1965 AMC 12

The number of real values of $x$ satisfying the equation $2^{2x^2-7x+5}=1$ is:

A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the shorter of the arcs intercepted by the side is:

The expression $81^{-(2^{-2})}$ has the same value as:

Line $l_2$ intersects line $l_1$ and line $l_3$ is parallel to $l_1.$ The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:

When the repeating decimal $0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is:

If $10^{\log_{10}9}=8x+5,$ then $x$ equals:

The sum of the reciprocals of the roots of the equation $ax^2+bx+c=0$ is:

One side of a given triangle is $18$ inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is:

The vertex of the parabola $y=x^2-8x+c$ will be a point on the $x$-axis if the value of $c$ is:

The statement $x^2-x-6\lt0$ is equivalent to the statement:

Consider the statements:

$$
\begin{aligned} \mathrm{I:}\quad &(\sqrt{-4})(\sqrt{-16})\\ &=\sqrt{(-4)(-16)}, \end{aligned}
$$

$$
\mathrm{II:}\quad \sqrt{(-4)(-16)}=\sqrt{64},
$$

and

$$
\mathrm{III:}\quad \sqrt{64}=8.
$$

Of these the following are *incorrect*:

A rhombus is inscribed in triangle $ABC$ in such a way that one of its vertices is $A$ and two of its sides lie along $AB$ and $AC.$ If $AC=6$ inches, $AB=12$ inches, and $BC=8$ inches, the side of the rhombus, in inches, is:

Let $n$ be the number of number-pairs $(x,y)$ which satisfy $5y-3x=15$ and $x^2+y^2\leq16.$ Then $n$ is:

The sum of the numerical coefficients in the complete expansion of $(x^2-2xy+y^2)^7$ is:

The symbol $25_b$ represents a two-digit number in the base $b.$ If the number $52_b$ is double the number $25_b,$ then $b$ is:

Let line $AC$ be perpendicular to line $CE.$ Connect $A$ to the midpoint $D$ of $CE,$ and connect $E$ to the midpoint $B$ of $AC.$ If $AD$ and $EB$ intersect in point $F,$ and $BC=CD=15$ inches, then the area of triangle $DFE,$ in square inches, is:

Given the true statement: The picnic on Sunday will not be held only if the weather is not fair. We can then conclude that:

If $1-y$ is used as an approximation to the value of $\dfrac1{1+y},$ $|y|\lt1,$ the ratio of the error made to the correct value is:

If $x^4+4x^3+6px^2+4qx+r$ is exactly divisible by $x^3+3x^2+9x+3,$ the value of $(p+q)r$ is:

For every $n$ the sum $S_n$ of $n$ terms of an arithmetic progression is $2n+3n^2.$ The $r$th term is: