Question bank AMC12 1986

$[x-(y-x)]-[(x-y)-x]=$

If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y=\frac23x+4,$ then an equation for $L$ is

In the figure, $\triangle ABC$ has a right angle at $C$ and $\angle A=20^\circ.$ If $BD$ is the bisector of $\angle ABC,$ then $\angle BDC=$

Let $S$ be the statement

“If the sum of the digits of the whole number $n$ is divisible by $6,$ then $n$ is divisible by $6.$”

A value of $n$ which shows $S$ to be false is

Simplify $\left(\sqrt[6]{27}-\sqrt{6\frac34}\right)^2.$

Using a table of a certain height, two identical blocks of wood are placed as shown in Figure $1.$ Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure $2,$ length $s$ is found to be $28$ inches. How high is the table?

The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5.$ The solution set for $x$ is

The population of the United States in $1980$ was $226{,}504{,}825.$ The area of the country is $3{,}615{,}122$ square miles. There are $(5280)^2$ square feet in one square mile. Which number below best approximates the average number of square feet per person?

The product $$
\begin{aligned} &\left(1-\frac1{2^2}\right) \left(1-\frac1{3^2}\right)\\ &\quad{}\cdots \left(1-\frac1{9^2}\right) \left(1-\frac1{10^2}\right) \end{aligned}
$$ equals

The $120$ permutations of AHSME are arranged in dictionary order, as if each were an ordinary five-letter word. The last letter of the $86$th word in this list is

In $\triangle ABC,$ $AB=13,$ $BC=14$ and $CA=15.$ Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC.$ The length of $HM$ is

John scores $93$ on this year’s AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. How many questions does he leave unanswered? (In the new scoring system that year, one received $5$ points for each correct answer, $0$ points for each wrong answer, and $2$ points for each problem left unanswered. In the previous scoring system, one started with $30$ points, received $4$ more for each correct answer, lost $1$ point for each wrong answer, and neither gained nor lost points for unanswered questions. There are $30$ questions in the $1986$ AHSME.)

A parabola $y=ax^2+bx+c$ has vertex $(4,2).$ If $(2,0)$ is on the parabola, then $abc$ equals

Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$ jumps equals $g$ meters, then one meter equals how many skips?

A student attempted to compute the average, $A,$ of $x,$ $y$ and $z$ by computing the average of $x$ and $y,$ and then computing the average of the result and $z.$ Whenever $x\lt y\lt z,$ the student’s final result is

In $\triangle ABC,$ $AB=8,$ $BC=7,$ $CA=6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA.$ The length of $PC$ is

A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)

A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse is $50\%$ longer than the minor axis, the length of the major axis is

A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?

Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%,$ then $y$ decreases by