Difference between revisions of "1961 AHSME Problems"

(Created page with "== Problem 1== When simplified, <math>(-\frac{1}{125})^{-2/3}</math> becomes: <math>\textbf{(A)}\ \frac{1}{25} \qquad \textbf{(B)}\ -\frac{1}{25} \qquad \textbf{(C)}\ 25\qquad ...")
 
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we say that the set is closed under that operation. Then <math>u</math> is closed under:
 
we say that the set is closed under that operation. Then <math>u</math> is closed under:
  
<math>\textbf{(A)}\ \text{Addition}\qquad
+
<math>\text{(A)}\ \text{Addition}\qquad
\textbf{(B)}\ \text{Multiplication} \qquad
+
\text{(B)}\ \text{Multiplication} \qquad
\textbf{(C)}\ \text{Division} \qquad\
+
\text{(C)}\ \text{Division} \qquad\
\textbf{(D)}\ \text{Extraction of a positive integral root}}\qquad
+
\text{(D)}\ \text{Extraction of a positive integral root}}\qquad
\textbf{(E)}\text{None of these}}    </math>
+
\text{(E)}\text{None of these}}    </math>
  
 
[[1961 AHSME Problems/Problem 4|Solution]]
 
[[1961 AHSME Problems/Problem 4|Solution]]
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<math>\textbf{(A)}\ C_1+C_2=A+B \qquad
 
<math>\textbf{(A)}\ C_1+C_2=A+B \qquad
\textbf{(B)}\ C_1-C_2=B-A \qquad
+
\textbf{(B)}\ C_1-C_2=B-A \qquad\
 
\textbf{(C)}\ C_1-C_2=A-B} \qquad
 
\textbf{(C)}\ C_1-C_2=A-B} \qquad
 
\textbf{(D)}\ C_1+C_2=B-A}\qquad
 
\textbf{(D)}\ C_1+C_2=B-A}\qquad
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Consider the graphs of <math>y=2\log{x}</math> and <math>y=\log{2x}</math>. We may say that:
 
Consider the graphs of <math>y=2\log{x}</math> and <math>y=\log{2x}</math>. We may say that:
  
<math>\textbf{(A)}\ \text{They do not intersect}\qquad\
+
<math>\text{(A)}\ \text{They do not intersect}\qquad \
\textbf{(B)}\ \text{They intersect at 1 point only}\qquad\
+
\text{(B)}\ \text{They intersect at 1 point only}\qquad \
\textbf{(C)}\ \text{They intersect at 2 points only} \qquad\
+
\text{(C)}\ \text{They intersect at 2 points only} \qquad \
\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }\qquad\
+
\text{(D)}\ \text{They intersect at a finite number of points but greater than 2 }\qquad \
\textbf{(E)}\ \text{They coincide} }  </math>   
+
\text{(E)}\ \text{They coincide} }  </math>   
  
 
[[1961 AHSME Problems/Problem 19|Solution]]
 
[[1961 AHSME Problems/Problem 19|Solution]]
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Thirty-one books are arranged from left to right in order of increasing prices.  
 
Thirty-one books are arranged from left to right in order of increasing prices.  
The price of each book differs by &#036;2 from that of each adjacent book.  
+
The price of each book differs by <math>\textdollar{2}</math> from that of each adjacent book.  
 
For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:
 
For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:
  
<math>\textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}\qquad
+
<math>\textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}\qquad \
\textbf{(B)}\ \text{The middle book sells for &#036;36 }\qquad
+
\textbf{(B)}\ \text{The middle book sells for \textdollar 36 }\qquad \
\textbf{(C)}\ \text{The cheapest book sells for &#036;4 }\qquad
+
\textbf{(C)}\ \text{The cheapest book sells for \textdollar4 }\qquad \
\textbf{(D)}\ \text{The most expensive book sells for &#036;64 }\qquad
+
\textbf{(D)}\ \text{The most expensive book sells for \textdollar64 } \qquad \
 
\textbf{(E)}\ \text{None of these is correct }  </math>   
 
\textbf{(E)}\ \text{None of these is correct }  </math>   
  
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<math>\textbf{(A)}\ x^2-bx-ac=0\qquad
 
<math>\textbf{(A)}\ x^2-bx-ac=0\qquad
\textbf{(B)}\ x^2-bx+ac=0 \qquad
+
\textbf{(B)}\ x^2-bx+ac=0 \qquad\
 
\textbf{(C)}\ x^2+3bx+ca+2b^2=0 \qquad
 
\textbf{(C)}\ x^2+3bx+ca+2b^2=0 \qquad
\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }\qquad
+
\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }\qquad\
 
\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} </math>     
 
\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} </math>     
  
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\textbf{(C)}\ 4:3 \qquad
 
\textbf{(C)}\ 4:3 \qquad
 
\textbf{(D)}\ 3:1 }\qquad
 
\textbf{(D)}\ 3:1 }\qquad
\textbf{(E)}\ 7:1 }     
+
\textbf{(E)}\ 7:1 }    </math>
  
 
[[1961 AHSME Problems/Problem 31|Solution]]
 
[[1961 AHSME Problems/Problem 31|Solution]]
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== Problem 32==
 
== Problem 32==
  
A regular polygon of </math>n<math> sides is inscribed in a circle of radius </math>R<math>. The area of the polygon is </math>3R^2<math>. Then </math>n<math> equals:  
+
A regular polygon of <math>n</math> sides is inscribed in a circle of radius <math>R</math>. The area of the polygon is <math>3R^2</math>. Then <math>n</math> equals:  
  
</math>\textbf{(A)}\ 8\qquad
+
<math>\textbf{(A)}\ 8\qquad
 
\textbf{(B)}\ 10\qquad
 
\textbf{(B)}\ 10\qquad
 
\textbf{(C)}\ 12\qquad
 
\textbf{(C)}\ 12\qquad
 
\textbf{(D)}\ 15}\qquad
 
\textbf{(D)}\ 15}\qquad
\textbf{(E)}\ 18}  <math>   
+
\textbf{(E)}\ 18}  </math>   
  
 
[[1961 AHSME Problems/Problem 32|Solution]]
 
[[1961 AHSME Problems/Problem 32|Solution]]
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== Problem 33==
 
== Problem 33==
  
The number of solutions of </math>2^{2x}-3^{2y}=55<math>, in which </math>x<math> and </math>y<math> are integers, is:
+
The number of solutions of <math>2^{2x}-3^{2y}=55</math>, in which <math>x</math> and <math>y</math> are integers, is:
  
</math>\textbf{(A)}\ 0\qquad
+
<math>\textbf{(A)}\ 0\qquad
 
\textbf{(B)}\ 1\qquad
 
\textbf{(B)}\ 1\qquad
 
\textbf{(C)}\ 2\qquad
 
\textbf{(C)}\ 2\qquad
 
\textbf{(D)}\ 3}\qquad
 
\textbf{(D)}\ 3}\qquad
\textbf{(E)}\ \text{More than three, but finite}} <math>   
+
\textbf{(E)}\ \text{More than three, but finite}} </math>   
  
 
[[1961 AHSME Problems/Problem 33|Solution]]
 
[[1961 AHSME Problems/Problem 33|Solution]]
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== Problem 34==
 
== Problem 34==
  
Let S be the set of values assumed by the fraction </math>\frac{2x+3}{x+2}<math>.
+
Let S be the set of values assumed by the fraction <math>\frac{2x+3}{x+2}</math>.
When </math>x<math> is any member of the interval </math>x \ge 0<math>. If there exists a number </math>M<math> such that no number of the set </math>S<math> is greater than </math>M<math>,  
+
When <math>x</math> is any member of the interval <math>x \ge 0</math>. If there exists a number <math>M</math> such that no number of the set <math>S</math> is greater than <math>M</math>,  
then </math>M<math> is an upper bound of </math>S<math>. If there exists a number </math>m<math> such that such that no number of the set </math>S<math> is less than </math>m<math>,  
+
then <math>M</math> is an upper bound of <math>S</math>. If there exists a number <math>m</math> such that such that no number of the set <math>S</math> is less than <math>m</math>,  
then </math>m<math> is a lower bound of </math>S<math>. We may then say:
+
then <math>m</math> is a lower bound of <math>S</math>. We may then say:
  
</math>\textbf{(A)}\ \text{m is in S, but M is not in S}\qquad\
+
<math>\textbf{(A)}\ \text{m is in S, but M is not in S}\qquad\
 
\textbf{(B)}\ \text{M is in S, but m is not in S}\qquad\
 
\textbf{(B)}\ \text{M is in S, but m is not in S}\qquad\
 
\textbf{(C)}\ \text{Both m and M are in S}\qquad\
 
\textbf{(C)}\ \text{Both m and M are in S}\qquad\
 
\textbf{(D)}\ \text{Neither m nor M are in S}\qquad\
 
\textbf{(D)}\ \text{Neither m nor M are in S}\qquad\
\textbf{(E)}\ \text{M does not exist either in or outside S}    <math>  
+
\textbf{(E)}\ \text{M does not exist either in or outside S}    </math>  
  
 
[[1961 AHSME Problems/Problem 34|Solution]]
 
[[1961 AHSME Problems/Problem 34|Solution]]
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== Problem 35==
 
== Problem 35==
  
The number </math>695<math> is to be written with a factorial base of numeration, that is, </math>695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!<math>  
+
The number <math>695</math> is to be written with a factorial base of numeration, that is, <math>695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!</math>  
where </math>a_1, a_2, a_3 ... a_n<math> are integers such that </math>0 \le a_k \le k,<math> and </math>n!<math> means </math>n(n-1)(n-2)...2 \times 1<math>. Find </math>a_4<math>  
+
where <math>a_1, a_2, a_3 ... a_n</math> are integers such that <math>0 \le a_k \le k,</math> and <math>n!</math> means <math>n(n-1)(n-2)...2 \times 1</math>. Find <math>a_4</math>  
  
</math>\textbf{(A)}\ 0\qquad
+
<math>\textbf{(A)}\ 0\qquad
 
\textbf{(B)}\ 1\qquad
 
\textbf{(B)}\ 1\qquad
 
\textbf{(C)}\ 2\qquad
 
\textbf{(C)}\ 2\qquad
 
\textbf{(D)}\ 3}\qquad
 
\textbf{(D)}\ 3}\qquad
\textbf{(E)}\ 4}    <math>
+
\textbf{(E)}\ 4}    </math>
  
 
[[1961 AHSME Problems/Problem 35|Solution]]
 
[[1961 AHSME Problems/Problem 35|Solution]]
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== Problem 36==
 
== Problem 36==
  
In </math>\triangle ABC<math> the median from </math>A<math> is given perpendicular to the median from </math>B<math>. If </math>BC=7<math> and </math>AC=6<math>, find the length of </math>AB<math>.  
+
In <math>\triangle ABC</math> the median from <math>A</math> is given perpendicular to the median from <math>B</math>. If <math>BC=7</math> and <math>AC=6</math>, find the length of <math>AB</math>.  
  
</math>\textbf{(A)}\ 4\qquad
+
<math>\textbf{(A)}\ 4\qquad
 
\textbf{(B)}\ \sqrt{17} \qquad
 
\textbf{(B)}\ \sqrt{17} \qquad
 
\textbf{(C)}\ 4.25\qquad
 
\textbf{(C)}\ 4.25\qquad
 
\textbf{(D)}\ 2\sqrt{5} }\qquad
 
\textbf{(D)}\ 2\sqrt{5} }\qquad
\textbf{(E)}\ 4.5}  <math>   
+
\textbf{(E)}\ 4.5}  </math>   
  
 
[[1961 AHSME Problems/Problem 36|Solution]]
 
[[1961 AHSME Problems/Problem 36|Solution]]
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== Problem 37==
 
