Difference between revisions of "1974 AHSME Problems"

(Created page with "==Problem 1== If <math> x\not=0 </math> or <math> 4 </math> and <math> y\not=0 </math> or <math> 6 </math>, then <math> \frac{2}{x}+\frac{3}{y}=\frac{1}{2} </math> is equivalent ...")
 
 
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{{AHSME Problems
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|year = 1974
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}}
 
==Problem 1==
 
==Problem 1==
 
If <math> x\not=0 </math> or <math> 4 </math> and <math> y\not=0 </math> or <math> 6 </math>, then <math> \frac{2}{x}+\frac{3}{y}=\frac{1}{2} </math> is equivalent to
 
If <math> x\not=0 </math> or <math> 4 </math> and <math> y\not=0 </math> or <math> 6 </math>, then <math> \frac{2}{x}+\frac{3}{y}=\frac{1}{2} </math> is equivalent to
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<math> \mathrm{(D) \  } \frac{4y}{y-6}=x \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(D) \  } \frac{4y}{y-6}=x \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 1|Solution]]
 
==Problem 2==
 
==Problem 2==
 
Let <math> x_1 </math> and <math> x_2 </math> be such that <math> x_1\not=x_2 </math> and <math> 3x_i^2-hx_i=b </math>, <math> i=1, 2 </math>. Then <math> x_1+x_2 </math> equals
 
Let <math> x_1 </math> and <math> x_2 </math> be such that <math> x_1\not=x_2 </math> and <math> 3x_i^2-hx_i=b </math>, <math> i=1, 2 </math>. Then <math> x_1+x_2 </math> equals
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<math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \  } \frac{b}{3} \qquad \mathrm{(D) \  } 2b \qquad \mathrm{(E) \  }-\frac{b}{3}  </math>
 
<math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \  } \frac{b}{3} \qquad \mathrm{(D) \  } 2b \qquad \mathrm{(E) \  }-\frac{b}{3}  </math>
  
 +
[[1974 AHSME Problems/Problem 2|Solution]]
 
==Problem 3==
 
==Problem 3==
 
The coefficient of <math> x^7 </math> in the polynomial expansion of  
 
The coefficient of <math> x^7 </math> in the polynomial expansion of  
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<math> \mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \  } 6 \qquad \mathrm{(D) \  } -12 \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \  } 6 \qquad \mathrm{(D) \  } -12 \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 3|Solution]]
 
==Problem 4==
 
==Problem 4==
 
What is the remainder when <math> x^{51}+51 </math> is divided by <math> x+1 </math>?
 
What is the remainder when <math> x^{51}+51 </math> is divided by <math> x+1 </math>?
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<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 49 \qquad \mathrm{(D) \  } 50 \qquad \mathrm{(E) \  }51  </math>
 
<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 49 \qquad \mathrm{(D) \  } 50 \qquad \mathrm{(E) \  }51  </math>
  
 +
[[1974 AHSME Problems/Problem 4|Solution]]
 
==Problem 5==
 
==Problem 5==
 
Given a quadrilateral <math> ABCD </math> inscribed in a circle with side <math> AB </math> extended beyond <math> B </math> to point <math> E </math>, if <math> \measuredangle BAD=92^\circ </math> and <math> \measuredangle ADC=68^\circ </math>, find <math> \measuredangle EBC </math>.
 
Given a quadrilateral <math> ABCD </math> inscribed in a circle with side <math> AB </math> extended beyond <math> B </math> to point <math> E </math>, if <math> \measuredangle BAD=92^\circ </math> and <math> \measuredangle ADC=68^\circ </math>, find <math> \measuredangle EBC </math>.
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<math> \mathrm{(A)\ } 66^\circ \qquad \mathrm{(B) \ }68^\circ \qquad \mathrm{(C) \  } 70^\circ \qquad \mathrm{(D) \  } 88^\circ \qquad \mathrm{(E) \  }92^\circ  </math>
 
<math> \mathrm{(A)\ } 66^\circ \qquad \mathrm{(B) \ }68^\circ \qquad \mathrm{(C) \  } 70^\circ \qquad \mathrm{(D) \  } 88^\circ \qquad \mathrm{(E) \  }92^\circ  </math>
  
 +
[[1974 AHSME Problems/Problem 5|Solution]]
 
==Problem 6==
 
==Problem 6==
 
For positive real numbers <math> x </math> and <math> y </math> define <math> x*y=\frac{x\cdot y}{x+y} </math>' then
 
For positive real numbers <math> x </math> and <math> y </math> define <math> x*y=\frac{x\cdot y}{x+y} </math>' then
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<math>\mathrm{(E) \  }\text{none of these} \qquad  </math>
 
<math>\mathrm{(E) \  }\text{none of these} \qquad  </math>
  
 +
[[1974 AHSME Problems/Problem 6|Solution]]
 
==Problem 7==
 
==Problem 7==
 
A town's population increased by <math> 1,200 </math> people, and then this new population decreased by <math> 11\% </math>. The town now had <math> 32 </math> less people than it did before the <math> 1,200 </math> increase. What is the original population?
 
