: <math>1\frac{1}{2}</math> cups flour, <math>2</math> eggs, <math>3</math> tablespoons butter, <math>\frac{3}{4}</math> cups sugar, and <math>1</math> package of chocolate drops. They will make only full recipes, not partial recipes.<br />
<br />
The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.)<br />
<br />
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math><br />
<br />
[[1999 AMC 8 Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.<br />
<br />
Which of the following is the front view for the stack map in Fig. 4?<br />
<br />
<asy><br />
unitsize(24);<br />
<br />
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);<br />
draw((1,0)--(1,2));<br />
draw((0,1)--(2,1));<br />
<br />
draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle);<br />
draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3));<br />
draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3));<br />
draw((17/3,7/3)--(14/3,7/3));<br />
draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0));<br />
draw((5,1)--(6,1)--(6,0));<br />
draw((20/3,4/3)--(6,4/3));<br />
draw((17/3,13/3)--(16/3,14/3));<br />
draw((17/3,10/3)--(16/3,11/3));<br />
draw((14/3,10/3)--(13/3,11/3));<br />
draw((5,2)--(13/3,8/3));<br />
draw((5,1)--(13/3,5/3));<br />
draw((6,2)--(17/3,7/3));<br />
<br />
draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle);<br />
draw((11,3)--(10,3)--(10,0));<br />
draw((11,2)--(9,2));<br />
draw((11,1)--(9,1));<br />
<br />
draw((13,0)--(16,0)--(16,2)--(13,2)--cycle);<br />
draw((13,1)--(16,1));<br />
draw((14,0)--(14,2));<br />
draw((15,0)--(15,2));<br />
<br />
label("Figure 1",(1,0),S);<br />
label("Figure 2",(17/3,0),S);<br />
label("Figure 3",(10,0),S);<br />
label("Figure 4",(14.5,0),S);<br />
<br />
label("$1$",(1.5,.2),N);<br />
label("$2$",(.5,.2),N);<br />
label("$3$",(.5,1.2),N);<br />
label("$4$",(1.5,1.2),N);<br />
<br />
label("$1$",(13.5,.2),N);<br />
label("$3$",(14.5,.2),N);<br />
label("$1$",(15.5,.2),N);<br />
label("$2$",(13.5,1.2),N);<br />
label("$2$",(14.5,1.2),N);<br />
label("$4$",(15.5,1.2),N);<br />
</asy><br />
<br />
<br /> <br /><br />
<br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle);<br />
draw((0,3)--(1,3));<br />
draw((0,2)--(1,2)--(1,0));<br />
draw((0,1)--(3,1));<br />
draw((2,0)--(2,2));<br />
<br />
draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle);<br />
draw((8,3)--(7,3)--(7,0));<br />
draw((8,2)--(6,2)--(6,0));<br />
draw((8,1)--(5,1));<br />
<br />
draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle);<br />
draw((12,3)--(11,3)--(11,0));<br />
draw((12,2)--(10,2));<br />
draw((12,1)--(10,1));<br />
<br />
draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle);<br />
draw((17,3)--(16,3));<br />
draw((17,2)--(16,2)--(16,0));<br />
draw((17,1)--(14,1));<br />
draw((15,0)--(15,2));<br />
<br />
draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle);<br />
draw((22,3)--(20,3));<br />
draw((22,2)--(20,2));<br />
draw((22,1)--(20,1)--(20,0));<br />
draw((21,0)--(21,4));<br />
<br />
label("(A)",(1.5,0),S);<br />
label("(B)",(6.5,0),S);<br />
label("(C)",(11,0),S);<br />
label("(D)",(15.5,0),S);<br />
label("(E)",(20.5,0),S);<br />
</asy><br />
<br />
[[1999 AMC 8 Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The degree measure of angle <math>A</math> is<br />
<br />
<asy><br />
unitsize(12);<br />
draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle);<br />
draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW));<br />
draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW));<br />
draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW));<br />
label(rotate(30)*"$40^\circ$",(2,-8.9),ENE);<br />
label("$100^\circ$",(21/3,-2/3),SE);<br />
label("$110^\circ$",(900/83,-317/83),NNW);<br />
label("$A$",(0,0),NW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 45</math><br />
<br />
[[1999 AMC 8 Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?<br />
<br />
<math>\text{(A)}\ \frac{3}{8} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ 2\frac{2}{3} \qquad \text{(E)}\ 3\frac{1}{3}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
Square <math>ABCD</math> has sides of length 3. Segments <math>CM</math> and <math>CN</math> divide the square's area into three equal parts. How long is segment <math>CM</math>?<br />
<br />
<asy><br />
pair A,B,C,D,M,N;<br />
A = (0,0);<br />
B = (0,3);<br />
C = (3,3);<br />
D = (3,0);<br />
M = (0,1);<br />
N = (1,0);<br />
draw(A--B--C--D--cycle);<br />
draw(M--C--N);<br />
label("$A$",A,SW);<br />
label("$M$",M,W);<br />
label("$B$",B,NW);<br />
label("$C$",C,NE);<br />
label("$D$",D,SE);<br />
label("$N$",N,S);<br />
</asy><br />
<br />
<math>\text{(A)}\ \sqrt{10} \qquad \text{(B)}\ \sqrt{12} \qquad \text{(C)}\ \sqrt{13} \qquad \text{(D)}\ \sqrt{14} \qquad \text{(E)}\ \sqrt{15}</math><br />
<br />
[[1999 AMC 8 Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
When <math>1999^{2000}</math> is divided by <math>5</math>, the remainder is <br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 0</math><br />
<br />
[[1999 AMC 8 Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
Points <math>B</math>, <math>D</math>, and <math>J</math> are midpoints of the sides of right triangle <math>ACG</math>. Points <math>K</math>, <math>E</math>, <math>I</math> are midpoints of the sides of triangle <math>JDG</math>, etc. If the dividing and shading process is done 100 times (the first three are shown) and <math>AC=CG=6</math>, then the total area of the shaded triangles is nearest<br />
<br />
<asy><br />
draw((0,0)--(6,0)--(6,6)--cycle);<br />
draw((3,0)--(3,3)--(6,3));<br />
draw((4.5,3)--(4.5,4.5)--(6,4.5));<br />
draw((5.25,4.5)--(5.25,5.25)--(6,5.25));<br />
fill((3,0)--(6,0)--(6,3)--cycle,black);<br />
fill((4.5,3)--(6,3)--(6,4.5)--cycle,black);<br />
fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black);<br />
<br />
label("$A$",(0,0),SW);<br />
label("$B$",(3,0),S);<br />
label("$C$",(6,0),SE);<br />
label("$D$",(6,3),E);<br />
label("$E$",(6,4.5),E);<br />
label("$F$",(6,5.25),E);<br />
label("$G$",(6,6),NE);<br />
label("$H$",(5.25,5.25),NW);<br />
label("$I$",(4.5,4.5),NW);<br />
label("$J$",(3,3),NW);<br />
label("$K$",(4.5,3),S);<br />
label("$L$",(5.25,4.5),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10</math><br />
<br />
[[1999 AMC 8 Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AMC8 box|year=1999|before=[[1998 AJHSME Problems|1998 AJHSME]]|after=[[2000 AMC 8 Problems|2000 AMC 8]]}}<br />
* [[AMC 8]]<br />
* [[AMC 8 Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1998_AJHSME_Problems&diff=725451998 AJHSME Problems2015-10-18T17:45:12Z<p>Rep'na: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
<br />
For <math>x=7</math>, which of the following is the smallest?<br />
<br />
<math>\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac{6}{x-1} \qquad \text{(D)}\ \dfrac{x}{6} \qquad \text{(E)}\ \dfrac{x+1}{6}</math><br />
<br />
[[1998 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
If <math>\begin{tabular}{r|l}a&b \\ \hline c&d\end{tabular} = \text{a}\cdot \text{d} - \text{b}\cdot \text{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?<br />
<br />
<math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2</math><br />
<br />
[[1998 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
<math>\dfrac{\dfrac{3}{8} + \dfrac{7}{8}}{\dfrac{4}{5}} = </math><br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)} \dfrac{25}{16} \qquad \text{(C)}\ 2 \qquad \text{(D)}\ \dfrac{43}{20} \qquad \text{(E)}\ \dfrac{47}{16}</math><br />
<br />
[[1998 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
How many triangles are in this figure? (Some triangles may overlap other triangles.)<br />
<br />
<asy><br />
draw((0,0)--(42,0)--(14,21)--cycle);<br />
draw((14,21)--(18,0)--(30,9));<br />
</asy><br />
<br />
<math>\text{(A)}\ 9 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 5</math><br />
<br />
[[1998 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Which of the following numbers is largest?<br />
<br />
<math>\text{(A)}\ 9.12344 \qquad \text{(B)}\ 9.123\overline{4} \qquad \text{(C)}\ 9.12\overline{34} \qquad \text{(D)}\ 9.1\overline{234} \qquad \text{(E)}\ 9.\overline{1234}</math><br />
<br />
[[1998 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is<br />
<br />
<asy><br />
for(int a=0; a<4; ++a)<br />
{<br />
for(int b=0; b<4; ++b)<br />
{<br />
dot((a,b));<br />
}<br />
}<br />
draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math><br />
<br />
[[1998 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
<math>100\times 19.98\times 1.998\times 1000=</math><br />
<br />
<math>\text{(A)}\ (1.998)^2 \qquad \text{(B)}\ (19.98)^2 \qquad \text{(C)}\ (199.8)^2 \qquad \text{(D)}\ (1998)^2 \qquad \text{(E)}\ (19980)^2</math><br />
<br />
[[1998 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days?<br />
<br />
<math>\text{(A)}\ 140 \qquad \text{(B)}\ 170 \qquad \text{(C)}\ 185 \qquad \text{(D)}\ 198.5 \qquad \text{(E)}\ 199.85</math><br />
<br />
[[1998 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
For a sale, a store owner reduces the price of a <math>\$</math>10 scarf by <math>20\% </math>. Later the price is lowered again, this time by one-half the reduced price. The price is now<br />
<br />
<math>\text{(A)}\ 2.00\text{ dollars} \qquad \text{(B)}\ 3.75\text{ dollars} \qquad \text{(C)}\ 4.00\text{ dollars} \qquad \text{(D)}\ 4.90\text{ dollars} \qquad \text{(E)}\ 6.40\text{ dollars}</math><br />
<br />
[[1998 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
Each of the letters <math>\text{W}</math>, <math>\text{X}</math>, <math>\text{Y}</math>, and <math>\text{Z}</math> represents a different integer in the set <math>\{ 1,2,3,4\}</math>, but not necessarily in that order. If <math>\dfrac{\text{W}}{\text{X}} - \dfrac{\text{Y}}{\text{Z}}=1</math>, then the sum of <math>\text{W}</math> and <math>\text{Y}</math> is<br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7</math><br />
<br />
[[1998 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Harry has 3 sisters and 5 brothers. His sister Harriet has <math>\text{S}</math> sisters and <math>\text{B}</math> brothers. What is the product of <math>\text{S}</math> and <math>\text{B}</math>?<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 18</math><br />
<br />
[[1998 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
<math>2\left(1-\dfrac{1}{2}\right) + 3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + \cdots + 10\left(1-\dfrac{1}{10}\right)=</math><br />
<br />
<math>\text{(A)}\ 45 \qquad \text{(B)}\ 49 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55</math><br />
<br />
[[1998 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)<br />
<br />
<asy><br />
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);<br />
draw((0,0)--(4,4));<br />
draw((0,4)--(3,1)--(3,3));<br />
draw((1,1)--(2,0)--(4,2));<br />
fill((1,1)--(2,0)--(3,1)--(2,2)--cycle,black);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{1}{7} \qquad \text{(C)}\ \dfrac{1}{8} \qquad \text{(D)}\ \dfrac{1}{12} \qquad \text{(E)}\ \dfrac{1}{16}</math><br />
<br />
[[1998 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
An Annville Junior High School, <math>30\%</math> of the students in the Math Club are in the Science Club, and <math>80\%</math> of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?<br />
<br />
<math>\text{(A)}\ 12 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40</math><br />
<br />
[[1998 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Don't Crowd the Isles==<br />
<br />
Problems 15, 16, and 17 all refer to the following:<br />
<br />
<center><br />
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.<br />
</center><br />
<br />
===Problem 15===<br />
<br />
Estimate the population of Nisos in the year 2050.<br />
<br />
<math>\text{(A)}\ 600 \qquad \text{(B)}\ 800 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 2000 \qquad \text{(E)}\ 3000</math><br />
<br />
[[1998 AJHSME Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
<br />
Estimate the year in which the population of Nisos will be approximately 6,000.<br />
<br />
<math>\text{(A)}\ 2050 \qquad \text{(B)}\ 2075 \qquad \text{(C)}\ 2100 \qquad \text{(D)}\ 2125 \qquad \text{(E)}\ 2150</math><br />
<br />
[[1998 AJHSME Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
<br />
In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?<br />
<br />
<math>\text{(A)}\ 50\text{ yrs.} \qquad \text{(B)}\ 75\text{ yrs.} \qquad \text{(C)}\ 100\text{ yrs.} \qquad \text{(D)}\ 125\text{ yrs.} \qquad \text{(E)}\ 150\text{ yrs.}</math><br />
<br />
[[1998 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded?<br />
<br />
<asy><br />
draw((2,0)--(2,1)--(4,1)--(4,0)--cycle);<br />
draw(circle((2.25,.75),.225));<br />
draw((2.05,.95)--(2.45,.55));<br />
draw((2.45,.95)--(2.05,.55));<br />
<br />
draw((0,2)--(4,2)--(4,3)--(0,3)--cycle);<br />
draw((2,2)--(2,3),dashed);<br />
draw((1.3,2.1)..(2,2.3)..(2.7,2.1),EndArrow);<br />
draw((1.3,3.1)..(2,3.3)..(2.7,3.1),EndArrow);<br />
<br />
draw((0,4)--(4,4)--(4,6)--(0,6)--cycle);<br />
draw((0,5)--(4,5),dashed);<br />
draw((-.1,4.3)..(-.3,5)..(-.1,5.7),EndArrow);<br />
draw((3.9,4.3)..(3.7,5)..(3.9,5.7),EndArrow);<br />
</asy><br />
<br />
<br /> <br /><br />
<br />
<asy><br />
unitsize(5);<br />
draw((0,0)--(16,0)--(16,8)--(0,8)--cycle);<br />
draw((0,4)--(16,4),dashed);<br />
draw((8,0)--(8,8),dashed);<br />
draw(circle((1,3),.9));<br />
draw(circle((7,7),.9));<br />
draw(circle((15,5),.9));<br />
draw(circle((9,1),.9));<br />
<br />
draw((24,0)--(40,0)--(40,8)--(24,8)--cycle);<br />
draw((24,4)--(40,4),dashed);<br />
draw((32,0)--(32,8),dashed);<br />
draw(circle((31,1),.9));<br />
draw(circle((33,1),.9));<br />
draw(circle((31,7),.9));<br />
draw(circle((33,7),.9));<br />
<br />
draw((48,0)--(64,0)--(64,8)--(48,8)--cycle);<br />
draw((48,4)--(64,4),dashed);<br />
draw((56,0)--(56,8),dashed);<br />
draw(circle((49,1),.9));<br />
draw(circle((49,7),.9));<br />
draw(circle((63,1),.9));<br />
draw(circle((63,7),.9));<br />
<br />
draw((72,0)--(88,0)--(88,8)--(72,8)--cycle);<br />
draw((72,4)--(88,4),dashed);<br />
draw((80,0)--(80,8),dashed);<br />
draw(circle((79,3),.9));<br />
draw(circle((79,5),.9));<br />
draw(circle((81,3),.9));<br />
draw(circle((81,5),.9));<br />
<br />
draw((96,0)--(112,0)--(112,8)--(96,8)--cycle);<br />
draw((96,4)--(112,4),dashed);<br />
draw((104,0)--(104,8),dashed);<br />
draw(circle((97,3),.9));<br />
draw(circle((97,5),.9));<br />
draw(circle((111,3),.9));<br />
draw(circle((111,5),.9));<br />
<br />
label("(A)",(8,10),N);<br />
label("(B)",(32,10),N);<br />
label("(C)",(56,10),N);<br />
label("(D)",(80,10),N);<br />
label("(E)",(104,10),N);<br />
</asy><br />
<br />
[[1998 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
Tamika selects two different numbers at random from the set <math>\{ 8,9,10 \}</math> and adds them. Carlos takes two different numbers at random from the set <math>\{3, 5, 6\}</math> and multiplies them. What is the probability that Tamika's result is greater than Carlos' result?<br />
<br />
<math>\text{(A)}\ \dfrac{4}{9} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math><br />
<br />
[[1998 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Let <math>PQRS</math> be a square piece of paper. <math>P</math> is folded onto <math>R</math> and then <math>Q</math> is folded onto <math>S</math>. The area of the resulting figure is 9 square inches. Find the perimeter of square <math>PQRS</math>.<br />
