1964 AMC 12

What is the value of $\left[\log_{10}\left(5\log_{10}100\right)\right]^2?$

The graph of $x^2-4y^2=0$ is:

When a positive integer $x$ is divided by a positive integer $y,$ the quotient is $u$ and the remainder is $v,$ where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y?$

The expression

$$
\frac{P+Q}{P-Q}-\frac{P-Q}{P+Q},
$$

where $P=x+y$ and $Q=x-y,$ is equivalent to:

If $y$ varies directly as $x,$ and if $y=8$ when $x=4,$ the value of $y$ when $x=-8$ is:

If $x,$ $2x+2,$ $3x+3,$ $\ldots$ are in geometric progression, the fourth term is:

Let $n$ be the number of real values of $p$ for which the roots of

$$
x^2-px+p=0
$$

are equal. Then $n$ equals:

The smaller root of the equation

$$
\begin{aligned} &\left(x-\frac34\right)\left(x-\frac34\right)\\ &\quad+\left(x-\frac34\right) \left(x-\frac12\right)=0 \end{aligned}
$$

is:

A jobber buys an article at $\$24$ less $12\dfrac12\%.$ He then wishes to sell the article at a gain of $33\dfrac13\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?

Given a square with side of length $s.$ On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is:

Given $2^x=8^{y+1}$ and $9^y=3^{x-9},$ find the value of $x+y.$

Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2\gt0?$

A circle is inscribed in a triangle with side lengths $8,$ $13,$ and $17.$ Let the segments of the side of length $8,$ made by a point of tangency, be $r$ and $s,$ with $r\lt s.$ What is the ratio $r:s?$

A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700.$ Based on the cost, the percent gain on the entire transaction is:

A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T.$ The equation of the line is:

Let $f(x)=x^2+3x+2$ and let $S$ be the set of integers $\{0,1,2,\ldots,25\}.$ The number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by $6$ is:

Given the distinct points $P(x_1,y_1),$ $Q(x_2,y_2)$ and $R(x_1+x_2,y_1+y_2).$ Line segments are drawn connecting these points to each other and to the origin $O.$ Of the three possibilities: $(1)$ parallelogram, $(2)$ straight line, $(3)$ trapezoid, figure $OPRQ,$ depending upon the location of the points $P,$ $Q,$ and $R,$ can be:

Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is:

If $2x-3y-z=0$ and $x+3y-14z=0,$ $z\ne0,$ the numerical value of $\dfrac{x^2+3xy}{y^2+z^2}$ is:

The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is: