1961 AMC 12

When simplified, $\left(-\dfrac{1}{125}\right)^{-\frac{2}{3}}$ becomes:

An automobile travels $\frac{a}{6}$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in the $3$ minutes?

If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:

Let the set consisting of the squares of the positive integers be called $u;$ thus $u$ is the set $1,$ $4,$ $9,$ $\ldots.$ If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then $u$ is closed under:

Let $S=(x-1)^4+4(x-1)^3$ $+6(x-1)^2+4(x-1)+1.$ Then $S$ equals:

When simplified, $\log 8\div\log\dfrac18$ becomes:

When simplified, the third term in the expansion of $$
\left(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2}\right)^6
$$ is:

Let the two base angles of a triangle be $A$ and $B,$ with $B$ larger than $A.$ The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2,$ with $C_2$ adjacent to side $a.$ Then:

Let $r$ be the result of doubling both the base and the exponent of $a^b,$ $b\ne0.$ If $r$ equals the product of $a^b$ by $x^b,$ then $x$ equals:

Each side of triangle $ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC,$ and $E$ is the midpoint of $AD.$ The length of $BE,$ in the same unit, is:

Two tangents are drawn to a circle from an exterior point $A;$ they touch the circle at points $B$ and $C,$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R,$ and touches the circle at $Q.$ If $AB=20,$ then the perimeter of triangle $APR$ is:

The first three terms of a geometric progression are $\sqrt2,$ $\sqrt[3]2,$ $\sqrt[6]2.$ Find the fourth term.

The symbol $|a|$ means $a$ if $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to:

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of $K,$ where $K$ is the area of the rhombus in square inches.

If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is:

An altitude $h$ of a triangle is increased by a length $m.$ How much must be taken from the corresponding base $b$ so that the area of the new triangle is one-half that of the original triangle?

In the base ten number system the number $526$ means $5\cdot10^2+2\cdot10+6.$ In the Land of Mathesis, however, numbers are written in the base $r.$ Jones purchases an automobile there for $440$ monetary units (abbreviated m.u.). He gives the salesman a $1000$ m.u. bill, and receives, in change, $340$ m.u. The base $r$ is:

The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. The net change over the four years, to the nearest percent, is:

Consider the graphs of $y=2\log x$ and $y=\log2x.$ We may say that:

The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants: