Question bank AMC12 1970

The fourth power of $\sqrt{1+\sqrt{1+\sqrt1}}$ is:

A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is:

If $x=1+2^p$ and $y=1+2^{-p},$ then $y$ in terms of $x$ is:

Let $S$ be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that:

If $f(x)=\dfrac{x^4+x^2}{x+1},$ then $f(i),$ where $i=\sqrt{-1},$ is equal to:

The smallest value of $x^2+8x$ for real values of $x$ is:

Inside square $ABCD$ with side $s,$ quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from side $CD?$

If $a=\log_8 225$ and $b=\log_2 15,$ then:

Points $P$ and $Q$ are on line segment $AB,$ and both points are on the same side of the midpoint of $AB.$ Point $P$ divides $AB$ in the ratio $2:3,$ and $Q$ divides $AB$ in the ratio $3:4.$ If $PQ=2,$ then the length of segment $AB$ is:

Let $F=0.4818181\ldots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by:

If two factors of $2x^3-hx+k$ are $x+2$ and $x-1,$ the value of $|2h-3k|$ is:

A circle with radius $r$ is tangent to sides $AB,$ $AD,$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC.$ The area of the rectangle, in terms of $r,$ is:

Given the binary operation $*$ defined by $a*b=a^b$ for all positive numbers $a$ and $b.$ Then for all positive $a,$ $b,$ $c,$ $n,$ we have:

Consider $x^2+px+q=0,$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1,$ then $p$ equals:

Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1).$ One of these lines has the equation:

If $F(n)$ is a function such that $F(1)=F(2)=F(3)=1,$ and such that $$
F(n+1)=\frac{F(n)F(n-1)+1}{F(n-2)}
$$ for $n\ge3,$ then $F(6)$ is equal to:

If $r\gt0,$ then for all $p$ and $q$ such that $pq\ne0$ and $pr\gt qr,$ we have:

$\sqrt{3+2\sqrt2}-\sqrt{3-2\sqrt2}$ is equal to:

The sum of an infinite geometric series with common ratio $r$ such that $|r|\lt1$ is $15,$ and the sum of the squares of the terms of this series is $45.$ The first term of the series is:

Lines $HK$ and $BC$ lie in a plane. $M$ is the midpoint of line segment $BC,$ and $BH$ and $CK$ are perpendicular to $HK.$ Then we: