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Difference between revisions of "1996 AJHSME Problems"
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== Problem 2 ==
 
== 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?
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 2|Solution]]
 
[[1996 AJHSME Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==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
 +
 +
<math>\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134</math>
  
 
[[1996 AJHSME Problems/Problem 3|Solution]]
 
[[1996 AJHSME Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
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 +
<math>\dfrac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=</math>
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 4|Solution]]
 
[[1996 AJHSME Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
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 +
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.
 +
 +
{{image}}
 +
 +
Which of the following expressions represents a negative number?
 +
 +
<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>
  
 
[[1996 AJHSME Problems/Problem 5|Solution]]
 
[[1996 AJHSME Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
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 +
What is the smallest result that can be obtained from the following process?
 +
 +
*Choose three different numbers from the set <math>\{3,5,7,11,13,17\}</math>.
 +
*Add two of these numbers.
 +
*Multiply their sum by the third number.
 +
 +
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56</math>
  
 
[[1996 AJHSME Problems/Problem 6|Solution]]
 
[[1996 AJHSME Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
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 +
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?
 +
 +
<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 7|Solution]]
 
[[1996 AJHSME Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
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 +
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
 +
 +
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 17</math>
  
 
[[1996 AJHSME Problems/Problem 8|Solution]]
 
[[1996 AJHSME Problems/Problem 8|Solution]]

Revision as of 10:37, 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

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

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
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