Question bank AMC12 1966

Given that the ratio of $3x-4$ to $y+15$ is constant, and $y=3$ when $x=2,$ then, when $y=12,$ $x$ equals:

When the base of a triangle is increased $10\%$ and the altitude to this base is decreased $10\%,$ the change in area is:

If the arithmetic mean of two numbers is $6$ and their geometric mean is $10,$ then an equation with the given two numbers as roots is:

Circle $\mathrm{I}$ is circumscribed about a given square and circle $\mathrm{II}$ is inscribed in the given square. If $r$ is the ratio of the area of circle $\mathrm{I}$ to that of circle $\mathrm{II},$ then $r$ equals:

The number of values of $x$ satisfying the equation

$$
\frac{2x^2-10x}{x^2-5x}=x-3
$$

is:

$AB$ is a diameter of a circle centered at $O.$ $C$ is a point on the circle such that angle $BOC$ is $60^\circ.$ If the diameter of the circle is $5$ inches, the length of chord $AC,$ expressed in inches, is:

Let

$$
\frac{35x-29}{x^2-3x+2} =\frac{N_1}{x-1}+\frac{N_2}{x-2}
$$

be an identity in $x.$ The numerical value of $N_1N_2$ is:

The length of the common chord of two intersecting circles is $16$ feet. If the radii are $10$ feet and $17$ feet, a possible value for the distance between the centers of the circles, expressed in feet, is:

If $x=(\log_8 2)^{(\log_2 8)},$ then $\log_3 x$ equals:

If the sum of two numbers is $1$ and their product is $1,$ then the sum of their cubes is, where $i=\sqrt{-1}:$

The sides of triangle $BAC$ are in the ratio $2:3:4.$ $BD$ is the angle-bisector drawn to the shortest side $AC,$ dividing it into segments $AD$ and $CD.$ If the length of $AC$ is $10,$ then the length of the longer segment of $AC$ is:

The number of real values of $x$ that satisfy the equation

$$
(2^{6x+3})(4^{3x+6})=8^{4x+5}
$$

is:

The number of points with positive rational coordinates selected from the set of points in the $xy$-plane such that $x+y\leq5,$ is:

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F.$ The area of triangle $BEF,$ expressed in square inches, is:

If $x-y\gt x$ and $x+y\lt y,$ then:

If

$$
\begin{gathered} \dfrac{4^x}{2^{x+y}}=8,\\ \dfrac{9^{x+y}}{3^{5y}}=243, \end{gathered}
$$

$x$ and $y$ are real numbers, then $xy$ equals:

The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is:

In a given arithmetic sequence the first term is $2,$ the last term is $29,$ and the sum of all the terms is $155.$ The common difference is:

Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,$ $12,$ $\ldots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,$ $19,$ $\ldots.$ Then $s_1=s_2$ for:

If the proposition “$a=0$” is true, the negation of the proposition “For real values of $a$ and $b,$ if $a=0,$ then $ab=0$” is: