Difference between revisions of "2000 AMC 8 Problems"

(New page: ==Problem 1== Aunt Anna is <math>42</math> years old. Caitlin is <math>5</math> years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? <math> \mathrm{(A)...)
 
Line 46: Line 46:
 
\mathrm{(D)}\ 5
 
\mathrm{(D)}\ 5
 
\qquad
 
\qquad
\mathrm{(E)}</math> infinitely many
+
\mathrm{(E)}\ infinitely\ many
 +
</math>
  
 
[[2000 AMC 8 Problems/Problem 3|Solution]]
 
[[2000 AMC 8 Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
In <math>1960</math> only <math>5%</math> of the working adults in Carlin City worked at home. By <math>1970</math> the "at-home" work force had increased to <math>8%</math>. In <math>1980</math> there were approximately <math>15%</math> working at home, and in <math>1990</math> there were <math>30%</math>. The graph that best illustrates this is:
+
In <math>1960</math> only <math>5\%</math> of the working adults in Carlin City worked at home. By <math>1970</math> the "at-home" work force had increased to <math>8\%</math>. In <math>1980</math> there were approximately <math>15\%</math> working at home, and in <math>1990</math> there were <math>30\%</math>. The graph that best illustrates this is:
 
 
<math>
 
</math>
 
  
 
[[2000 AMC 8 Problems/Problem 4|Solution]]
 
[[2000 AMC 8 Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
Each principal of Lincoln High School serves exactly one 3-year term. What is the maximum number of principals this school could have during an 8 year period?
+
Each principal of Lincoln High School serves exactly one <math>3</math>-year term. What is the maximum number of principals this school could have during an <math>8</math>-year period?
  
 
<math>
 
<math>
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==Problem 6==
 
==Problem 6==
Figure <math>ABCD</math> is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is
+
Figure <math>ABCD</math> is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded <math>L</math>-shaped region is
  
 
<math>
 
<math>
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==Problem 7==
 
==Problem 7==
What is the minimum possible product of three different numbers of the set <math>{-8.-6,-4,0,3,5,7}?</math>
+
What is the minimum possible product of three different numbers of the set <math>\{-8.-6,-4,0,3,5,7\}?</math>
  
 
<math>
 
<math>
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==Problem 8==
 
==Problem 8==
Three dice with faces numbered <math>1</math> through <math>6</math> are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots <math>NOT</math> visible in the view is
+
Three dice with faces numbered <math>1</math> through <math>6</math> are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots <math>NOT</math> visible in this view is
  
 
<math>
 
<math>
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</math>
 
</math>
  
[[2000 AMC 8 Problems/Problem 7|Solution]]
+
[[2000 AMC 8 Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
 +
Three-digit powers of <math>2</math> and <math>5</math> are used in this <math>cross-number</math> puzzle. What is the only possible digit for the outlined square?
 +
 +
<math>ACROSS\ DOWN</math>
 +
 +
<math>2)\ 2^m \qquad\ 1)\ 5^n</math>
 +
 +
 +
<math>
 +
\mathrm{(A)}\ 0
 +
\qquad
 +
\mathrm{(B)}\ 2
 +
\qquad
 +
\mathrm{(C)}\ 4
 +
\qquad
 +
\mathrm{(D)}\ 6
 +
\qquad
 +
\mathrm{(E)}\ 8
 +
</math>
 +
 +
[[2000 AMC 8 Problems/Problem 9|Solution]]
 +
 +
==Problem 10==
 +
Ara and Shea were once the same height. Since then Shea has grown <math>20\%</math> while Ara has grow half as many inches as Shea. Shea is now <math>60</math> inches tall. How tall, in inches, is Ara now?
 +
 +
<math>
 +
\mathrm{(A)}\ 48
 +
\qquad
 +
\mathrm{(B)}\ 51
 +
\qquad
 +
\mathrm{(C)}\ 52
 +
\qquad
 +
\mathrm{(D)}\ 54
 +
\qquad
 +
\mathrm{(E)}\ 55
 +
</math>
 +
 +
[[2000 AMC 8 Problems/Problem 10|Solution]]
 +
 +
==Problem 11==
 +
The number <math>64</math> has the property that it is divisible by its units digit. How many whole numbers between <math>10</math> and <math>50</math> have this property?
 +
 +
 +
<math>
 +
\mathrm{(A)}\ 15
 +
\qquad
 +
\mathrm{(B)}\ 16
 +
\qquad
 +
\mathrm{(C)}\ 17
 +
\qquad
 +
\mathrm{(D)}\ 18
 +
\qquad
 +
\mathrm{(E)}\ 20
 +
</math>
 +
 +
[[2000 AMC 8 Problems/Problem 11|Solution]]
 +
 +
==Problem 12==
 +
==Problem 13==
 +
==Problem 14==
 +
What is the units digit of <math>19^{19} + 99^{99}?</math>
 +
 +
<math>
 +
\mathrm{(A)}\ 0
 +
\qquad
 +
\mathrm{(B)}\ 1
 +
\qquad
 +
\mathrm{(C)}\ 2
 +
\qquad
 +
\mathrm{(D)}\ 8
 +
\qquad
 +
\mathrm{(E)}\ 9
 +
</math>
 +
 +
[[2000 AMC 8 Problems/Problem 14|Solution]]
 +
 +
==Problem 15==
 +
==Problem 16==
 +
In order for Mateen to walk a kilometer <math>(1000m)</math> in his rectangular backyard, he must walk the length <math>25</math> times or walk its perimeter <math>10</math> times. What is the area of Mateen's backyard in square meters?
 +
 +
<math>
 +
\mathrm{(A)}\ 40
 +
\qquad
 +
\mathrm{(B)}\ 200
 +
\qquad
 +
\mathrm{(C)}\ 400
 +
\qquad
 +
\mathrm{(D)}\ 500
 +
\qquad
 +
\mathrm{(E)}\ 1000
 +
</math>
 +
 +
[[2000 AMC 8 Problems/Problem 16|Solution]]

