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Difference between revisions of "1997 AHSME Problems"
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{{AHSME Problems
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|year = 1997
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}}
 
== Problem 1 ==
 
== Problem 1 ==
 
If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which
 
If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which
  
−
<math> \begin{tabular}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular} </math>
+
<math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </math>
  
 
then <math>\texttt{a+b =}</math>
 
then <math>\texttt{a+b =}</math>
Line 98: Line 101:
  
 
[[1997 AHSME Problems/Problem 8|Solution]]
 
[[1997 AHSME Problems/Problem 8|Solution]]
 +
 +
 +
==Problem 9==
 +
 +
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
 +
 +
<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);
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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>
 +
 +
<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>
 +
 +
[[1997 AHSME Problems/Problem 9|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 <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?
 +
 +
<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>
 +
 +
[[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?
 +
 +
<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 </math>
 +
 +
[[1997 AHSME Problems/Problem 11|Solution]]
 +
 +
==Problem 12==
 +
 +
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
 +
 +
<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>
 +
 +
[[1997 AHSME Problems/Problem 12|Solution]]
 +
 +
==Problem 13==
 +
 +
How many two-digit positive integers <math>N</math> have the property that the sum of <math>N</math> and the number obtained by reversing the order of the digits of <math>N</math> is a perfect square?
 +
 +
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math>
 +
 +
[[1997 AHSME Problems/Problem 13|Solution]]
 +
 +
==Problem 14==
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The number of geese in a flock increases so that the difference between the populations in year <math>n+2</math> and year <math>n</math> is directly proportional to the population in year <math>n+1</math>. If the populations in the years <math>1994</math>, <math>1995</math>, and <math>1997</math> were <math>39</math>, <math>60</math>, and <math>123</math>, respectively, then the population in <math>1996</math> was
 +
 +
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 84\qquad\textbf{(C)}\ 87\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 102 </math>
 +
 +
[[1997 AHSME Problems/Problem 14|Solution]]
 +
 +
==Problem 15==
 +
 +
Medians <math>BD</math> and <math>CE</math> of triangle <math>ABC</math> are perpendicular, <math>BD=8</math>, and <math>CE=12</math>.  The area of triangle <math>ABC</math> is
 +
 +
<asy>
 +
defaultpen(linewidth(.8pt));
 +
dotfactor=4;
 +
pair A = origin;
 +
pair B = (1.25,1);
 +
pair C = (2,0);
 +
pair D = midpoint(A--C);
 +
pair E = midpoint(A--B);
 +
pair G = intersectionpoint(E--C,B--D);
 +
dot(A);dot(B);dot(C);dot(D);dot(E);dot(G);
 +
label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE);
 +
draw(A--B--C--cycle);
 +
draw(B--D);
 +
draw(E--C);
 +
draw(rightanglemark(C,G,D,3));</asy>
 +
 +
<math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 48\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96 </math>
 +
 +
[[1997 AHSME Problems/Problem 15|Solution]]
 +
 +
==Problem 16==
 +
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The three row sums and the three column sums of the array
 +
 +
<cmath> \left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right] </cmath>
 +
 +
are the same. What is the least number of entries that must be altered to make all six sums different from one another?
 +
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<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math>
 +
 +
[[1997 AHSME Problems/Problem 16|Solution]]
 +
 +
==Problem 17==
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A line <math>x=k</math> intersects the graph of <math>y=\log_5 x</math> and the graph of <math>y=\log_5 (x + 4)</math>. The distance between the points of intersection is <math>0.5</math>. Given that <math>k = a + \sqrt{b}</math>, where <math>a</math> and <math>b</math> are integers, what is <math>a+b</math>?
 +
 +
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 </math>
 +
 +
[[1997 AHSME Problems/Problem 17|Solution]]
 +
 +
==Problem 18==
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 +
A list of integers has mode <math>32</math> and mean <math>22</math>. The smallest number in the list is <math>10</math>. The median <math>m</math> of the list is a member of the list. If the list member <math>m</math> were replaced by <math>m+10</math>, the mean and median of the new list would be <math>24</math> and <math>m+10</math>, respectively. If were <math>m</math> instead replaced by <math>m-8</math>, the median of the new list would be <math>m-4</math>. What is <math>m</math>?
 +
 +
<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 20 </math>
 +
 +
[[1997 AHSME Problems/Problem 18|Solution]]
 +
 +
==Problem 19==
 +
 +
A circle with center <math>O</math> is tangent to the coordinate axes and to the hypotenuse of the <math> 30^\circ </math>-<math> 60^\circ </math>-<math> 90^\circ </math> triangle <math>ABC</math> as shown, where <math>AB=1</math>. To the nearest hundredth, what is the radius of the circle?
 +
 +
 +
<asy>
 +
defaultpen(linewidth(.8pt));
 +
dotfactor=3;
 +
pair A = origin;
 +
pair B = (1,0);
 +
pair C = (0,sqrt(3));
 +
pair O = (2.33,2.33);
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dot(A);dot(B);dot(C);dot(O);
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label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,W);label("$O$",O,NW);
 +
label("$1$",midpoint(A--B),S);label("$60^\circ$",B,2W + N);
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draw((3,0)--A--(0,3));
 +
draw(B--C);
 +
draw(Arc(O,2.33,163,288.5));</asy>
 +
 +
<math> \textbf{(A)}\ 2.18\qquad\textbf{(B)}\ 2.24\qquad\textbf{(C)}\ 2.31\qquad\textbf{(D)}\ 2.37\qquad\textbf{(E)}\ 2.41 </math>
 +
 +
[[1997 AHSME Problems/Problem 19|Solution]]
 +
 +
==Problem 20==
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 +
Which one of the following integers can be expressed as the sum of <math>100</math> consecutive positive integers?
 +
 +
<math> \textbf{(A)}\ 1,\!627,\!384,\!950\qquad\textbf{(B)}\ 2,\!345,\!678,\!910\qquad\textbf{(C)}\ 3,\!579,\!111,\!300\qquad\textbf{(D)}\ 4,\!692,\!581,\!470\qquad\textbf{(E)}\ 5,\!815,\!937,\!260 </math>
 +
 +
[[1997 AHSME Problems/Problem 20|Solution]]
 +
 +
==Problem 21==
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For any positive integer <math>n</math>, let
 +
 +
<math> f(n) =\left\{\begin{matrix}\log_{8}{n}, &\text{if }\log_{8}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right. </math>
 +
 +
What is <math> \sum_{n = 1}^{1997}{f(n)} </math>?
 +
 +
<math> \textbf{(A)}\ \log_{8}{2047}\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ \frac{55}{3}\qquad\textbf{(D)}\ \frac{58}{3}\qquad\textbf{(E)}\ 585 </math>
 +
 +
[[1997 AHSME Problems/Problem 21|Solution]]
 +
 +
==Problem 22==
 +
 +
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had <math>56</math> dollars. The absolute difference between the amounts Ashley and Betty had to spend was <math>19</math> dollars. The absolute difference between the amounts Betty and Carlos had was <math>7</math> dollars, between Carlos and Dick was <math>5</math> dollars, between Dick and Elgin was <math>4</math> dollars, and between Elgin and Ashley was <math>11</math> dollars. How many dollars did Elgin have?
 +
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<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 </math>
 +
 +
[[1997 AHSME Problems/Problem 22|Solution]]
 +
 +
==Problem 23==
 +
 +
<asy>
 +
defaultpen(linewidth(.8pt)+fontsize(10pt));
 +
draw((-1,1)--(2,1));
 +
draw((-1,0)--(1,0));
 +
draw((-1,1)--(-1,0));
 +
draw((0,-1)--(0,3));
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draw((1,2)--(1,0));
 +
draw((-1,1)--(1,1));
 +
draw((0,2)--(1,2));
 +
draw((0,3)--(1,2));
 +
draw((0,-1)--(2,1));
 +
draw((0,-1)--((0,-1) + sqrt(2)*dir(-15)));
 +
draw(((0,-1) + sqrt(2)*dir(-15))--(1,0));
 +
label("$\textbf{A}$",foot((0,2),(0,3),(1,2)),SW);
 +
label("$\textbf{B}$",midpoint((0,1)--(1,2)));
 +
label("$\textbf{C}$",midpoint((-1,0)--(0,1)));
 +
label("$\textbf{D}$",midpoint((0,0)--(1,1)));
 +
label("$\textbf{E}$",midpoint((1,0)--(2,1)),NW);
 +
label("$\textbf{F}$",midpoint((0,-1)--(1,0)),NW);
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label("$\textbf{G}$",midpoint((0,-1)--(1,0)),2SE);</asy>
 +
 +
In the figure, polygons <math>A</math>, <math>E</math>, and <math>F</math> are isosceles right triangles; <math>B</math>, <math>C</math>, and <math>D</math> are squares with sides of length <math>1</math>; and <math>G</math> is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is
 +
 +
<math> \textbf{(A)}\ 1/2\qquad\textbf{(B)}\ 2/3\qquad\textbf{(C)}\ 3/4\qquad\textbf{(D)}\ 5/6\qquad\textbf{(E)}\ 4/3 </math>
 +
 +
[[1997 AHSME Problems/Problem 23|Solution]]
 +
 +
==Problem 24==
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A rising number, such as <math>34689</math>, is a positive integer each digit of which is larger than each of the digits to its left. There are <math>\binom{9}{5} = 126</math> five-digit rising numbers. When these numbers are arranged from smallest to largest, the <math>97^{th}</math> number in the list does not contain the digit
 +
 +
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math>
 +
 +
[[1997 AHSME Problems/Problem 24|Solution]]
 +
 +
==Problem 25==
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Let <math>ABCD</math> be a parallelogram and let <math>\overrightarrow{AA^\prime}</math>, <math>\overrightarrow{BB^\prime}</math>, <math>\overrightarrow{CC^\prime}</math>, and <math>\overrightarrow{DD^\prime}</math> be parallel rays in space on the same side of the plane determined by <math>ABCD</math>. If <math>AA^\prime = 10</math>, <math>BB^\prime = 8</math>, <math>CC^\prime = 18</math>, and <math>DD^\prime = 22</math> and <math>M</math> and <math>N</math> are the midpoints of <math>A^\prime C^\prime</math> and <math>B^\prime D^\prime</math>, respectively, then <math>MN = </math>
 +
 +
<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4 </math>
 +
 +
[[1997 AHSME Problems/Problem 25|Solution]]
 +
 +
==Problem 26==
 +
 +
Triangle <math>ABC</math> and point <math>P</math> in the same plane are given. Point <math>P</math> is equidistant from <math>A</math> and <math>B</math>, angle <math>APB</math> is twice angle <math>ACB</math>, and <math>\overline{AC}</math> intersects <math>\overline{BP}</math> at point <math>D</math>. If <math>PB = 3</math> and <math>PD= 2</math>, then <math>AD\cdot CD =</math>
 +
 +
<asy>
 +
defaultpen(linewidth(.8pt));
 +
dotfactor=4;
 +
pair A = origin;
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pair B = (2,0);
 +
pair C = (3,1);
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pair P = (1,2.25);
 +
pair D = intersectionpoint(P--B,C--A);
 +
dot(A);dot(B);dot(C);dot(P);dot(D);
 +
label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE + N);label("$P$",P,N);
 +
draw(A--B--P--cycle);
 +
draw(A--C--B--cycle);</asy>
 +
 +
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math>
 +
 +
[[1997 AHSME Problems/Problem 26|Solution]]
 +
 +
==Problem 27==
 +
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Consider those functions <math>f</math> that satisfy <math> f(x+4)+f(x-4) = f(x) </math> for all real <math>x</math>. Any such function is periodic, and there is a least common positive period <math>p</math> for all of them. Find <math>p</math>.
 +
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<math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 32 </math>
 +
 +
[[1997 AHSME Problems/Problem 27|Solution]]
 +
 +
==Problem 28==
 +
 +
How many ordered triples of integers <math>(a,b,c)</math> satisfy <math> |a+b|+c = 19 </math> and <math> ab+|c| = 97 </math>?
 +
 +
<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 </math>
 +
 +
[[1997 AHSME Problems/Problem 28|Solution]]
 +
 +
==Problem 29==
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Call a positive real number special if it has a decimal representation that consists entirely of digits <math>0</math> and <math>7</math>. For example, <math> \frac{700}{99}= 7.\overline{07}= 7.070707\cdots </math> and <math> 77.007 </math> are special numbers. What is the smallest <math>n</math> such that <math>1</math> can be written as a sum of <math>n</math> special numbers?
 +
 +
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\\ \textbf{(E)}\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.} </math>
 +
 +
[[1997 AHSME Problems/Problem 29|Solution]]
 +
 +
==Problem 30==
 +
 +
For positive integers <math>n</math>, denote <math>D(n)</math> by the number of pairs of different adjacent digits in the binary (base two) representation of <math>n</math>. For example, <math> D(3) = D(11_{2}) = 0 </math>, <math> D(21) = D(10101_{2}) = 4 </math>, and <math> D(97) = D(1100001_{2}) = 2 </math>. For how many positive integers less than or equal to <math>97</math> does <math>D(n) = 2</math>?
 +
 +
<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 </math>
 +
 +
[[1997 AHSME Problems/Problem 30|Solution]]
 +
 +
== See also ==
 +
 +
* [[AMC 12 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
 +
{{AHSME box|year=1997|before=[[1996 AHSME]]|after=[[1998 AHSME]]}} 
 +
 +
{{MAA Notice}}

