1990 AMC 12

If $\dfrac{\frac{x}{4}}{2}=\dfrac4{\frac{x}{2}},$ then $x=$

$\left(\dfrac14\right)^{-\frac{1}{4}}=$

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is $75^\circ,$ then the largest angle is

Let $ABCD$ be a parallelogram with $\angle ABC=120^\circ,$ $AB=16,$ and $BC=10.$ Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4.$ If $\overline{BE}$ intersects $\overline{AD}$ at $F,$ then $FD$ is closest to

Which of these numbers is largest?

Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B?$

A triangle with integral sides has perimeter $8.$ The area of the triangle is

The number of real solutions of the equation $$
|x-2|+|x-3|=1
$$ is

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is

An $11\times11\times11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?

How many positive integers less than $50$ have an odd number of positive integer divisors?

Let $f$ be the function defined by $f(x)=ax^2-\sqrt2$ for some positive $a.$ If $f(f(\sqrt2))=-\sqrt2,$ then $a=$

If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5?$

$1.$ START $X$ AT $3$ AND $S$ AT $0.$

$2.$ INCREASE THE VALUE OF $X$ BY $2.$

$3.$ INCREASE THE VALUE OF $S$ BY THE VALUE OF $X.$

$4.$ IF $S$ IS AT LEAST $10000,$ THEN GO TO INSTRUCTION $5;$ OTHERWISE, GO TO INSTRUCTION $2$ AND PROCEED FROM THERE.

$5.$ PRINT THE VALUE OF $X.$

$6.$ STOP.

An acute isosceles triangle, $ABC,$ is inscribed in a circle. Through $B$ and $C,$ tangents to the circle are drawn, meeting at point $D.$ If $\angle ABC=\angle ACB=2\angle D$ and $x$ is the radian measure of $\angle A,$ then $x=$

Four whole numbers, when added three at a time, give the sums $180,$ $197,$ $208,$ and $222.$ What is the largest of the four numbers?

At one of George Washington’s parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If $13$ married couples attended, how many handshakes were there among these $26$ people?

How many of the numbers $100,$ $101,$ $\ldots,$ $999$ have three different digits in increasing order or in decreasing order?

First $a$ is chosen at random from the set $\{1,2,3,\ldots,99,100\},$ and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is

For how many integers $N$ between $1$ and $1990$ is the improper fraction $$
\frac{N^2+7}{N+4}
$$ **not** in lowest terms?

In the figure, $ABCD$ is a quadrilateral with right angles at $A$ and $C.$ Points $E$ and $F$ are on $\overline{AC},$ and $\overline{DE}$ and $\overline{BF}$ are perpendicular to $\overline{AC}.$ If $AE=3,$ $DE=5,$ and $CE=7,$ then $BF=$