Question bank AMC12 1989

$(-1)^{5^2}+1^{2^5}=$

$\sqrt{\frac19+\frac1{16}}=$

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is $24,$ then the area of the original square is

In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD=BC=5,$ $AB=4,$ and $DC=10.$ The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in the right triangle $DEF.$ Then $CF=$

Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is $20$ toothpicks high and $10$ toothpicks wide, then the number of toothpicks used is

If $a,$ $b\gt0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area $6,$ then $ab=$

In $\triangle ABC,$ $\angle A=100^\circ,$ $\angle B=50^\circ,$ $\angle C=30^\circ,$ $AH$ is an altitude, and $BM$ is a median. Then $\angle MHC=$

For how many integers $n$ between $1$ and $100$ does $x^2+x-n$ factor into the product of two linear factors with integer coefficients?

Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letters repeated. How many such monograms are possible?

Consider the sequence defined recursively by $u_1=a$ (any positive number), and $u_{n+1}=-\frac{1}{u_n+1},$ $n=1,$ $2,$ $3,$ $\ldots.$ For which of the following values of $n$ must $u_n=a?$

Let $a,$ $b,$ $c$ and $d$ be integers with $a\lt2b,$ $b\lt3c,$ and $c\lt4d.$ If $d\lt100,$ the largest possible value for $a$ is

The traffic on a certain east-west highway moves at a constant speed of $60$ miles per hour in both directions. An eastbound driver passes $20$ westbound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a $100$-mile section of highway?

Two strips of width $1$ overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is

$\cot10+\tan5=$

In $\triangle ABC,$ $AB=5,$ $BC=7,$ $AC=9$ and $D$ is on $\overline{AC}$ with $BD=5.$ Find the ratio $AD:DC.$

A *lattice point* is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)?$ (Include both endpoints of the segment in your count.)

The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989$ cm. The length of each side of the triangle exceeds the length of each side of the square by $d$ cm. The square has perimeter greater than $0.$ How many positive integers are **not** possible values for $d?$

The set of all real numbers $x$ for which $$
x+\sqrt{x^2+1}-\frac1{x+\sqrt{x^2+1}}
$$ is a rational number is the set of all

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3,$ $4,$ and $5.$ What is the area of the triangle?

Let $x$ be a real number selected uniformly at random between $100$ and $200.$ If $\lfloor\sqrt{x}\rfloor=12,$ find the probability that $\lfloor\sqrt{100x}\rfloor=120.$ ($\lfloor v\rfloor$ means the greatest integer less than or equal to $v.$)