A town's population increased by <math> 1,200 </math> people, and then this new population decreased by <math> 11\% </math>. The town now had <math> 32 </math> less people than it did before the <math> 1,200 </math> increase. What is the original population?
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<math> \mathrm{(A)\ } 1,200 \qquad \mathrm{(B) \ }11,200 \qquad \mathrm{(C) \  } 9,968 \qquad \mathrm{(D) \  } 10,000 \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } 1,200 \qquad \mathrm{(B) \ }11,200 \qquad \mathrm{(C) \  } 9,968 \qquad \mathrm{(D) \  } 10,000 \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 7|Solution]]
 
==Problem 8==
 
==Problem 8==
 
What is the smallest prime number dividing the sum <math> 3^{11}+5^{13} </math>?
 
What is the smallest prime number dividing the sum <math> 3^{11}+5^{13} </math>?
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<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 5 \qquad \mathrm{(D) \  } 3^{11}+5^{13} \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 5 \qquad \mathrm{(D) \  } 3^{11}+5^{13} \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 8|Solution]]
 
==Problem 9==
 
==Problem 9==
 
The integers greater than one are arranged in five columns as follows:
 
The integers greater than one are arranged in five columns as follows:
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In which column will the number <math> 1,000 </math> fall?
 
In which column will the number <math> 1,000 </math> fall?
  
<math> \mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \  } \text{third} \qquad \mathrm{(D) \  } 3\text{fourth} \qquad \mathrm{(E) \  }\text{fifth}  </math>
+
<math> \mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \  } \text{third} \qquad \mathrm{(D) \  } \text{fourth} \qquad \mathrm{(E) \  }\text{fifth}  </math>
 +
 
 +
[[1974 AHSME Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
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<math> \mathrm{(A)\ } -1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \  } 3 \qquad \mathrm{(D) \  } 4 \qquad \mathrm{(E) \  }5  </math>
 
<math> \mathrm{(A)\ } -1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \  } 3 \qquad \mathrm{(D) \  } 4 \qquad \mathrm{(E) \  }5  </math>
 +
 +
[[1974 AHSME Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
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<math> \mathrm{(D) \  } |a-c|(1+m^2) \qquad \mathrm{(E) \  }|a-c|\,|m|  </math>
 
<math> \mathrm{(D) \  } |a-c|(1+m^2) \qquad \mathrm{(E) \  }|a-c|\,|m|  </math>
  
 +
[[1974 AHSME Problems/Problem 11|Solution]]
 
==Problem 12==
 
==Problem 12==
 
If <math> g(x)=1-x^2 </math> and <math> f(g(x))=\frac{1-x^2}{x^2} </math> when <math> x\not=0 </math>, then <math> f(1/2) </math> equals  
 
If <math> g(x)=1-x^2 </math> and <math> f(g(x))=\frac{1-x^2}{x^2} </math> when <math> x\not=0 </math>, then <math> f(1/2) </math> equals  
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<math> \mathrm{(A)\ } 3/4 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 3 \qquad \mathrm{(D) \  } \sqrt{2}/2 \qquad \mathrm{(E) \  }\sqrt{2}  </math>
 
<math> \mathrm{(A)\ } 3/4 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 3 \qquad \mathrm{(D) \  } \sqrt{2}/2 \qquad \mathrm{(E) \  }\sqrt{2}  </math>
  
 +
[[1974 AHSME Problems/Problem 12|Solution]]
 
==Problem 13==
 
==Problem 13==
 
Which of the following is equivalent to "If P is true, then Q is false."?
 
Which of the following is equivalent to "If P is true, then Q is false."?
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<math>  \mathrm{(E) \  }\text{``If Q is true then P is true."} \qquad  </math>
 
<math>  \mathrm{(E) \  }\text{``If Q is true then P is true."} \qquad  </math>
  
 +
[[1974 AHSME Problems/Problem 13|Solution]]
 
==Problem 14==
 
==Problem 14==
 
Which statement is correct?
 
Which statement is correct?
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<math> \qquad \mathrm{(E) \  }\text{If } x<1, \text{then } x^2<x.  </math>
 
<math> \qquad \mathrm{(E) \  }\text{If } x<1, \text{then } x^2<x.  </math>
  
 +
[[1974 AHSME Problems/Problem 14|Solution]]
 
==Problem 15==
 
==Problem 15==
 
If <math> x<-2 </math>, then <math> |1-|1+x|| </math> equals  
 
If <math> x<-2 </math>, then <math> |1-|1+x|| </math> equals  
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<math> \mathrm{(A)\ } 2+x \qquad \mathrm{(B) \ }-2-x \qquad \mathrm{(C) \  } x \qquad \mathrm{(D) \  } -x \qquad \mathrm{(E) \  }-2  </math>
 
<math> \mathrm{(A)\ } 2+x \qquad \mathrm{(B) \ }-2-x \qquad \mathrm{(C) \  } x \qquad \mathrm{(D) \  } -x \qquad \mathrm{(E) \  }-2  </math>
  
 +
[[1974 AHSME Problems/Problem 15|Solution]]
 
==Problem 16==
 
==Problem 16==
 
A circle of radius <math> r </math> is inscribed in a right isosceles triangle, and a circle of radius <math> R </math> is circumscribed about the triangle. Then <math> R/r </math> equals
 
A circle of radius <math> r </math> is inscribed in a right isosceles triangle, and a circle of radius <math> R </math> is circumscribed about the triangle. Then <math> R/r </math> equals
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<math> \mathrm{(A)\ } 1+\sqrt{2} \qquad \mathrm{(B) \ }\frac{2+\sqrt{2}}{2} \qquad \mathrm{(C) \  } \frac{\sqrt{2}-1}{2} \qquad \mathrm{(D) \  } \frac{1+\sqrt{2}}{2} \qquad \mathrm{(E) \  }2(2-\sqrt{2})  </math>
 
<math> \mathrm{(A)\ } 1+\sqrt{2} \qquad \mathrm{(B) \ }\frac{2+\sqrt{2}}{2} \qquad \mathrm{(C) \  } \frac{\sqrt{2}-1}{2} \qquad \mathrm{(D) \  } \frac{1+\sqrt{2}}{2} \qquad \mathrm{(E) \  }2(2-\sqrt{2})  </math>
  
 +
[[1974 AHSME Problems/Problem 16|Solution]]
 
==Problem 17==
 
==Problem 17==
 
If <math> i^2=-1 </math>, then <math> (1+i)^{20}-(1-i)^{20} </math> equals
 
If <math> i^2=-1 </math>, then <math> (1+i)^{20}-(1-i)^{20} </math> equals
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<math> \mathrm{(A)\ } -1024 \qquad \mathrm{(B) \ }-1024i \qquad \mathrm{(C) \  } 0 \qquad \mathrm{(D) \  } 1024 \qquad \mathrm{(E) \  }1024i  </math>
 
<math> \mathrm{(A)\ } -1024 \qquad \mathrm{(B) \ }-1024i \qquad \mathrm{(C) \  } 0 \qquad \mathrm{(D) \  } 1024 \qquad \mathrm{(E) \  }1024i  </math>
  
 +
[[1974 AHSME Problems/Problem 17|Solution]]
 
==Problem 18==
 
==Problem 18==
 
If <math> \log_8{3}=p </math> and <math> \log_3{5}=q </math>, then, in terms of <math> p </math> and <math> q </math>, <math> \log_{10}{5} </math> equals
 
If <math> \log_8{3}=p </math> and <math> \log_3{5}=q </math>, then, in terms of <math> p </math> and <math> q </math>, <math> \log_{10}{5} </math> equals
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<math> \mathrm{(A)\ } pq \qquad \mathrm{(B) \ }\frac{3p+q}{5} \qquad \mathrm{(C) \  } \frac{1+3pq}{p+q} \qquad \mathrm{(D) \  } \frac{3pq}{1+3pq} \qquad \mathrm{(E) \  }p^2+q^2  </math>
 
<math> \mathrm{(A)\ } pq \qquad \mathrm{(B) \ }\frac{3p+q}{5} \qquad \mathrm{(C) \  } \frac{1+3pq}{p+q} \qquad \mathrm{(D) \  } \frac{3pq}{1+3pq} \qquad \mathrm{(E) \  }p^2+q^2  </math>
  
 +
[[1974 AHSME Problems/Problem 18|Solution]]
 
==Problem 19==
 
==Problem 19==
 
In the adjoining figure <math> ABCD </math> is a square and <math> CMN </math> is an equilateral triangle. If the area of <math> ABCD </math> is one square inch, then the area of <math> CMN </math> in square inches is  
 
In the adjoining figure <math> ABCD </math> is a square and <math> CMN </math> is an equilateral triangle. If the area of <math> ABCD </math> is one square inch, then the area of <math> CMN </math> in square inches is  
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<math> \mathrm{(A)\ } 2\sqrt{3}-3 \qquad \mathrm{(B) \ }1-\frac{\sqrt{3}}{3} \qquad \mathrm{(C) \  } \frac{\sqrt{3}}{4} \qquad \mathrm{(D) \  } \frac{\sqrt{2}}{3} \qquad \mathrm{(E) \  }4-2\sqrt{3}  </math>
 
<math> \mathrm{(A)\ } 2\sqrt{3}-3 \qquad \mathrm{(B) \ }1-\frac{\sqrt{3}}{3} \qquad \mathrm{(C) \  } \frac{\sqrt{3}}{4} \qquad \mathrm{(D) \  } \frac{\sqrt{2}}{3} \qquad \mathrm{(E) \  }4-2\sqrt{3}  </math>
  
 +
[[1974 AHSME Problems/Problem 19|Solution]]
 
==Problem 20==
 
==Problem 20==
 
Let
 
Let
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<math> \mathrm{(E) \  }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}  </math>
 
<math> \mathrm{(E) \  }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}  </math>
  
 +
[[1974 AHSME Problems/Problem 20|Solution]]
 
==Problem 21==
 
==Problem 21==
 
In a geometric series of positive terms the difference between the fifth and fourth terms is <math> 576 </math>, and the difference between the second and first terms is <math> 9 </math>. What is the sum of the first five terms of this series?
 
In a geometric series of positive terms the difference between the fifth and fourth terms is <math> 576 </math>, and the difference between the second and first terms is <math> 9 </math>. What is the sum of the first five terms of this series?
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<math> \mathrm{(A)\ } 1061 \qquad \mathrm{(B) \ }1023 \qquad \mathrm{(C) \  } 1024 \qquad \mathrm{(D) \  } 768 \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } 1061 \qquad \mathrm{(B) \ }1023 \qquad \mathrm{(C) \  } 1024 \qquad \mathrm{(D) \  } 768 \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 21|Solution]]
 
==Problem 22==
 
==Problem 22==
 
The minimum of <math> \sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2} </math> is attained when <math> A </math> is  
 
The minimum of <math> \sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2} </math> is attained when <math> A </math> is  
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<math> \mathrm{(A)\ } -180^\circ \qquad \mathrm{(B) \ }60^\circ \qquad \mathrm{(C) \  } 120^\circ \qquad \mathrm{(D) \  } 0^\circ \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } -180^\circ \qquad \mathrm{(B) \ }60^\circ \qquad \mathrm{(C) \  } 120^\circ \qquad \mathrm{(D) \  } 0^\circ \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 22|Solution]]
 
==Problem 23==
 
==Problem 23==
 
In the adjoining figure <math> TP </math> and <math> T'Q </math> are parallel tangents to a circle of radius <math> r </math>, with <math> T </math> and <math> T' </math> the points of tangency. <math> PT''Q </math> is a third tangent with <math> T''' </math> as a point of tangency. If <math> TP=4 </math> and <math> T'Q=9 </math> then <math> r </math> is
 
In the adjoining figure <math> TP </math> and <math> T'Q </math> are parallel tangents to a circle of radius <math> r </math>, with <math> T </math> and <math> T' </math> the points of tangency. <math> PT''Q </math> is a third tangent with <math> T''' </math> as a point of tangency. If <math> TP=4 </math> and <math> T'Q=9 </math> then <math> r </math> is
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<math> \mathrm{(E) \  }\text{not determinable from the given information}  </math>
 
<math> \mathrm{(E) \  }\text{not determinable from the given information}  </math>
  
 +
[[1974 AHSME Problems/Problem 23|Solution]]
 
==Problem 24==
 
==Problem 24==
 
A fair die is rolled six times. The probability of rolling at least a five at least five times is  
 
A fair die is rolled six times. The probability of rolling at least a five at least five times is  
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<math> \mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{12}{729} \qquad \mathrm{(C) \  } \frac{2}{729} \qquad \mathrm{(D) \  } \frac{3}{729} \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{12}{729} \qquad \mathrm{(C) \  } \frac{2}{729} \qquad \mathrm{(D) \  } \frac{3}{729} \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 24|Solution]]
 
==Problem 25==
 
==Problem 25==
 
In parallelogram <math> ABCD </math> of the accompanying diagram, line <math> DP </math> is drawn bisecting <math> BC </math> at <math> N </math> and meeting <math> AB </math> (extended) at <math> P </math>. From vertex <math> C </math>, line <math> CQ </math> is drawn bisecting side <math> AD </math> at <math> M </math> and meeting <math> AB </math> (extended) at <math> Q </math>. Lines <math> DP </math> and <math> CQ </math> meet at <math> O </math>. If the area of parallelogram <math> ABCD </math> is <math> k </math>, then the area of the triangle <math> QPO </math> is equal to  
 
In parallelogram <math> ABCD </math> of the accompanying diagram, line <math> DP </math> is drawn bisecting <math> BC </math> at <math> N </math> and meeting <math> AB </math> (extended) at <math> P </math>. From vertex <math> C </math>, line <math> CQ </math> is drawn bisecting side <math> AD </math> at <math> M </math> and meeting <math> AB </math> (extended) at <math> Q </math>. Lines <math> DP </math> and <math> CQ </math> meet at <math> O </math>. If the area of parallelogram <math> ABCD </math> is <math> k </math>, then the area of the triangle <math> QPO </math> is equal to  
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<math> \mathrm{(A)\ } k \qquad \mathrm{(B) \ }\frac{6k}{5} \qquad \mathrm{(C) \  } \frac{9k}{8} \qquad \mathrm{(D) \  } \frac{5k}{4} \qquad \mathrm{(E) \  }2k  </math>
 
<math> \mathrm{(A)\ } k \qquad \mathrm{(B) \ }\frac{6k}{5} \qquad \mathrm{(C) \  } \frac{9k}{8} \qquad \mathrm{(D) \  } \frac{5k}{4} \qquad \mathrm{(E) \  }2k  </math>
  
 +
[[1974 AHSME Problems/Problem 25|Solution]]
 
==Problem 26==
 
==Problem 26==
 
The number of distinct positive integral divisors of <math> (30)^4 </math> excluding <math> 1 </math> and <math> (30)^4 </math> is  
 
The number of distinct positive integral divisors of <math> (30)^4 </math> excluding <math> 1 </math> and <math> (30)^4 </math> is  
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<math> \mathrm{(A)\ } 100 \qquad \mathrm{(B) \ }125 \qquad \mathrm{(C) \  } 123 \qquad \mathrm{(D) \  } 30 \qquad \mathrm{(E) \  }\text{none of these}  </math>
 
<math> \mathrm{(A)\ } 100 \qquad \mathrm{(B) \ }125 \qquad \mathrm{(C) \  } 123 \qquad \mathrm{(D) \  } 30 \qquad \mathrm{(E) \  }\text{none of these}  </math>
  
 +
[[1974 AHSME Problems/Problem 26|Solution]]
 
==Problem 27==
 
==Problem 27==
 
If <math> f(x)=3x+2 </math> for all real <math> x </math>, then the statement:
 
If <math> f(x)=3x+2 </math> for all real <math> x </math>, then the statement:
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<math> \mathrm{(A)}\ b\le a/3\qquad\mathrm{(B)}\ b > a/3\qquad\mathrm{(C)}\ a\le b/3\qquad\mathrm{(D)}\ a > b/3\\ \qquad\mathrm{(E)}\ \text{The statement is never true.} </math>
 
<math> \mathrm{(A)}\ b\le a/3\qquad\mathrm{(B)}\ b > a/3\qquad\mathrm{(C)}\ a\le b/3\qquad\mathrm{(D)}\ a > b/3\\ \qquad\mathrm{(E)}\ \text{The statement is never true.} </math>
  
 +
[[1974 AHSME Problems/Problem 27|Solution]]
 
==Problem 28==
 
==Problem 28==
 
Which of the following is satisfied by all numbers <math> x </math> of the form
 
Which of the following is satisfied by all numbers <math> x </math> of the form
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<math> \mathrm{(D) \  } 0\le x<1/3 \text{ or }2/3\le x<1 \qquad \mathrm{(E) \  }1/2\le x\le 3/4  </math>
 
<math> \mathrm{(D) \  } 0\le x<1/3 \text{ or }2/3\le x<1 \qquad \mathrm{(E) \  }1/2\le x\le 3/4  </math>
  
 +
[[1974 AHSME Problems/Problem 28|Solution]]
 
==Problem 29==
 
==Problem 29==
 
For <math> p=1, 2, \cdots, 10 </math> let <math> S_p </math> be the sum of the first <math> 40 </math> terms of the arithmetic progression whose first term is <math> p </math> and whose common difference is <math> 2p-1 </math>; then <math> S_1+S_2+\cdots+S_{10} </math> is
 
For <math> p=1, 2, \cdots, 10 </math> let <math> S_p </math> be the sum of the first <math> 40 </math> terms of the arithmetic progression whose first term is <math> p </math> and whose common difference is <math> 2p-1 </math>; then <math> S_1+S_2+\cdots+S_{10} </math> is
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<math> \mathrm{(A)\ } 80000 \qquad \mathrm{(B) \ }80200 \qquad \mathrm{(C) \  } 80400 \qquad \mathrm{(D) \  } 80600 \qquad \mathrm{(E) \  }80800  </math>
 
<math> \mathrm{(A)\ } 80000 \qquad \mathrm{(B) \ }80200 \qquad \mathrm{(C) \  } 80400 \qquad \mathrm{(D) \  } 80600 \qquad \mathrm{(E) \  }80800  </math>
  
 +
[[1974 AHSME Problems/Problem 29|Solution]]
 
==Problem 30==
 
==Problem 30==
 
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If <math> R </math> is the ratio of the lesser part to the greater part, then the value of  
 
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If <math> R </math> is the ratio of the lesser part to the greater part, then the value of  
Line 248: Line 282:
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }2R \qquad \mathrm{(C) \  } R^{-1} \qquad \mathrm{(D) \  } 2+R^{-1} \qquad \mathrm{(E) \  }2+R  </math>
 
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }2R \qquad \mathrm{(C) \  } R^{-1} \qquad \mathrm{(D) \  } 2+R^{-1} \qquad \mathrm{(E) \  }2+R  </math>
  
==See Also==
+
[[1974 AHSME Problems/Problem 30|Solution]]
*[[AHSME]]
+
== See also ==
*[[1974 AHSME]]
+
 
 +
* [[AMC 12 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
 
 +
{{AHSME box|year=1974|before=[[1973 AHSME]]|after=[[1975 AHSME]]}} 
 +
 
 +
{{MAA Notice}}

Latest revision as of 13:04, 19 February 2020

1974 AHSME (Answer Key)
Printable version: Wiki | AoPS ResourcesPDF

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 $x\not=0$ or $4$ and $y\not=0$ or $6$, then $\frac{2}{x}+\frac{3}{y}=\frac{1}{2}$ is equivalent to

$\mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \  } \frac{x}{2}+\frac{y}{3}=2 \qquad$

$\mathrm{(D) \  } \frac{4y}{y-6}=x \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 2

Let $x_1$ and $x_2$ be such that $x_1\not=x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals

$\mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \  } \frac{b}{3} \qquad \mathrm{(D) \  } 2b \qquad \mathrm{(E) \  }-\frac{b}{3}$

Solution

Problem 3

The coefficient of $x^7$ in the polynomial expansion of

\[(1+2x-x^2)^4\]

is

$\mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \  } 6 \qquad \mathrm{(D) \  } -12 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 4

What is the remainder when $x^{51}+51$ is divided by $x+1$?

$\mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 49 \qquad \mathrm{(D) \  } 50 \qquad \mathrm{(E) \  }51$

Solution

Problem 5

Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^\circ$ and $\measuredangle ADC=68^\circ$, find $\measuredangle EBC$.

$\mathrm{(A)\ } 66^\circ \qquad \mathrm{(B) \ }68^\circ \qquad \mathrm{(C) \  } 70^\circ \qquad \mathrm{(D) \  } 88^\circ \qquad \mathrm{(E) \  }92^\circ$

Solution

Problem 6

For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$' then

$\mathrm{(A)\ } \text{``*" is commutative but not associative} \qquad$

$\mathrm{(B) \ }\text{``*" is associative but not commutative} \qquad$

$\mathrm{(C) \  } \text{``*" is neither commutative nor associative} \qquad$

$\mathrm{(D) \  } \text{``*" is commutative and associative} \qquad$

$\mathrm{(E) \  }\text{none of these} \qquad$

Solution

Problem 7

A town's population increased by $1,200$ people, and then this new population decreased by $11\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?

$\mathrm{(A)\ } 1,200 \qquad \mathrm{(B) \ }11,200 \qquad \mathrm{(C) \  } 9,968 \qquad \mathrm{(D) \  } 10,000 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 8

What is the smallest prime number dividing the sum $3^{11}+5^{13}$?

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 5 \qquad \mathrm{(D) \  } 3^{11}+5^{13} \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 9

The integers greater than one are arranged in five columns as follows:

\[\begin{tabular}{c c c c c}\ & 2 & 3 & 4 & 5\\ 9 & 8 & 7 & 6 &\ \\ \ & 10 & 11 & 12 & 13\\ 17 & 16 & 15 & 14 &\ \\ \ & . & . & . & .\\ \end{tabular}\]

(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)

In which column will the number $1,000$ fall?

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

Solution

Problem 10

What is the smallest integral value of $k$ such that

\[2x(kx-4)-x^2+6=0\]

has no real roots?

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

Solution

Problem 11

If $(a, b)$ and $(c, d)$ are two points on the line whose equation is $y=mx+k$, then the distance between $(a, b)$ and $(c, d)$, in terms of $a, c,$ and $m$ is

$\mathrm{(A)\ } |a-c|\sqrt{1+m^2} \qquad \mathrm{(B) \ }|a+c|\sqrt{1+m^2} \qquad \mathrm{(C) \  } \frac{|a-c|}{\sqrt{1+m^2}} \qquad$

$\mathrm{(D) \  } |a-c|(1+m^2) \qquad \mathrm{(E) \  }|a-c|\,|m|$

Solution

Problem 12

If $g(x)=1-x^2$ and $f(g(x))=\frac{1-x^2}{x^2}$ when $x\not=0$, then $f(1/2)$ equals

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

Solution

Problem 13

Which of the following is equivalent to "If P is true, then Q is false."?

$\mathrm{(A)\ } \text{``P is true or Q is false."} \qquad$

$\mathrm{(B) \ }\text{``If Q is false then P is true."} \qquad$

$\mathrm{(C) \  } \text{``If P is false then Q is true."} \qquad$

$\mathrm{(D) \  } \text{``If Q is true then P is false."} \qquad$

$\mathrm{(E) \  }\text{``If Q is true then P is true."} \qquad$

Solution

Problem 14

Which statement is correct?

$\mathrm{(A)\ } \text{If } x<0, \text{then } x^2>x. \qquad \mathrm{(B) \ } \text{If } x^2>0, \text{then } x>0.$

$\qquad \mathrm{(C) \  } \text{If } x^2>x, \text{then } x>0. \qquad \mathrm{(D) \  } \text{If } x^2>x, \text{then } x<0.$

$\qquad \mathrm{(E) \  }\text{If } x<1, \text{then } x^2<x.$

Solution

Problem 15

If $x<-2$, then $|1-|1+x||$ equals

$\mathrm{(A)\ } 2+x \qquad \mathrm{(B) \ }-2-x \qquad \mathrm{(C) \  } x \qquad \mathrm{(D) \  } -x \qquad \mathrm{(E) \  }-2$

Solution

Problem 16

A circle of radius $r$ is inscribed in a right isosceles triangle, and a circle of radius $R$ is circumscribed about the triangle. Then $R/r$ equals

$\mathrm{(A)\ } 1+\sqrt{2} \qquad \mathrm{(B) \ }\frac{2+\sqrt{2}}{2} \qquad \mathrm{(C) \  } \frac{\sqrt{2}-1}{2} \qquad \mathrm{(D) \  } \frac{1+\sqrt{2}}{2} \qquad \mathrm{(E) \  }2(2-\sqrt{2})$

Solution

Problem 17

If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals

$\mathrm{(A)\ } -1024 \qquad \mathrm{(B) \ }-1024i \qquad \mathrm{(C) \  } 0 \qquad \mathrm{(D) \  } 1024 \qquad \mathrm{(E) \  }1024i$

Solution

Problem 18

If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals

$\mathrm{(A)\ } pq \qquad \mathrm{(B) \ }\frac{3p+q}{5} \qquad \mathrm{(C) \  } \frac{1+3pq}{p+q} \qquad \mathrm{(D) \  } \frac{3pq}{1+3pq} \qquad \mathrm{(E) \  }p^2+q^2$

Solution

Problem 19

In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is

[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((.82,0)--(1,1)--(0,.76)--cycle); label("A", (0,0), S); label("B", (1,0), S); label("C", (1,1), N); label("D", (0,1), N); label("M", (0,.76), W); label("N", (.82,0), S);[/asy]

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

Solution

Problem 20

Let

\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\]

Then

$\mathrm{(A)\ } T<1 \qquad \mathrm{(B) \ }T=1 \qquad \mathrm{(C) \  } 1<T<2 \qquad \mathrm{(D) \  } T>2 \qquad$

$\mathrm{(E) \  }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}$

Solution

Problem 21

In a geometric series of positive terms the difference between the fifth and fourth terms is $576$, and the difference between the second and first terms is $9$. What is the sum of the first five terms of this series?

$\mathrm{(A)\ } 1061 \qquad \mathrm{(B) \ }1023 \qquad \mathrm{(C) \  } 1024 \qquad \mathrm{(D) \  } 768 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 22

The minimum of $\sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2}$ is attained when $A$ is

$\mathrm{(A)\ } -180^\circ \qquad \mathrm{(B) \ }60^\circ \qquad \mathrm{(C) \  } 120^\circ \qquad \mathrm{(D) \  } 0^\circ \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 23

In the adjoining figure $TP$ and $T'Q$ are parallel tangents to a circle of radius $r$, with $T$ and $T'$ the points of tangency. $PT''Q$ is a third tangent with $T'''$ as a point of tangency. If $TP=4$ and $T'Q=9$ then $r$ is

[asy] unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label("O",O,W); label("T",T,N); label("T'",T1,S); label("T''",T2,NE); label("P",P,NE); label("Q",Q,S); draw(O--T2); label("$r$",midpoint(O--T2),NW); draw(T--P); label("4",midpoint(T--P),N); draw(T1--Q); label("9",midpoint(T1--Q),S); draw(P--Q);[/asy]

$\mathrm{(A)\ } 25/6 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \  } 25/4 \qquad$

$\mathrm{(D) \  } \text{a number other than }25/6, 6, 25/4 \qquad$

$\mathrm{(E) \  }\text{not determinable from the given information}$

Solution

Problem 24

A fair die is rolled six times. The probability of rolling at least a five at least five times is

$\mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{12}{729} \qquad \mathrm{(C) \  } \frac{2}{729} \qquad \mathrm{(D) \  } \frac{3}{729} \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 25

In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to

[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75));[/asy]

$\mathrm{(A)\ } k \qquad \mathrm{(B) \ }\frac{6k}{5} \qquad \mathrm{(C) \  } \frac{9k}{8} \qquad \mathrm{(D) \  } \frac{5k}{4} \qquad \mathrm{(E) \  }2k$

Solution

Problem 26

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is

$\mathrm{(A)\ } 100 \qquad \mathrm{(B) \ }125 \qquad \mathrm{(C) \  } 123 \qquad \mathrm{(D) \  } 30 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 27

If $f(x)=3x+2$ for all real $x$, then the statement: "$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$" is true when

$\mathrm{(A)}\ b\le a/3\qquad\mathrm{(B)}\ b > a/3\qquad\mathrm{(C)}\ a\le b/3\qquad\mathrm{(D)}\ a > b/3\\ \qquad\mathrm{(E)}\ \text{The statement is never true.}$

Solution

Problem 28

Which of the following is satisfied by all numbers $x$ of the form

\[x=\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{25}}{3^{25}}\]

where $a_1$ is $0$ or $2$, $a_2$ is $0$ or $2$,...,$a_{25}$ is $0$ or $2$?

$\mathrm{(A)\ } 0\le x<1/3 \qquad \mathrm{(B) \ } 1/3\le x<2/3 \qquad \mathrm{(C) \  } 2/3\le x<1 \qquad$

$\mathrm{(D) \  } 0\le x<1/3 \text{ or }2/3\le x<1 \qquad \mathrm{(E) \  }1/2\le x\le 3/4$

Solution

Problem 29

For $p=1, 2, \cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\cdots+S_{10}$ is

$\mathrm{(A)\ } 80000 \qquad \mathrm{(B) \ }80200 \qquad \mathrm{(C) \  } 80400 \qquad \mathrm{(D) \  } 80600 \qquad \mathrm{(E) \  }80800$

Solution

Problem 30

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $R$ is the ratio of the lesser part to the greater part, then the value of

\[R^{[R^{(R^2+R^{-1})}+R^{-1}]}+R^{-1}\]

is

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }2R \qquad \mathrm{(C) \  } R^{-1} \qquad \mathrm{(D) \  } 2+R^{-1} \qquad \mathrm{(E) \  }2+R$

Solution

See also

1974 AHSME (ProblemsAnswer KeyResources)
Preceded by
1973 AHSME
Followed by
1975 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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