Difference between revisions of "1993 AJHSME Problems"
5849206328x (talk | contribs) |
(→Problem 1) |
||
(13 intermediate revisions by 8 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AJHSME Problems | ||
+ | |year = 1993 | ||
+ | }} | ||
==Problem 1== | ==Problem 1== | ||
Line 35: | Line 38: | ||
Which one of the following bar graphs could represent the data from the circle graph? | Which one of the following bar graphs could represent the data from the circle graph? | ||
− | { | + | <asy> |
+ | unitsize(36); | ||
+ | draw(circle((0,0),1),gray); | ||
+ | fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray); | ||
+ | fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black); | ||
+ | </asy> | ||
+ | |||
+ | <asy> | ||
+ | unitsize(4); | ||
+ | |||
+ | fill((1,0)--(1,15)--(5,15)--(5,0)--cycle,gray); | ||
+ | fill((6,0)--(6,15)--(10,15)--(10,0)--cycle,black); | ||
+ | draw((11,0)--(11,20)--(15,20)--(15,0)); | ||
+ | |||
+ | fill((26,0)--(26,15)--(30,15)--(30,0)--cycle,gray); | ||
+ | fill((31,0)--(31,15)--(35,15)--(35,0)--cycle,black); | ||
+ | draw((36,0)--(36,15)--(40,15)--(40,0)); | ||
+ | |||
+ | fill((51,0)--(51,10)--(55,10)--(55,0)--cycle,gray); | ||
+ | fill((56,0)--(56,10)--(60,10)--(60,0)--cycle,black); | ||
+ | draw((61,0)--(61,20)--(65,20)--(65,0)); | ||
+ | |||
+ | fill((76,0)--(76,10)--(80,10)--(80,0)--cycle,gray); | ||
+ | fill((81,0)--(81,15)--(85,15)--(85,0)--cycle,black); | ||
+ | draw((86,0)--(86,20)--(90,20)--(90,0)); | ||
+ | |||
+ | fill((101,0)--(101,15)--(105,15)--(105,0)--cycle,gray); | ||
+ | fill((106,0)--(106,10)--(110,10)--(110,0)--cycle,black); | ||
+ | draw((111,0)--(111,20)--(115,20)--(115,0)); | ||
+ | |||
+ | for(int a = 0; a < 5; ++a) | ||
+ | { | ||
+ | draw((25*a,21)--(25*a,0)--(25*a+16,0)); | ||
+ | } | ||
+ | |||
+ | label("(A)",(8,21),N); | ||
+ | label("(B)",(33,21),N); | ||
+ | label("(C)",(58,21),N); | ||
+ | label("(D)",(83,21),N); | ||
+ | label("(E)",(108,21),N); | ||
+ | </asy> | ||
[[1993 AJHSME Problems/Problem 5|Solution]] | [[1993 AJHSME Problems/Problem 5|Solution]] | ||
Line 43: | Line 86: | ||
A can of soup can feed <math>3</math> adults or <math>5</math> children. If there are <math>5</math> cans of soup and <math>15</math> children are fed, then how many adults would the remaining soup feed? | A can of soup can feed <math>3</math> adults or <math>5</math> children. If there are <math>5</math> cans of soup and <math>15</math> children are fed, then how many adults would the remaining soup feed? | ||
− | <math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ | + | <math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 25</math> |
[[1993 AJHSME Problems/Problem 6|Solution]] | [[1993 AJHSME Problems/Problem 6|Solution]] | ||
Line 85: | Line 128: | ||
This line graph represents the price of a trading card during the first <math>6</math> months of <math>1993</math>. | This line graph represents the price of a trading card during the first <math>6</math> months of <math>1993</math>. | ||
− | {{ | + | <asy> |
+ | unitsize(18); | ||
+ | for (int a = 0; a <= 6; ++a) | ||
+ | { | ||
+ | draw((4*a,0)--(4*a,10)); | ||
+ | } | ||
+ | for (int a = 0; a <= 5; ++a) | ||
+ | { | ||
+ | draw((0,2*a)--(24,2*a)); | ||
+ | } | ||
+ | draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5)); | ||
+ | |||
+ | label("$Jan$",(2,0),S); | ||
+ | label("$Feb$",(6,0),S); | ||
+ | label("$Mar$",(10,0),S); | ||
+ | label("$Apr$",(14,0),S); | ||
+ | label("$May$",(18,0),S); | ||
+ | label("$Jun$",(22,0),S); | ||
+ | label("$\textbf{1993 PRICES FOR A TRADING CARD}$",(12,10),N); | ||
+ | |||
+ | label("$\begin{tabular}{c}\textbf{P} <br /> \textbf{R} <br /> \textbf{I} <br /> \textbf{C} <br /> \textbf{E} \end{tabular}$",(-2,5),W); | ||
+ | label("$1$",(0,2),W); | ||
+ | label("$2$",(0,4),W); | ||
+ | label("$3$",(0,6),W); | ||
+ | label("$4$",(0,8),W); | ||
+ | label("$5$",(0,10),W); | ||
+ | </asy> | ||
The greatest monthly drop in price occurred during | The greatest monthly drop in price occurred during | ||
Line 97: | Line 166: | ||
Consider this histogram of the scores for <math>81</math> students taking a test: | Consider this histogram of the scores for <math>81</math> students taking a test: | ||
− | {{ | + | <asy> |
+ | unitsize(12); | ||
+ | draw((0,0)--(26,0)); | ||
+ | draw((1,1)--(25,1)); | ||
+ | draw((3,2)--(25,2)); | ||
+ | draw((5,3)--(23,3)); | ||
+ | draw((5,4)--(21,4)); | ||
+ | draw((7,5)--(21,5)); | ||
+ | draw((9,6)--(21,6)); | ||
+ | draw((11,7)--(19,7)); | ||
+ | draw((11,8)--(19,8)); | ||
+ | draw((11,9)--(19,9)); | ||
+ | draw((11,10)--(19,10)); | ||
+ | draw((13,11)--(19,11)); | ||
+ | draw((13,12)--(19,12)); | ||
+ | draw((13,13)--(17,13)); | ||
+ | draw((13,14)--(17,14)); | ||
+ | draw((15,15)--(17,15)); | ||
+ | draw((15,16)--(17,16)); | ||
+ | |||
+ | draw((1,0)--(1,1)); | ||
+ | draw((3,0)--(3,2)); | ||
+ | draw((5,0)--(5,4)); | ||
+ | draw((7,0)--(7,5)); | ||
+ | draw((9,0)--(9,6)); | ||
+ | draw((11,0)--(11,10)); | ||
+ | draw((13,0)--(13,14)); | ||
+ | draw((15,0)--(15,16)); | ||
+ | draw((17,0)--(17,16)); | ||
+ | draw((19,0)--(19,12)); | ||
+ | draw((21,0)--(21,6)); | ||
+ | draw((23,0)--(23,3)); | ||
+ | draw((25,0)--(25,2)); | ||
+ | |||
+ | for (int a = 1; a < 13; ++a) | ||
+ | { | ||
+ | draw((2*a,-.25)--(2*a,.25)); | ||
+ | } | ||
+ | |||
+ | label("$40$",(2,-.25),S); | ||
+ | label("$45$",(4,-.25),S); | ||
+ | label("$50$",(6,-.25),S); | ||
+ | label("$55$",(8,-.25),S); | ||
+ | label("$60$",(10,-.25),S); | ||
+ | label("$65$",(12,-.25),S); | ||
+ | label("$70$",(14,-.25),S); | ||
+ | label("$75$",(16,-.25),S); | ||
+ | label("$80$",(18,-.25),S); | ||
+ | label("$85$",(20,-.25),S); | ||
+ | label("$90$",(22,-.25),S); | ||
+ | label("$95$",(24,-.25),S); | ||
+ | |||
+ | label("$1$",(2,1),N); | ||
+ | label("$2$",(4,2),N); | ||
+ | label("$4$",(6,4),N); | ||
+ | label("$5$",(8,5),N); | ||
+ | label("$6$",(10,6),N); | ||
+ | label("$10$",(12,10),N); | ||
+ | label("$14$",(14,14),N); | ||
+ | label("$16$",(16,16),N); | ||
+ | label("$12$",(18,12),N); | ||
+ | label("$6$",(20,6),N); | ||
+ | label("$3$",(22,3),N); | ||
+ | label("$2$",(24,2),N); | ||
+ | |||
+ | label("Number",(4,8),N); | ||
+ | label("of Students",(4,7),N); | ||
+ | |||
+ | label("$\textbf{STUDENT TEST SCORES}$",(14,18),N); | ||
+ | </asy> | ||
The median is in the interval labeled | The median is in the interval labeled | ||
Line 121: | Line 259: | ||
The word "'''HELP'''" in block letters is painted in black with strokes <math>1</math> unit wide on a <math>5</math> by <math>15</math> rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is | The word "'''HELP'''" in block letters is painted in black with strokes <math>1</math> unit wide on a <math>5</math> by <math>15</math> rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is | ||
− | {{ | + | <asy> |
+ | unitsize(12); | ||
+ | fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black); | ||
+ | fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black); | ||
+ | fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black); | ||
+ | fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black); | ||
+ | fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white); | ||
+ | draw((0,0)--(15,0)--(15,5)--(0,5)--cycle); | ||
+ | label("$5\left\{ \begin{tabular}{c} <br /> <br /> <br /> <br /> \end{tabular}\right.$",(1,2.5),W); | ||
+ | label(rotate(90)*"$\{$",(0.5,0.1),S); | ||
+ | label("$1$",(0.5,-0.6),S); | ||
+ | label(rotate(90)*"$\{$",(3.5,0.1),S); | ||
+ | label("$1$",(3.5,-0.6),S); | ||
+ | label(rotate(90)*"$\{$",(7.5,0.1),S); | ||
+ | label("$1$",(7.5,-0.6),S); | ||
+ | label(rotate(90)*"$\{$",(11.5,0.1),S); | ||
+ | label("$1$",(11.5,-0.6),S); | ||
+ | label(rotate(270)*"$\left\{ \begin{tabular}{c} <br /> <br /> \end{tabular}\right.$",(1.5,4),N); | ||
+ | label("$3$",(1.5,5.8),N); | ||
+ | label(rotate(270)*"$\left\{ \begin{tabular}{c} <br /> <br /> \end{tabular}\right.$",(5.5,4),N); | ||
+ | label("$3$",(5.5,5.8),N); | ||
+ | label(rotate(270)*"$\left\{ \begin{tabular}{c} <br /> <br /> \end{tabular}\right.$",(9.5,4),N); | ||
+ | label("$3$",(9.5,5.8),N); | ||
+ | label(rotate(270)*"$\left\{ \begin{tabular}{c} <br /> <br /> \end{tabular}\right.$",(13.5,4),N); | ||
+ | label("$3$",(13.5,5.8),N); | ||
+ | label("$\left. \begin{tabular}{c} <br /> \end{tabular}\right\} 2$",(14,1),E); | ||
+ | </asy> | ||
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38</math> | <math>\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38</math> | ||
Line 161: | Line 325: | ||
Square corners, <math>5</math> units on a side, are removed from a <math>20</math> unit by <math>30</math> unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is | Square corners, <math>5</math> units on a side, are removed from a <math>20</math> unit by <math>30</math> unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is | ||
− | + | <asy> | |
+ | fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); | ||
+ | fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); | ||
+ | draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); | ||
+ | draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); | ||
+ | draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); | ||
+ | draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); | ||
+ | draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); | ||
+ | </asy> | ||
<math>\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000</math> | <math>\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000</math> | ||
Line 171: | Line 343: | ||
The rectangle shown has length <math>AC=32</math>, width <math>AE=20</math>, and <math>B</math> and <math>F</math> are midpoints of <math>\overline{AC}</math> and <math>\overline{AE}</math>, respectively. The area of quadrilateral <math>ABDF</math> is | The rectangle shown has length <math>AC=32</math>, width <math>AE=20</math>, and <math>B</math> and <math>F</math> are midpoints of <math>\overline{AC}</math> and <math>\overline{AE}</math>, respectively. The area of quadrilateral <math>ABDF</math> is | ||
− | + | <asy> | |
+ | pair A,B,C,D,EE,F; | ||
+ | A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10); | ||
+ | draw(A--C--D--EE--cycle); | ||
+ | draw(B--D--F); | ||
+ | dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,N); | ||
+ | label("$C$",C,NE); | ||
+ | label("$D$",D,SE); | ||
+ | label("$E$",EE,SW); | ||
+ | label("$F$",F,W); | ||
+ | </asy> | ||
<math>\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340</math> | <math>\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340</math> | ||
Line 221: | Line 405: | ||
