1969 AMC 12

When $x$ is added to both the numerator and the denominator of the fraction $\frac{a}{b},$ $a\ne b,$ $b\ne0,$ the value of the fraction is changed to $\frac{c}{d}.$ Then $x$ equals:

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 $y:x$ is:

If $N,$ written in base $2,$ is $11000,$ the integer immediately preceding $N,$ written in base $2,$ is:

Let a binary operation $*$ on ordered pairs of integers be defined by $(a,b)*(c,d)=(a-c,b+d).$ Then, if $(3,2)*(0,0)$ and $(x,y)*(3,2)$ represent identical pairs, $x$ equals:

If a number $N,$ $N\ne0,$ 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:

The area of the ring between two concentric circles is $12\dfrac12\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is:

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:

Triangle $ABC$ is inscribed in a circle. The measures 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, in degrees, is:

The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is:

The number of points equidistant from a circle and two parallel tangents to the circle is:

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:

Let $F=\dfrac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x.$ Then $m$ has a particular value between:

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:

The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}\gt0$ is the set of all $x$ such that:

In a circle with center at $O$ 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:

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:

The equation $2^{2x}-8\cdot2^x+12=0$ is satisfied by:

The number of points common to the graphs of

$$
(x-y+2)(3x+y-4)=0
$$

and

$$
(x+y-2)(2x-5y+7)=0
$$

is:

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:

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: