Question bank AMC12 1984

$\frac{1000^2}{252^2-248^2}$ equals

If $x,$ $y$ and $y-\frac1x$ are not $0,$ then $$
\frac{x-\frac1y}{y-\frac1x}
$$ equals

Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10.$ Then

A rectangle intersects a circle as shown: $AB=4,$ $BC=5$ and $DE=3.$ Then $EF$ equals

The largest integer $n$ for which $n^{200}\lt5^{300}$ is

In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters $b,$ $g,$ $t$ to represent the number of boys, girls and teachers, respectively, then the total number of boys, girls and teachers can be represented by the expression

When Dave walks to school, he averages $90$ steps per minute, each of his steps $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school?

Figure $ABCD$ is a trapezoid with $AB\parallel DC,$ $AB=5,$ $BC=3\sqrt2,$ $\angle BCD=45^\circ$ and $\angle CDA=60^\circ.$ The length of $DC$ is

The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i,$ $-2+i$ and $-1-2i.$ The fourth number is

A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result if one starts with an entry $x\ne0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then $y$ equals

If the sequence $\{a_n\}$ is defined by $$
\begin{aligned} a_1&=2,\\ a_{n+1}&=a_n+2n\quad(n\ge1), \end{aligned}
$$ then $a_{100}$ equals

$\frac{2\sqrt6}{\sqrt2+\sqrt3+\sqrt5}$ equals

The product of all real roots of the equation $x^{\log_{10}x}=10$ is

If $\sin2x\sin3x=\cos2x\cos3x,$ then one value for $x$ is

The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x.$ If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is

A right triangle $ABC$ with hypotenuse $AB$ has side $AC=15.$ Altitude $CH$ divides $AB$ into segments $AH$ and $HB,$ with $HB=16.$ The area of $\triangle ABC$ is

A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the $x$-axis, the $y$-axis, and the line $x+y=2.$ Then $x$ is

A box contains $11$ balls, numbered $1,$ $2,$ $3,$ $\ldots,$ $11.$ If $6$ balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?

The number of distinct solutions of the equation $\left|x-|2x+1|\right|=3$ is