== Problem 37==
  
In racing over a distance </math>d<math> at uniform speed, </math>A<math> can beat </math>B<math> by </math>20<math> yards, </math>B<math> can beat </math>C<math> by </math>10<math> yards,  
+
In racing over a distance <math>d</math> at uniform speed, <math>A</math> can beat <math>B</math> by <math>20</math> yards, <math>B</math> can beat <math>C</math> by <math>10</math> yards,  
and </math>A<math> can beat </math>C<math> by </math>28<math> yards. Then </math>d<math>, in yards, equals:
+
and <math>A</math> can beat <math>C</math> by <math>28</math> yards. Then <math>d</math>, in yards, equals:
  
</math>\textbf{(A)}\ \text{Not determined by the given information}\qquad
+
<math>\textbf{(A)}\ \text{Not determined by the given information}\qquad
 
\textbf{(B)}\ 58\qquad
 
\textbf{(B)}\ 58\qquad
 
\textbf{(C)}\ 100\qquad
 
\textbf{(C)}\ 100\qquad
 
\textbf{(D)}\ 116}\qquad
 
\textbf{(D)}\ 116}\qquad
\textbf{(E)}\ 120} <math>   
+
\textbf{(E)}\ 120} </math>   
  
 
[[1961 AHSME Problems/Problem 37|Solution]]
 
[[1961 AHSME Problems/Problem 37|Solution]]
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== Problem 38==
 
== Problem 38==
  
</math>\triangle ABC<math> is inscribed in a semicircle of radius </math>r<math> so that its base </math>AB<math> coincides with diameter </math>AB<math>.  
+
<math>\triangle ABC</math> is inscribed in a semicircle of radius <math>r</math> so that its base <math>AB</math> coincides with diameter <math>AB</math>.  
Point </math>C<math> does not coincide with either </math>A<math> or </math>B<math>. Let </math>s=AC+BC<math>. Then, for all permissible positions of </math>C<math>:
+
Point <math>C</math> does not coincide with either <math>A</math> or <math>B</math>. Let <math>s=AC+BC</math>. Then, for all permissible positions of <math>C</math>:
  
</math>\textbf{(A)}\ s^2\le8r^2\qquad
+
<math>\textbf{(A)}\ s^2\le8r^2\qquad
 
\textbf{(B)}\ s^2=8r^2 \qquad
 
\textbf{(B)}\ s^2=8r^2 \qquad
\textbf{(C)}\ s^2 \ge 8r^2 \qquad
+
\textbf{(C)}\ s^2 \ge 8r^2 \qquad\
 
\textbf{(D)}\ s^2\le4r^2 }\qquad
 
\textbf{(D)}\ s^2\le4r^2 }\qquad
\textbf{(E)}\ x^2=4r^2 }  <math>   
+
\textbf{(E)}\ x^2=4r^2 }  </math>   
  
 
[[1961 AHSME Problems/Problem 38|Solution]]
 
[[1961 AHSME Problems/Problem 38|Solution]]
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== Problem 39==
 
== Problem 39==
  
Any five points are taken inside or on a square with side length </math>1<math>. Let a be the smallest possible number with the  
+
Any five points are taken inside or on a square with side length <math>1</math>. Let a be the smallest possible number with the  
 
property that it is always possible to select one pair of points from these five such that the distance between them  
 
property that it is always possible to select one pair of points from these five such that the distance between them  
is equal to or less than </math>a<math>. Then </math>a<math> is:
+
is equal to or less than <math>a</math>. Then <math>a</math> is:
  
</math>\textbf{(A)}\ \sqrt{3}/3\qquad
+
<math>\textbf{(A)}\ \sqrt{3}/3\qquad
 
\textbf{(B)}\ \sqrt{2}/2\qquad
 
\textbf{(B)}\ \sqrt{2}/2\qquad
 
\textbf{(C)}\ 2\sqrt{2}/3\qquad
 
\textbf{(C)}\ 2\sqrt{2}/3\qquad
 
\textbf{(D)}\ 1 }\qquad
 
\textbf{(D)}\ 1 }\qquad
\textbf{(E)}\ \sqrt{2}} <math>
+
\textbf{(E)}\ \sqrt{2}} </math>
  
 
[[1961 AHSME Problems/Problem 39|Solution]]
 
[[1961 AHSME Problems/Problem 39|Solution]]
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== Problem 40==
 
== Problem 40==
  
Find the minimum value of </math>\sqrt{x^2+y^2}<math> if </math>5x+12y=60<math>.  
+
Find the minimum value of <math>\sqrt{x^2+y^2}</math> if <math>5x+12y=60</math>.  
  
</math>\textbf{(A)}\ \frac{60}{13}\qquad
+
<math>\textbf{(A)}\ \frac{60}{13}\qquad
 
\textbf{(B)}\ \frac{13}{5}\qquad
 
\textbf{(B)}\ \frac{13}{5}\qquad
 
\textbf{(C)}\ \frac{13}{12}\qquad
 
\textbf{(C)}\ \frac{13}{12}\qquad
 
\textbf{(D)}\ 1}\qquad
 
\textbf{(D)}\ 1}\qquad
\textbf{(E)}\ 0 } $
+
\textbf{(E)}\ 0 } </math>
  
 
[[1961 AHSME Problems/Problem 40|Solution]]
 
[[1961 AHSME Problems/Problem 40|Solution]]

Revision as of 08:41, 10 October 2014

Problem 1

When simplified, $(-\frac{1}{125})^{-2/3}$ becomes:

$\textbf{(A)}\ \frac{1}{25} \qquad \textbf{(B)}\ -\frac{1}{25} \qquad \textbf{(C)}\ 25\qquad \textbf{(D)}\ -25}\qquad \textbf{(E)}\ 25\sqrt{-1}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 2

An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?

$\textbf{(A)}\ \frac{a}{1080r}\qquad \textbf{(B)}\ \frac{30r}{a}\qquad \textbf{(C)}\ \frac{30a}{r}\qquad \textbf{(D)}\ \frac{10r}{a}}\qquad \textbf{(E)}\ \frac{10a}{r}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 3

If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:

$\textbf{(A)}\ \pm \frac{2}{3} \qquad \textbf{(B)}\ -\frac{2}{3}\qquad \textbf{(C)}\ -\frac{3}{2} \qquad \textbf{(D)}\ 6}\qquad \textbf{(E)}\ -6}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 4

Let the set consisting of the squares of the positive integers be called $u$; thus $u$ is the set $1, 4, 9, 16 \ldots$. If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then $u$ is closed under:

$\text{(A)}\ \text{Addition}\qquad \text{(B)}\ \text{Multiplication} \qquad \text{(C)}\ \text{Division} \qquad\ \text{(D)}\ \text{Extraction of a positive integral root}}\qquad \text{(E)}\text{None of these}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 5

Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:

$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 }\qquad \textbf{(E)}\ x^4+1}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 6

When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes:

$\textbf{(A)}\ 6\log{2} \qquad \textbf{(B)}\ \log{2} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0}\qquad \textbf{(E)}\ -1}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 7

When simplified, the third term in the expansion of $(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2})^6$ is:

$\textbf{(A)}\ \frac{15}{x}\qquad \textbf{(B)}\ -\frac{15}{x}\qquad \textbf{(C)}\ -\frac{6x^2}{a^9} \qquad \textbf{(D)}\ \frac{20}{a^3}}\qquad \textbf{(E)}\ -\frac{20}{a^3}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 8

Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$. The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Then:

$\textbf{(A)}\ C_1+C_2=A+B \qquad \textbf{(B)}\ C_1-C_2=B-A \qquad\ \textbf{(C)}\ C_1-C_2=A-B} \qquad \textbf{(D)}\ C_1+C_2=B-A}\qquad \textbf{(E)}\ C_1-C_2=A+B}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 9

Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$. If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals:

$\textbf{(A)} a\qquad \textbf{(B)}\ 2a \qquad \textbf{(C)}\ 4a \qquad \textbf{(D)}\ 2}\qquad \textbf{(E)}\ 4}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 10

Each side of $\triangle ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, in the same unit, is:

$\textbf{(A)}\ \sqrt{18} \qquad \textbf{(B)}\ \sqrt{28} \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \sqrt{63} }\qquad \textbf{(E)}\ \sqrt{98}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 11

Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\triangle APR$ is

$\textbf{(A)}\ 42\qquad \textbf{(B)}\ 40.5 \qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 39\frac{7}{8} }\qquad \textbf{(E)}\ \text{not determined by the given information}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 12

The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term.