<br />
<asy><br />
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);<br />
label("$P$",(0,2),SE);<br />
label("$Q$",(2,2),SW);<br />
label("$R$",(2,0),NW);<br />
label("$S$",(0,0),NE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 9 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 36</math><br />
<br />
[[1998 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
A <math>4\times 4\times 4</math> cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box?<br />
<br />
<math>\text{(A)}\ 48 \qquad \text{(B)}\ 52 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80</math><br />
<br />
[[1998 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. <br />
<br />
<br /><br />
<br />
Rule 1: If the integer is less than 10, multiply it by 9.<br />
<br />
Rule 2: If the integer is even and greater than 9, divide it by 2.<br />
<br />
Rule 3: If the integer is odd and greater than 9, subtract 5 from it.<br />
<br />
<br /><br />
<br />
A sample sequence: <math>23, 18, 9, 81, 76, \ldots .</math><br />
<br />
Find the <math>98^\text{th}</math> term of the sequence that begins <math>98, 49, \ldots .</math><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 54</math><br />
<br />
[[1998 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
If the pattern in the diagram continues, what fraction of the interior would be shaded in the eighth triangle?<br />
<br />
<asy><br />
unitsize(10);<br />
draw((0,0)--(12,0)--(6,6sqrt(3))--cycle);<br />
<br />
draw((15,0)--(27,0)--(21,6sqrt(3))--cycle);<br />
fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black);<br />
<br />
draw((30,0)--(42,0)--(36,6sqrt(3))--cycle);<br />
fill((34,0)--(32,2sqrt(3))--(36,2sqrt(3))--cycle,black);<br />
fill((38,0)--(36,2sqrt(3))--(40,2sqrt(3))--cycle,black);<br />
fill((36,2sqrt(3))--(34,4sqrt(3))--(38,4sqrt(3))--cycle,black);<br />
<br />
draw((45,0)--(57,0)--(51,6sqrt(3))--cycle);<br />
fill((48,0)--(46.5,1.5sqrt(3))--(49.5,1.5sqrt(3))--cycle,black);<br />
fill((51,0)--(49.5,1.5sqrt(3))--(52.5,1.5sqrt(3))--cycle,black);<br />
fill((54,0)--(52.5,1.5sqrt(3))--(55.5,1.5sqrt(3))--cycle,black);<br />
fill((49.5,1.5sqrt(3))--(48,3sqrt(3))--(51,3sqrt(3))--cycle,black);<br />
fill((52.5,1.5sqrt(3))--(51,3sqrt(3))--(54,3sqrt(3))--cycle,black);<br />
fill((51,3sqrt(3))--(49.5,4.5sqrt(3))--(52.5,4.5sqrt(3))--cycle,black);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{3}{8} \qquad \text{(B)}\ \dfrac{5}{27} \qquad \text{(C)}\ \dfrac{7}{16} \qquad \text{(D)}\ \dfrac{9}{16} \qquad \text{(E)}\ \dfrac{11}{45}</math><br />
<br />
[[1998 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?<br />
<br />
<asy><br />
unitsize(24);<br />
for(int a = 0; a < 10; ++a)<br />
{<br />
draw((0,a)--(8,a));<br />
}<br />
for (int b = 0; b < 9; ++b)<br />
{<br />
draw((b,0)--(b,9));<br />
}<br />
draw((0,0)--(0,-.5));<br />
draw((1,0)--(1,-1.5));<br />
draw((.5,-1)--(1.5,-1));<br />
draw((2,0)--(2,-.5));<br />
draw((4,0)--(4,-.5));<br />
draw((5,0)--(5,-1.5));<br />
draw((4.5,-1)--(5.5,-1));<br />
draw((6,0)--(6,-.5));<br />
draw((8,0)--(8,-.5));<br />
<br />
fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black);<br />
fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black);<br />
fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black);<br />
fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black);<br />
fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black);<br />
<br />
label("$2$",(1.5,8.2),N);<br />
label("$4$",(3.5,8.2),N);<br />
label("$5$",(4.5,8.2),N);<br />
label("$7$",(6.5,8.2),N);<br />
label("$8$",(7.5,8.2),N);<br />
label("$9$",(0.5,7.2),N);<br />
label("$11$",(2.5,7.2),N);<br />
label("$12$",(3.5,7.2),N);<br />
label("$13$",(4.5,7.2),N);<br />
label("$14$",(5.5,7.2),N);<br />
label("$16$",(7.5,7.2),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ 36 \qquad \text{(B)}\ 64 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 120</math><br />
<br />
[[1998 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
Three generous friends, each with some cash, redistribute their money as follows: Ami gives enough money to Jan and Toy to double the amount that each has. Jan then gives enough to Ami and Toy to double their amounts. Finally, Toy gives Ami and Jan enough to double their amounts. If Toy has \$36 when they begin and \$36 when they end, what is the total amount that all three friends have?<br />
<br />
<math>\text{(A)}\ 108\text{ dollars} \qquad \text{(B)}\ 180\text{ dollars} \qquad \text{(C)}\ 216\text{ dollars} \qquad \text{(D)}\ 252\text{ dollars} \qquad \text{(E)}\ 288\text{ dollars}</math><br />
<br />
[[1998 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1998|before=[[1997 AJHSME Problems|1997 AJHSME]]|after=[[1999 AMC 8 Problems|1999 AMC 8]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1997_AJHSME_Problems&diff=724991997 AJHSME Problems2015-10-17T14:58:34Z<p>Rep'na: /* Problem 22 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = </math><br />
<br />
<math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997</math><br />
<br />
[[1997 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?<br />
<br />
<math>\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398</math><br />
<br />
[[1997 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which of the following numbers is the largest?<br />
<br />
<math>\text{(A)}\ 0.97 \qquad \text{(B)}\ 0.979 \qquad \text{(C)}\ 0.9709 \qquad \text{(D)}\ 0.907 \qquad \text{(E)}\ 0.9089</math><br />
<br />
[[1997 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?<br />
<br />
<math>\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650</math><br />
<br />
[[1997 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is<br />
<br />
<math>\text{(A)}\ 119 \qquad \text{(B)}\ 126 \qquad \text{(C)}\ 140 \qquad \text{(D)}\ 175 \qquad \text{(E)}\ 189</math><br />
<br />
[[1997 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
In the number <math>74982.1035</math> the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?<br />
<br />
<math>\text{(A)}\ 1,000 \qquad \text{(B)}\ 10,000 \qquad \text{(C)}\ 100,000 \qquad \text{(D)}\ 1,000,000 \qquad \text{(E)}\ 10,000,000</math><br />
<br />
[[1997 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The area of the smallest square that will contain a circle of radius 4 is<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128</math><br />
<br />
[[1997 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?<br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 75 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 120</math><br />
<br />
[[1997 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?<br />
<br />
<math>\text{(A)}\ \dfrac{1}{12} \qquad \text{(B)}\ \dfrac{1}{9} \qquad \text{(C)}\ \dfrac{1}{6} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math><br />
<br />
[[1997 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.<br />
<br />
<asy><br />
unitsize(8);<br />
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);<br />
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);<br />
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);<br />
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);<br />
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);<br />
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);<br />
draw((0,6)--(0,0)--(6,0));<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{5}{12} \qquad \text{(B)}\ \dfrac{1}{2} \qquad \text{(C)}\ \dfrac{7}{12} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[1997 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>\boxed{N}</math> mean the number of whole number divisors of <math>N</math>. For example, <math>\boxed{3}=2</math> because 3 has two divisors, 1 and 3. Find the value of<br />
<br />
<cmath>\boxed{\boxed{11}\times\boxed{20}}.</cmath><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 24</math><br />
<br />
[[1997 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
<math>\angle 1 + \angle 2 = 180^\circ </math><br />
<br />
<math>\angle 3 = \angle 4</math><br />
<br />
Find <math>\angle 4.</math><br />
<br />
<asy><br />
pair H,I,J,K,L;<br />
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);<br />
draw(H--I--J--cycle);<br />
draw(K--L--J);<br />
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);<br />
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);<br />
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);<br />
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math><br />
<br />
[[1997 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are <math>50\%</math>, <math>25\%</math>, and <math>20\%</math>, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?<br />
<br />
<math>\text{(A)}\ 31\% \qquad \text{(B)}\ 32\% \qquad \text{(C)}\ 33\% \qquad \text{(D)}\ 35\% \qquad \text{(E)}\ 95\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?<br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);<br />
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));<br />
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));<br />
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));<br />
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));<br />
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{2}{3} \qquad \text{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \text{(E)}\ \dfrac{7}{9}</math><br />
<br />
[[1997 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Penni Precisely buys \$100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then<br />
<br />
<math>\text{(A)}\ A=B=C \qquad \text{(B)}\ A=B<C \qquad \text{(C)}\ C<B=A</math><br />
<br />
<math>\text{(D)}\ A<B<C \qquad \text{(E)}\ B<A<C</math><br />
<br />
[[1997 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
A cube has eight vertices (corners) and twelve edges. A segment, such as <math>x</math>, which joins two vertices not joined by an edge is called a diagonal. Segment <math>y</math> is also a diagonal. How many diagonals does a cube have?<br />
<br />
<asy><br />
draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));<br />
draw((3,0)--(3,3));<br />
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);<br />
draw((2.5,4)--(2.5,1),dashed);<br />
label("$x$",(2.75,3.5),NNE);<br />
label("$y$",(4.125,1.5),NNE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16</math><br />
<br />
[[1997 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for \$5. This week they are on sale at 5 boxes for \$4. The percent decrease in the price per box during the sale was closest to<br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 65\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
If the product <math>\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9</math>, what is the sum of <math>a</math> and <math>b</math>?<br />
<br />
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 37</math><br />
<br />
[[1997 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is<br />
<br />
<math>\text{(A)}\ \dfrac{5}{32} \qquad \text{(B)}\ \dfrac{11}{64} \qquad \text{(C)}\ \dfrac{3}{16} \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{1}{2}</math><br />
<br />
[[1997 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Each corner cube is removed from this <math>3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}</math> cube. The surface area of the remaining figure is<br />
<br />
<asy><br />
draw((2.7,3.99)--(0,3)--(0,0));<br />
draw((3.7,3.99)--(1,3)--(1,0));<br />
draw((4.7,3.99)--(2,3)--(2,0));<br />
draw((5.7,3.99)--(3,3)--(3,0));<br />
<br />
draw((0,0)--(3,0)--(5.7,0.99));<br />
draw((0,1)--(3,1)--(5.7,1.99));<br />
draw((0,2)--(3,2)--(5.7,2.99));<br />
draw((0,3)--(3,3)--(5.7,3.99));<br />
<br />
draw((0,3)--(3,3)--(3,0));<br />
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));<br />
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));<br />
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));<br />
</asy><br />
<br />
<math>\text{(A)}\ 19\text{ sq.cm} \qquad \text{(B)}\ 24\text{ sq.cm} \qquad \text{(C)}\ 30\text{ sq.cm} \qquad \text{(D)}\ 54\text{ sq.cm} \qquad \text{(E)}\ 72\text{ sq.cm}</math><br />
<br />
[[1997 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth \$200. How much is a three-inch cube of silver worth?<br />
<br />
<math>\text{(A)}\ 300\text{ dollars} \qquad \text{(B)}\ 375\text{ dollars} \qquad \text{(C)}\ 450\text{ dollars} \qquad \text{(D)}\ 560\text{ dollars} \qquad \text{(E)}\ 675\text{ dollars}</math><br />
<br />
[[1997 AJHSME Problems/Problem 22|Solution]]<br />
I<br />
<br />
==Problem 23==<br />
<br />
There are positive integers that have these properties:<br />
<br />
* the sum of the squares of their digits is 50, and<br />
* each digit is larger than the one to its left.<br />
<br />
The product of the digits of the largest integer with both properties is<br />
<br />
<math>\text{(A)}\ 7 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 60</math><br />
<br />
[[1997 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Diameter <math>ACE</math> is divided at <math>C</math> in the ratio <math>2:3</math>. The two semicircles, <math>ABC</math> and <math>CDE</math>, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is<br />
<br />
<asy><br />
pair A,B,C,D,EE;<br />
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);<br />
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);<br />
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));<br />
draw(circle((5,0),5));<br />
<br />
dot(A); dot(B); dot(C); dot(D); dot(EE);<br />
label("$A$",A,W);<br />
label("$B$",B,N);<br />
label("$C$",C,E);<br />
label("$D$",D,N);<br />
label("$E$",EE,W);<br />
</asy><br />
<br />
<math>\text{(A)}\ 2:3 \qquad \text{(B)}\ 1:1 \qquad \text{(C)}\ 3:2 \qquad \text{(D)}\ 9:4 \qquad \text{(E)}\ 5:2</math><br />
<br />
[[1997 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1997|before=[[1996 AJHSME Problems|1996 AJHSME]]|after=[[1998 AJHSME Problems|1998 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1997_AJHSME_Problems&diff=724981997 AJHSME Problems2015-10-17T14:54:02Z<p>Rep'na: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = </math><br />
<br />
<math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997</math><br />
<br />
[[1997 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?<br />
<br />
<math>\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398</math><br />
<br />
[[1997 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which of the following numbers is the largest?<br />
<br />
<math>\text{(A)}\ 0.97 \qquad \text{(B)}\ 0.979 \qquad \text{(C)}\ 0.9709 \qquad \text{(D)}\ 0.907 \qquad \text{(E)}\ 0.9089</math><br />
<br />
[[1997 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?<br />
<br />
<math>\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650</math><br />
<br />
[[1997 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is<br />
<br />
<math>\text{(A)}\ 119 \qquad \text{(B)}\ 126 \qquad \text{(C)}\ 140 \qquad \text{(D)}\ 175 \qquad \text{(E)}\ 189</math><br />
<br />
[[1997 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
In the number <math>74982.1035</math> the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?<br />
<br />
<math>\text{(A)}\ 1,000 \qquad \text{(B)}\ 10,000 \qquad \text{(C)}\ 100,000 \qquad \text{(D)}\ 1,000,000 \qquad \text{(E)}\ 10,000,000</math><br />
<br />
[[1997 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The area of the smallest square that will contain a circle of radius 4 is<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128</math><br />
