Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1985 AHSME Problems"

m (Fixed wording of Problem 8)
m (Improved LaTeX for Problem 30)
 
(26 intermediate revisions by the same user not shown)
Line 72: Line 72:
 
Let <math>a,a',b,b'</math> be real numbers with <math>a</math> and <math>a'</math> nonzero. The solution to <math>ax+b=0</math> is less than the solution to <math>a'x+b'=0</math> if and only if  
 
Let <math>a,a',b,b'</math> be real numbers with <math>a</math> and <math>a'</math> nonzero. The solution to <math>ax+b=0</math> is less than the solution to <math>a'x+b'=0</math> if and only if  
  
−
<math> \mathrm{(A)\ } a'b < ab' \qquad \mathrm{(B) \ }ab' < a'b \qquad \mathrm{(C) \  } ab < a'b' \qquad \mathrm{(D) \  } \frac{b}{a} < \frac{b'}{a'} \qquad </math>
+
<math> \mathrm{(A)\ } a'b < ab' \qquad \mathrm{(B) \ }ab' < a'b \qquad \mathrm{(C) \  } ab < a'b' \qquad \mathrm{(D) \  } \frac{b}{a} < \frac{b'}{a'} \qquad \mathrm{(E) \  }\frac{b'}{a'} < \frac{b}{a}  </math>
−
 
 
−
<math> \mathrm{(E) \  }\frac{b'}{a'} < \frac{b}{a}  </math>
 
  
 
[[1985 AHSME Problems/Problem 8|Solution]]
 
[[1985 AHSME Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
−
The odd positive integers <math> 1, 3, 5, 7, \cdots </math>, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which <math> 1985 </math> appears in is the
+
The odd positive integers <math>1, 3, 5, 7, \ldots</math>, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which <math>1985</math> appears is the
  
 
<asy>
 
<asy>
Line 104: Line 102:
  
 
==Problem 10==
 
==Problem 10==
−
An arbitrary [[circle]] can intersect the [[graph]] of <math> y=\sin x </math> in
+
An arbitrary circle can intersect the graph of <math>y = \sin x</math> in
  
−
<math> \mathrm{(A)\  } \text{at most }2\text{ points} \qquad \mathrm{(B)\  }\text{at most }4\text{ points} \qquad \mathrm{(C) \  } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}\qquad \mathrm{(E) \  }\text{more than }16\text{ points}  </math>
+
<math> \mathrm{(A)\  } \text{at most }2\text{ points} \qquad \mathrm{(B)\  }\text{at most }4\text{ points} \qquad \mathrm{(C) \  } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}</math>
 +
<math>\mathrm{(E) \  }\text{more than }16\text{ points}  </math>
  
 
[[1985 AHSME Problems/Problem 10|Solution]]
 
[[1985 AHSME Problems/Problem 10|Solution]]
 
==Problem 11==
 
==Problem 11==
−
How many distinguishable rearrangements of the letters in <math>CONTEST</math> have both the vowels first? (For instance, <math>OETCNST</math> is one such arrangement but <math>OTETSNC</math> is not.)
+
How many distinguishable rearrangements of the letters in <math>CONTEST</math> have both the vowels first? (For instance, <math>OETCNST</math> is one such arrangement, but <math>OTETSNC</math> is not.)
  
 
<math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \  } 240 \qquad \mathrm{(D) \  } 720 \qquad \mathrm{(E) \  }2520 </math>
 
<math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \  } 240 \qquad \mathrm{(D) \  } 720 \qquad \mathrm{(E) \  }2520 </math>
Line 116: Line 115:
 
[[1985 AHSME Problems/Problem 11|Solution]]
 
[[1985 AHSME Problems/Problem 11|Solution]]
 
==Problem 12==
 
==Problem 12==
−
Let <math> p, q </math> and <math> r </math> be distinct [[prime number]]s, where <math> 1 </math> is not considered a prime. Which of the following is the smallest positive [[perfect cube]] having <math> n=pq^2r^4 </math> as a [[divisor]]?
+
Let <math>p</math>, <math>q</math> and <math>r</math> be distinct prime numbers, where <math>1</math> is not considered a prime. Which of the following is the smallest positive perfect cube having <math>n = pq^2r^4</math> as a divisor?
  
−
<math> \mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) \ }(pq^2r^2)^3 \qquad \mathrm{(C) \ } (p^2q^2r^2)^3 \qquad \mathrm{(D) \ } (pqr^2)^3 \qquad \mathrm{(E) \  }4p^3q^3r^3 </math>
+
<math> \mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) }\left(pq^2r^2\right)^3 \qquad \mathrm{(C) } \left(p^2q^2r^2\right)^3 \qquad \mathrm{(D) } \left(pqr^2\right)^3 \qquad \mathrm{(E) \  }4p^3q^3r^3 </math>
  
 
[[1985 AHSME Problems/Problem 12|Solution]]
 
[[1985 AHSME Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
−
Pegs are put in a board <math> 1 </math> unit apart both horizontally and vertically. A rubber band is stretched over <math> 4 </math> pegs as shown in the figure, forming a [[quadrilateral]]. Its [[area]] in square units is
+
Pegs are put in a board <math>1</math> unit apart both horizontally and vertically. A rubber band is stretched over <math>4</math> pegs as shown in the figure, forming a quadrilateral. Its area in square units is
  
 
<asy>
 
<asy>
Line 137: Line 136:
 
[[1985 AHSME Problems/Problem 13|Solution]]
 
[[1985 AHSME Problems/Problem 13|Solution]]
 
==Problem 14==
 
==Problem 14==
−
Exactly three of the interior angles of a convex [[polygon]] are obtuse. What is the maximum number of sides of such a polygon?
+
Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?
  
 
<math> \mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \  } 6 \qquad \mathrm{(D) \  } 7 \qquad \mathrm{(E) \  }8 </math>
 
<math> \mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \  } 6 \qquad \mathrm{(D) \  } 7 \qquad \mathrm{(E) \  }8 </math>
Line 143: Line 142:
 
[[1985 AHSME Problems/Problem 14|Solution]]
 
[[1985 AHSME Problems/Problem 14|Solution]]
 
==Problem 15==
 
==Problem 15==
−
If <math> a </math> and <math> b </math> are positive numbers such that <math> a^b=b^a </math> and <math> b=9a </math>, then the value of <math> a </math> is
+
If <math>a</math> and <math>b</math> are positive numbers such that <math>a^b = b^a</math> and <math>b = 9a</math>, then the value of <math>a</math> is
  
 
<math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \  } \sqrt[9]{9} \qquad \mathrm{(D) \  } \sqrt[3]{9} \qquad \mathrm{(E) \  }\sqrt[4]{3} </math>
 
<math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \  } \sqrt[9]{9} \qquad \mathrm{(D) \  } \sqrt[3]{9} \qquad \mathrm{(E) \  }\sqrt[4]{3} </math>
Line 149: Line 148:
 
[[1985 AHSME Problems/Problem 15|Solution]]
 
[[1985 AHSME Problems/Problem 15|Solution]]
 
==Problem 16==
 
==Problem 16==
−
If <math> A=20^\circ </math> and <math> B=25^\circ </math>, then the value of <math> (1+\tan A)(1+\tan B) </math> is
+
If <math>\usepackage{gensymb} A = 20 \degree</math> and <math>\usepackage{gensymb} B = 25 \degree</math>, then the value of <math>\left(1+\tan A\right)\left(1+\tan B\right)</math> is
  
−
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \  } 1+\sqrt{2} \qquad \mathrm{(D) \  } 2(\tan A+\tan B) \qquad \mathrm{(E) \  }\text{none of these} </math>
+
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \  } 1+\sqrt{2} \qquad \mathrm{(D) \  } 2\left(\tan A+\tan B\right) \qquad \mathrm{(E) \  }\text{none of these} </math>
  
 
[[1985 AHSME Problems/Problem 16|Solution]]
 
[[1985 AHSME Problems/Problem 16|Solution]]
 
==Problem 17==
 
==Problem 17==
−
[[Diagonal]] <math> DB </math> of [[rectangle]] <math> ABCD </math> is divided into three segments of length <math> 1 </math> by [[parallel]] lines <math> L </math> and <math> L' </math> that pass through <math> A </math> and <math> C </math> and are [[perpendicular]] to <math> DB </math>. The area of <math> ABCD </math>, rounded to the one decimal place, is  
+
Diagonal <math>DB</math> of rectangle <math>ABCD</math> is divided into three segments of length <math>1</math> by parallel lines <math>L</math> and <math>L'</math> that pass through <math>A</math> and <math>C</math> and are perpendicular to <math>DB</math>. The area of <math>ABCD</math>, rounded to the one decimal place, is  
  
 
<asy>
 
<asy>
Line 181: Line 180:
 
[[1985 AHSME Problems/Problem 17|Solution]]
 
[[1985 AHSME Problems/Problem 17|Solution]]
 
==Problem 18==
 
==Problem 18==
−
Six bags of marbles contain <math> 18, 19, 21, 23, 25 </math> and <math> 34 </math> marbles, respectively. One bag contains chipped marbles only. The other <math> 5 </math> bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
+
Six bags of marbles contain <math>18</math>, <math>19</math>, <math>21</math>, <math>23</math>, <math>25</math> and <math>34</math> marbles, respectively. One bag contains chipped marbles only. The other <math>5</math> bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
  
 
<math> \mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \  } 21 \qquad \mathrm{(D) \  } 23 \qquad \mathrm{(E) \  }25 </math>
 
<math> \mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \  } 21 \qquad \mathrm{(D) \  } 23 \qquad \mathrm{(E) \  }25 </math>
Line 188: Line 187:
  
 
==Problem 19==
 
==Problem 19==
−
Consider the graphs <math> y=Ax^2 </math> and <math> y^2+3=x^2+4y </math>, where <math> A </math> is a positive constant and <math> x </math> and <math> y </math> are real variables. In how many points do the two graphs intersect?
+
Consider the graphs of <math>y = Ax^2</math> and <math>y^2+3 = x^2+4y</math>, where <math>A</math> is a positive constant and <math>x</math> and <math>y</math> are real variables. In how many points do the two graphs intersect?
  
 
<math> \mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad </math>  
 
<math> \mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad </math>  
Line 198: Line 197:
 
[[1985 AHSME Problems/Problem 19|Solution]]
 
[[1985 AHSME Problems/Problem 19|Solution]]
 
==Problem 20==
 
==Problem 20==
−
A wooden [[cube]] with edge length <math> n </math> units (where <math> n </math> is an integer <math> >2 </math>) is painted black all over. By slices parallel to its faces, the cube is cut into <math> n^3 </math> smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is <math> n </math>?
+
A wooden cube with edge length <math>n</math> units (where <math>n</math> is an integer <math>>2</math>) is painted black all over. By slices parallel to its faces, the cube is cut into <math>n^3</math> smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is <math>n</math>?
  
 
<math> \mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \  } 7 \qquad \mathrm{(D) \  } 8 \qquad \mathrm{(E) \  }\text{none of these} </math>
 
<math> \mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \  } 7 \qquad \mathrm{(D) \  } 8 \qquad \mathrm{(E) \  }\text{none of these} </math>
Line 204: Line 203:
 
[[1985 AHSME Problems/Problem 20|Solution]]
 
[[1985 AHSME Problems/Problem 20|Solution]]
 
==Problem 21==
 
==Problem 21==
−
How many integers <math> x </math> satisfy the equation <math> (x^2-x-1)^{x+2}=1? </math>
+
How many integers <math>x</math> satisfy the equation <cmath>\left(x^2-x-1\right)^{x+2} = 1?</cmath>
  
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 4 \qquad \mathrm{(D) \  } 5 \qquad \mathrm{(E) \  }\text{none of these} </math>
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 4 \qquad \mathrm{(D) \  } 5 \qquad \mathrm{(E) \  }\text{none of these} </math>
Line 210: Line 209:
 
[[1985 AHSME Problems/Problem 21|Solution]]
 
[[1985 AHSME Problems/Problem 21|Solution]]
 
==Problem 22==
 
==Problem 22==
−
In a circle with center <math> O </math>, <math> AD </math> is a [[diameter]], <math> ABC </math> is a [[chord]], <math> BO=5 </math> and <math> \angle ABO = \ \stackrel{\frown}{CD} \ =60^\circ </math>. Then the length of <math> BC </math> is
+
In a circle with center <math>O</math>, <math>AD</math> is a diameter, <math>ABC</math> is a chord, <math>BO = 5</math> and <math>\angle ABO = \ \stackrel{\frown}{CD} \ = 60^{\circ}</math>. Then the length of <math>BC</math> is
  
 
<asy>
 
<asy>
Line 230: Line 229:
 
[[1985 AHSME Problems/Problem 22|Solution]]
 
[[1985 AHSME Problems/Problem 22|Solution]]
 
==Problem 23==
 
==Problem 23==
−
If <math> x=\frac{-1+i\sqrt{3}}{2} </math> and <math> y=\frac{-1-i\sqrt{3}}{2} </math>, where <math> i^2=-1 </math>, then which of the following is ''not'' correct?
+
If <cmath>x = \frac{-1+i\sqrt{3}}{2} \qquad\text{and}\qquad y = \frac{-1-i\sqrt{3}}{2},</cmath> where <math>i^2 = -1</math>, then which of the following is not correct?
  
−
<math> \mathrm{(A)\ } x^5+y^5=-1 \qquad \mathrm{(B) \ }x^7+y^7=-1 \qquad \mathrm{(C) \  } x^9+y^9=-1 \qquad </math>  
+
<math> \mathrm{(A)\ } x^5+y^5 = -1 \qquad \mathrm{(B) \ }x^7+y^7 = -1 \qquad \mathrm{(C) \  } x^9+y^9 = -1 \qquad </math>  
  
−
<math> \mathrm{(D) \  } x^{11}+y^{11}=-1 \qquad \mathrm{(E) \  }x^{13}+y^{13}=-1 </math>
+
<math> \mathrm{(D) \  } x^{11}+y^{11} = -1 \qquad \mathrm{(E) \  }x^{13}+y^{13} = -1 </math>
  
 
[[1985 AHSME Problems/Problem 23|Solution]]
 
[[1985 AHSME Problems/Problem 23|Solution]]
 
==Problem 24==
 
==Problem 24==
−
A non-zero [[digit]] is chosen in such a way that the probability of choosing digit <math> d </math> is <math> \log_{10}{(d+1)}-\log_{10}{d} </math>. The probability that the digit <math> 2 </math> is chosen is exactly <math> \frac{1}{2} </math> the probability that the digit chosen is in the set
+
A non-zero digit is chosen in such a way that the probability of choosing digit <math>d</math> is <math>\log_{10}{(d+1)}-\log_{10}{d}</math>. The probability that the digit <math>2</math> is chosen is exactly <math>1/2</math> the probability that the digit chosen is in the set
  
−
<math> \mathrm{(A)\ } \{2, 3\} \qquad \mathrm{(B) \ }\{3, 4\} \qquad \mathrm{(C) \  } \{4, 5, 6, 7, 8\} \qquad \mathrm{(D) \  } \{5, 6, 7, 8, 9\} \qquad \mathrm{(E) \  }\{4, 5, 6, 7, 8, 9\} </math>
+
<math> \mathrm{(A)\ } \{2,3\} \qquad \mathrm{(B) \ }\{3,4\} \qquad \mathrm{(C) \  } \{4,5,6,7,8\} \qquad \mathrm{(D) \  } \{5,6,7,8,9\} \qquad \mathrm{(E) \  }\{4,5,6,7,8,9\} </math>
  
 
[[1985 AHSME Problems/Problem 24|Solution]]
 
[[1985 AHSME Problems/Problem 24|Solution]]
 
==Problem 25==
 
==Problem 25==
−
The [[volume]] of a certain rectangular solid is <math> 8 \text{cm}^3 </math>, its total [[surface area]] is <math> 32 \text{cm}^2 </math>, and its three dimensions are in [[geometric progression]]. The sums of the lengths in cm of all the edges of this solid is
+
The volume of a certain rectangular solid is <math>8</math> cm<sup>3</sup>, its total surface area is <math>32</math> cm<sup>2</sup>, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
  
 
<math> \mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \  } 36 \qquad \mathrm{(D) \  } 40 \qquad \mathrm{(E) \  }44 </math>
 
<math> \mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \  } 36 \qquad \mathrm{(D) \  } 40 \qquad \mathrm{(E) \  }44 </math>
Line 250: Line 249:
 
[[1985 AHSME Problems/Problem 25|Solution]]
 
[[1985 AHSME Problems/Problem 25|Solution]]
 
==Problem 26==
 
==Problem 26==
−
Find the least [[positive integer]] <math> n </math> for which <math> \frac{n-13}{5n+6} </math> is a non-zero reducible fraction.
+
Find the least positive integer <math>n</math> for which <math>\frac{n-13}{5n+6}</math> is a non-zero reducible fraction.
  
 
<math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \  } 155 \qquad \mathrm{(D) \  } 226 \qquad \mathrm{(E) \  }\text{none of these} </math>
 
<math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \  } 155 \qquad \mathrm{(D) \  } 226 \qquad \mathrm{(E) \  }\text{none of these} </math>
  
 
[[1985 AHSME Problems/Problem 26|Solution]]
 
[[1985 AHSME Problems/Problem 26|Solution]]
 +
 
==Problem 27==
 
==Problem 27==
−
Consider a sequence <math> x_1, x_2, x_3, \cdots </math> defined by
+
Consider a sequence <math>x_1,x_2,x_3,\dotsc</math> defined by:
−
 
+
<cmath>\begin{align*}&x_1 = \sqrt[3]{3}, \\ &x_2 = \left(\sqrt[3]{3}\right)^{\sqrt[3]{3}},\end{align*}</cmath>
−
<math> x_1=\sqrt[3]{3} </math>
 
−
 
 
−
<math> x_2=(\sqrt[3]{3})^{\sqrt[3]{3}} </math>
 
−
 
 
 
and in general
 
and in general
 +
<cmath>x_n = \left(x_{n-1}\right)^{\sqrt[3]{3}} \text{ for } n > 1.</cmath>
  
−
<math> x_n=(x_{n-1})^{\sqrt[3]{3}} </math> for <math> n>1 </math>.
+
What is the smallest value of <math>n</math> for which <math>x_n</math> is an integer?
−
 
 
−
What is the smallest value of <math> n </math> for which <math> x_n </math> is an [[integer]]?
 
  
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 4 \qquad \mathrm{(D) \  } 9 \qquad \mathrm{(E) \  }27 </math>
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 4 \qquad \mathrm{(D) \  } 9 \qquad \mathrm{(E) \  }27 </math>
Line 273: Line 268:
  
 
==Problem 28==
 
==Problem 28==
−
In <math> \triangle ABC </math>, we have <math> \angle C=3\angle A, a=27 </math> and <math> c=48 </math>. What is <math> b </math>?
+
In <math>\triangle ABC</math>, we have <math>\angle C = 3\angle A</math>, <math>a = 27</math> and <math>c = 48</math>. What is <math>b</math>?
  
 
<asy>
 
<asy>
Line 289: Line 284:
  
 
[[1985 AHSME Problems/Problem 28|Solution]]
 
[[1985 AHSME Problems/Problem 28|Solution]]
 +
 
==Problem 29==
 
==Problem 29==
−
In their base <math> 10 </math> representations, the integer <math> a </math> consists of a sequence of <math> 1985 </math> eights and the integer <math> b </math> consists of a sequence of <math> 1985 </math> fives. What is the sum of the digits of the base <math> 10 </math> representation of <math> 9ab </math>?
+
In their base <math>10</math> representations, the integer <math>a</math> consists of a sequence of <math>1985</math> eights and the integer <math>b</math> consists of a sequence of <math>1985</math> fives. What is the sum of the digits of the base <math>10</math> representation of the integer <math>9ab</math>?
  
