1982 AMC 12

When the polynomial $x^3-2$ is divided by the polynomial $x^2-2,$ the remainder is

If a number eight times as large as $x$ is increased by two, then one fourth of the result equals

Evaluate $(x^x)^{(x^x)}$ at $x=2.$

The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is

Two positive numbers $x$ and $y$ are in the ratio $a:b,$ where $0\lt a\lt b.$ If $x+y=c,$ then the smaller of $x$ and $y$ is

The sum of all but one of the interior angles of a convex polygon equals $2570^\circ.$ The remaining angle is

If the operation $x*y$ is defined by $x*y=(x+1)(y+1)-1,$ then which one of the following is false?

By definition $r!=r(r-1)\cdots1$ and $\binom jk=\frac{j!}{k!(j-k)!},$ where $r,$ $j,$ $k$ are positive integers and $k\lt j.$ If $\binom n1,$ $\binom n2,$ $\binom n3$ form an arithmetic progression with $n\gt3,$ then $n$ equals

A vertical line divides the triangle with vertices $(0,0),$ $(1,1)$ and $(9,1)$ in the $xy$-plane into two regions of equal area. The equation of the line is $x=$

In the adjoining diagram, $BO$ bisects $\angle CBA,$ $CO$ bisects $\angle ACB,$ and $MN$ is parallel to $BC.$ If $AB=12,$ $BC=24,$ and $AC=18,$ then the perimeter of $\triangle AMN$ is

How many integers with four different digits are there between $1{,}000$ and $9{,}999$ such that the absolute value of the difference between the first digit and the last digit is $2?$

Let $f(x)=ax^7+bx^3+cx-5,$ where $a,$ $b,$ and $c$ are constants. If $f(-7)=7,$ then $f(7)$ equals

If $a\gt1,$ $b\gt1$ and $p=\frac{\log_b(\log_ba)}{\log_ba},$ then $a^p$ equals

In the adjoining figure, points $B$ and $C$ lie on line segment $AD,$ and $AB,$ $BC,$ and $CD$ are diameters of circles $O,$ $N,$ and $P,$ respectively. Circles $O,$ $N,$ and $P$ all have radius $15,$ and the line $AG$ is tangent to circle $P$ at $G.$ If $AG$ intersects circle $N$ at points $E$ and $F,$ then chord $EF$ has length

Let $\lfloor z\rfloor$ denote the greatest integer not exceeding $z.$ Let $x$ and $y$ satisfy the simultaneous equations $$
\begin{aligned} y&=2\lfloor x\rfloor+3,\\ y&=3\lfloor x-2\rfloor+5. \end{aligned}
$$ If $x$ is not an integer, then $x+y$ is

In the adjoining figure, a wooden cube has edges of length $3$ meters. Square holes of side one meter, centered in each face, are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is

How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^x+3=0?$

In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ.$ Find the cosine of $\angle BHD.$

Let $f(x)=|x-2|+|x-4|$ $-|2x-6|,$ for $2\le x\le8.$ The sum of the largest and smallest values of $f(x)$ is

The number of pairs of positive integers $(x,y)$ which satisfy the equation $x^2+y^2=x^3$ is