Difference between revisions of "1993 AJHSME Problems"

Revision as of 02:41, 15 February 2010

Problem 1

Which pair of numbers does NOT have a product equal to $36$ ?

$\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}$

Solution

Problem 2

When the fraction $\dfrac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be

$\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133$

Solution

Problem 3

Which of the following numbers has the largest prime factor?

$\text{(A)}\ 39 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 77 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 121$

Solution

Problem 4

$1000\times 1993 \times 0.1993 \times 10 =$

$\text{(A)}\ 1.993\times 10^3 \qquad \text{(B)}\ 1993.1993 \qquad \text{(C)}\ (199.3)^2 \qquad \text{(D)}\ 1,993,001.993 \qquad \text{(E)}\ (1993)^2$

Solution

Problem 5

Which one of the following bar graphs could represent the data from the circle graph?

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

Solution

Problem 6

A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$

Solution

Problem 7

$3^3+3^3+3^3 =$

$\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}$

Solution

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 $60$ pills, then the supply of medicine would last approximately

$\text{(A)}\ 1\text{ month} \qquad \text{(B)}\ 4\text{ months} \qquad \text{(C)}\ 6\text{ months} \qquad \text{(D)}\ 8\text{ months} \qquad \text{(E)}\ 1\text{ year}$

Solution

Problem 9

Consider the operation $*$ defined by the following table:

$\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{tabular}$

For example, $3*2=1$ . Then $(2*4)*(1*3)=$

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

Solution

Problem 10

This line graph represents the price of a trading card during the first $6$ months of $1993$ .

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

The greatest monthly drop in price occurred during

$\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}$

Solution

Problem 11

Consider this histogram of the scores for $81$ students taking a test:

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

The median is in the interval labeled

$\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80$

Solution

Problem 12

If each of the three operation signs, $+$ , $-$ , $\times$ , is used exactly ONCE in one of the blanks in the expression

$5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3$

then the value of the result could equal

$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 19$

Solution

Problem 13

The word "HELP" in block letters is painted in black with strokes $1$ unit wide on a $5$ by $15$ rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is

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

$\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38$

Solution

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 $1,2,3$ . Then $A+B=$

$\begin{tabular}{|c|c|c|} \hline 1 & & \\ \hline & 2 & A \\ \hline & & B \\ \hline \end{tabular}$

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

Solution

Problem 15

The arithmetic mean (average) of four numbers is $85$ . If the largest of these numbers is $97$ , then the mean of the remaining three numbers is

$\text{(A)}\ 81.0 \qquad \text{(B)}\ 82.7 \qquad \text{(C)}\ 83.0 \qquad \text{(D)}\ 84.0 \qquad \text{(E)}\ 84.3$

Solution

Problem 16

$\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{3}}} =$

$\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{3}{10} \qquad \text{(C)}\ \dfrac{7}{10} \qquad \text{(D)}\ \dfrac{5}{6} \qquad \text{(E)}\ \dfrac{10}{3}$

Solution

Problem 17

Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ 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

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

$\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$

Solution

Problem 18

The rectangle shown has length $AC=32$ , width $AE=20$ , and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$ , respectively. The area of quadrilateral $ABDF$ is

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

$\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340$

Solution

Problem 19

$(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) =$

$\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$

Solution

Problem 20

When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is

$\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833$

Solution

Problem 21

If the length of a rectangle is increased by $20\%$ and its width is increased by $50\%$ , then the area is increased by

$\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\%$

Solution

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?

$\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199$

Solution

Problem 23

Five runners, $P$ , $Q$ , $R$ , $S$ , $T$ , have a race, and $P$ beats $Q$ , $P$ beats $R$ , $Q$ beats $S$ , and $T$ finishes after $P$ and before $Q$ . Who could NOT have finished third in the race?

$\text{(A)}\ P\text{ and }Q \qquad \text{(B)}\ P\text{ and }R \qquad \text{(C)}\ P\text{ and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T$

Solution

Problem 24

What number is directly above $142$ in this array of numbers?

\[\begin{tabular}{cccccc}
 & & & 1 & & \\
& & 2 & 3 & 4 & \\
& 5 & 6 & 7 & 8 & 9 \\
10 & 11 & 12 & \cdots & & \\
\end{tabular}\] (Error compiling LaTeX. Unknown error_msg)

$\text{(A)}\ 99 \qquad \text{(B)}\ 119 \qquad \text{(C)}\ 120 \qquad \text{(D)}\ 121 \qquad \text{(E)}\ 122$

Solution

Problem 25

A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is

$\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}$

Solution

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

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Difference between revisions of "1993 AJHSME Problems"

Revision as of 02:41, 15 February 2010

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 1: / Line 1: @@
 ==Problem 1==
+Which pair of numbers does NOT have a product equal to <math>36</math>?
+<math>\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}</math>
 [[1993 AJHSME Problems/Problem 1|Solution]]
 == Problem 2 ==
+When the fraction <math>\dfrac{49}{84}</math> is expressed in simplest form, then the sum of the numerator and the denominator will be
+<math>\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133</math>
 [[1993 AJHSME Problems/Problem 2|Solution]]
 == Problem 3 ==
+Which of the following numbers has the largest prime factor?
+<math>\text{(A)}\ 39 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 77 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 121</math>
 [[1993 AJHSME Problems/Problem 3|Solution]]
 == Problem 4 ==
+<math>1000\times 1993 \times 0.1993 \times 10 = </math>
+<math>\text{(A)}\ 1.993\times 10^3 \qquad \text{(B)}\ 1993.1993 \qquad \text{(C)}\ (199.3)^2 \qquad \text{(D)}\ 1,993,001.993 \qquad \text{(E)}\ (1993)^2</math>
 [[1993 AJHSME Problems/Problem 4|Solution]]
 == Problem 5 ==
+Which one of the following bar graphs could represent the data from the circle graph?
+{{image}}
 [[1993 AJHSME Problems/Problem 5|Solution]]
 == Problem 6 ==
+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)}\ 10</math>
 [[1993 AJHSME Problems/Problem 6|Solution]]
 == Problem 7 ==
+<math>3^3+3^3+3^3 = </math>
+<math>\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}</math>
 [[1993 AJHSME Problems/Problem 7|Solution]]
 == 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 <math>60</math> pills, then the supply of medicine would last approximately
+<math>\text{(A)}\ 1\text{ month} \qquad \text{(B)}\ 4\text{ months} \qquad \text{(C)}\ 6\text{ months} \qquad \text{(D)}\ 8\text{ months} \qquad \text{(E)}\ 1\text{ year}</math>
 [[1993 AJHSME Problems/Problem 8|Solution]]
 == Problem 9 ==
+Consider the operation <math>*</math> defined by the following table:
+<cmath>\begin{tabular}{c|cccc}
+* & 1 & 2 & 3 & 4 \\ \hline
+& 1 & 2 & 3 & 4 \\
+& 2 & 4 & 1 & 3 \\
+& 3 & 1 & 4 & 2 \\
+& 4 & 3 & 2 & 1
+\end{tabular}</cmath>
+For example, <math>3*2=1</math>.  Then <math>(2*4)*(1*3)=</math>
+<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math>
 [[1993 AJHSME Problems/Problem 9|Solution]]
 == Problem 10 ==
+This line graph represents the price of a trading card during the first <math>6</math> months of <math>1993</math>.
+{{image}}
+The greatest monthly drop in price occurred during
+<math>\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}</math>
 [[1993 AJHSME Problems/Problem 10|Solution]]
 == Problem 11 ==
+Consider this histogram of the scores for <math>81</math> students taking a test:
+{{image}}
+The median is in the interval labeled
+<math>\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80</math>
 [[1993 AJHSME Problems/Problem 11|Solution]]
 == Problem 12 ==
+If each of the three operation signs, <math>+</math>, <math>-</math>, <math>\times </math>, is used exactly ONCE in one of the blanks in the expression
+<cmath>5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3</cmath>