Latest revision as of 12:38, 19 February 2020

1997 AHSME (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. You will have 90 minutes working time to complete the test.
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 • 26 • 27 • 28 • 29 • 30

Problem 1

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

$\begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array}$

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 $2 \times 2$ 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

Problem 13

How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square?

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

Solution

Problem 14

The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $1996$ was

$\textbf{(A)}\ 81\qquad\textbf{(B)}\ 84\qquad\textbf{(C)}\ 87\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 102$

Solution

Problem 15

Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is

[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(C);dot(D);dot(E);dot(G); label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE); draw(A--B--C--cycle); draw(B--D); draw(E--C); draw(rightanglemark(C,G,D,3));[/asy]

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 48\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96$

Solution

Problem 16

The three row sums and the three column sums of the array

\[\left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right]\]

are the same. What is the least number of entries that must be altered to make all six sums different from one another?

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

Solution

Problem 17

A line $x=k$ intersects the graph of $y=\log_5 x$ and the graph of $y=\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Solution

Problem 18

A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m+10$, the mean and median of the new list would be $24$ and $m+10$, respectively. If were $m$ instead replaced by $m-8$, the median of the new list would be $m-4$. What is $m$?

$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 20$

Solution

Problem 19

A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?


[asy] defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); dot(A);dot(B);dot(C);dot(O); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,W);label("$O$",O,NW); label("$1$",midpoint(A--B),S);label("$60^\circ$",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]

$\textbf{(A)}\ 2.18\qquad\textbf{(B)}\ 2.24\qquad\textbf{(C)}\ 2.31\qquad\textbf{(D)}\ 2.37\qquad\textbf{(E)}\ 2.41$

Solution

Problem 20

Which one of the following integers can be expressed as the sum of $100$ consecutive positive integers?

$\textbf{(A)}\ 1,\!627,\!384,\!950\qquad\textbf{(B)}\ 2,\!345,\!678,\!910\qquad\textbf{(C)}\ 3,\!579,\!111,\!300\qquad\textbf{(D)}\ 4,\!692,\!581,\!470\qquad\textbf{(E)}\ 5,\!815,\!937,\!260$

Solution

Problem 21

For any positive integer $n$, let

$f(n) =\left\{\begin{matrix}\log_{8}{n}, &\text{if }\log_{8}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.$

What is $\sum_{n = 1}^{1997}{f(n)}$?

$\textbf{(A)}\ \log_{8}{2047}\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ \frac{55}{3}\qquad\textbf{(D)}\ \frac{58}{3}\qquad\textbf{(E)}\ 585$

Solution

Problem 22

Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7$ dollars, between Carlos and Dick was $5$ dollars, between Dick and Elgin was $4$ dollars, and between Elgin and Ashley was $11$ dollars. How many dollars did Elgin have?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Solution

Problem 23

[asy] defaultpen(linewidth(.8pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label("$\textbf{A}$",foot((0,2),(0,3),(1,2)),SW); label("$\textbf{B}$",midpoint((0,1)--(1,2))); label("$\textbf{C}$",midpoint((-1,0)--(0,1))); label("$\textbf{D}$",midpoint((0,0)--(1,1))); label("$\textbf{E}$",midpoint((1,0)--(2,1)),NW); label("$\textbf{F}$",midpoint((0,-1)--(1,0)),NW); label("$\textbf{G}$",midpoint((0,-1)--(1,0)),2SE);[/asy]

In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is

$\textbf{(A)}\ 1/2\qquad\textbf{(B)}\ 2/3\qquad\textbf{(C)}\ 3/4\qquad\textbf{(D)}\ 5/6\qquad\textbf{(E)}\ 4/3$

Solution

Problem 24

A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{th}$ number in the list does not contain the digit

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

Solution

Problem 25

Let $ABCD$ be a parallelogram and let $\overrightarrow{AA^\prime}$, $\overrightarrow{BB^\prime}$, $\overrightarrow{CC^\prime}$, and $\overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^\prime = 10$, $BB^\prime = 8$, $CC^\prime = 18$, and $DD^\prime = 22$ and $M$ and $N$ are the midpoints of $A^\prime C^\prime$ and $B^\prime D^\prime$, respectively, then $MN =$

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

Solution

Problem 26

Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\overline{AC}$ intersects $\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\cdot CD =$

[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (2,0); pair C = (3,1); pair P = (1,2.25); pair D = intersectionpoint(P--B,C--A); dot(A);dot(B);dot(C);dot(P);dot(D); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE + N);label("$P$",P,N); draw(A--B--P--cycle); draw(A--C--B--cycle);[/asy]

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Solution

Problem 27

Consider those functions $f$ that satisfy $f(x+4)+f(x-4) = f(x)$ for all real $x$. Any such function is periodic, and there is a least common positive period $p$ for all of them. Find $p$.

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 32$

Solution

Problem 28

How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

Solution

Problem 29

Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\frac{700}{99}= 7.\overline{07}= 7.070707\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special numbers?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\\ \textbf{(E)}\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.}$

Solution

Problem 30

For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$?

$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35$

Solution

See also

1997 AHSME (Problems • Answer Key • Resources)
Preceded by
1996 AHSME 		Followed by
1998 AHSME
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions


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