 
<math> \mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \  } 17865 \qquad \mathrm{(D) \  } 17874 \qquad \mathrm{(E) \  }19851 </math>
 
<math> \mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \  } 17865 \qquad \mathrm{(D) \  } 17874 \qquad \mathrm{(E) \  }19851 </math>
Line 297: Line 293:
  
 
==Problem 30==
 
==Problem 30==
−
Let <math> \lfloor x \rfloor </math> be the greatest integer less than or equal to <math> x </math>. Then the number of real solutions to <math> 4x^2-40\lfloor x \rfloor +51=0 </math> is
+
Let <math>\left\lfloor x\right\rfloor</math> be the greatest integer less than or equal to <math>x</math>. Then the number of real solutions to <math>4x^2-40\left\lfloor x\right\rfloor+51 = 0</math> is
  
 
<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 2 \qquad \mathrm{(D) \  } 3 \qquad \mathrm{(E) \  }4 </math>
 
<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 2 \qquad \mathrm{(D) \  } 3 \qquad \mathrm{(E) \  }4 </math>

Latest revision as of 02:26, 20 March 2024

1985 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 $2x+1=8$, then $4x+1=$

$\mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 18 \qquad \mathrm{(E) \ }19$

Solution

Problem 2

In an arcade game, the "monster" is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $\usepackage{gensymb} 60\degree$. What is the perimeter of the monster in cm?

[asy] size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("$60^\circ$", (1,0));[/asy]

$\mathrm{(A)\ } \pi+2 \qquad \mathrm{(B) \ }2\pi \qquad \mathrm{(C) \ } \frac{5}{3}\pi \qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2$

Solution

Problem 3

In right $\triangle ABC$ with legs $5$ and $12$, arcs of circles are drawn, one with center $A$ and radius $12$, the other with center $B$ and radius $5$. They intersect the hypotenuse in $M$ and $N$. Then $MN$ has length

[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); real r=degrees(B); draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270)); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$12$", (6,0), S); label("$5$", (12,3.5), E);[/asy]

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }\frac{13}{5} \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 4 \qquad \mathrm{(E) \ }\frac{24}{5}$

Solution

Problem 4

A large bag of coins contains pennies, dimes and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is

$\mathrm{(A)\ } $306 \qquad \mathrm{(B) \ } $333 \qquad \mathrm{(C)\ } $342 \qquad \mathrm{(D) \ } $348 \qquad \mathrm{(E) \ } $360$

Solution

Problem 5

Which terms must be removed from the sum

\[\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}\]

if the sum of the remaining terms is to equal $1$?

$\mathrm{(A)\ } \frac{1}{4}\text{ and }\frac{1}{8} \qquad \mathrm{(B) \ }\frac{1}{4}\text{ and }\frac{1}{12} \qquad \mathrm{(C) \ } \frac{1}{8}\text{ and }\frac{1}{12} \qquad \mathrm{(D) \ } \frac{1}{6}\text{ and }\frac{1}{10} \qquad \mathrm{(E) \ }\frac{1}{8}\text{ and }\frac{1}{10}$

Solution

Problem 6

One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $\frac{2}{3}$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is

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

Solution

Problem 7

In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from right to left. Thus, $a \times b - c$ in such languages means the same as $a(b-c)$ in ordinary algebraic notation. If $a\div b-c+d$ is evaluated in such a language, the result in ordinary algebraic notation would be

$\mathrm{(A)\ } \frac{a}{b}-c+d \qquad \mathrm{(B) \ }\frac{a}{b}-c-d \qquad \mathrm{(C) \ } \frac{d+c-b}{a} \qquad \mathrm{(D) \ } \frac{a}{b-c+d} \qquad \mathrm{(E) \ }\frac{a}{b-c-d}$

Solution

Problem 8

Let $a,a',b,b'$ be real numbers with $a$ and $a'$ nonzero. The solution to $ax+b=0$ is less than the solution to $a'x+b'=0$ if and only if

$\mathrm{(A)\ } a'b < ab' \qquad \mathrm{(B) \ }ab' < a'b \qquad \mathrm{(C) \ } ab < a'b' \qquad \mathrm{(D) \ } \frac{b}{a} < \frac{b'}{a'} \qquad \mathrm{(E) \ }\frac{b'}{a'} < \frac{b}{a}$

Solution

Problem 9

The odd positive integers $1, 3, 5, 7, \ldots$, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which $1985$ appears is the

[asy] int i,j; for(i=0; i<4; i=i+1) { label(string(16*i+1), (2*1,-2*i)); label(string(16*i+3), (2*2,-2*i)); label(string(16*i+5), (2*3,-2*i)); label(string(16*i+7), (2*4,-2*i)); } for(i=0; i<3; i=i+1) { for(j=0; j<4; j=j+1) { label(string(16*i+15-2*j), (2*j,-2*i-1)); }} dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10)); for(i=-10; i<-6; i=i+1) { for(j=1; j<4; j=j+1) { dot((2*j,i)); }}[/asy]

$\mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \ } \text{third} \qquad \mathrm{(D) \ } \text{fourth} \qquad \mathrm{(E) \ }\text{fifth}$

Solution

Problem 10

An arbitrary circle can intersect the graph of $y = \sin x$ in

$\mathrm{(A)\ } \text{at most }2\text{ points} \qquad \mathrm{(B)\ }\text{at most }4\text{ points} \qquad \mathrm{(C) \ } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}$ $\mathrm{(E) \ }\text{more than }16\text{ points}$

Solution

Problem 11

How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement, but $OTETSNC$ is not.)

$\mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520$

Solution

Problem 12

Let $p$, $q$ and $r$ be distinct prime numbers, where $1$ is not considered a prime. Which of the following is the smallest positive perfect cube having $n = pq^2r^4$ as a divisor?

$\mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) }\left(pq^2r^2\right)^3 \qquad \mathrm{(C) } \left(p^2q^2r^2\right)^3 \qquad \mathrm{(D) } \left(pqr^2\right)^3 \qquad \mathrm{(E) \ }4p^3q^3r^3$

Solution

Problem 13

Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is

[asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));[/asy]

$\mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }4.5 \qquad \mathrm{(C) \ } 5 \qquad \mathrm{(D) \ } 5.5 \qquad \mathrm{(E) \ }6$

Solution

Problem 14

Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?

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

Solution

Problem 15

If $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 9a$, then the value of $a$ is

$\mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \ } \sqrt[9]{9} \qquad \mathrm{(D) \ } \sqrt[3]{9} \qquad \mathrm{(E) \ }\sqrt[4]{3}$

Solution

Problem 16

If $\usepackage{gensymb} A = 20 \degree$ and $\usepackage{gensymb} B = 25 \degree$, then the value of $\left(1+\tan A\right)\left(1+\tan B\right)$ is

$\mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2\left(\tan A+\tan B\right) \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Problem 17

Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. The area of $ABCD$, rounded to the one decimal place, is

[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("$L$", (x/2,0), SW); label("$L^\prime$", C, SW); label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle);[/asy]

$\mathrm{(A)\ } 4.1 \qquad \mathrm{(B) \ }4.2 \qquad \mathrm{(C) \ } 4.3 \qquad \mathrm{(D) \ } 4.4 \qquad \mathrm{(E) \ }4.5$

Solution

Problem 18

Six bags of marbles contain $18$, $19$, $21$, $23$, $25$ and $34$ marbles, respectively. One bag contains chipped marbles only. The other $5$ bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?

$\mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \ } 21 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ }25$

Solution

Problem 19

Consider the graphs of $y = Ax^2$ and $y^2+3 = x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?

$\mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad$

$\mathrm{(C) \ }\text{at least }1,\text{ but the number varies for different positive values of }A \qquad$

$\mathrm{(D) \ }0\text{ for at least one positive value of }A \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Problem 20

A wooden cube with edge length $n$ units (where $n$ is an integer $>2$) is painted black all over. By slices parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $n$?

$\mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Problem 21

How many integers $x$ satisfy the equation \[\left(x^2-x-1\right)^{x+2} = 1?\]

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Problem 22

In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$ and $\angle ABO = \ \stackrel{\frown}{CD} \ = 60^{\circ}$. Then the length of $BC$ is

[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C); draw(C--A--D^^B--O^^Circle(O,1)); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$O$", O, dir(305)); label("$5$", B--O, dir(O--B)*dir(90)); label("$60^\circ$", dir(185), dir(185)); label("$60^\circ$", B+0.05*dir(-25), dir(-25));[/asy]

$\mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }3+\sqrt{3} \qquad \mathrm{(C) \ } 5-\frac{\sqrt{3}}{2} \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of the above}$

Solution

Problem 23

If \[x = \frac{-1+i\sqrt{3}}{2} \qquad\text{and}\qquad y = \frac{-1-i\sqrt{3}}{2},\] where $i^2 = -1$, then which of the following is not correct?

$\mathrm{(A)\ } x^5+y^5 = -1 \qquad \mathrm{(B) \ }x^7+y^7 = -1 \qquad \mathrm{(C) \ } x^9+y^9 = -1 \qquad$

$\mathrm{(D) \ } x^{11}+y^{11} = -1 \qquad \mathrm{(E) \ }x^{13}+y^{13} = -1$

Solution

Problem 24

A non-zero digit is chosen in such a way that the probability of choosing digit $d$ is $\log_{10}{(d+1)}-\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $1/2$ the probability that the digit chosen is in the set

$\mathrm{(A)\ } \{2,3\} \qquad \mathrm{(B) \ }\{3,4\} \qquad \mathrm{(C) \ } \{4,5,6,7,8\} \qquad \mathrm{(D) \ } \{5,6,7,8,9\} \qquad \mathrm{(E) \ }\{4,5,6,7,8,9\}$

Solution

Problem 25

The volume of a certain rectangular solid is $8$ cm3, its total surface area is $32$ cm2, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is

$\mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 40 \qquad \mathrm{(E) \ }44$

Solution

Problem 26

Find the least positive integer $n$ for which $\frac{n-13}{5n+6}$ is a non-zero reducible fraction.

$\mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Problem 27

Consider a sequence $x_1,x_2,x_3,\dotsc$ defined by: \begin{align*}&x_1 = \sqrt[3]{3}, \\ &x_2 = \left(\sqrt[3]{3}\right)^{\sqrt[3]{3}},\end{align*} and in general \[x_n = \left(x_{n-1}\right)^{\sqrt[3]{3}} \text{ for } n > 1.\]

What is the smallest value of $n$ for which $x_n$ is an integer?

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27$

Solution

Problem 28

In $\triangle ABC$, we have $\angle C = 3\angle A$, $a = 27$ and $c = 48$. What is $b$?

[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(14,0), C=(10,6); draw(A--B--C--cycle); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$a$", B--C, dir(B--C)*dir(-90)); label("$b$", A--C, dir(C--A)*dir(-90)); label("$c$", A--B, dir(A--B)*dir(-90));[/asy]

$\mathrm{(A)\ } 33 \qquad \mathrm{(B) \ }35 \qquad \mathrm{(C) \ } 37 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ }\text{not uniquely determined}$

Solution

Problem 29

In their base $10$ representations, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. What is the sum of the digits of the base $10$ representation of the integer $9ab$?

$\mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \ } 17865 \qquad \mathrm{(D) \ } 17874 \qquad \mathrm{(E) \ }19851$

Solution

Problem 30

Let $\left\lfloor x\right\rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\left\lfloor x\right\rfloor+51 = 0$ is

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

Solution

See also

1986 AHSME (Problems • Answer Key • Resources)
Preceded by
1984 AHSME 		Followed by
1986 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


The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems&oldid=217186"