Revision as of 09:37, 10 April 2009

Problem 1

Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?

$\mathrm{(A)}\ 15 \qquad \mathrm{(B)}\ 16 \qquad \mathrm{(C)}\ 17 \qquad \mathrm{(D)}\ 21 \qquad \mathrm{(E)}\ 37$

Solution

Problem 2

Which of these numbers is less than its reciprocal?

$\mathrm{(A)}\ -2 \qquad \mathrm{(B)}\ -1 \qquad \mathrm{(C)}\ 0 \qquad \mathrm{(D)}\ 1 \qquad \mathrm{(E)}\ 2$

Solution

Problem 3

How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi?$


$\mathrm{(A)}\ 2 \qquad \mathrm{(B)}\ 3 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 5 \qquad \mathrm{(E)}\ infinitely\ many$

Solution

Problem 4

In $1960$ only $5\%$ of the working adults in Carlin City worked at home. By $1970$ the "at-home" work force had increased to $8\%$. In $1980$ there were approximately $15\%$ working at home, and in $1990$ there were $30\%$. The graph that best illustrates this is:

Solution

Problem 5

Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period?

$\mathrm{(A)}\ 2 \qquad \mathrm{(B)}\ 3 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 5 \qquad \mathrm{(E)}\ 8$

Solution

Problem 6

Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded $L$-shaped region is

$\mathrm{(A)}\ 7 \qquad \mathrm{(B)}\ 10 \qquad \mathrm{(C)}\ 12.5 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 15$


Solution

Problem 7

What is the minimum possible product of three different numbers of the set $\{-8.-6,-4,0,3,5,7\}?$

$\mathrm{(A)}\ -336 \qquad \mathrm{(B)}\ -280 \qquad \mathrm{(C)}\ -210 \qquad \mathrm{(D)}\ -192 \qquad \mathrm{(E)}\ 0$

Solution

Problem 8

Three dice with faces numbered $1$ through $6$ are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots $NOT$ visible in this view is

$\mathrm{(A)}\ 21 \qquad \mathrm{(B)}\ 22 \qquad \mathrm{(C)}\ 31 \qquad \mathrm{(D)}\ 41 \qquad \mathrm{(E)}\ 53$

Solution

Problem 9

Three-digit powers of $2$ and $5$ are used in this $cross-number$ puzzle. What is the only possible digit for the outlined square?

$ACROSS\ DOWN$

$2)\ 2^m \qquad\ 1)\ 5^n$


$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 6 \qquad \mathrm{(E)}\ 8$

Solution

Problem 10

Ara and Shea were once the same height. Since then Shea has grown $20\%$ while Ara has grow half as many inches as Shea. Shea is now $60$ inches tall. How tall, in inches, is Ara now?

$\mathrm{(A)}\ 48 \qquad \mathrm{(B)}\ 51 \qquad \mathrm{(C)}\ 52 \qquad \mathrm{(D)}\ 54 \qquad \mathrm{(E)}\ 55$

Solution

Problem 11

The number $64$ has the property that it is divisible by its units digit. How many whole numbers between $10$ and $50$ have this property?


$\mathrm{(A)}\ 15 \qquad \mathrm{(B)}\ 16 \qquad \mathrm{(C)}\ 17 \qquad \mathrm{(D)}\ 18 \qquad \mathrm{(E)}\ 20$

Solution

Problem 12

Problem 13

Problem 14

What is the units digit of $19^{19} + 99^{99}?$

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 8 \qquad \mathrm{(E)}\ 9$

Solution

Problem 15

Problem 16

In order for Mateen to walk a kilometer $(1000m)$ in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters?

$\mathrm{(A)}\ 40 \qquad \mathrm{(B)}\ 200 \qquad \mathrm{(C)}\ 400 \qquad \mathrm{(D)}\ 500 \qquad \mathrm{(E)}\ 1000$

Solution

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