What number is directly above <math>142</math> in this array of numbers? | What number is directly above <math>142</math> in this array of numbers? | ||
− | <cmath>\begin{ | + | <cmath>\begin{array}{cccccc} |
& & & 1 & & \\ | & & & 1 & & \\ | ||
& & 2 & 3 & 4 & \\ | & & 2 & 3 & 4 & \\ | ||
& 5 & 6 & 7 & 8 & 9 \\ | & 5 & 6 & 7 & 8 & 9 \\ | ||
10 & 11 & 12 & \cdots & & \\ | 10 & 11 & 12 & \cdots & & \\ | ||
− | \end{ | + | \end{array}</cmath> |
<math>\text{(A)}\ 99 \qquad \text{(B)}\ 119 \qquad \text{(C)}\ 120 \qquad \text{(D)}\ 121 \qquad \text{(E)}\ 122</math> | <math>\text{(A)}\ 99 \qquad \text{(B)}\ 119 \qquad \text{(C)}\ 120 \qquad \text{(D)}\ 121 \qquad \text{(E)}\ 122</math> | ||
Line 245: | Line 429: | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | |||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 00:31, 19 September 2020
1993 AJHSME (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
Which pair of numbers does NOT have a product equal to ?
Problem 2
When the fraction is expressed in simplest form, then the sum of the numerator and the denominator will be
Problem 3
Which of the following numbers has the largest prime factor?
Problem 4
Problem 5
Which one of the following bar graphs could represent the data from the circle graph?
Problem 6
A can of soup can feed adults or children. If there are cans of soup and children are fed, then how many adults would the remaining soup feed?
Problem 7
Problem 8
To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains pills, then the supply of medicine would last approximately
Problem 9
Consider the operation defined by the following table:
For example, . Then
Problem 10
This line graph represents the price of a trading card during the first months of .
The greatest monthly drop in price occurred during
Problem 11
Consider this histogram of the scores for students taking a test:
The median is in the interval labeled
Problem 12
If each of the three operation signs, , , , is used exactly ONCE in one of the blanks in the expression
then the value of the result could equal
Problem 13
The word "HELP" in block letters is painted in black with strokes unit wide on a by rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is
Problem 14
The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers . Then
Problem 15
The arithmetic mean (average) of four numbers is . If the largest of these numbers is , then the mean of the remaining three numbers is
Problem 16
Problem 17
Square corners, units on a side, are removed from a unit by unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is
Problem 18
The rectangle shown has length , width , and and are midpoints of and , respectively. The area of quadrilateral is
Problem 19
Problem 20
When is expressed as a single whole number, the sum of the digits is
Problem 21
If the length of a rectangle is increased by and its width is increased by , then the area is increased by
Problem 22
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?
Problem 23
Five runners, , , , , , have a race, and beats , beats , beats , and finishes after and before . Who could NOT have finished third in the race?
Problem 24
What number is directly above in this array of numbers?
Problem 25
A checkerboard consists of one-inch squares. A square card, inches on a side, is placed on the board so that it covers part or all of the area of each of squares. The maximum possible value of is
See also
1993 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by 1992 AJHSME |
Followed by 1994 AJHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.