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Difference between revisions of "1997 AHSME Problems"
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[[1997 AHSME Problems/Problem 8|Solution]]
 
[[1997 AHSME Problems/Problem 8|Solution]]
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==Problem 9==
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In the figure, <math>ABCD</math> is a <math>2 \times \2</math> square, <math>E</math> is the midpoint of <math>\overline{AD}</math>, and <math>F</math> is on <math>\overline{BE}</math>.  If <math>\overline{CF}</math> is perpendicular to <math>\overline{BE}</math>, then the area of quadrilateral <math>CDEF</math> is
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<asy>
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defaultpen(linewidth(.8pt));
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dotfactor=4;
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pair A = (0,2);
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pair B = origin;
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pair C = (2,0);
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pair D = (2,2);
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pair E = midpoint(A--D);
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pair F = foot(C,B,E);
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dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
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label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW);
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draw(A--B--C--D--cycle);
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draw(B--E);
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draw(C--F);
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draw(rightanglemark(B,F,C,4));</asy>
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<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3-\frac{\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{11}{5}\qquad\textbf{(D)}\ \sqrt{5}\qquad\textbf{(E)}\ \frac{9}{4} </math>
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[[1997 AHSME Problems/Problem 9|Solution]]
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==Problem 10==
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Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the <math>4</math> replaced by <math>3</math> and the other die has the <math>3</math> replaced by <math>4</math> . When these dice are rolled, what is the probability that the sum is an odd number?
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<math> \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{4}{9}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{5}{9}\qquad\textbf{(E)}\ \frac{11}{18} </math>
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[[1997 AHSME Problems/Problem 10|Solution]]
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==Problem 11==
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In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored <math>23</math>,<math>14</math>, <math>11</math>, and <math>20</math> points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than <math>18</math>, what is the least number of points she could have scored in the tenth game?
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<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 </math>
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[[1997 AHSME Problems/Problem 11|Solution]]
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==Problem 12==
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If <math>m</math> and <math>b</math> are real numbers and <math>mb>0</math>, then the line whose equation is <math>y=mx+b</math> ''cannot'' contain the point
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<math> \textbf{(A)}\ (0,1997)\qquad\textbf{(B)}\ (0,-1997)\qquad\textbf{(C)}\ (19,97)\qquad\textbf{(D)}\ (19,-97)\qquad\textbf{(E)}\ (1997,0) </math>
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[[1997 AHSME Problems/Problem 12|Solution]]
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[[1997 AHSME Problems/Problem 13|Solution]]
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[[1997 AHSME Problems/Problem 14|Solution]]
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[[1997 AHSME Problems/Problem 15|Solution]]
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[[1997 AHSME Problems/Problem 16|Solution]]
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[[1997 AHSME Problems/Problem 17|Solution]]
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[[1997 AHSME Problems/Problem 18|Solution]]
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[[1997 AHSME Problems/Problem 19|Solution]]
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[[1997 AHSME Problems/Problem 20|Solution]]

Revision as of 19:10, 8 August 2011

Problem 1

If $\texttt{a}$ and $\texttt{b}$ are digits for which

$\begin{tabular}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular}$ (Error compiling LaTeX. Unknown error_msg)

then $\texttt{a+b =}$

$\mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }12$

Solution

Problem 2

The adjacent sides of the decagon shown meet at right angles. What is its perimeter?

[asy] defaultpen(linewidth(.8pt)); dotfactor=4; dot(origin);dot((12,0));dot((12,1));dot((9,1));dot((9,7));dot((7,7));dot((7,10));dot((3,10));dot((3,8));dot((0,8)); draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8)--(0,8)--cycle); label("$8$",midpoint(origin--(0,8)),W); label("$2$",midpoint((3,8)--(3,10)),W); label("$12$",midpoint(origin--(12,0)),S);[/asy]

$\mathrm{(A)\ } 22 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \ } 34 \qquad \mathrm{(D) \ } 44 \qquad \mathrm{(E) \ }50$

Solution

Problem 3

If $x$, $y$, and $z$ are real numbers such that

$(x-3)^2 + (y-4)^2 + (z-5)^2 = 0$,

then $x + y + z =$

$\mathrm{(A)\ } -12 \qquad \mathrm{(B) \ }0 \qquad \mathrm{(C) \ } 8 \qquad \mathrm{(D) \ } 12 \qquad \mathrm{(E) \ }50$

Solution


Problem 4

If $a$ is $50\%$ larger than $c$, and $b$ is $25\%$ larger than $c$, then $a$ is what percent larger than $b$?

$\mathrm{(A)\ } 20\% \qquad \mathrm{(B) \ }25\% \qquad \mathrm{(C) \ } 50\% \qquad \mathrm{(D) \ } 100\% \qquad \mathrm{(E) \ }200\%$

Solution

Problem 5

A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? [asy] defaultpen(linewidth(.8pt)); draw(origin--(0,3)--(4,3)--(4,0)--cycle); draw((0,1)--(4,1)); draw((2,0)--midpoint((0,1)--(4,1))); real r = 4/3; draw((r,3)--foot((r,3),(0,1),(4,1))); draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]

$\mathrm{(A)\ } 35.2 \qquad \mathrm{(B) \ }76 \qquad \mathrm{(C) \ } 80 \qquad \mathrm{(D) \ } 84 \qquad \mathrm{(E) \ }86$

Solution


Problem 6

Consider the sequence

$1,-2,3,-4,5,-6,\ldots,$

whose $n$th term is $(-1)^{n+1}\cdot n$. What is the average of the first $200$ terms of the sequence?

$\textbf{(A)}-\!1\qquad\textbf{(B)}-\!0.5\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 0.5\qquad\textbf{(E)}\ 1$

Solution


Problem 7

The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Solution


Problem 8

Mientka Publishing Company prices its bestseller Where's Walter? as follows:

$C(n) =\left\{\begin{matrix}12n, &\text{if }1\le n\le 24\\ 11n, &\text{if }25\le n\le 48\\ 10n, &\text{if }49\le n\end{matrix}\right.$

where $n$ is the number of books ordered, and $C(n)$ is the cost in dollars of $n$ books. Notice that $25$ books cost less than $24$ books. For how many values of $n$ is it cheaper to buy more than $n$ books than to buy exactly $n$ books?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$

Solution


Problem 9

In the figure, $ABCD$ is a $2 \times \2$ (Error compiling LaTeX. Unknown error_msg) square, $E$ is the midpoint of $\overline{AD}$, and $F$ is on $\overline{BE}$. If $\overline{CF}$ is perpendicular to $\overline{BE}$, then the area of quadrilateral $CDEF$ is

[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3-\frac{\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{11}{5}\qquad\textbf{(D)}\ \sqrt{5}\qquad\textbf{(E)}\ \frac{9}{4}$

Solution

Problem 10

Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $4$ replaced by $3$ and the other die has the $3$ replaced by $4$ . When these dice are rolled, what is the probability that the sum is an odd number?

$\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{4}{9}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{5}{9}\qquad\textbf{(E)}\ \frac{11}{18}$

Solution

Problem 11

In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$,$14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of points she could have scored in the tenth game?

$\textbf{(A)}\ 26\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30$

Solution

Problem 12

If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point

$\textbf{(A)}\ (0,1997)\qquad\textbf{(B)}\ (0,-1997)\qquad\textbf{(C)}\ (19,97)\qquad\textbf{(D)}\ (19,-97)\qquad\textbf{(E)}\ (1997,0)$

Solution Solution Solution Solution Solution Solution Solution Solution Solution

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