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \sqrt[7]{2}\qquad \textbf{(C)}\ \sqrt[8]{2}\qquad \textbf{(D)}\ \sqrt[9]{2}}\qquad \textbf{(E)}\ \sqrt[10]{2}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 13

The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to?

$\textbf{(A)}\ t^3\qquad \textbf{(B)}\ t^2+t\qquad \textbf{(C)}\ |t^2+t|\qquad \textbf{(D)}\ t\sqrt{t^2+1}}\qquad \textbf{(E)}\ |t|\sqrt{1+t^2}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 14

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches.

$\textbf{(A)}\ \sqrt{K}\qquad \textbf{(B)}\ \frac{1}{2}\sqrt{2K}\qquad \textbf{(C)}\ \frac{1}{3}\sqrt{3K}\qquad \textbf{(D)}\ \frac{1}{4}\sqrt{4K}}\qquad \textbf{(E)}\ \text{None of these are correct}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 15

If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is:

$\textbf{(A)}\ \frac{x^3}{y^2}\qquad \textbf{(B)}\ \frac{y^3}{x^2}\qquad \textbf{(C)}\ \frac{x^2}{y^3}\qquad \textbf{(D)}\ \frac{y^2}{x^3}}\qquad \textbf{(E)}\ y}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 16

An altitude $h$ of a triangle is increased by a length $m$. How much must be taken from the corresponding base $b$ so that the area of the new triangle is one-half that of the original triangle?

$\textbf{(A)}\ \frac{bm}{h+m}\qquad \textbf{(B)}\ \frac{bh}{2h+2m}\qquad \textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad \textbf{(D)}\ \frac{b(m+h)}{2m+h}}\qquad \textbf{(E)}\ \frac{b(2m+h)}{2(h+m)}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 17

In the base ten number system the number $526$ means $5 \times 10^2+2 \times 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. King Rusczyk purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8}\qquad \textbf{(E)}\ 12}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 18

The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is:

$\textbf{(A)}\ -12 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1}\qquad \textbf{(E)}\ 12}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 19

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that:

$\text{(A)}\ \text{They do not intersect}\qquad \ \text{(B)}\ \text{They intersect at 1 point only}\qquad \ \text{(C)}\ \text{They intersect at 2 points only} \qquad \ \text{(D)}\ \text{They intersect at a finite number of points but greater than 2 }\qquad \ \text{(E)}\ \text{They coincide} }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 20

The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:

$\textbf{(A)}\ \text{I and II}\qquad \textbf{(B)}\ \text{II and III}\qquad \textbf{(C)}\ \text{I and III}\qquad \textbf{(D)}\ \text{III and IV}}\qquad \textbf{(E)}\ \text{I and IV}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 21

Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals:

$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{8}\qquad \textbf{(C)}\ \frac{1}{9}\qquad \textbf{(D)}\ \frac{1}{12}}\qquad \textbf{(E)}\ \frac{1}{16}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 22

If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by:

$\textbf{(A)}\ 3x^2-x+4\qquad \textbf{(B)}\ 3x^2-4\qquad \textbf{(C)}\ 3x^2+4\qquad \textbf{(D)}\ 3x-4 }\qquad \textbf{(E)}\ 3x+4 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 23

Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is:

$\textbf{(A)}\ 60\qquad \textbf{(B)}\ 70\qquad \textbf{(C)}\ 75\qquad \textbf{(D)}\ 80}\qquad \textbf{(E)}\ 85}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 24

Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by $\textdollar{2}$ from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:

$\textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}\qquad \\ \textbf{(B)}\ \text{The middle book sells for \textdollar 36 }\qquad \\ \textbf{(C)}\ \text{The cheapest book sells for \textdollar4 }\qquad \\ \textbf{(D)}\ \text{The most expensive book sells for \textdollar64 } \qquad \\ \textbf{(E)}\ \text{None of these is correct }$

Solution

Problem 25

$\triangle ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in $\angle B$ is:

$\textbf{(A)}\ 25\frac{5}{7}\qquad \textbf{(B)}\ 26\frac{1}{3}\qquad \textbf{(C)}\ 30\qquad \textbf{(D)}\ 40}\qquad \textbf{(E)}\ \text{Not determined by the information given}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 26

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:

$\textbf{(A)}\ -1221 \qquad \textbf{(B)}\ -21.5 \qquad \textbf{(C)}\ -20.5 \qquad \textbf{(D)}\ 3 }\qquad \textbf{(E)}\ 3.5 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 27

Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is:

$\textbf{(A)}\ \infty\qquad \textbf{(B)}\ \text{finite, but greater than 2}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 0$

Solution

Problem 28

If $2137^{753}$ is multiplied out, the units' digit in the final product in the final product is:

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7}\qquad \textbf{(E)}\ 9}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 29

Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:

$\textbf{(A)}\ x^2-bx-ac=0\qquad \textbf{(B)}\ x^2-bx+ac=0 \qquad\ \textbf{(C)}\ x^2+3bx+ca+2b^2=0 \qquad \textbf{(D)}\ x^2+3bx-ca+2b^2=0 }\qquad\ \textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 30

If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=$?

$\textbf{(A)}\ \frac{a+b}{a+1}\qquad \textbf{(B)}\ \frac{2a+b}{a+1}\qquad \textbf{(C)}\ \frac{a+2b}{1+a}\qquad \textbf{(D)}\ \frac{2a+b}{1-a}}\qquad \textbf{(E)}\ \frac{a+2b}{1-a}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 31

In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:

$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 }\qquad \textbf{(E)}\ 7:1 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 32

A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals:

$\textbf{(A)}\ 8\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 15}\qquad \textbf{(E)}\ 18}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 33

The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3}\qquad \textbf{(E)}\ \text{More than three, but finite}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 34

Let S be the set of values assumed by the fraction $\frac{2x+3}{x+2}$. When $x$ is any member of the interval $x \ge 0$. If there exists a number $M$ such that no number of the set $S$ is greater than $M$, then $M$ is an upper bound of $S$. If there exists a number $m$ such that such that no number of the set $S$ is less than $m$, then $m$ is a lower bound of $S$. We may then say:

$\textbf{(A)}\ \text{m is in S, but M is not in S}\qquad\\ \textbf{(B)}\ \text{M is in S, but m is not in S}\qquad\\ \textbf{(C)}\ \text{Both m and M are in S}\qquad\\ \textbf{(D)}\ \text{Neither m nor M are in S}\qquad\\ \textbf{(E)}\ \text{M does not exist either in or outside S}$

Solution

Problem 35

The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k,$ and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3}\qquad \textbf{(E)}\ 4}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 36

In $\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.

$\textbf{(A)}\ 4\qquad \textbf{(B)}\ \sqrt{17} \qquad \textbf{(C)}\ 4.25\qquad \textbf{(D)}\ 2\sqrt{5} }\qquad \textbf{(E)}\ 4.5}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 37

In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, and $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals:

$\textbf{(A)}\ \text{Not determined by the given information}\qquad \textbf{(B)}\ 58\qquad \textbf{(C)}\ 100\qquad \textbf{(D)}\ 116}\qquad \textbf{(E)}\ 120}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 38

$\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:

$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\ \textbf{(D)}\ s^2\le4r^2 }\qquad \textbf{(E)}\ x^2=4r^2 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 39

Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:

$\textbf{(A)}\ \sqrt{3}/3\qquad \textbf{(B)}\ \sqrt{2}/2\qquad \textbf{(C)}\ 2\sqrt{2}/3\qquad \textbf{(D)}\ 1 }\qquad \textbf{(E)}\ \sqrt{2}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 40

Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$.

$\textbf{(A)}\ \frac{60}{13}\qquad \textbf{(B)}\ \frac{13}{5}\qquad \textbf{(C)}\ \frac{13}{12}\qquad \textbf{(D)}\ 1}\qquad \textbf{(E)}\ 0 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

See also

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