AHSME 1969

When $x$ is added to both the numerator and denominator of the fraction $\frac{a}{b},a \ne b,b \ne 0$, the value of the fraction is changed to $\frac{c}{d}$. Then $x$ equals: $\text{(A) } \frac{1}{c-d} \text{(B) } \frac{ad-bc}{c-d} \text{(C) } \frac{ad-bc}{c+d} \text{(D) }\frac{bc-ad}{c-d} \text{(E) } \frac{bc+ad}{c-d}$

If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio of $y:x$ is: $\text{(A) } 23:17 \text{(B) } 17y:23 \text{(C) } 23x:17 \\ \text{(D) dependent upon the cost} \text{(E) none of these.}$

If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is: $\text{(A) } 10001 \text{(B) } 10010 \text{(C) } 10011 \text{(D) } 10110 \text{(E) } 10111$

Let a binary operation $\star$ on ordered pairs of integers be defined by $(a,b)\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\star (0,0)$ and $(x,y)\star (3,2)$ represent identical pairs, $x$ equals: $\text{(A) } -3 \text{(B) } 0 \text{(C) } 2 \text{(D) } 3 \text{(E) } 6$

If a number $N,N \ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is $\text{(A) } \frac{1}{R} \text{(B) } R \text{(C) } 4 \text{(D) } \frac{1}{4} \text{(E) } -R$

The area of the ring between two concentric circles is $12\tfrac{1}{2}\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is: $\text{(A) } \frac{5}{\sqrt{2}} \text{(B) } 5 \text{(C) } 5\sqrt{2} \text{(D) } 10 \text{(E) } 10\sqrt{2}$

If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: $\text{(A) } -3 \text{(B) } 0 \text{(C) } 3 \text{(D) } \sqrt{ac} \text{(E) } \frac{a+c}{2}$

Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$ and $CA$ are, respectively, $x+75^{\circ} , 2x+25^{\circ},3x-22^{\circ}$. Then one interior angle of the triangle is: $\text{(A) } 57\tfrac{1}{2}^{\circ} \text{(B) } 59^{\circ} \text{(C) } 60^{\circ} \text{(D) } 61^{\circ} \text{(E) } 122^{\circ}$

The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning at 2 is: $\text{(A) } 27 \text{(B) } 27\tfrac{1}{4} \text{(C) } 27\tfrac{1}{2} \text{(D) } 28 \text{(E) } 27\tfrac{1}{2}$

The number of points equidistant from a circle and two parallel tangents to the circle is: $\text{(A) } 0 \text{(B) } 2 \text{(C) } 3 \text{(D) } 4 \text{(E) } \infty$

Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals: $\text{(A) } -\tfrac{3}{5} \text{(B) } -\tfrac{2}{5} \text{(C) } -\tfrac{1}{5} \text{(D) } \tfrac{1}{5} \text{(E) either }-\tfrac{1}{5}\text{ or} \tfrac{1}{5}.$

Let $F=\frac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between: $\text{(A) } 3 \text{ and } 4 \text{(B) } 4 \text{ and } 5 \text{(C) } 5 \text{ and } 6 \text{(D) } -4 \text{ and } -3 \text{(E) } -6 \text{ and } -5$

A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $\frac{a}{b}$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals: $\text{(A) }\sqrt{a}:\sqrt{b} \text{(B) } \sqrt{a}:\sqrt{a-b} \text{(C) } \sqrt{b}:\sqrt{a-b} \text{(D) } a:\sqrt{a-b} \text{(E) } b:\sqrt{a-b}$

The complete set of $x$-values satisfying the inequality $\frac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that: $\text{(A) } x>2 \text{ or } x2 \text{ or } x1 \text{ or } x1 \text{ or } x<-1 \\ \text{(E) } x \text{ is any real number except 1 or -1}$

In a circle with center $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$, a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is: $\text{(A) } \frac{3r^2}{16} \text{(B) } \frac{\pi r^2}{16} \text{(C) } \frac{\pi r^2\sqrt{2}}{8} \text{(D) } \frac{r^2\sqrt{3}}{32} \text{(E) } \frac{r^2\sqrt{6}}{48}$

When $(a-b)^n,n\ge2,ab\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: $\text{(A) } \tfrac{1}{2}k(k-1) \text{(B) } \tfrac{1}{2}k(k+1) \text{(C) } 2k-1 \text{(D) } 2k \text{(E) } 2k+1$

The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by: $\text{(A) } log(3) \text{(B) } \tfrac{1}{2}log(6) \text{(C) } 1+log(\tfrac{3}{2}) \text{(D) } 1+\frac{log(3)}{log(2)} \text{(E) none of these}$

The number of points common to the graphs of $(x-y+2)(3x+y-4)=0 \text{ and } (x+y-2)(2x-5y+7)=0$ is: $\text{(A) } 2 \text{(B) } 4 \text{(C) } 6 \text{(D) } 16 \text{(E) } \infty$

The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is: $\text{(A) } 0 \text{(B) } 3 \text{(C) } 4 \text{(D) } 12 \text{(E) } \infty$

Let $P$ equal the product of 3,659,893,456,789,325,678 and 342,973,489,379,256. The number of digits in $P$ is: $\text{(A) } 36 \text{(B) } 35 \text{(C) } 34 \text{(D) } 33 \text{(E) } 32$

If the graph of $x^2+y^2=m$ is tangent to that of $x+y=\sqrt{2m}$, then: $\text{(A) m must equal } \tfrac{1}{2} \text{(B) m must equal } \frac{1}{\sqrt{2}} \\ \text{(C) m must equal } \sqrt{2} \text{(D) m must equal } 2 \\ \text{(E) m may be any non-negative real number}$

Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by $$f={(x,y) | y=x \text{ when } 0 \le x \le 5, y=2x-5 \text{ when } 5 \le x \le 8}.$$ Then $K$ is: $\text{(A) } 21.5 \text{(B) } 36.4 \text{(C) } 36.5 \text{(D) } 44 \text{(E) less than 44 but arbitrarily close to it}$

For any integer $n>1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\text{(A) } 0 \text{(B) } 1 \\ \text{(C) } \frac{n}{2} \text{ for n even, } \frac{n+1}{2} \text{ for n odd} \\ \text{(D) } n-1 \text{(E) } n$

When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then: $\text{(A) } r>r' \text{ always} \text{(B) } rr' \text{ sometimes and } rr' \text{ sometimes and } r=r' \text{ sometimes} \\ \text{(E) } r=r' \text{ always}$

If it is known that $\log_2(a)+\log_2(b) \ge 6$, then the least value that can be taken on by $a+b$ is: $\text{(A) } 2\sqrt{6} \text{(B) } 6 \text{(C) } 8\sqrt{2} \text{(D) } 16 \text{(E) none of these}$

A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is: $\text{(A) } 1 \text{(B) } 15 \text{(C) } 15\tfrac{1}{3} \text{(D) } 15\tfrac{1}{2} \text{(E) } 15\tfrac{3}{4}$

A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: $\text{(A) } \frac{2}{n-1} \text{(B) } \frac{n-1}{2} \text{(C) } \frac{2}{n} \text{(D) } 2n \text{(E) } 2(n-1)$

Let $n$ be the number of points $P$ interior to the region bounded by a circle with radius $1$, such that the sum of squares of the distances from $P$ to the endpoints of a given diameter is $3$. Then $n$ is: $\text{(A) } 0 \text{(B) } 1 \text{(C) } 2 \text{(D) } 4 \text{(E) } \infty$

If $x=t^{1/(t-1)}$ and $y=t^{t/(t-1)},t>0,t \ne 1$, a relation between $x$ and $y$ is: $\text{(A) } y^x=x^{1/y} \text{(B) } y^{1/x}=x^{y} \text{(C) } y^x=x^y \text{(D) } x^x=y^y \text{(E) none of these}$

Let $P$ be a point of hypotenuse $AB$ (or its extension) of isosceles right triangle $ABC$. Let $s=AP^2+PB^2$. Then: $\text{(A) } s2CP^2 \text{ if P is a trisection point of AB}$

Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$, and $v=xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: $\text{(A) 3} \text{(B) 4} \text{(C) 5} \text{(D) 6} \text{(E) 11}$

Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is: $\text{(A) } 4:3 \text{(B) } 3:2 \text{(C) } 7:4 \text{(D) } 78:71 \text{(E) undetermined}$

The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as:

$\text{(A) }2^{100}-1 \text{(B) } 2^{100}(x-1)-(x-2) \text{(C) } 2^{200}(x-3) \\ \text{(D) } x(2^{100}-1)+2(2^{99}-1) \text{(E) } 2^{100}(x+1)-(x+2)$

Let $L(m)$ be the $x$ coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: $\text{(A) arbitrarily close to } 0 \\ \text{(B) arbitrarily close to }\frac{1}{\sqrt{6}} \text{(C) arbitrarily close to }\frac{2}{\sqrt{6}} \text{(D) arbitrarily large} \text{(E) undetermined}$