AHSME 1974

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 $\mathrm{(A)\ } 4x+3y=xy \mathrm{(B) \ }y=\frac{4x}{6-y} \mathrm{(C) \ } \frac{x}{2}+\frac{y}{3}=2$ $\mathrm{(D) \ } \frac{4y}{y-6}=x \mathrm{(E) \ }\text{none of these}$

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

The coefficient of $x^7$ in the polynomial expansion of $$(1+2x-x^2)^4$$ is $\mathrm{(A)\ } -8 \mathrm{(B) \ }12 \mathrm{(C) \ } 6 \mathrm{(D) \ } -12 \mathrm{(E) \ }\text{none of these}$

What is the remainder when $x^{51}+51$ is divided by $x+1$? $\mathrm{(A)\ } 0 \mathrm{(B) \ }1 \mathrm{(C) \ } 49 \mathrm{(D) \ } 50 \mathrm{(E) \ }51$

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

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

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

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

What is the smallest prime number dividing the sum $3^{11}+5^{13}$? $\mathrm{(A)\ } 2 \mathrm{(B) \ }3 \mathrm{(C) \ } 5 \mathrm{(D) \ } 3^{11}+5^{13} \mathrm{(E) \ }\text{none of these}$

What is the smallest integral value of $k$ such that $$2x(kx-4)-x^2+6=0$$ has no real roots? $\mathrm{(A)\ } -1 \mathrm{(B) \ }2 \mathrm{(C) \ } 3 \mathrm{(D) \ } 4 \mathrm{(E) \ }5$

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

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

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

Which statement is correct? $\mathrm{(A)\ } \text{If } xx. \mathrm{(B) \ } \text{If } x^2>0, \text{then } x>0.$ $\mathrm{(C) \ } \text{If } x^2>x, \text{then } x>0. \mathrm{(D) \ } \text{If } x^2>x, \text{then } x<0.$ $\mathrm{(E) \ }\text{If } x<1, \text{then } x^2<x.$

If $x<-2$, then $|1-|1+x||$ equals $\mathrm{(A)\ } 2+x \mathrm{(B) \ }-2-x \mathrm{(C) \ } x \mathrm{(D) \ } -x \mathrm{(E) \ }-2$

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

If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals $\mathrm{(A)\ } -1024 \mathrm{(B) \ }-1024i \mathrm{(C) \ } 0 \mathrm{(D) \ } 1024 \mathrm{(E) \ }1024i$

If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals $\mathrm{(A)\ } pq \mathrm{(B) \ }\frac{3p+q}{5} \mathrm{(C) \ } \frac{1+3pq}{p+q} \mathrm{(D) \ } \frac{3pq}{1+3pq} \mathrm{(E) \ }p^2+q^2$

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

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

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 $\mathrm{(A)\ } T2$ $\mathrm{(E) \ }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}$

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

The minimum of $\sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2}$ is attained when $A$ is $\mathrm{(A)\ } -180^\circ \mathrm{(B) \ }60^\circ \mathrm{(C) \ } 120^\circ \mathrm{(D) \ } 0^\circ \mathrm{(E) \ }\text{none of these}$

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

OTT'T''PQ$r$49

$\mathrm{(A)\ } 25/6 \mathrm{(B) \ } 6 \mathrm{(C) \ } 25/4$ $\mathrm{(D) \ } \text{a number other than }25/6, 6, 25/4$ $\mathrm{(E) \ }\text{not determinable from the given information}$

A fair die is rolled six times. The probability of rolling at least a five at least five times is $\mathrm{(A)\ } \frac{13}{729} \mathrm{(B) \ }\frac{12}{729} \mathrm{(C) \ } \frac{2}{729} \mathrm{(D) \ } \frac{3}{729} \mathrm{(E) \ }\text{none of these}$

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

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

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $\mathrm{(A)\ } 100 \mathrm{(B) \ }125 \mathrm{(C) \ } 123 \mathrm{(D) \ } 30 \mathrm{(E) \ }\text{none of these}$

If $f(x)=3x+2$ for all real $x$, then the statement: "$|f(x)+4|0$ and $b>0$" is true when $\mathrm{(A)}\ b\le a/3 \mathrm{(B)}\ b > a/3 \mathrm{(C)}\ a\le b/3 \mathrm{(D)}\ a > b/3\\ \mathrm{(E)}\ \text{The statement is never true.}$

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

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

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