Question bank AMC12 1978

If $1-\frac4x+\frac4{x^2}=0,$ then $\frac2x$ equals

If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is

For all nonzero numbers $x$ and $y$ such that $x=\frac{1}{y},$ $$
\left(x-\frac1x\right)\left(y+\frac1y\right)
$$ equals

If $a=1,$ $b=10,$ $c=100$ and $d=1000,$ then $$
\begin{aligned} &(a+b+c-d)+(a+b-c+d)\\ &\quad+(a-b+c+d)\\ &\quad+(-a+b+c+d) \end{aligned}
$$ is equal to

Four boys bought a boat for $\$60.$ The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay?

The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: $$
\begin{aligned} x&=x^2+y^2,\\ y&=2xy. \end{aligned}
$$ is

Opposite sides of a regular hexagon are $12$ inches apart. The length of each side, in inches, is

If $x\ne y$ and the sequences $x,$ $a_1,$ $a_2,$ $y$ and $x,$ $b_1,$ $b_2,$ $b_3,$ $y$ each are in arithmetic progression, then $\frac{a_2-a_1}{b_2-b_1}$ equals

If $x\lt0,$ then $\left|x-\sqrt{(x-1)^2}\right|$ equals

If $B$ is a point on circle $C$ with center $P,$ then the set of all points $A$ in the plane of circle $C$ such that the distance between $A$ and $B$ is less than or equal to the distance between $A$ and any other point on circle $C$ is

If $r$ is positive and the line whose equation is $x+y=r$ is tangent to the circle whose equation is $x^2+y^2=r,$ then $r$ equals

In $\triangle ADE,$ $\angle ADE=140^\circ,$ points $B$ and $C$ lie on sides $AD$ and $AE,$ respectively, and points $A,$ $B,$ $C,$ $D,$ $E$ are distinct. If lengths $AB,$ $BC,$ $CD,$ and $DE$ are all equal, then the measure of $\angle EAD$ is

If $a,$ $b,$ $c,$ and $d$ are nonzero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0,$ then $a+b+c+d$ equals

If an integer $n,$ greater than $8,$ is a solution of the equation $x^2-ax+b=0$ and the representation of $a$ in the base $n$ numeration system is $18,$ then the base $n$ representation of $b$ is

If $\sin x+\cos x=\frac15$ and $0\le x\lt\pi,$ then $\tan x$ is

In a room containing $N$ people, $N\gt3,$ at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else?

If $k$ is a positive number and $f$ is a function such that, for every positive number $x,$ $$
\left[f(x^2+1)\right]^{\sqrt x}=k,
$$ then, for every positive number $y,$ $$
\left[f\left(\frac{9+y^2}{y^2}\right)\right]^{\sqrt{\frac{12}{y}}}
$$ is equal to

What is the smallest positive integer $n$ such that $\sqrt n-\sqrt{n-1}\lt0.01?$

A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le50,$ then the probability of choosing $n$ is $p,$ and if $n\gt50,$ then the probability of choosing $n$ is $3p.$ The probability that a perfect square is chosen is

If $a,$ $b,$ $c$ are nonzero real numbers such that $$
\begin{aligned} \frac{a+b-c}{c} &=\frac{a-b+c}{b}\\ &=\frac{-a+b+c}{a}, \end{aligned}
$$ and $$
x=\frac{(a+b)(b+c)(c+a)}{abc},
$$ and $x\lt0,$ then $x$ equals