Difference between revisions of "1974 AHSME Problems"

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

\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\] (Error making remote request. Unexpected URL sent back)

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

@@ Line 6: / Line 6: @@
 <math> \mathrm{(D) \  } \frac{4y}{y-6}=x \qquad \mathrm{(E) \  }\text{none of these}  </math>
+[[1974 AHSME Problems/Problem 1|Solution]]
 ==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
@@ Line 11: / Line 12: @@
 <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==
 The coefficient of <math> x^7 </math> in the polynomial expansion of
@@ Line 20: / Line 22: @@
 <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==
 What is the remainder when <math> x^{51}+51 </math> is divided by <math> x+1 </math>?
@@ Line 25: / Line 28: @@
 <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==
 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>.
@@ Line 30: / Line 34: @@
 <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==
 For positive real numbers <math> x </math> and <math> y </math> define <math> x*y=\frac{x\cdot y}{x+y} </math>' then
@@ Line 43: / Line 48: @@
 <math>\mathrm{(E) \  }\text{none of these} \qquad  </math>
+[[1974 AHSME Problems/Problem 6|Solution]]
 ==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?
@@ Line 48: / Line 54: @@
 <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==
 What is the smallest prime number dividing the sum <math> 3^{11}+5^{13} </math>?
@@ Line 53: / Line 60: @@
 <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==
 The integers greater than one are arranged in five columns as follows:
@@ Line 64: / Line 72: @@
 <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==
 What is the smallest integral value of <math> k </math> such that
@@ Line 73: / Line 82: @@
 <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==
 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
@@ Line 80: / Line 90: @@
 <math> \mathrm{(D) \  } |a-c|(1+m^2) \qquad \mathrm{(E) \  }|a-c|\,|m|  </math>
+[[1974 AHSME Problems/Problem 11|Solution]]
 ==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
@@ Line 85: / Line 96: @@
 <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==
 Which of the following is equivalent to "If P is true, then Q is false."?
@@ Line 98: / Line 110: @@
 <math>  \mathrm{(E) \  }\text{``If Q is true then P is true."} \qquad  </math>
+[[1974 AHSME Problems/Problem 13|Solution]]
 ==Problem 14==
 Which statement is correct?
@@ Line 107: / Line 120: @@
 <math> \qquad \mathrm{(E) \  }\text{If } x<1, \text{then } x^2<x.  </math>
+[[1974 AHSME Problems/Problem 14|Solution]]
 ==Problem 15==
 If <math> x<-2 </math>, then <math> |1-|1+x|| </math> equals
@@ Line 112: / Line 126: @@
 <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==
 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
@@ Line 117: / Line 132: @@
 <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==
 If <math> i^2=-1 </math>, then <math> (1+i)^{20}-(1-i)^{20} </math> equals
@@ Line 122: / Line 138: @@
 <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==
 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
@@ Line 127: / Line 144: @@
 <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==
 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
@@ Line 142: / Line 160: @@
 <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==
 Let
@@ Line 153: / Line 172: @@
 <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==
 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?
@@ Line 158: / Line 178: @@
 <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==
 The minimum of <math> \sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2} </math> is attained when <math> A </math> is
@@ Line 163: / Line 184: @@
 <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==
 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
@@ Line 187: / Line 209: @@
 <math> \mathrm{(E) \  }\text{not determinable from the given information}  </math>
+[[1974 AHSME Problems/Problem 23|Solution]]
 ==Problem 24==
 A fair die is rolled six times. The probability of rolling at least a five at least five times is
@@ Line 192: / Line 215: @@
 <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==
 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
@@ Line 211: / Line 235: @@
 <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==
 The number of distinct positive integral divisors of <math> (30)^4 </math> excluding <math> 1 </math> and <math> (30)^4 </math> is
@@ Line 216: / Line 241: @@
 <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==
 If <math> f(x)=3x+2 </math> for all real <math> x </math>, then the statement:
@@ Line 223: / Line 249: @@
 <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==
 Which of the following is satisfied by all numbers <math> x </math> of the form
@@ Line 234: / Line 261: @@
 <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==
 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
@@ Line 239: / Line 267: @@
 <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==
 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>
+[[1974 AHSME Problems/Problem 30|Solution]]
 ==See Also==
 *[[AHSME]]
 *[[1974 AHSME]]

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Difference between revisions of "1974 AHSME Problems"

Revision as of 15:36, 26 May 2012

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

See Also