Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1996 AJHSME Problems"
Line 72: Line 72:
  
 
==Problem 9==
 
==Problem 9==
 +
 +
If 5 times a number is 2, then 100 times the reciprocal of the number is
 +
 +
<math>\text{(A)}\ 2.5 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 250 \qquad \text{(E)}\ 500</math>
  
 
[[1996 AJHSME Problems/Problem 9|Solution]]
 
[[1996 AJHSME Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
 +
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 <dollar/>10.  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
 +
 +
<math>\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5</math>
  
 
[[1996 AJHSME Problems/Problem 10|Solution]]
 
[[1996 AJHSME Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
 +
Let <math>x</math> be the number
 +
<cmath>0.\underbrace{0000...0000}_{1996\text{ zeros}}1,</cmath>
 +
where there are 1996 zeros after the decimal point.  Which of the following expressions represents the largest number?
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 11|Solution]]
 
[[1996 AJHSME Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
 +
What number should be removed from the list
 +
<cmath>1,2,3,4,5,6,7,8,9,10,11</cmath>
 +
so that the average of the remaining numbers is <math>6.1</math>?
 +
 +
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math>
  
 
[[1996 AJHSME Problems/Problem 12|Solution]]
 
[[1996 AJHSME Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
 +
In the fall of 1996, a total of 800 students participated in an annual school clean-up day.  The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year.  The number of participants the organizers will expect in the fall of 1999 is
 +
 +
<math>\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700</math>
  
 
[[1996 AJHSME Problems/Problem 13|Solution]]
 
[[1996 AJHSME Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
 +
Six different digits from the set
 +
<cmath>\{ 1,2,3,4,5,6,7,8,9\}</cmath>
 +
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.
 +
The sum of the six digits used is
 +
 +
{{image}}
 +
 +
<math>\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 35</math>
  
 
[[1996 AJHSME Problems/Problem 14|Solution]]
 
[[1996 AJHSME Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
 +
The remainder when the product <math>1492\cdot 1776\cdot 1812\cdot 1996</math> is divided by 5 is
 +
 +
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4</math>
  
 
[[1996 AJHSME Problems/Problem 15|Solution]]
 
[[1996 AJHSME Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
 +
<math>1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=</math>
 +
 +
<math>\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998</math>
  
 
[[1996 AJHSME Problems/Problem 16|Solution]]
 
[[1996 AJHSME Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
 +
 +
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>?
 +
 +
{{image}}
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 17|Solution]]
 
[[1996 AJHSME Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
 +
 +
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
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 18|Solution]]
 
[[1996 AJHSME Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
 +
 +
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School.  The total number of students at East is 2000 and at West, 2500.  In the two schools combined, the percent of students who prefer tennis is
 +
 +
{{image}}
 +
 +
<math>\text{(A)}\ 30\% \qquad \text{(B)}\ 31\% \qquad \text{(C)}\ 32\% \qquad \text{(D)}\ 33\% \qquad \text{(E)}\ 34\%</math>
  
 
[[1996 AJHSME Problems/Problem 19|Solution]]
 
[[1996 AJHSME Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
 +
 +
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
 +
 +
<math>\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5</math>
  
 
[[1996 AJHSME Problems/Problem 20|Solution]]
 
[[1996 AJHSME Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
 +
 +
How many subsets containing three different numbers can be selected from the set
 +
<cmath>\{ 89,95,99,132, 166,173 \}</cmath>
 +
so that the sum of the three numbers is even?
 +
 +
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24</math>
  
 
[[1996 AJHSME Problems/Problem 21|Solution]]
 
[[1996 AJHSME Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
 +
 +
The horizontal and vertical distances between adjacent points equal 1 unit.  The area of triangle <math>ABC</math> is
 +
 +
{{image}}
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 22|Solution]]
 
[[1996 AJHSME Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
 +
 +
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
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 23|Solution]]
 
[[1996 AJHSME Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
 +
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
 +
 +
{{image}}
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 24|Solution]]
 
[[1996 AJHSME Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
 +
 +
A point is chosen at random from within a circular region.  What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 25|Solution]]
 
[[1996 AJHSME Problems/Problem 25|Solution]]

Revision as of 15:41, 17 August 2010

Problem 1

How many positive factors of 36 are also multiples of 4?

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

Solution

Problem 2

Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?

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

Solution

Problem 3

The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be

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

Solution

Problem 4

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

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

Solution

Problem 5

The letters $P$, $Q$, $R$, $S$, and $T$ represent numbers located on the number line as shown.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

Which of the following expressions represents a negative number?

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

Solution

Problem 6

What is the smallest result that can be obtained from the following process?

  • Choose three different numbers from the set $\{3,5,7,11,13,17\}$.
  • Add two of these numbers.
  • Multiply their sum by the third number.

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

Solution

Problem 7

Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?

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

Solution

Problem 8

Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is

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

Solution

Problem 9

If 5 times a number is 2, then 100 times the reciprocal of the number is

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

Solution

Problem 10

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 <dollar/>10. 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

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

Solution

Problem 11

Let $x$ be the number \[0.\underbrace{0000...0000}_{1996\text{ zeros}}1,\] where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?

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

Solution

Problem 12

What number should be removed from the list \[1,2,3,4,5,6,7,8,9,10,11\] so that the average of the remaining numbers is $6.1$?

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

Solution

Problem 13

In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers will expect in the fall of 1999 is

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

Solution

Problem 14

Six different digits from the set \[\{ 1,2,3,4,5,6,7,8,9\}\] are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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

Solution

Problem 15

The remainder when the product $1492\cdot 1776\cdot 1812\cdot 1996$ is divided by 5 is

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

Solution

Problem 16

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

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

Solution

Problem 17

Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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

Solution

Problem 18

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

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

Solution

Problem 19

The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis is

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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

Solution

Problem 20

Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display

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

Solution

Problem 21

How many subsets containing three different numbers can be selected from the set \[\{ 89,95,99,132, 166,173 \}\] so that the sum of the three numbers is even?

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

Solution

Problem 22

The horizontal and vertical distances between adjacent points equal 1 unit. The area of triangle $ABC$ is

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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

Solution

Problem 23

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

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

Solution

Problem 24

The measure of angle $ABC$ is $50^\circ$, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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

Solution

Problem 25

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

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

Solution

See also

1996 AJHSME (Problems • Answer Key • Resources)
Preceded by
1995 AJHSME 		Followed by
1997 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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1996_AJHSME_Problems&oldid=35659"