Question bank AMC12 1979

If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD,$ respectively, then the area of rectangle $DEFG$ in square meters is

For all nonzero real numbers $x$ and $y$ such that $x-y=xy,$ $\frac1x-\frac1y$ equals

In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD.$ What is the measure of $\angle AED$ in degrees?

For all real numbers $x,$ $x[x\{x(2-x)-4\}+10]+1$ equals

Find the sum of the digits of the largest even three digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged.

$\frac32+\frac54+\frac98+\frac{17}{16}$$+\frac{33}{32}+\frac{65}{64}-7=$

The square of an integer is called a *perfect square*. If $x$ is a perfect square, the next larger perfect square is

Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4.$

The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals

If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to the midpoint of a side) is $2,$ and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,$ $2,$ $3,$ $4,$ then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is

Find a positive integral solution to the equation $$
\frac{\substack{1+3+5+\cdots\\{}+(2n-1)}} {\substack{2+4+6+\cdots\\{}+2n}}=\frac{115}{116}.
$$

In the adjoining figure, $CD$ is the diameter of a semicircle with center $O.$ Point $A$ lies on the extension of $DC$ past $C;$ point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$ equals length $OD,$ and the measure of $\angle EOD$ is $45^\circ,$ then the measure of $\angle BAO$ is

The inequality $y-x\lt\sqrt{x^2}$ is satisfied if and only if

In a certain sequence of numbers, the first number is $1,$ and, for all $n\ge2,$ the product of the first $n$ numbers in the sequence is $n^2.$ The sum of the third and the fifth numbers in the sequence is

Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being $p:1$ in one jar and $q:1$ in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2.$ If the radius of the larger circle is $3,$ and if $A_1,$ $A_2,$ $A_1+A_2$ is an arithmetic progression, then the radius of the smaller circle is

Points $A,$ $B,$ $C,$ and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB,$ $AC,$ and $AD$ have lengths $x,$ $y,$ and $z,$ respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C,$ respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied? $$
\begin{aligned} &\text{I. }x\lt\frac z2\\ &\text{II. }y\lt x+\frac z2\\ &\text{III. }y\lt\frac z2 \end{aligned}
$$

To the nearest thousandth, $\log_{10}2$ is $0.301$ and $\log_{10}3$ is $0.477.$ Which of the following is the best approximation of $\log_5 10?$

Find the sum of the squares of all real numbers satisfying the equation $x^{256}-256^{32}=0.$

If $a=\frac12$ and $(a+1)(b+1)=2,$ then the radian measure of $\operatorname{Arctan}a+\operatorname{Arctan}b$ equals