+then the value of the result could equal
+<math>\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 19</math>
 [[1993 AJHSME Problems/Problem 12|Solution]]
 == Problem 13 ==
+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
+{{image}}
+<math>\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38</math>
 [[1993 AJHSME Problems/Problem 13|Solution]]
 == 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 <math>1,2,3</math>.  Then <math>A+B=</math>
+<cmath>\begin{tabular}{|c|c|c|} \hline
+& & \\ \hline
+ & 2 & A \\ \hline
+ & & B \\ \hline
+\end{tabular}</cmath>
+<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math>
 [[1993 AJHSME Problems/Problem 14|Solution]]
 == Problem 15 ==
+The arithmetic mean (average) of four numbers is <math>85</math>.  If the largest of these numbers is <math>97</math>, then the mean of the remaining three numbers is
+<math>\text{(A)}\ 81.0 \qquad \text{(B)}\ 82.7 \qquad \text{(C)}\ 83.0 \qquad \text{(D)}\ 84.0 \qquad \text{(E)}\ 84.3</math>
 [[1993 AJHSME Problems/Problem 15|Solution]]
 == Problem 16 ==
+<math>\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{3}}} =</math>
+<math>\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{3}{10} \qquad \text{(C)}\ \dfrac{7}{10} \qquad \text{(D)}\ \dfrac{5}{6} \qquad \text{(E)}\ \dfrac{10}{3}</math>
 [[1993 AJHSME Problems/Problem 16|Solution]]
 == Problem 17 ==
+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
+{{image}}
+<math>\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000</math>
 [[1993 AJHSME Problems/Problem 17|Solution]]
 == Problem 18 ==
+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
+{{image}}
+<math>\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340</math>
 [[1993 AJHSME Problems/Problem 18|Solution]]
 == Problem 19 ==
+<math>(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) = </math>
+<math>\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142</math>
 [[1993 AJHSME Problems/Problem 19|Solution]]
 == Problem 20 ==
+When <math>10^{93}-93</math> is expressed as a single whole number, the sum of the digits is
+<math>\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833</math>
 [[1993 AJHSME Problems/Problem 20|Solution]]
 == Problem 21 ==
+If the length of a rectangle is increased by <math>20\% </math> and its width is increased by <math>50\% </math>, then the area is increased by
+<math>\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% </math>
 [[1993 AJHSME Problems/Problem 21|Solution]]
 == 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?
+<math>\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199</math>
 [[1993 AJHSME Problems/Problem 22|Solution]]
 == Problem 23 ==
+Five runners, <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, <math>T</math>, have a race, and <math>P</math> beats <math>Q</math>, <math>P</math> beats <math>R</math>, <math>Q</math> beats <math>S</math>, and <math>T</math> finishes after <math>P</math> and before <math>Q</math>.  Who could NOT have finished third in the race?
+<math>\text{(A)}\ P\text{ and }Q \qquad \text{(B)}\ P\text{ and }R \qquad \text{(C)}\ P\text{ and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T</math>
 [[1993 AJHSME Problems/Problem 23|Solution]]
 == Problem 24 ==
+What number is directly above <math>142</math> in this array of numbers?
+<cmath>\begin{tabular}{cccccc}
+ & & & 1 & & \\
+& & 2 & 3 & 4 & \\
+& 5 & 6 & 7 & 8 & 9 \\
+& 11 & 12 & \cdots & & \\
+\end{tabular}</cmath>
+<math>\text{(A)}\ 99 \qquad \text{(B)}\ 119 \qquad \text{(C)}\ 120 \qquad \text{(D)}\ 121 \qquad \text{(E)}\ 122</math>
 [[1993 AJHSME Problems/Problem 24|Solution]]
 == Problem 25 ==
+A checkerboard consists of one-inch squares.  A square card, <math>1.5</math> inches on a side, is placed on the board so that it covers part or all of the area of each of <math>n</math> squares.  The maximum possible value of <math>n</math> is
+<math>\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}</math>
 [[1993 AJHSME Problems/Problem 25|Solution]]