<br />
[[1997 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?<br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 75 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 120</math><br />
<br />
[[1997 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?<br />
<br />
<math>\text{(A)}\ \dfrac{1}{12} \qquad \text{(B)}\ \dfrac{1}{9} \qquad \text{(C)}\ \dfrac{1}{6} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math><br />
<br />
[[1997 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.<br />
<br />
<asy><br />
unitsize(8);<br />
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);<br />
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);<br />
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);<br />
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);<br />
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);<br />
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);<br />
draw((0,6)--(0,0)--(6,0));<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{5}{12} \qquad \text{(B)}\ \dfrac{1}{2} \qquad \text{(C)}\ \dfrac{7}{12} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[1997 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>\boxed{N}</math> mean the number of whole number divisors of <math>N</math>. For example, <math>\boxed{3}=2</math> because 3 has two divisors, 1 and 3. Find the value of<br />
<br />
<cmath>\boxed{\boxed{11}\times\boxed{20}}.</cmath><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 24</math><br />
<br />
[[1997 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
<math>\angle 1 + \angle 2 = 180^\circ </math><br />
<br />
<math>\angle 3 = \angle 4</math><br />
<br />
Find <math>\angle 4.</math><br />
<br />
<asy><br />
pair H,I,J,K,L;<br />
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);<br />
draw(H--I--J--cycle);<br />
draw(K--L--J);<br />
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);<br />
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);<br />
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);<br />
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math><br />
<br />
[[1997 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are <math>50\%</math>, <math>25\%</math>, and <math>20\%</math>, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?<br />
<br />
<math>\text{(A)}\ 31\% \qquad \text{(B)}\ 32\% \qquad \text{(C)}\ 33\% \qquad \text{(D)}\ 35\% \qquad \text{(E)}\ 95\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?<br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);<br />
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));<br />
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));<br />
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));<br />
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));<br />
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{2}{3} \qquad \text{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \text{(E)}\ \dfrac{7}{9}</math><br />
<br />
[[1997 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Penni Precisely buys \$100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then<br />
<br />
<math>\text{(A)}\ A=B=C \qquad \text{(B)}\ A=B<C \qquad \text{(C)}\ C<B=A</math><br />
<br />
<math>\text{(D)}\ A<B<C \qquad \text{(E)}\ B<A<C</math><br />
<br />
[[1997 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
A cube has eight vertices (corners) and twelve edges. A segment, such as <math>x</math>, which joins two vertices not joined by an edge is called a diagonal. Segment <math>y</math> is also a diagonal. How many diagonals does a cube have?<br />
<br />
<asy><br />
draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));<br />
draw((3,0)--(3,3));<br />
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);<br />
draw((2.5,4)--(2.5,1),dashed);<br />
label("$x$",(2.75,3.5),NNE);<br />
label("$y$",(4.125,1.5),NNE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16</math><br />
<br />
[[1997 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for \$5. This week they are on sale at 5 boxes for \$4. The percent decrease in the price per box during the sale was closest to<br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 65\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
If the product <math>\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9</math>, what is the sum of <math>a</math> and <math>b</math>?<br />
<br />
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 37</math><br />
<br />
[[1997 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is<br />
<br />
<math>\text{(A)}\ \dfrac{5}{32} \qquad \text{(B)}\ \dfrac{11}{64} \qquad \text{(C)}\ \dfrac{3}{16} \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{1}{2}</math><br />
<br />
[[1997 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Each corner cube is removed from this <math>3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}</math> cube. The surface area of the remaining figure is<br />
<br />
<asy><br />
draw((2.7,3.99)--(0,3)--(0,0));<br />
draw((3.7,3.99)--(1,3)--(1,0));<br />
draw((4.7,3.99)--(2,3)--(2,0));<br />
draw((5.7,3.99)--(3,3)--(3,0));<br />
<br />
draw((0,0)--(3,0)--(5.7,0.99));<br />
draw((0,1)--(3,1)--(5.7,1.99));<br />
draw((0,2)--(3,2)--(5.7,2.99));<br />
draw((0,3)--(3,3)--(5.7,3.99));<br />
<br />
draw((0,3)--(3,3)--(3,0));<br />
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));<br />
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));<br />
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));<br />
</asy><br />
<br />
<math>\text{(A)}\ 19\text{ sq.cm} \qquad \text{(B)}\ 24\text{ sq.cm} \qquad \text{(C)}\ 30\text{ sq.cm} \qquad \text{(D)}\ 54\text{ sq.cm} \qquad \text{(E)}\ 72\text{ sq.cm}</math><br />
<br />
[[1997 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth <dollar/>200. How much is a three-inch cube of silver worth?<br />
<br />
<math>\text{(A)}\ 300\text{ dollars} \qquad \text{(B)}\ 375\text{ dollars} \qquad \text{(C)}\ 450\text{ dollars} \qquad \text{(D)}\ 560\text{ dollars} \qquad \text{(E)}\ 675\text{ dollars}</math><br />
<br />
[[1997 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
There are positive integers that have these properties:<br />
<br />
* the sum of the squares of their digits is 50, and<br />
* each digit is larger than the one to its left.<br />
<br />
The product of the digits of the largest integer with both properties is<br />
<br />
<math>\text{(A)}\ 7 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 60</math><br />
<br />
[[1997 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Diameter <math>ACE</math> is divided at <math>C</math> in the ratio <math>2:3</math>. The two semicircles, <math>ABC</math> and <math>CDE</math>, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is<br />
<br />
<asy><br />
pair A,B,C,D,EE;<br />
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);<br />
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);<br />
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));<br />
draw(circle((5,0),5));<br />
<br />
dot(A); dot(B); dot(C); dot(D); dot(EE);<br />
label("$A$",A,W);<br />
label("$B$",B,N);<br />
label("$C$",C,E);<br />
label("$D$",D,N);<br />
label("$E$",EE,W);<br />
</asy><br />
<br />
<math>\text{(A)}\ 2:3 \qquad \text{(B)}\ 1:1 \qquad \text{(C)}\ 3:2 \qquad \text{(D)}\ 9:4 \qquad \text{(E)}\ 5:2</math><br />
<br />
[[1997 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1997|before=[[1996 AJHSME Problems|1996 AJHSME]]|after=[[1998 AJHSME Problems|1998 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1997_AJHSME_Problems&diff=724971997 AJHSME Problems2015-10-17T14:53:24Z<p>Rep'na: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = </math><br />
<br />
<math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997</math><br />
<br />
[[1997 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?<br />
<br />
<math>\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398</math><br />
<br />
[[1997 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which of the following numbers is the largest?<br />
<br />
<math>\text{(A)}\ 0.97 \qquad \text{(B)}\ 0.979 \qquad \text{(C)}\ 0.9709 \qquad \text{(D)}\ 0.907 \qquad \text{(E)}\ 0.9089</math><br />
<br />
[[1997 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?<br />
<br />
<math>\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650</math><br />
<br />
[[1997 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is<br />
<br />
<math>\text{(A)}\ 119 \qquad \text{(B)}\ 126 \qquad \text{(C)}\ 140 \qquad \text{(D)}\ 175 \qquad \text{(E)}\ 189</math><br />
<br />
[[1997 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
In the number <math>74982.1035</math> the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?<br />
<br />
<math>\text{(A)}\ 1,000 \qquad \text{(B)}\ 10,000 \qquad \text{(C)}\ 100,000 \qquad \text{(D)}\ 1,000,000 \qquad \text{(E)}\ 10,000,000</math><br />
<br />
[[1997 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The area of the smallest square that will contain a circle of radius 4 is<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128</math><br />
<br />
[[1997 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?<br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 75 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 120</math><br />
<br />
[[1997 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?<br />
<br />
<math>\text{(A)}\ \dfrac{1}{12} \qquad \text{(B)}\ \dfrac{1}{9} \qquad \text{(C)}\ \dfrac{1}{6} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math><br />
<br />
[[1997 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.<br />
<br />
<asy><br />
unitsize(8);<br />
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);<br />
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);<br />
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);<br />
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);<br />
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);<br />
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);<br />
draw((0,6)--(0,0)--(6,0));<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{5}{12} \qquad \text{(B)}\ \dfrac{1}{2} \qquad \text{(C)}\ \dfrac{7}{12} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[1997 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>\boxed{N}</math> mean the number of whole number divisors of <math>N</math>. For example, <math>\boxed{3}=2</math> because 3 has two divisors, 1 and 3. Find the value of<br />
<br />
<cmath>\boxed{\boxed{11}\times\boxed{20}}.</cmath><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 24</math><br />
<br />
[[1997 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
<math>\angle 1 + \angle 2 = 180^\circ </math><br />
<br />
<math>\angle 3 = \angle 4</math><br />
<br />
Find <math>\angle 4.</math><br />
<br />
<asy><br />
pair H,I,J,K,L;<br />
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);<br />
draw(H--I--J--cycle);<br />
draw(K--L--J);<br />
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);<br />
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);<br />
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);<br />
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math><br />
<br />
[[1997 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are <math>50\%</math>, <math>25\%</math>, and <math>20\%</math>, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?<br />
<br />
<math>\text{(A)}\ 31\% \qquad \text{(B)}\ 32\% \qquad \text{(C)}\ 33\% \qquad \text{(D)}\ 35\% \qquad \text{(E)}\ 95\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?<br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);<br />
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));<br />
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));<br />
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));<br />
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));<br />
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{2}{3} \qquad \text{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \text{(E)}\ \dfrac{7}{9}</math><br />
<br />
[[1997 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Penni Precisely buys \$100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then<br />
<br />
<math>\text{(A)}\ A=B=C \qquad \text{(B)}\ A=B<C \qquad \text{(C)}\ C<B=A</math><br />
<br />
<math>\text{(D)}\ A<B<C \qquad \text{(E)}\ B<A<C</math><br />
<br />
[[1997 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
A cube has eight vertices (corners) and twelve edges. A segment, such as <math>x</math>, which joins two vertices not joined by an edge is called a diagonal. Segment <math>y</math> is also a diagonal. How many diagonals does a cube have?<br />
<br />
<asy><br />
draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));<br />
draw((3,0)--(3,3));<br />
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);<br />
draw((2.5,4)--(2.5,1),dashed);<br />
label("$x$",(2.75,3.5),NNE);<br />
label("$y$",(4.125,1.5),NNE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16</math><br />
<br />
[[1997 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for <math>5. This week they are on sale at 5 boxes for </math>4. The percent decrease in the price per box during the sale was closest to<br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 65\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
If the product <math>\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9</math>, what is the sum of <math>a</math> and <math>b</math>?<br />
<br />
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 37</math><br />
<br />
[[1997 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is<br />
<br />
<math>\text{(A)}\ \dfrac{5}{32} \qquad \text{(B)}\ \dfrac{11}{64} \qquad \text{(C)}\ \dfrac{3}{16} \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{1}{2}</math><br />
<br />
[[1997 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Each corner cube is removed from this <math>3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}</math> cube. The surface area of the remaining figure is<br />
<br />
<asy><br />
draw((2.7,3.99)--(0,3)--(0,0));<br />
draw((3.7,3.99)--(1,3)--(1,0));<br />
draw((4.7,3.99)--(2,3)--(2,0));<br />
draw((5.7,3.99)--(3,3)--(3,0));<br />
<br />
draw((0,0)--(3,0)--(5.7,0.99));<br />
draw((0,1)--(3,1)--(5.7,1.99));<br />
draw((0,2)--(3,2)--(5.7,2.99));<br />
draw((0,3)--(3,3)--(5.7,3.99));<br />
<br />
draw((0,3)--(3,3)--(3,0));<br />
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));<br />
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));<br />
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));<br />
</asy><br />
<br />
<math>\text{(A)}\ 19\text{ sq.cm} \qquad \text{(B)}\ 24\text{ sq.cm} \qquad \text{(C)}\ 30\text{ sq.cm} \qquad \text{(D)}\ 54\text{ sq.cm} \qquad \text{(E)}\ 72\text{ sq.cm}</math><br />
<br />
[[1997 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth <dollar/>200. How much is a three-inch cube of silver worth?<br />
<br />
<math>\text{(A)}\ 300\text{ dollars} \qquad \text{(B)}\ 375\text{ dollars} \qquad \text{(C)}\ 450\text{ dollars} \qquad \text{(D)}\ 560\text{ dollars} \qquad \text{(E)}\ 675\text{ dollars}</math><br />
<br />
[[1997 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
There are positive integers that have these properties:<br />
<br />
* the sum of the squares of their digits is 50, and<br />
* each digit is larger than the one to its left.<br />
<br />
The product of the digits of the largest integer with both properties is<br />
<br />
<math>\text{(A)}\ 7 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 60</math><br />
<br />
[[1997 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Diameter <math>ACE</math> is divided at <math>C</math> in the ratio <math>2:3</math>. The two semicircles, <math>ABC</math> and <math>CDE</math>, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is<br />
<br />
<asy><br />
pair A,B,C,D,EE;<br />
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);<br />
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);<br />
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));<br />
draw(circle((5,0),5));<br />
<br />
dot(A); dot(B); dot(C); dot(D); dot(EE);<br />
label("$A$",A,W);<br />
label("$B$",B,N);<br />
label("$C$",C,E);<br />
label("$D$",D,N);<br />
label("$E$",EE,W);<br />
</asy><br />
<br />
<math>\text{(A)}\ 2:3 \qquad \text{(B)}\ 1:1 \qquad \text{(C)}\ 3:2 \qquad \text{(D)}\ 9:4 \qquad \text{(E)}\ 5:2</math><br />
<br />
[[1997 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1997|before=[[1996 AJHSME Problems|1996 AJHSME]]|after=[[1998 AJHSME Problems|1998 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1997_AJHSME_Problems&diff=724941997 AJHSME Problems2015-10-16T15:53:42Z<p>Rep'na: /* Problem 16 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = </math><br />
<br />
<math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997</math><br />
<br />
[[1997 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?<br />
<br />
<math>\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398</math><br />
<br />
[[1997 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which of the following numbers is the largest?<br />
<br />
<math>\text{(A)}\ 0.97 \qquad \text{(B)}\ 0.979 \qquad \text{(C)}\ 0.9709 \qquad \text{(D)}\ 0.907 \qquad \text{(E)}\ 0.9089</math><br />
<br />
[[1997 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?<br />
<br />
<math>\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650</math><br />
<br />
[[1997 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is<br />
<br />
<math>\text{(A)}\ 119 \qquad \text{(B)}\ 126 \qquad \text{(C)}\ 140 \qquad \text{(D)}\ 175 \qquad \text{(E)}\ 189</math><br />
<br />
[[1997 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
In the number <math>74982.1035</math> the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?<br />
<br />
<math>\text{(A)}\ 1,000 \qquad \text{(B)}\ 10,000 \qquad \text{(C)}\ 100,000 \qquad \text{(D)}\ 1,000,000 \qquad \text{(E)}\ 10,000,000</math><br />
<br />
[[1997 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The area of the smallest square that will contain a circle of radius 4 is<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128</math><br />
<br />
[[1997 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?<br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 75 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 120</math><br />
<br />
[[1997 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?<br />
<br />
<math>\text{(A)}\ \dfrac{1}{12} \qquad \text{(B)}\ \dfrac{1}{9} \qquad \text{(C)}\ \dfrac{1}{6} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math><br />
<br />
[[1997 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.<br />
<br />
<asy><br />
unitsize(8);<br />
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);<br />
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);<br />
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);<br />
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);<br />
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);<br />
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);<br />
draw((0,6)--(0,0)--(6,0));<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{5}{12} \qquad \text{(B)}\ \dfrac{1}{2} \qquad \text{(C)}\ \dfrac{7}{12} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[1997 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>\boxed{N}</math> mean the number of whole number divisors of <math>N</math>. For example, <math>\boxed{3}=2</math> because 3 has two divisors, 1 and 3. Find the value of<br />
<br />
<cmath>\boxed{\boxed{11}\times\boxed{20}}.</cmath><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 24</math><br />
<br />
[[1997 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
<math>\angle 1 + \angle 2 = 180^\circ </math><br />
<br />
<math>\angle 3 = \angle 4</math><br />
<br />
Find <math>\angle 4.</math><br />
<br />
<asy><br />
pair H,I,J,K,L;<br />
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);<br />
draw(H--I--J--cycle);<br />
draw(K--L--J);<br />
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);<br />
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);<br />
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);<br />
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math><br />
<br />
[[1997 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are <math>50\%</math>, <math>25\%</math>, and <math>20\%</math>, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?<br />
<br />
<math>\text{(A)}\ 31\% \qquad \text{(B)}\ 32\% \qquad \text{(C)}\ 33\% \qquad \text{(D)}\ 35\% \qquad \text{(E)}\ 95\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?<br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);<br />
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));<br />
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));<br />
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));<br />
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));<br />
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{2}{3} \qquad \text{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \text{(E)}\ \dfrac{7}{9}</math><br />
<br />
[[1997 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Penni Precisely buys \$100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then<br />
<br />
<math>\text{(A)}\ A=B=C \qquad \text{(B)}\ A=B<C \qquad \text{(C)}\ C<B=A</math><br />
<br />
<math>\text{(D)}\ A<B<C \qquad \text{(E)}\ B<A<C</math><br />
<br />
[[1997 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
A cube has eight vertices (corners) and twelve edges. A segment, such as <math>x</math>, which joins two vertices not joined by an edge is called a diagonal. Segment <math>y</math> is also a diagonal. How many diagonals does a cube have?<br />
<br />
<asy><br />
draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));<br />
draw((3,0)--(3,3));<br />
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);<br />
draw((2.5,4)--(2.5,1),dashed);<br />
label("$x$",(2.75,3.5),NNE);<br />
label("$y$",(4.125,1.5),NNE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16</math><br />
<br />
[[1997 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for <dollar/>5. This week they are on sale at 5 boxes for <dollar/>4. The percent decrease in the price per box during the sale was closest to<br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 65\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
If the product <math>\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9</math>, what is the sum of <math>a</math> and <math>b</math>?<br />
<br />
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 37</math><br />
<br />
[[1997 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is<br />
<br />
<math>\text{(A)}\ \dfrac{5}{32} \qquad \text{(B)}\ \dfrac{11}{64} \qquad \text{(C)}\ \dfrac{3}{16} \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{1}{2}</math><br />
<br />
[[1997 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Each corner cube is removed from this <math>3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}</math> cube. The surface area of the remaining figure is<br />
<br />
<asy><br />
draw((2.7,3.99)--(0,3)--(0,0));<br />
draw((3.7,3.99)--(1,3)--(1,0));<br />
draw((4.7,3.99)--(2,3)--(2,0));<br />
draw((5.7,3.99)--(3,3)--(3,0));<br />
<br />
draw((0,0)--(3,0)--(5.7,0.99));<br />
draw((0,1)--(3,1)--(5.7,1.99));<br />
draw((0,2)--(3,2)--(5.7,2.99));<br />
draw((0,3)--(3,3)--(5.7,3.99));<br />
<br />
draw((0,3)--(3,3)--(3,0));<br />
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));<br />
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));<br />
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));<br />
</asy><br />
<br />
<math>\text{(A)}\ 19\text{ sq.cm} \qquad \text{(B)}\ 24\text{ sq.cm} \qquad \text{(C)}\ 30\text{ sq.cm} \qquad \text{(D)}\ 54\text{ sq.cm} \qquad \text{(E)}\ 72\text{ sq.cm}</math><br />
<br />
[[1997 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth <dollar/>200. How much is a three-inch cube of silver worth?<br />
<br />
<math>\text{(A)}\ 300\text{ dollars} \qquad \text{(B)}\ 375\text{ dollars} \qquad \text{(C)}\ 450\text{ dollars} \qquad \text{(D)}\ 560\text{ dollars} \qquad \text{(E)}\ 675\text{ dollars}</math><br />
<br />
[[1997 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
There are positive integers that have these properties:<br />
<br />
* the sum of the squares of their digits is 50, and<br />
* each digit is larger than the one to its left.<br />
<br />
The product of the digits of the largest integer with both properties is<br />
<br />
<math>\text{(A)}\ 7 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 60</math><br />
<br />
[[1997 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Diameter <math>ACE</math> is divided at <math>C</math> in the ratio <math>2:3</math>. The two semicircles, <math>ABC</math> and <math>CDE</math>, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is<br />
<br />
<asy><br />
pair A,B,C,D,EE;<br />
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);<br />
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);<br />
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));<br />
draw(circle((5,0),5));<br />
<br />
dot(A); dot(B); dot(C); dot(D); dot(EE);<br />
label("$A$",A,W);<br />
label("$B$",B,N);<br />
label("$C$",C,E);<br />
label("$D$",D,N);<br />
label("$E$",EE,W);<br />
</asy><br />
<br />
<math>\text{(A)}\ 2:3 \qquad \text{(B)}\ 1:1 \qquad \text{(C)}\ 3:2 \qquad \text{(D)}\ 9:4 \qquad \text{(E)}\ 5:2</math><br />
<br />
[[1997 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1997|before=[[1996 AJHSME Problems|1996 AJHSME]]|after=[[1998 AJHSME Problems|1998 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1997_AJHSME_Problems&diff=724931997 AJHSME Problems2015-10-16T15:52:47Z<p>Rep'na: /* Problem 16 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = </math><br />
<br />
<math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997</math><br />
<br />
[[1997 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?<br />
<br />
<math>\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398</math><br />
<br />
[[1997 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
Which of the following numbers is the largest?<br />
<br />
<math>\text{(A)}\ 0.97 \qquad \text{(B)}\ 0.979 \qquad \text{(C)}\ 0.9709 \qquad \text{(D)}\ 0.907 \qquad \text{(E)}\ 0.9089</math><br />
<br />
[[1997 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?<br />
<br />
<math>\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650</math><br />
<br />
[[1997 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is<br />
<br />
<math>\text{(A)}\ 119 \qquad \text{(B)}\ 126 \qquad \text{(C)}\ 140 \qquad \text{(D)}\ 175 \qquad \text{(E)}\ 189</math><br />
<br />
[[1997 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
In the number <math>74982.1035</math> the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?<br />
<br />
<math>\text{(A)}\ 1,000 \qquad \text{(B)}\ 10,000 \qquad \text{(C)}\ 100,000 \qquad \text{(D)}\ 1,000,000 \qquad \text{(E)}\ 10,000,000</math><br />
<br />
[[1997 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
The area of the smallest square that will contain a circle of radius 4 is<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128</math><br />
<br />
[[1997 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?<br />
<br />
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 75 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 120</math><br />
<br />
[[1997 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?<br />
<br />
<math>\text{(A)}\ \dfrac{1}{12} \qquad \text{(B)}\ \dfrac{1}{9} \qquad \text{(C)}\ \dfrac{1}{6} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math><br />
<br />
[[1997 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.<br />
<br />
<asy><br />
unitsize(8);<br />
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);<br />
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);<br />
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);<br />
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);<br />
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);<br />
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);<br />
draw((0,6)--(0,0)--(6,0));<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{5}{12} \qquad \text{(B)}\ \dfrac{1}{2} \qquad \text{(C)}\ \dfrac{7}{12} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{5}{6}</math><br />
<br />
[[1997 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>\boxed{N}</math> mean the number of whole number divisors of <math>N</math>. For example, <math>\boxed{3}=2</math> because 3 has two divisors, 1 and 3. Find the value of<br />
<br />
<cmath>\boxed{\boxed{11}\times\boxed{20}}.</cmath><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 24</math><br />
<br />
[[1997 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
<math>\angle 1 + \angle 2 = 180^\circ </math><br />
<br />
<math>\angle 3 = \angle 4</math><br />
<br />
Find <math>\angle 4.</math><br />
<br />
<asy><br />
pair H,I,J,K,L;<br />
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);<br />
draw(H--I--J--cycle);<br />
draw(K--L--J);<br />
draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE);<br />
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S);<br />
label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE);<br />
label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 25^\circ \qquad \text{(C)}\ 30^\circ \qquad \text{(D)}\ 35^\circ \qquad \text{(E)}\ 40^\circ</math><br />
<br />
[[1997 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are <math>50\%</math>, <math>25\%</math>, and <math>20\%</math>, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?<br />
<br />
<math>\text{(A)}\ 31\% \qquad \text{(B)}\ 32\% \qquad \text{(C)}\ 33\% \qquad \text{(D)}\ 35\% \qquad \text{(E)}\ 95\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?<br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is<br />
<br />
<asy><br />
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);<br />
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));<br />
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));<br />
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));<br />
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));<br />
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{2}{3} \qquad \text{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \text{(E)}\ \dfrac{7}{9}</math><br />
<br />
[[1997 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Penni Precisely buys <math>100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then<br />
<br />
</math>\text{(A)}\ A=B=C \qquad \text{(B)}\ A=B<C \qquad \text{(C)}\ C<B=A<math><br />
<br />
</math>\text{(D)}\ A<B<C \qquad \text{(E)}\ B<A<C$<br />
<br />
[[1997 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
A cube has eight vertices (corners) and twelve edges. A segment, such as <math>x</math>, which joins two vertices not joined by an edge is called a diagonal. Segment <math>y</math> is also a diagonal. How many diagonals does a cube have?<br />
<br />
<asy><br />
draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));<br />
draw((3,0)--(3,3));<br />
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);<br />
draw((2.5,4)--(2.5,1),dashed);<br />
label("$x$",(2.75,3.5),NNE);<br />
label("$y$",(4.125,1.5),NNE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16</math><br />
<br />
[[1997 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for <dollar/>5. This week they are on sale at 5 boxes for <dollar/>4. The percent decrease in the price per box during the sale was closest to<br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 65\%</math><br />
<br />
[[1997 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
If the product <math>\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9</math>, what is the sum of <math>a</math> and <math>b</math>?<br />
<br />
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 37</math><br />
<br />
[[1997 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is<br />
<br />
<math>\text{(A)}\ \dfrac{5}{32} \qquad \text{(B)}\ \dfrac{11}{64} \qquad \text{(C)}\ \dfrac{3}{16} \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{1}{2}</math><br />
<br />
[[1997 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Each corner cube is removed from this <math>3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}</math> cube. The surface area of the remaining figure is<br />
<br />
<asy><br />
draw((2.7,3.99)--(0,3)--(0,0));<br />
draw((3.7,3.99)--(1,3)--(1,0));<br />
draw((4.7,3.99)--(2,3)--(2,0));<br />
draw((5.7,3.99)--(3,3)--(3,0));<br />
<br />
draw((0,0)--(3,0)--(5.7,0.99));<br />
draw((0,1)--(3,1)--(5.7,1.99));<br />
draw((0,2)--(3,2)--(5.7,2.99));<br />
draw((0,3)--(3,3)--(5.7,3.99));<br />
<br />
draw((0,3)--(3,3)--(3,0));<br />
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));<br />
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));<br />
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));<br />
</asy><br />
<br />
<math>\text{(A)}\ 19\text{ sq.cm} \qquad \text{(B)}\ 24\text{ sq.cm} \qquad \text{(C)}\ 30\text{ sq.cm} \qquad \text{(D)}\ 54\text{ sq.cm} \qquad \text{(E)}\ 72\text{ sq.cm}</math><br />
<br />
[[1997 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth <dollar/>200. How much is a three-inch cube of silver worth?<br />
<br />
<math>\text{(A)}\ 300\text{ dollars} \qquad \text{(B)}\ 375\text{ dollars} \qquad \text{(C)}\ 450\text{ dollars} \qquad \text{(D)}\ 560\text{ dollars} \qquad \text{(E)}\ 675\text{ dollars}</math><br />
<br />
[[1997 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
There are positive integers that have these properties:<br />
<br />
* the sum of the squares of their digits is 50, and<br />
* each digit is larger than the one to its left.<br />
<br />
The product of the digits of the largest integer with both properties is<br />
<br />
<math>\text{(A)}\ 7 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 60</math><br />
<br />
[[1997 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Diameter <math>ACE</math> is divided at <math>C</math> in the ratio <math>2:3</math>. The two semicircles, <math>ABC</math> and <math>CDE</math>, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is<br />
<br />
<asy><br />
pair A,B,C,D,EE;<br />
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);<br />
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);<br />
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));<br />
draw(circle((5,0),5));<br />
<br />
dot(A); dot(B); dot(C); dot(D); dot(EE);<br />
label("$A$",A,W);<br />
label("$B$",B,N);<br />
label("$C$",C,E);<br />
label("$D$",D,N);<br />
label("$E$",EE,W);<br />
</asy><br />
<br />
<math>\text{(A)}\ 2:3 \qquad \text{(B)}\ 1:1 \qquad \text{(C)}\ 3:2 \qquad \text{(D)}\ 9:4 \qquad \text{(E)}\ 5:2</math><br />
<br />
[[1997 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math><br />
<br />
[[1997 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1997|before=[[1996 AJHSME Problems|1996 AJHSME]]|after=[[1998 AJHSME Problems|1998 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1996_AJHSME_Problems&diff=724751996 AJHSME Problems2015-10-15T16:28:18Z<p>Rep'na: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
<br />
How many positive factors of 36 are also multiples of 4?<br />
<br />
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math><br />
<br />
[[1996 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
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?<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
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<br />
<br />
<math>\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134</math><br />
<br />
[[1996 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
<math>\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=</math><br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
The letters <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, and <math>T</math> represent numbers located on the number line as shown.<br />
<br />
<asy><br />
unitsize(36);<br />
draw((-4,0)--(4,0));<br />
draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1));<br />
draw((3.9,0.1)--(4,0)--(3.9,-0.1));<br />
<br />
for (int i = -3; i <= 3; ++i)<br />
{<br />
draw((i,-0.1)--(i,0));<br />
}<br />
label("$-3$",(-3,-0.1),S);<br />
label("$-2$",(-2,-0.1),S);<br />
label("$-1$",(-1,-0.1),S);<br />
label("$0$",(0,-0.1),S);<br />
label("$1$",(1,-0.1),S);<br />
label("$2$",(2,-0.1),S);<br />
label("$3$",(3,-0.1),S);<br />
<br />
draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1));<br />
draw((-3.6,0)--(-3.6,0.25));<br />
label("$P$",(-3.6,0.25),N);<br />
draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1));<br />
draw((-1.2,0)--(-1.2,0.25));<br />
label("$Q$",(-1.2,0.25),N);<br />
draw((0.1,0.1)--(0.2,0)--(0.3,0.1));<br />
draw((0.2,0)--(0.2,0.25));<br />
label("$R$",(0.2,0.25),N);<br />
draw((0.8,0.1)--(0.9,0)--(1,0.1));<br />
draw((0.9,0)--(0.9,0.25));<br />
label("$S$",(0.9,0.25),N);<br />
draw((1.4,0.1)--(1.5,0)--(1.6,0.1));<br />
draw((1.5,0)--(1.5,0.25));<br />
label("$T$",(1.5,0.25),N);<br />
</asy><br />
<br />
Which of the following expressions represents a negative number?<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
What is the smallest result that can be obtained from the following process?<br />
<br />
*Choose three different numbers from the set <math>\{3,5,7,11,13,17\}</math>.<br />
*Add two of these numbers.<br />
*Multiply their sum by the third number.<br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56</math><br />
<br />
[[1996 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
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?<br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1996 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Points <math>A</math> and <math>B</math> are 10 units apart. Points <math>B</math> and <math>C</math> are 4 units apart. Points <math>C</math> and <math>D</math> are 3 units apart. If <math>A</math> and <math>D</math> are as close as possible, then the number of units between them is<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 17</math><br />
<br />
[[1996 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
If 5 times a number is 2, then 100 times the reciprocal of the number is<br />
<br />
<math>\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500</math><br />
<br />
[[1996 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
When Walter drove up to the gasoline pump, he noticed that his gasoline tank was 1/8 full. He purchased 7.5 gallons of gasoline for <math>\$10</math>. With this additional gasoline, his gasoline tank was then 5/8 full. The number of gallons of gasoline his tank holds when it is full is<br />
<br />
<math>\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5</math><br />
<br />
[[1996 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>x</math> be the number<br />
<cmath>0.\underbrace{0000...0000}_{1996\text{ zeros}}1,</cmath><br />
where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?<br />
<br />
<math>\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</math><br />
<br />
[[1996 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
What number should be removed from the list<br />
<cmath>1,2,3,4,5,6,7,8,9,10,11</cmath><br />
so that the average of the remaining numbers is <math>6.1</math>?<br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1996 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
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<br />
<br />
<math>\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700</math><br />
<br />
[[1996 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Six different digits from the set<br />
<cmath>\{ 1,2,3,4,5,6,7,8,9\}</cmath><br />
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.<br />
The sum of the six digits used is<br />
<br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(1,0)--(1,1)--(4,1)--(4,2)--(1,2)--(1,3)--(0,3)--cycle);<br />
draw((0,1)--(1,1)--(1,2)--(0,2));<br />
draw((2,1)--(2,2));<br />
draw((3,1)--(3,2));<br />
label("$23$",(0.5,0),S);<br />
label("$12$",(4,1.5),E);<br />
</asy><br />
<br />
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 35</math><br />
<br />
[[1996 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
The remainder when the product <math>1492\cdot 1776\cdot 1812\cdot 1996</math> is divided by 5 is<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4</math><br />
<br />
[[1996 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
<math>1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=</math><br />
<br />
<math>\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998</math><br />
<br />
[[1996 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Figure <math>OPQR</math> is a square. Point <math>O</math> is the origin, and point <math>Q</math> has coordinates (2,2). What are the coordinates for <math>T</math> so that the area of triangle <math>PQT</math> equals the area of square <math>OPQR</math>?<br />
<br />
<asy><br />
pair O,P,Q,R,T;<br />
O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0);<br />
draw((-5,0)--(3,0)); draw((0,-1)--(0,3));<br />
draw(P--Q--R);<br />
draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8));<br />
draw((-0.2,2.8)--(0,3)--(0.2,2.8));<br />
draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2));<br />
draw((2.8,-0.2)--(3,0)--(2.8,0.2));<br />
draw(Q--T);<br />
<br />
label("$O$",O,SW);<br />
label("$P$",P,S);<br />
label("$Q$",Q,NE);<br />
label("$R$",R,W);<br />
label("$T$",T,S);<br />
</asy><br />
<br />
<center>NOT TO SCALE</center><br />
<br />
<math>\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)</math><br />
<br />
[[1996 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
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<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
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<br />
<br />
<asy><br />
unitsize(18);<br />
draw(circle((0,0),4));<br />
draw(circle((9,0),4));<br />
draw((-4,0)--(0,0)--4*dir(352.8));<br />
draw((0,0)--4*dir(100.8));<br />
draw((5,0)--(9,0)--(4*dir(324)+(9,0)));<br />
draw((9,0)--(4*dir(50.4)+(9,0)));<br />
<br />
label("$48\%$",(0,-1),S);<br />
label("bowling",(0,-2),S);<br />
label("$30\%$",(1.5,1.5),N);<br />
label("golf",(1.5,0.5),N);<br />
label("$22\%$",(-2,1.5),N);<br />
label("tennis",(-2,0.5),N);<br />
<br />
label("$40\%$",(8.5,-1),S);<br />
label("tennis",(8.5,-2),S);<br />
label("$24\%$",(10.5,0.5),E);<br />
label("golf",(10.5,-0.5),E);<br />
label("$36\%$",(7.8,1.7),N);<br />
label("bowling",(7.8,0.7),N);<br />
<br />
label("$\textbf{East JHS}$",(0,-4),S);<br />
label("$\textbf{2000 students}$",(0,-5),S);<br />
label("$\textbf{West MS}$",(9,-4),S);<br />
label("$\textbf{2500 students}$",(9,-5),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%</math><br />
<br />
[[1996 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Suppose there is a special key on a calculator that replaces the number <math>x</math> currently displayed with the number given by the formula <math>1/(1-x)</math>. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since <math>1/(1-2)=-1</math>. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display <br />
<br />
<math>\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5</math><br />
<br />
[[1996 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
How many subsets containing three different numbers can be selected from the set <br />
<cmath>\{ 89,95,99,132, 166,173 \}</cmath><br />
so that the sum of the three numbers is even?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24</math><br />
<br />
[[1996 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
The horizontal and vertical distances between adjacent points equal 1 unit. The area of triangle <math>ABC</math> is<br />
<br />
<asy><br />
for (int a = 0; a < 5; ++a)<br />
{<br />
for (int b = 0; b < 4; ++b)<br />
{<br />
dot((a,b));<br />
}<br />
}<br />
draw((0,0)--(3,2)--(4,3)--cycle);<br />
label("$A$",(0,0),SW);<br />
label("$B$",(3,2),SE);<br />
label("$C$",(4,3),NE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4</math><br />
<br />
[[1996 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
The manager of a company planned to distribute a <math>\$50</math> bonus to each employee from the company fund, but the fund contained <math>\$5</math> less than what was needed. Instead the manager gave each employee a <math>\$45</math> bonus and kept the remaining <math>\$95</math> in the company fund. The amount of money in the company fund before any bonuses were paid was<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
The measure of angle <math>ABC</math> is <math>50^\circ </math>, <math>\overline{AD}</math> bisects angle <math>BAC</math>, and <math>\overline{DC}</math> bisects angle <math>BCA</math>. The measure of angle <math>ADC</math> is<br />
<br />
<asy><br />
pair A,B,C,D;<br />
A = (0,0); B = (9,10); C = (10,0); D = (6.66,3);<br />
dot(A); dot(B); dot(C); dot(D);<br />
draw(A--B--C--cycle);<br />
draw(A--D--C);<br />
<br />
label("$A$",A,SW);<br />
label("$B$",B,N);<br />
label("$C$",C,SE);<br />
label("$D$",D,N);<br />
label("$50^\circ $",(9.4,8.8),SW);<br />
</asy><br />
<br />
<math>\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 </math><br />
<br />
[[1996 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
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?<br />
<br />
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4</math><br />
<br />
[[1996 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1996|before=[[1995 AJHSME Problems|1995 AJHSME]]|after=[[1997 AJHSME Problems|1997 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1996_AJHSME_Problems&diff=724741996 AJHSME Problems2015-10-15T16:13:57Z<p>Rep'na: /* Problem 23 */</p>
<hr />
<div>==Problem 1==<br />
<br />
How many positive factors of 36 are also multiples of 4?<br />
<br />
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math><br />
<br />
[[1996 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
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?<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
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<br />
<br />
<math>\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134</math><br />
<br />
[[1996 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
<math>\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=</math><br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
The letters <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, and <math>T</math> represent numbers located on the number line as shown.<br />
<br />
<asy><br />
unitsize(36);<br />
draw((-4,0)--(4,0));<br />
draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1));<br />
draw((3.9,0.1)--(4,0)--(3.9,-0.1));<br />
<br />
for (int i = -3; i <= 3; ++i)<br />
{<br />
draw((i,-0.1)--(i,0));<br />
}<br />
label("$-3$",(-3,-0.1),S);<br />
label("$-2$",(-2,-0.1),S);<br />
label("$-1$",(-1,-0.1),S);<br />
label("$0$",(0,-0.1),S);<br />
label("$1$",(1,-0.1),S);<br />
label("$2$",(2,-0.1),S);<br />
label("$3$",(3,-0.1),S);<br />
<br />
draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1));<br />
draw((-3.6,0)--(-3.6,0.25));<br />
label("$P$",(-3.6,0.25),N);<br />
draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1));<br />
draw((-1.2,0)--(-1.2,0.25));<br />
label("$Q$",(-1.2,0.25),N);<br />
draw((0.1,0.1)--(0.2,0)--(0.3,0.1));<br />
draw((0.2,0)--(0.2,0.25));<br />
label("$R$",(0.2,0.25),N);<br />
draw((0.8,0.1)--(0.9,0)--(1,0.1));<br />
draw((0.9,0)--(0.9,0.25));<br />
label("$S$",(0.9,0.25),N);<br />
draw((1.4,0.1)--(1.5,0)--(1.6,0.1));<br />
draw((1.5,0)--(1.5,0.25));<br />
label("$T$",(1.5,0.25),N);<br />
</asy><br />
<br />
Which of the following expressions represents a negative number?<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
What is the smallest result that can be obtained from the following process?<br />
<br />
*Choose three different numbers from the set <math>\{3,5,7,11,13,17\}</math>.<br />
*Add two of these numbers.<br />
*Multiply their sum by the third number.<br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56</math><br />
<br />
[[1996 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
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?<br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1996 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Points <math>A</math> and <math>B</math> are 10 units apart. Points <math>B</math> and <math>C</math> are 4 units apart. Points <math>C</math> and <math>D</math> are 3 units apart. If <math>A</math> and <math>D</math> are as close as possible, then the number of units between them is<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 17</math><br />
<br />
[[1996 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
If 5 times a number is 2, then 100 times the reciprocal of the number is<br />
<br />
<math>\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500</math><br />
<br />
[[1996 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
When Walter drove up to the gasoline pump, he noticed that his gasoline tank was 1/8 full. He purchased 7.5 gallons of gasoline for <math>\$10</math>. With this additional gasoline, his gasoline tank was then 5/8 full. The number of gallons of gasoline his tank holds when it is full is<br />
<br />
<math>\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5</math><br />
<br />
[[1996 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>x</math> be the number<br />
<cmath>0.\underbrace{0000...0000}_{1996\text{ zeros}}1,</cmath><br />
where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?<br />
<br />
<math>\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</math><br />
<br />
[[1996 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
What number should be removed from the list<br />
<cmath>1,2,3,4,5,6,7,8,9,10,11</cmath><br />
so that the average of the remaining numbers is <math>6.1</math>?<br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1996 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
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<br />
<br />
<math>\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700</math><br />
<br />
[[1996 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Six different digits from the set<br />
<cmath>\{ 1,2,3,4,5,6,7,8,9\}</cmath><br />
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.<br />
The sum of the six digits used is<br />
<br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(1,0)--(1,1)--(4,1)--(4,2)--(1,2)--(1,3)--(0,3)--cycle);<br />
draw((0,1)--(1,1)--(1,2)--(0,2));<br />
draw((2,1)--(2,2));<br />
draw((3,1)--(3,2));<br />
label("$23$",(0.5,0),S);<br />
label("$12$",(4,1.5),E);<br />
</asy><br />
<br />
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 35</math><br />
<br />
[[1996 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
The remainder when the product <math>1492\cdot 1776\cdot 1812\cdot 1996</math> is divided by 5 is<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4</math><br />
<br />
[[1996 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
<math>1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=</math><br />
<br />
<math>\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998</math><br />
<br />
[[1996 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Figure <math>OPQR</math> is a square. Point <math>O</math> is the origin, and point <math>Q</math> has coordinates (2,2). What are the coordinates for <math>T</math> so that the area of triangle <math>PQT</math> equals the area of square <math>OPQR</math>?<br />
<br />
<asy><br />
pair O,P,Q,R,T;<br />
O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0);<br />
draw((-5,0)--(3,0)); draw((0,-1)--(0,3));<br />
draw(P--Q--R);<br />
draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8));<br />
draw((-0.2,2.8)--(0,3)--(0.2,2.8));<br />
draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2));<br />
draw((2.8,-0.2)--(3,0)--(2.8,0.2));<br />
draw(Q--T);<br />
<br />
label("$O$",O,SW);<br />
label("$P$",P,S);<br />
label("$Q$",Q,NE);<br />
label("$R$",R,W);<br />
label("$T$",T,S);<br />
</asy><br />
<br />
<center>NOT TO SCALE</center><br />
<br />
<math>\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)</math><br />
<br />
[[1996 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
Ana's monthly salary was <dollar/>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<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
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<br />
<br />
<asy><br />
unitsize(18);<br />
draw(circle((0,0),4));<br />
draw(circle((9,0),4));<br />
draw((-4,0)--(0,0)--4*dir(352.8));<br />
draw((0,0)--4*dir(100.8));<br />
draw((5,0)--(9,0)--(4*dir(324)+(9,0)));<br />
draw((9,0)--(4*dir(50.4)+(9,0)));<br />
<br />
label("$48\%$",(0,-1),S);<br />
label("bowling",(0,-2),S);<br />
label("$30\%$",(1.5,1.5),N);<br />
label("golf",(1.5,0.5),N);<br />
label("$22\%$",(-2,1.5),N);<br />
label("tennis",(-2,0.5),N);<br />
<br />
label("$40\%$",(8.5,-1),S);<br />
label("tennis",(8.5,-2),S);<br />
label("$24\%$",(10.5,0.5),E);<br />
label("golf",(10.5,-0.5),E);<br />
label("$36\%$",(7.8,1.7),N);<br />
label("bowling",(7.8,0.7),N);<br />
<br />
label("$\textbf{East JHS}$",(0,-4),S);<br />
label("$\textbf{2000 students}$",(0,-5),S);<br />
label("$\textbf{West MS}$",(9,-4),S);<br />
label("$\textbf{2500 students}$",(9,-5),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%</math><br />
<br />
[[1996 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Suppose there is a special key on a calculator that replaces the number <math>x</math> currently displayed with the number given by the formula <math>1/(1-x)</math>. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since <math>1/(1-2)=-1</math>. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display <br />
<br />
<math>\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5</math><br />
<br />
[[1996 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
How many subsets containing three different numbers can be selected from the set <br />
<cmath>\{ 89,95,99,132, 166,173 \}</cmath><br />
so that the sum of the three numbers is even?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24</math><br />
<br />
[[1996 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
The horizontal and vertical distances between adjacent points equal 1 unit. The area of triangle <math>ABC</math> is<br />
<br />
<asy><br />
for (int a = 0; a < 5; ++a)<br />
{<br />
for (int b = 0; b < 4; ++b)<br />
{<br />
dot((a,b));<br />
}<br />
}<br />
draw((0,0)--(3,2)--(4,3)--cycle);<br />
label("$A$",(0,0),SW);<br />
label("$B$",(3,2),SE);<br />
label("$C$",(4,3),NE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4</math><br />
<br />
[[1996 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
The manager of a company planned to distribute a <math>\$50</math> bonus to each employee from the company fund, but the fund contained <math>\$5</math> less than what was needed. Instead the manager gave each employee a <math>\$45</math> bonus and kept the remaining <math>\$95</math> in the company fund. The amount of money in the company fund before any bonuses were paid was<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
The measure of angle <math>ABC</math> is <math>50^\circ </math>, <math>\overline{AD}</math> bisects angle <math>BAC</math>, and <math>\overline{DC}</math> bisects angle <math>BCA</math>. The measure of angle <math>ADC</math> is<br />
<br />
<asy><br />
pair A,B,C,D;<br />
A = (0,0); B = (9,10); C = (10,0); D = (6.66,3);<br />
dot(A); dot(B); dot(C); dot(D);<br />
draw(A--B--C--cycle);<br />
draw(A--D--C);<br />
<br />
label("$A$",A,SW);<br />
label("$B$",B,N);<br />
label("$C$",C,SE);<br />
label("$D$",D,N);<br />
label("$50^\circ $",(9.4,8.8),SW);<br />
</asy><br />
<br />
<math>\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 </math><br />
<br />
[[1996 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
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?<br />
<br />
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4</math><br />
<br />
[[1996 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1996|before=[[1995 AJHSME Problems|1995 AJHSME]]|after=[[1997 AJHSME Problems|1997 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1996_AJHSME_Problems&diff=724731996 AJHSME Problems2015-10-15T15:46:40Z<p>Rep'na: /* Problem 10 */</p>
<hr />
<div>==Problem 1==<br />
<br />
How many positive factors of 36 are also multiples of 4?<br />
<br />
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math><br />
<br />
[[1996 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
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?<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
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<br />
<br />
<math>\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134</math><br />
<br />
[[1996 AJHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
<math>\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=</math><br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
The letters <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, and <math>T</math> represent numbers located on the number line as shown.<br />
<br />
<asy><br />
unitsize(36);<br />
draw((-4,0)--(4,0));<br />
draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1));<br />
draw((3.9,0.1)--(4,0)--(3.9,-0.1));<br />
<br />
for (int i = -3; i <= 3; ++i)<br />
{<br />
draw((i,-0.1)--(i,0));<br />
}<br />
label("$-3$",(-3,-0.1),S);<br />
label("$-2$",(-2,-0.1),S);<br />
label("$-1$",(-1,-0.1),S);<br />
label("$0$",(0,-0.1),S);<br />
label("$1$",(1,-0.1),S);<br />
label("$2$",(2,-0.1),S);<br />
label("$3$",(3,-0.1),S);<br />
<br />
draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1));<br />
draw((-3.6,0)--(-3.6,0.25));<br />
label("$P$",(-3.6,0.25),N);<br />
draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1));<br />
draw((-1.2,0)--(-1.2,0.25));<br />
label("$Q$",(-1.2,0.25),N);<br />
draw((0.1,0.1)--(0.2,0)--(0.3,0.1));<br />
draw((0.2,0)--(0.2,0.25));<br />
label("$R$",(0.2,0.25),N);<br />
draw((0.8,0.1)--(0.9,0)--(1,0.1));<br />
draw((0.9,0)--(0.9,0.25));<br />
label("$S$",(0.9,0.25),N);<br />
draw((1.4,0.1)--(1.5,0)--(1.6,0.1));<br />
draw((1.5,0)--(1.5,0.25));<br />
label("$T$",(1.5,0.25),N);<br />
</asy><br />
<br />
Which of the following expressions represents a negative number?<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
What is the smallest result that can be obtained from the following process?<br />
<br />
*Choose three different numbers from the set <math>\{3,5,7,11,13,17\}</math>.<br />
*Add two of these numbers.<br />
*Multiply their sum by the third number.<br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56</math><br />
<br />
[[1996 AJHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
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?<br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1996 AJHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
Points <math>A</math> and <math>B</math> are 10 units apart. Points <math>B</math> and <math>C</math> are 4 units apart. Points <math>C</math> and <math>D</math> are 3 units apart. If <math>A</math> and <math>D</math> are as close as possible, then the number of units between them is<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 17</math><br />
<br />
[[1996 AJHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
If 5 times a number is 2, then 100 times the reciprocal of the number is<br />
<br />
<math>\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500</math><br />
<br />
[[1996 AJHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
When Walter drove up to the gasoline pump, he noticed that his gasoline tank was 1/8 full. He purchased 7.5 gallons of gasoline for <math>\$10</math>. With this additional gasoline, his gasoline tank was then 5/8 full. The number of gallons of gasoline his tank holds when it is full is<br />
<br />
<math>\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5</math><br />
<br />
[[1996 AJHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Let <math>x</math> be the number<br />
<cmath>0.\underbrace{0000...0000}_{1996\text{ zeros}}1,</cmath><br />
where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?<br />
<br />
<math>\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</math><br />
<br />
[[1996 AJHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
What number should be removed from the list<br />
<cmath>1,2,3,4,5,6,7,8,9,10,11</cmath><br />
so that the average of the remaining numbers is <math>6.1</math>?<br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math><br />
<br />
[[1996 AJHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
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<br />
<br />
<math>\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700</math><br />
<br />
[[1996 AJHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Six different digits from the set<br />
<cmath>\{ 1,2,3,4,5,6,7,8,9\}</cmath><br />
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.<br />
The sum of the six digits used is<br />
<br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(1,0)--(1,1)--(4,1)--(4,2)--(1,2)--(1,3)--(0,3)--cycle);<br />
draw((0,1)--(1,1)--(1,2)--(0,2));<br />
draw((2,1)--(2,2));<br />
draw((3,1)--(3,2));<br />
label("$23$",(0.5,0),S);<br />
label("$12$",(4,1.5),E);<br />
</asy><br />
<br />
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 35</math><br />
<br />
[[1996 AJHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
The remainder when the product <math>1492\cdot 1776\cdot 1812\cdot 1996</math> is divided by 5 is<br />
<br />
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4</math><br />
<br />
[[1996 AJHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
<math>1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=</math><br />
<br />
<math>\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998</math><br />
<br />
[[1996 AJHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Figure <math>OPQR</math> is a square. Point <math>O</math> is the origin, and point <math>Q</math> has coordinates (2,2). What are the coordinates for <math>T</math> so that the area of triangle <math>PQT</math> equals the area of square <math>OPQR</math>?<br />
<br />
<asy><br />
pair O,P,Q,R,T;<br />
O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0);<br />
draw((-5,0)--(3,0)); draw((0,-1)--(0,3));<br />
draw(P--Q--R);<br />
draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8));<br />
draw((-0.2,2.8)--(0,3)--(0.2,2.8));<br />
draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2));<br />
draw((2.8,-0.2)--(3,0)--(2.8,0.2));<br />
draw(Q--T);<br />
<br />
label("$O$",O,SW);<br />
label("$P$",P,S);<br />
label("$Q$",Q,NE);<br />
label("$R$",R,W);<br />
label("$T$",T,S);<br />
</asy><br />
<br />
<center>NOT TO SCALE</center><br />
<br />
<math>\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)</math><br />
<br />
[[1996 AJHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
Ana's monthly salary was <dollar/>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<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
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<br />
<br />
<asy><br />
unitsize(18);<br />
draw(circle((0,0),4));<br />
draw(circle((9,0),4));<br />
draw((-4,0)--(0,0)--4*dir(352.8));<br />
draw((0,0)--4*dir(100.8));<br />
draw((5,0)--(9,0)--(4*dir(324)+(9,0)));<br />
draw((9,0)--(4*dir(50.4)+(9,0)));<br />
<br />
label("$48\%$",(0,-1),S);<br />
label("bowling",(0,-2),S);<br />
label("$30\%$",(1.5,1.5),N);<br />
label("golf",(1.5,0.5),N);<br />
label("$22\%$",(-2,1.5),N);<br />
label("tennis",(-2,0.5),N);<br />
<br />
label("$40\%$",(8.5,-1),S);<br />
label("tennis",(8.5,-2),S);<br />
label("$24\%$",(10.5,0.5),E);<br />
label("golf",(10.5,-0.5),E);<br />
label("$36\%$",(7.8,1.7),N);<br />
label("bowling",(7.8,0.7),N);<br />
<br />
label("$\textbf{East JHS}$",(0,-4),S);<br />
label("$\textbf{2000 students}$",(0,-5),S);<br />
label("$\textbf{West MS}$",(9,-4),S);<br />
label("$\textbf{2500 students}$",(9,-5),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%</math><br />
<br />
[[1996 AJHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Suppose there is a special key on a calculator that replaces the number <math>x</math> currently displayed with the number given by the formula <math>1/(1-x)</math>. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since <math>1/(1-2)=-1</math>. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display <br />
<br />
<math>\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5</math><br />
<br />
[[1996 AJHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
How many subsets containing three different numbers can be selected from the set <br />
<cmath>\{ 89,95,99,132, 166,173 \}</cmath><br />
so that the sum of the three numbers is even?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24</math><br />
<br />
[[1996 AJHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
The horizontal and vertical distances between adjacent points equal 1 unit. The area of triangle <math>ABC</math> is<br />
<br />
<asy><br />
for (int a = 0; a < 5; ++a)<br />
{<br />
for (int b = 0; b < 4; ++b)<br />
{<br />
dot((a,b));<br />
}<br />
}<br />
draw((0,0)--(3,2)--(4,3)--cycle);<br />
label("$A$",(0,0),SW);<br />
label("$B$",(3,2),SE);<br />
label("$C$",(4,3),NE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4</math><br />
<br />
[[1996 AJHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
The manager of a company planned to distribute a <dollar/>50 bonus to each employee from the company fund, but the fund contained <dollar/>5 less than what was needed. Instead the manager gave each employee a <dollar/>45 bonus and kept the remaining <dollar/>95 in the company fund. The amount of money in the company fund before any bonuses were paid was<br />
<br />
<math>\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}</math><br />
<br />
[[1996 AJHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
The measure of angle <math>ABC</math> is <math>50^\circ </math>, <math>\overline{AD}</math> bisects angle <math>BAC</math>, and <math>\overline{DC}</math> bisects angle <math>BCA</math>. The measure of angle <math>ADC</math> is<br />
<br />
<asy><br />
pair A,B,C,D;<br />
A = (0,0); B = (9,10); C = (10,0); D = (6.66,3);<br />
dot(A); dot(B); dot(C); dot(D);<br />
draw(A--B--C--cycle);<br />
draw(A--D--C);<br />
<br />
label("$A$",A,SW);<br />
label("$B$",B,N);<br />
label("$C$",C,SE);<br />
label("$D$",D,N);<br />
label("$50^\circ $",(9.4,8.8),SW);<br />
</asy><br />
<br />
<math>\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 </math><br />
<br />
[[1996 AJHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
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?<br />
<br />
<math>\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4</math><br />
<br />
[[1996 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1996|before=[[1995 AJHSME Problems|1995 AJHSME]]|after=[[1997 AJHSME Problems|1997 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1992_AJHSME_Problems&diff=724101992 AJHSME Problems2015-10-09T14:56:45Z<p>Rep'na: /* Problem 4 */</p>
<hr />
<div>==Problem 1==<br />
<br />
<math>\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=</math><br />
<br />
<math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10</math><br />
<br />
[[1992 AJHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
Which of the following is not equal to <math>\dfrac{5}{4}</math>?<br />
<br />
<math>\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfrac{3}{12} \qquad \text{(D)}\ 1\dfrac{1}{5} \qquad \text{(E)}\ 1\dfrac{10}{40}</math><br />
<br />
[[1992 AJHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
What is the largest difference that can be formed by subtracting two numbers chosen from the set <math>\{ -16,-4,0,2,4,12 \}</math>?<br />
<br />
<math>\text{(A)}\ 10 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 28 \qquad \text{(E)}\ 48</math><br />
<br />
[[1992 AJHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
During the softball season, Judy had <math>35</math> hits. Among her hits were <math>1</math> home run, <math>1</math> triple and <math>5</math> doubles. The rest of her hits were singles. What percent of her hits were singles?<br />
<br />
<math>\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\% </math><br />
<br />
[[1992 AJHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
A circle of diameter <math>1</math> is removed from a <math>2\times 3</math> rectangle, as shown. Which whole number is closest to the area of the shaded region?<br />
<br />
<asy><br />
fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray);<br />
draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1));<br />
fill(circle((1,5/4),1/2),white);<br />
draw(circle((1,5/4),1/2),linewidth(1));<br />
</asy><br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math><br />
<br />
[[1992 AJHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
Suppose that <br />
<asy> <br />
unitsize(18);<br />
draw((0,0)--(2,0)--(1,sqrt(3))--cycle);<br />
label("$a$",(1,sqrt(3)-0.2),S);<br />
label("$b$",(sqrt(3)/10,0.1),ENE);<br />
label("$c$",(2-sqrt(3)/10,0.1),WNW);<br />
</asy><br />
means <math>a+b-c</math>.<br />
For example, <br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(2,0)--(1,sqrt(3))--cycle);<br />
label("$5$",(1,sqrt(3)-0.2),S);<br />
label("$4$",(sqrt(3)/10,0.1),ENE);<br />
label("$6$",(2-sqrt(3)/10,0.1),WNW);<br />
</asy><br />
is <math>5+4-6 = 3</math>.<br />
Then the sum <br />
<asy><br />
unitsize(18);<br />
draw((0,0)--(2,0)--(1,sqrt(3))--cycle);<br />
label("$1$",(1,sqrt(3)-0.2),S);<br />
label("$3$",(sqrt(3)/10,0.1),ENE);<br />
label("$4$",(2-sqrt(3)/10,0.1),WNW);<br />
draw((3,0)--(5,0)--(4,sqrt(3))--cycle);<br />
label("$2$",(4,sqrt(3)-0.2),S);<br />
label("$5$",(3+sqrt(3)/10,0.1),ENE);<br />
label("$6$",(5-sqrt(3)/10,0.1),WNW);<br />
label("$+$",(2.5,-0.1),N);<br />
</asy><br />
is<br />
<br />
<math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2</math><br />
<br />
[[1992 AJHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
The digit-sum of <math>998</math> is <math>9+9+8=26</math>. How many 3-digit whole numbers, whose digit-sum is <math>26</math>, are even?<br />
<br />
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math><br />
<br />
[[1992 AJHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
A store owner bought <math>1500</math> pencils at <math>\$0.10</math> each. If he sells them for <math>\$0.25</math> each, how many of them must he sell to make a profit of exactly <math>\$100.00</math>?<br />
<br />
<math>\text{(A)}\ 400 \qquad \text{(B)}\ 667 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 1500 \qquad \text{(E)}\ 1900</math><br />
<br />
[[1992 AJHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
The population of a small town is <math>480</math>. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town?<br />
<br />
<asy><br />
draw((0,13)--(0,0)--(20,0));<br />
<br />
draw((3,0)--(3,10)--(8,10)--(8,0));<br />
draw((3,5)--(8,5));<br />
draw((11,0)--(11,5)--(16,5)--(16,0));<br />
<br />
label("$\textbf{POPULATION}$",(10,11),N);<br />
label("$\textbf{F}$",(5.5,0),S); <br />
label("$\textbf{M}$",(13.5,0),S);<br />
</asy><br />
<br />
<math>\text{(A)}\ 120 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 360</math><br />
<br />
[[1992 AJHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
An isosceles right triangle with legs of length <math>8</math> is partitioned into <math>16</math> congruent triangles as shown. The shaded area is<br />
<br />
<asy><br />
for (int a=0; a <= 3; ++a)<br />
{<br />
for (int b=0; b <= 3-a; ++b)<br />
{<br />
fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey);<br />
}<br />
}<br />
for (int c=0; c <= 3; ++c)<br />
{<br />
draw((c,0)--(c,4-c),linewidth(1));<br />
draw((0,c)--(4-c,c),linewidth(1));<br />
draw((c+1,0)--(0,c+1),linewidth(1));<br />
}<br />
<br />
label("$8$",(2,0),S); <br />
label("$8$",(0,2),W);<br />
</asy><br />
<br />
<math>\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 64</math><br />
<br />
[[1992 AJHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
The bar graph shows the results of a survey on color preferences. What percent preferred blue?<br />
<br />
<asy><br />
for (int a = 1; a <= 6; ++a)<br />
{<br />
draw((-1.5,4*a)--(1.5,4*a));<br />
}<br />
draw((0,28)--(0,0)--(32,0));<br />
draw((3,0)--(3,20)--(6,20)--(6,0));<br />
draw((9,0)--(9,24)--(12,24)--(12,0));<br />
draw((15,0)--(15,16)--(18,16)--(18,0));<br />
draw((21,0)--(21,24)--(24,24)--(24,0));<br />
draw((27,0)--(27,16)--(30,16)--(30,0));<br />
<br />
label("$20$",(-1.5,8),W);<br />
label("$40$",(-1.5,16),W);<br />
label("$60$",(-1.5,24),W);<br />
<br />
label("$\textbf{COLOR SURVEY}$",(16,26),N);<br />
label("$\textbf{F}$",(-6,25),W);<br />
label("$\textbf{r}$",(-6.75,22.4),W);<br />
label("$\textbf{e}$",(-6.75,19.8),W);<br />
label("$\textbf{q}$",(-6.75,17.2),W);<br />
label("$\textbf{u}$",(-6.75,15),W);<br />
label("$\textbf{e}$",(-6.75,12.4),W);<br />
label("$\textbf{n}$",(-6.75,9.8),W);<br />
label("$\textbf{c}$",(-6.75,7.2),W);<br />
label("$\textbf{y}$",(-6.75,4.6),W);<br />
<br />
label("D",(4.5,.2),N);<br />
label("E",(4.5,3),N);<br />
label("R",(4.5,5.8),N);<br />
<br />
label("E",(10.5,.2),N);<br />
label("U",(10.5,3),N);<br />
label("L",(10.5,5.8),N);<br />
label("B",(10.5,8.6),N);<br />
<br />
label("N",(16.5,.2),N);<br />
label("W",(16.5,3),N);<br />
label("O",(16.5,5.8),N);<br />
label("R",(16.5,8.6),N);<br />
label("B",(16.5,11.4),N);<br />
<br />
label("K",(22.5,.2),N);<br />
label("N",(22.5,3),N);<br />
label("I",(22.5,5.8),N);<br />
label("P",(22.5,8.6),N);<br />
<br />
label("N",(28.5,.2),N);<br />
label("E",(28.5,3),N);<br />
label("E",(28.5,5.8),N);<br />
label("R",(28.5,8.6),N);<br />
label("G",(28.5,11.4),N);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20\% \qquad \text{(B)}\ 24\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 36\% \qquad \text{(E)}\ 42\% </math><br />
<br />
[[1992 AJHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first <math>30,000</math> miles the car traveled. For how many miles was each tire used?<br />
<br />
<math>\text{(A)}\ 6000 \qquad \text{(B)}\ 7500 \qquad \text{(C)}\ 24,000 \qquad \text{(D)}\ 30,000 \qquad \text{(E)}\ 37,500</math><br />
<br />
[[1992 AJHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
Five test scores have a mean (average score) of <math>90</math>, a median (middle score) of <math>91</math> and a mode (most frequent score) of <math>94</math>. The sum of the two lowest test scores is<br />
<br />
<math>\text{(A)}\ 170 \qquad \text{(B)}\ 171 \qquad \text{(C)}\ 176 \qquad \text{(D)}\ 177 \qquad \text{(E)}\ \text{not determined by the information given}</math><br />
<br />
[[1992 AJHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 48</math><br />
<br />
[[1992 AJHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
What is the <math>1992^\text{nd}</math> letter in this sequence?<br />
<br />
<cmath>\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\cdots </cmath><br />
<br />
<math>\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}</math><br />
<br />
[[1992 AJHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
<asy><br />
draw(ellipse((0,-5),10,3));<br />
fill((-10,-5)--(10,-5)--(10,5)--(-10,5)--cycle,white);<br />
draw(ellipse((0,0),10,3));<br />
draw((10,0)--(10,-5));<br />
draw((-10,0)--(-10,-5));<br />
<br />
draw((0,0)--(7,-3*sqrt(51)/10));<br />
label("10",(7/2,-3*sqrt(51)/20),NE);<br />
label("5",(-10,-3),E);<br />
</asy><br />
<br />
Which cylinder has twice the volume of the cylinder shown above?<br />
<br />
<asy><br />
unitsize(4);<br />
<br />
draw(ellipse((0,-5),20,6));<br />
fill((-20,-5)--(20,-5)--(20,5)--(-20,5)--cycle,white);<br />
draw(ellipse((0,0),20,6));<br />
draw((20,0)--(20,-5));<br />
draw((-20,0)--(-20,-5));<br />
draw((0,0)--(14,-3*sqrt(51)/5));<br />
label("20",(7,-3*sqrt(51)/10),NE);<br />
label("5",(-20,-4),E);<br />
label("(A)",(0,6),N);<br />
<br />
draw(ellipse((31,-7),10,3));<br />
fill((21,-7)--(41,-7)--(41,7)--(21,7)--cycle,white);<br />
draw(ellipse((31,3),10,3));<br />
draw((41,3)--(41,-7));<br />
draw((21,3)--(21,-7));<br />
draw((31,3)--(38,3-3*sqrt(51)/10));<br />
label("10",(34.5,3-3*sqrt(51)/20),NE);<br />
label("10",(21,-4),E);<br />
label("(B)",(31,6),N);<br />
<br />
draw(ellipse((47,-15.5),5,3/2));<br />
fill((42,-15.5)--(42,-15.5)--(42,15.5)--(42,15.5)--cycle,white);<br />
draw(ellipse((47,4.5),5,3/2));<br />
draw((42,4.5)--(42,-15.5));<br />
draw((52,4.5)--(52,-15.5));<br />
draw((47,4.5)--(50.5,4.5-3*sqrt(51)/20));<br />
label("5",(48.75,4.5-3*sqrt(51)/40),NE);<br />
label("10",(42,-6),E);<br />
label("(C)",(47,6),N);<br />
<br />
draw(ellipse((73,-10),20,6));<br />
fill((53,-10)--(93,-10)--(93,5)--(53,5)--cycle,white);<br />
draw(ellipse((73,0),20,6));<br />
draw((53,0)--(53,-10));<br />
draw((93,0)--(93,-10));<br />
draw((73,0)--(87,-3*sqrt(51)/5));<br />
label("20",(80,-3*sqrt(51)/10),NE);<br />
label("10",(53,-6),E);<br />
label("(D)",(73,6),N);<br />
</asy><br />
<br />
<math>\text{(E)}\ \text{None of the above}</math><br />
<br />
[[1992 AJHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
The sides of a triangle have lengths <math>6.5</math>, <math>10</math>, and <math>s</math>, where <math>s</math> is a whole number. What is the smallest possible value of <math>s</math>?<br />
<br />
<asy><br />
pair A,B,C;<br />
A=origin; B=(10,0); C=6.5*dir(15);<br />
dot(A); dot(B); dot(C);<br />
draw(B--A--C);<br />
draw(B--C,dashed);<br />
label("$6.5$",3.25*dir(15),NNW);<br />
label("$10$",(5,0),S);<br />
label("$s$",(8,1),NE);<br />
</asy><br />
<br />
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7</math><br />
<br />
[[1992 AJHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
On a trip, a car traveled <math>80</math> miles in an hour and a half, then was stopped in traffic for <math>30</math> minutes, then traveled <math>100</math> miles during the next <math>2</math> hours. What was the car's average speed in miles per hour for the <math>4</math>-hour trip?<br />
<br />
<math>\text{(A)}\ 45 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90</math><br />
<br />
[[1992 AJHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
The distance between the <math>5^\text{th}</math> and <math>26^\text{th}</math> exits on an interstate highway is <math>118</math> miles. If any two exits are at least <math>5</math> miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the <math>5^\text{th}</math> and <math>26^\text{th}</math> exits?<br />
<br />
<math>\text{(A)}\ 8 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 47 \qquad \text{(E)}\ 98</math><br />
<br />
[[1992 AJHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Which pattern of identical squares could NOT be folded along the lines shown to form a cube?<br />
<br />
<asy><br />
unitsize(12);<br />
<br />
draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle);<br />
draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3));<br />
<br />
draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle);<br />
draw((7,-1)--(8,-1)--(8,-2)--(7,-2));<br />
draw((9,-1)--(9,-2));<br />
draw((10,-1)--(10,-2));<br />
<br />
draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle);<br />
draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1));<br />
<br />
draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle);<br />
draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3));<br />
draw((24,-2)--(24,-3));<br />
<br />
draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle);<br />
draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1));<br />
draw((29,-1)--(29,-2));<br />
<br />
label("(A)",(0,-0.5),W);<br />
label("(B)",(7,-0.5),W);<br />
label("(C)",(14,-0.5),W);<br />
label("(D)",(21,-0.5),W);<br />
label("(E)",(28,-0.5),W);<br />
</asy><br />
<br />
[[1992 AJHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?<br />
<br />
<asy><br />
unitsize(12);<br />
<br />
fill((2,0)--(2,9)--(3,9)--(3,0)--cycle,lightgray);<br />
draw((3,0)--(3,9)--(2,9)--(2,0));<br />
draw((2,7)--(1,7)--(1,0));<br />
draw((2,8)--(3,8));<br />
draw((2,7)--(3,7));<br />
for (int a = 1; a <= 6; ++a)<br />
{<br />
draw((1,a)--(3,a));<br />
}<br />
<br />
fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,lightgray);<br />
draw((6,0)--(6,3)--(5,3)--(5,0));<br />
draw((5,3)--(5,5)--(4,5)--(4,0));<br />
draw((4,4)--(5,4));<br />
draw((4,3)--(5,3));<br />
draw((4,2)--(6,2));<br />
draw((4,1)--(6,1));<br />
<br />
fill((8,0)--(8,6)--(9,6)--(9,0)--cycle,lightgray);<br />
draw((9,0)--(9,6)--(8,6)--(8,0));<br />
draw((8,6)--(8,9)--(7,9)--(7,0));<br />
draw((7,8)--(8,8));<br />
draw((7,7)--(8,7));<br />
draw((7,6)--(8,6));<br />
for (int a = 1; a <= 5; ++a)<br />
{<br />
draw((7,a)--(9,a));<br />
}<br />
<br />
fill((11,0)--(11,12)--(12,12)--(12,0)--cycle,lightgray);<br />
draw((12,0)--(12,12)--(11,12)--(11,0));<br />
draw((11,9)--(10,9)--(10,0));<br />
draw((11,11)--(12,11));<br />
draw((11,10)--(12,10));<br />
draw((11,9)--(12,9));<br />
for (int a = 1; a <= 8; ++a)<br />
{<br />
draw((10,a)--(12,a));<br />
}<br />
<br />
fill((14,0)--(14,10)--(15,10)--(15,0)--cycle,lightgray);<br />
draw((15,0)--(15,10)--(14,10)--(14,0));<br />
draw((14,8)--(13,8)--(13,0));<br />
draw((14,9)--(15,9));<br />
draw((14,8)--(15,8));<br />
for (int a = 1; a <= 7; ++a)<br />
{<br />
draw((13,a)--(15,a));<br />
}<br />
<br />
draw((16,0)--(0,0)--(0,13),black);<br />
label("Jan",(2,0),S); <br />
label("Feb",(5,0),S);<br />
label("Mar",(8,0),S);<br />
label("Apr",(11,0),S);<br />
label("May",(14,0),S);<br />
label("$\textbf{MONTHLY SALES}$",(8,14),N);<br />
label("S",(0,8),W);<br />
label("A",(0,7),W);<br />
label("L",(0,6),W);<br />
label("E",(0,5),W);<br />
label("S",(0,4),W);<br />
<br />
draw((4,12.5)--(4,13.5)--(5,13.5)--(5,12.5)--cycle);<br />
label("Drums",(4,13),W);<br />
fill((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle,lightgray);<br />
draw((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle);<br />
label("Bugles",(15,13),W);<br />
</asy><br />
<br />
<math>\text{(A)}\ \text{Jan} \qquad \text{(B)}\ \text{Feb} \qquad \text{(C)}\ \text{Mar} \qquad \text{(D)}\ \text{Apr} \qquad \text{(E)}\ \text{May}</math><br />
<br />
[[1992 AJHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
Eight <math>1\times 1</math> square tiles are arranged as shown so their outside edges form a polygon with a perimeter of <math>14</math> units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?<br />
<br />
<asy><br />
for (int a=1; a <= 4; ++a)<br />
{<br />
draw((a,0)--(a,2));<br />
}<br />
draw((0,0)--(4,0));<br />
draw((0,1)--(5,1));<br />
draw((1,2)--(5,2));<br />
draw((0,0)--(0,1));<br />
draw((5,1)--(5,2));<br />
</asy><br />
<br />
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20</math><br />
<br />
[[1992 AJHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
If two dice are tossed, the probability that the product of the numbers showing on the tops of the dice is greater than <math>10</math> is<br />
<br />
<math>\text{(A)}\ \dfrac{3}{7} \qquad \text{(B)}\ \dfrac{17}{36} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{5}{8} \qquad \text{(E)}\ \dfrac{11}{12}</math><br />
<br />
[[1992 AJHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
Four circles of radius <math>3</math> are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to<br />
<br />
<asy><br />
fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray);<br />
fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white);<br />
fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white);<br />
fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white);<br />
fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white);<br />
<br />
draw(circle((3,3),3));<br />
draw(circle((3,-3),3));<br />
draw(circle((-3,-3),3));<br />
draw(circle((-3,3),3));<br />
draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle);<br />
</asy><br />
<br />
<math>\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27</math><br />
<br />
[[1992 AJHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?<br />
<br />
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10</math><br />
<br />
[[1992 AJHSME Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
{{AJHSME box|year=1992|before=[[1991 AJHSME Problems|1991 AJHSME]]|after=[[1993 AJHSME Problems|1993 AJHSME]]}}<br />
* [[AJHSME]]<br />
* [[AJHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
<br />
<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1991_AJHSME_Problems/Problem_19&diff=724061991 AJHSME Problems/Problem 192015-10-08T16:16:40Z<p>Rep'na: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
The average (arithmetic mean) of <math>10</math> different positive whole numbers is <math>10</math>. The largest possible value of any of these numbers is<br />
<br />
<math>\text{(A)}\ 10 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 55 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 91</math><br />
<br />
==Solution==<br />
<br />
If the average of the numbers is <math>10</math>, then their sum is <math>10\times 10=100</math>. <br />
<br />
To maximize the largest number of the ten, we minimize the other nine. Since they must be distinct, positive whole numbers, we let them be <math>1,2,3,4,5,6,7,8,9</math>. Their sum is <math>45</math>.<br />
<br />
The sum of nine of the numbers is <math>45</math>, and the sum of all ten is <math>100</math> so the last number must be <math>100-45=55\rightarrow \boxed{\text{C}}</math>.<br />
<br />
==See Also==<br />
<br />
{{AJHSME box|year=1991|num-b=18|num-a=20}}<br />
[[Category:Introductory Algebra Problems]]<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1987_AJHSME_Problems/Problem_24&diff=723591987 AJHSME Problems/Problem 242015-10-04T06:42:29Z<p>Rep'na: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>A multiple choice examination consists of <math>20</math> questions. The scoring is <math>+5</math> for each correct answer, <math>-2</math> for each incorrect answer, and <math>0</math> for each unanswered question. John's score on the examination is <math>48</math>. What is the maximum number of questions he could have answered correctly?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude><br />
<br />
<math>\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 16</math><br />
<br />
==Solution==<br />
===Solution 1===<br />
<br />
Let <math>c</math> be the number of questions correct, <math>w</math> be the number of questions wrong, and <math>b</math> be the number of questions left blank. We are given that <br />
<cmath>\begin{align}<br />
c+w+b &= 20 \\<br />
5c-2w &= 48 <br />
\end{align}</cmath><br />
<br />
Adding equation <math>(2)</math> to double equation <math>(1)</math>, we get <cmath>7c+2b=88</cmath><br />
<br />
Since we want to maximize the value of <math>c</math>, we try to find the largest multiple of <math>7</math> less than <math>88</math>. This is <math>84=7\times 12</math>, so let <math>c=12</math>. Then we have <cmath>7(12)+2b=88\Rightarrow b=2</cmath> <br />
<br />
Finally, we have <math>w=20-12-2=6</math>. We want <math>c</math>, so the answer is <math>12</math>, or <math>\boxed{\text{D}}</math>.<br />
<br />
===Solution 2===<br />
<br />
If John answered 16 questions correctly, then he answered at most 4 questions incorrectly, giving him at least <math>16 \cdot 5 - 4 \cdot 2 = 72</math> points. Therefore, John did not answer 16 questions correctly. If he answered 12 questions correctly and 6 questions incorrectly (leaving 2 questions unanswered), then he scored <math>12 \cdot 5 - 6 \cdot 2 = 48</math> points. As all other options are less than 12, we conclude that 12 is the most questions John could have answered correctly, and the answer is <math>\boxed{\text{D}}</math>.<br />
<br />
==See Also==<br />
<br />
{{AJHSME box|year=1987|num-b=23|num-a=25}}<br />
[[Category:Introductory Algebra Problems]]<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1999_AMC_8_Problems/Problem_24&diff=721991999 AMC 8 Problems/Problem 242015-09-27T17:19:06Z<p>Rep'na: </p>
<hr />
<div>==Problem==<br />
<br />
When <math>1999^{2000}</math> is divided by <math>5</math>, the remainder is <br />
<br />
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 0</math><br />
<br />
<br />
==Solution 1==<br />
Note that the units digits of the powers of 9 have a pattern: <math>9^1 = {\bf 9}</math>,<math>9^2 = 8{\bf 1}</math>,<math>9^3 = 72{\bf 9}</math>,<math>9^4 = 656{\bf 1}</math>, and so on. Since all natural numbers with the same last digit have the same remainder when divided by 5, the entire number doesn't matter, just the last digit. For even powers of <math>9</math>, the number ends in a <math>1</math>. Since the exponent is even, the final digit is <math>1</math>. Note that all natural numbers that end in <math>1</math> have a remainder of <math>1</math> when divided by <math>5</math>. So, our answer is <math>\boxed{\text{(D)}\ 1}</math>.<br />
<br />
==Solution 2==<br />
<br />
Write <math>1999</math> as <math>2000-1</math>. We are taking <math>(2000-1)^{2000} \mod{10}</math>. Using the binomial theorem, we see that ALL terms in this expansion are divisible by <math>2000</math> except for the very last term, which is just <math>(-1)^{2000}</math>. This is clear because the binomial expansion is just choosing how many <math>2000</math>s and how many <math>-1</math>s there are for each term. Using this, we can take the entire polynomial <math>\mod{10}</math>, which leaves just <math>(-1)^{2000}=\boxed{\text{(D)}\ 1}</math>.<br />
<br />
==Solution 3==<br />
As <math>1999 \equiv -1 \pmod{5}</math>, we have <math>1999^{2000} \equiv (-1)^{2000} \equiv 1 \pmod{5}</math>. Thus, the answer is <math>\boxed{\text{(D)}\ 1}</math>.<br />
<br />
==See Also==<br />
{{AMC8 box|year=1999|num-b=23|num-a=25}}<br />
{{MAA Notice}}</div>Rep'nahttps://artofproblemsolving.com/wiki/index.php?title=1999_AMC_8_Problems/Problem_21&diff=721981999 AMC 8 Problems/Problem 212015-09-27T16:56:45Z<p>Rep'na: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
The degree measure of angle <math>A</math> is<br />
<br />
<asy><br />
unitsize(12);<br />
draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle);<br />
draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW));<br />
draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW));<br />
draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW));<br />
label(rotate(30)*"$40^\circ$",(2,-8.9),ENE);<br />
label("$100^\circ$",(21/3,-2/3),SE);<br />
label("$110^\circ$",(900/83,-317/83),NNW);<br />
label("$A$",(0,0),NW);<br />
</asy><br />
<br />
<math>\text{(A)}\ 20 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 45</math><br />
<br />
==Solution==<br />
===Solution 1===<br />
<br />
Angle-chasing using the small triangles:<br />
<br />
Use the line below and to the left of the <math>110^\circ</math> angle to find that the rightmost angle in the small lower-left triangle is <math>180 - 110 = 70^\circ</math>.<br />
<br />
Then use the small lower-left triangle to find that the remaining angle in that triangle is <math>180 - 70 - 40 = 70^\circ</math>.<br />
<br />
Use congruent vertical angles to find that the lower angle in the smallest triangle containing <math>A</math> is also <math>70^\circ</math>. <br />
<br />
Next, use line segment <math>AB</math> to find that the other angle in the smallest triangle containing <math>A</math> is <math>180 - 100 = 80^\circ</math>.<br />
<br />
The small triangle containing <math>A</math> has a <math>70^\circ</math> angle and an <math>80^\circ</math> angle. The remaining angle must be <math>180 - 70 - 80 = \boxed{30^\circ, B}</math><br />
<br />
===Solution 2===<br />
The third angle of the triangle containing the <math>100^\circ</math> angle and the <math>40^\circ</math> angle is <math>180^\circ - 100^\circ - 40^\circ = 40^\circ</math>. It follows that <math>A</math> is the third angle of the triangle consisting of the found <math>40^\circ</math> angle and the given <math>110^\circ</math> angle. Thus, <math>A</math> is a <math>180^\circ - 110^\circ - 40^\circ = 30^\circ</math> angle, and so the answer is <math>\boxed{30^\circ, \textbf{B}}</math>.<br />
<br />
==See Also==<br />
{{AMC8 box|year=1999|num-b=20|num-a=22}}<br />
{{MAA Notice}}</div>Rep'na