Question bank AMC12 1983

If $x\ne0,$ $\frac{x}{2}=y^2$ and $\frac{x}{4}=4y,$ then $x$ equals

Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3$ cm from $P?$

Three primes, $p,$ $q$ and $r,$ satisfy $p+q=r$ and $1\lt p\lt q.$ Then $p$ equals

In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $FE,$ and sides $BC$ and $ED.$ Each side has length $1.$ Also, $\angle FAB=\angle BCD=60^\circ.$ The area of the figure is

Triangle $ABC$ has a right angle at $C.$ If $\sin A=\frac23,$ then $\tan B$ is

When $x^5,$ $x+\frac1x$ and $1+\frac2x+\frac3{x^3}$ are multiplied, the product is a polynomial of degree

Alice sells an item at $\$10$ less than the list price and receives $10\%$ of her selling price as her commission. Bob sells the same item at $\$20$ less than the list price and receives $20\%$ of his selling price as his commission. If they both get the same commission, then the list price is

Let $f(x)=\frac{x+1}{x-1}.$ Then for $x^2\ne1,$ $f(-x)$ is

In a certain population the ratio of the number of women to the number of men is $11$ to $10.$ If the average (arithmetic mean) age of the women is $34$ and the average age of the men is $32,$ then the average age of the population is

Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC.$ The circle also intersects $AC$ and $BC$ at points $D$ and $E,$ respectively. The length of $AE$ is

Simplify $\sin(x-y)\cos y$${}+\cos(x-y)\sin y.$

If $\log_7(\log_3(\log_2x))=0,$ then $x^{-\frac{1}{2}}$ equals

If $xy=a,$ $xz=b,$ and $yz=c,$ and none of these quantities is $0,$ then $x^2+y^2+z^2$ equals

The units digit of $3^{1001}7^{1002}13^{1003}$ is

Three balls marked $1,$ $2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6,$ what is the probability that the ball numbered $2$ was drawn all three times?

Let $$
\begin{aligned} x&=0.123456789101112\\ &\phantom{={}}\ldots998999, \end{aligned}
$$ where the digits are obtained by writing the integers $1$ through $999$ in order. The $1983$rd digit to the right of the decimal point is

The diagram to the right shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of $F.$ Which one?

Let $f$ be a polynomial function such that, for all real $x,$ $$
f(x^2+1)=x^4+5x^2+3.
$$ For all real $x,$ $f(x^2-1)$ is

Point $D$ is on side $CB$ of triangle $ABC.$ If $\angle CAD=\angle DAB=60^\circ,$ $AC=3$ and $AB=6,$ then the length of $AD$ is

If $\tan\alpha$ and $\tan\beta$ are the roots of $x^2-px+q=0,$ and $\cot\alpha$ and $\cot\beta$ are the roots of $x^2-rx+s=0,$ then $rs$ is necessarily