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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<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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<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) \  } 3\text{fourth} \qquad \mathrm{(E) \  }\text{fifth}  </math>
  
 +
[[1974 AHSME Problems/Problem 9|Solution]]
 
==Problem 10==
 
==Problem 10==
 
What is the smallest integral value of <math> k </math> such that  
 
What is the smallest integral value of <math> k </math> such that  
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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==
 
If <math> (a, b) </math> and <math> (c, d) </math> are two points on the line whose equation is <math> y=mx+k </math>, then the distance between <math> (a, b) </math> and <math> (c, d) </math>, in terms of <math> a, c, </math> and <math> m </math> is  
 
If <math> (a, b) </math> and <math> (c, d) </math> are two points on the line whose equation is <math> y=mx+k </math>, then the distance between <math> (a, b) </math> and <math> (c, d) </math>, in terms of <math> a, c, </math> and <math> m </math> is  
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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 277:
 
<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>
  
 +
[[1974 AHSME Problems/Problem 30|Solution]]
 
==See Also==
 
==See Also==
 
*[[AHSME]]
 
*[[AHSME]]
 
*[[1974 AHSME]]
 
*[[1974 AHSME]]

Revision as of 15:36, 26 May 2012

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?

$\mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \  } \text{third} \qquad \mathrm{(D) \  } 3\text{fourth} \qquad \mathrm{(E) \  }\text{fifth}$

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

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