Question bank AMC12 1957

The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:

In the equation $2x^2-hx+2k=0,$ the sum of the roots is $4$ and the product of the roots is $-3.$ Then $h$ and $k$ have the values, respectively:

The simplest form of $1-\dfrac{1}{1+\dfrac{a}{1-a}}$ is:

The first step in finding the product $(3x+2)(x-5)$ by use of the distributive property in the form $a(b+c)=ab+ac$ is:

Through the use of theorems on logarithms, $\log\dfrac ab+\log\dfrac bc+\log\dfrac cd-\log\dfrac{ay}{dx}$ can be reduced to:

An open box is constructed by starting with a rectangular sheet of metal $10$ in. by $14$ in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:

The area of a circle inscribed in an equilateral triangle is $48\pi.$ The perimeter of this triangle is:

The numbers $x,$ $y,$ $z$ are proportional to $2,$ $3,$ $5.$ The sum of $x,$ $y,$ and $z$ is $100.$ The number $y$ is given by the equation $y=ax-10.$ Then $a$ is:

The value of $x-y^{x-y}$ when $x=2$ and $y=-2$ is:

The graph of $y=2x^2+4x+3$ has its:

The angle formed by the hands of a clock at $2:15$ is:

Comparing the numbers $10^{-49}$ and $2\cdot10^{-50},$ we may say:

A rational number between $\sqrt2$ and $\sqrt3$ is:

If $y=\sqrt{x^2-2x+1}$ ${}+\sqrt{x^2+2x+1},$ then $y$ is:

The table below shows the distance $s$ in feet a ball rolls down an inclined plane in $t$ seconds.

$t$ $0$ $1$ $2$ $3$ $4$ $5$ $s$ $0$ $10$ $40$ $90$ $160$ $250$

The distance $s$ for $t=2.5$ is:

Goldfish are sold at $15$ cents each. The rectangular coordinate graph showing the cost of $1$ to $12$ goldfish is:

A cube is made by soldering twelve $3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:

Circle $O$ has diameters $\overline{AB}$ and $\overline{CD}$ perpendicular to each other. $\overline{AM}$ is any chord intersecting $\overline{CD}$ at $P.$ Then $AP\cdot AM$ is equal to:

The base of the decimal number system is ten, meaning, for example, that $123=1\cdot10^2+2\cdot10+3.$ In the binary system, which has base two, the first five positive integers are $1,$ $10,$ $11,$ $100,$ $101.$ The numeral $10011$ in the binary system would then be written in the decimal system as:

A man makes a trip by automobile at an average speed of $50$ mph. He returns over the same route at an average speed of $45$ mph. His average speed for the entire trip is: