Question bank AMC12 1981

If $\sqrt{x+2}=2,$ then $(x+2)^2$ equals

Point $E$ is on side $AB$ of square $ABCD.$ If $EB$ has length one and $EC$ has length two, then the area of the square is

For $x\ne0,$ $\frac1x+\frac1{2x}+\frac1{3x}$ equals

If three times the larger of two numbers is four times the smaller and the difference between the numbers is $8,$ then the larger of the two numbers is

In trapezoid $ABCD,$ sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $\angle DCB=110^\circ$ and $\angle CBD=30^\circ,$ then $\angle ADB=$

If $\frac{x}{x-1}=\frac{y^2+2y-1}{y^2+2y-2},$ then $x$ equals

How many of the first one hundred positive integers are divisible by all of the numbers $2,$ $3,$ $4,$ $5?$

For all positive numbers $x,$ $y,$ $z,$ the product $$
\begin{aligned} &(x+y+z)^{-1}\\ &\quad\cdot(x^{-1}+y^{-1}+z^{-1})\\ &\quad\cdot(xy+yz+zx)^{-1}\\ &\quad\cdot\big((xy)^{-1}+(yz)^{-1}+(zx)^{-1}\big) \end{aligned}
$$ equals

In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a,$ then the surface area of the cube is

The lines $L$ and $K$ are symmetric to each other with respect to the line $y=x.$ If the equation of line $L$ is $y=ax+b$ with $a\ne0$ and $b\ne0,$ then the equation of $K$ is $y=$

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length

If $p,$ $q,$ and $M$ are positive numbers and $q\lt100,$ then the number obtained by increasing $M$ by $p\%$ and decreasing the result by $q\%$ exceeds $M$ if and only if

Suppose that at the end of any year, a unit of money has lost $10\%$ of the value it had at the beginning of that year. Find the smallest integer $n$ such that after $n$ years the unit of money will have lost at least $90\%$ of its value. (To the nearest thousandth $\log_{10}3$ is $0.477.$)

In a geometric sequence of real numbers, the sum of the first two terms is $7,$ and the sum of the first six terms is $91.$ The sum of the first four terms is

If $b\gt1,$ $x\gt0,$ and $(2x)^{\log_b2}-(3x)^{\log_b3}=0,$ then $x$ is

The base three representation of $x$ is $$
12112211122211112222.
$$ The first digit (on the left) of the base nine representation of $x$ is

The function $f$ is not defined for $x=0,$ but, for all nonzero real numbers $x,$ $$
f(x)+2f\left(\frac1x\right)=3x.
$$ The equation $f(x)=f(-x)$ is satisfied by

The number of real solutions to the equation $$
\frac{x}{100}=\sin x
$$ is

In $\triangle ABC,$ $M$ is the midpoint of side $BC,$ $AN$ bisects $\angle BAC,$ $BN\perp AN,$ and $\theta$ is the measure of $\angle BAC.$ If sides $AB$ and $AC$ have lengths $14$ and $19,$ respectively, then length $MN$ equals

A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD,$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path to $A.$ (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n=3.$) If $\angle CDA=8^\circ,$ what is the largest value $n$ can have?