Question bank AMC12 1968

Let $P$ units be the increase in the circumference of a circle resulting from an increase of $\pi$ units in the diameter. Then $P$ equals:

The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:

A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0.$ Its equation is:

Define an operation $*$ for positive real numbers by $a*b=\dfrac{ab}{a+b}.$ Then $4*(4*4)$ equals:

If $f(n)=\dfrac13n(n+1)(n+2),$ then $f(r)-f(r-1)$ equals:

Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D,$ and let side $BC$ be extended through $C,$ to meet in point $E.$ Let $S$ represent the degree-sum of angles $CDE$ and $DCE,$ and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=\frac{S}{S'},$ then:

Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ If $OQ$ is $3$ inches, then $OP,$ in inches, is:

A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is:

The sum of the real values of $x$ satisfying $\lvert x+2\rvert=2\lvert x-2\rvert$ is:

Assume that, for a certain school, it is true that

$\mathrm{I}:$ Some students are not honest.

$\mathrm{II}:$ All fraternity members are honest.

A necessary conclusion is:

If an arc of $60^\circ$ on circle $\mathrm{I}$ has the same length as an arc of $45^\circ$ on circle $\mathrm{II},$ the ratio of the area of circle $\mathrm{I}$ to that of circle $\mathrm{II}$ is:

A circle passes through the vertices of a triangle with side-lengths $7\dfrac12,$ $10,$ $12\dfrac12.$ The radius of the circle is:

If $m$ and $n$ are the roots of $x^2+mx+n=0,$ $m\ne0,$ $n\ne0,$ then the sum of the roots is:

If $x$ and $y$ are nonzero numbers such that $x=1+\dfrac1y$ and $y=1+\dfrac1x,$ then $y$ equals:

Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is:

If $x$ is such that $\dfrac1x\lt2$ and $\dfrac1x\gt-3,$ then:

Let $f(n)=\dfrac{x_1+x_2+\cdots+x_n}{n},$ where $n$ is a positive integer. If $x_k=(-1)^k,$ $k=1,$ $2,$ $\ldots,$ $n,$ the set of possible values of $f(n)$ is:

Side $AB$ of triangle $ABC$ has length $8$ inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC,$ and $E$ is on segment $BC.$ Line $AE$ extended bisects angle $FEC.$ If $DE$ has length $5$ inches, then the length of $CE,$ in inches, is:

Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals:

The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^\circ$ and the largest angle is $160^\circ,$ then $n$ equals: