Question bank AMC12 1974

If $x\ne0$ or $4$ and $y\ne0$ or $6,$ then $\frac2x+\frac3y=\frac12$ is equivalent to

Let $x_1$ and $x_2$ be such that $x_1\ne x_2$ and $3x_i^2-hx_i=b,$ $i=1,$ $2.$ Then $x_1+x_2$ equals

The coefficient of $x^7$ in the polynomial expansion of $$
(1+2x-x^2)^4
$$ is

What is the remainder when $x^{51}+51$ is divided by $x+1?$

Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E,$ if $\angle BAD=92^\circ$ and $\angle ADC=68^\circ,$ find $\angle EBC.$

For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y};$ then

A town’s population increased by $1{,}200$ people, and then this new population decreased by $11\%.$ The town now had $32$ less people than it did before the $1{,}200$ increase. What is the original population?

What is the smallest prime number dividing the sum $3^{11}+5^{13}?$

The integers greater than one are arranged in five columns as follows:

$2$ $3$ $4$ $5$ $9$ $8$ $7$ $6$ $10$ $11$ $12$ $13$ $17$ $16$ $15$ $14$ $\vdots$ $\vdots$ $\vdots$ $\vdots$

(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)

In which column will the number $1{,}000$ fall?

What is the smallest integral value of $k$ such that $$
2x(kx-4)-x^2+6=0
$$ has no real roots?

If $(a,b)$ and $(c,d)$ are two points on the line whose equation is $y=mx+k,$ then the distance between $(a,b)$ and $(c,d),$ in terms of $a,$ $c,$ and $m,$ is

If $g(x)=1-x^2$ and $f(g(x))=\frac{1-x^2}{x^2}$ when $x\ne0,$ then $f(\frac{1}{2})$ equals

Which of the following is equivalent to “If $P$ is true then $Q$ is false”?

Which statement is correct?

If $x\lt-2$ then $\bigl|1-|1+x|\bigr|$ equals

A circle of radius $r$ is inscribed in a right isosceles triangle, and a circle of radius $R$ is circumscribed about the triangle. Then $\frac{R}{r}$ equals

If $i^2=-1,$ then $(1+i)^{20}-(1-i)^{20}$ equals

If $\log_8 3=p$ and $\log_3 5=q,$ then, in terms of $p$ and $q,$ $\log_{10}5$ equals

In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is

Let $$
\begin{aligned} T={}&\frac1{3-\sqrt8} -\frac1{\sqrt8-\sqrt7}\\ &+\frac1{\sqrt7-\sqrt6} -\frac1{\sqrt6-\sqrt5}\\ &+\frac1{\sqrt5-2}; \end{aligned}
$$ then