Question bank AMC12 1967

The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9.$ If $5b9$ is divisible by $9,$ then $a+b$ equals:

An equivalent of the expression

$$
\begin{aligned} &\left(\frac{x^2+1}{x}\right) \left(\frac{y^2+1}{y}\right)\\ &\quad+ \left(\frac{x^2-1}{y}\right) \left(\frac{y^2-1}{x}\right),\\ &\qquad xy\ne0, \end{aligned}
$$

is:

The side of an equilateral triangle is $s.$ A circle is inscribed in the triangle and a square is inscribed in the circle. The area of the square is:

Given $\dfrac{\log a}{p}=\dfrac{\log b}{q}=\dfrac{\log c}{r}=\log x,$ all logarithms to the same base and $x\ne1.$ If $\dfrac{b^2}{ac}=x^y,$ then $y$ is:

A triangle is circumscribed about a circle of radius $r$ inches. If the perimeter of the triangle is $P$ inches and the area is $K$ square inches, then $\frac{P}{K}$ is:

If $f(x)=4^x,$ then $f(x+1)-f(x)$ equals:

If $\dfrac ab\lt-\dfrac cd,$ where $a,$ $b,$ $c,$ $d$ are real numbers and $bd\ne0,$ then:

To $m$ ounces of an $m\%$ solution of acid, $x$ ounces of water are added to yield an $(m-10)\%$ solution. If $m\gt25,$ then $x$ is:

Let $K,$ in square units, be the area of a trapezoid such that the shorter base, the altitude, and the longer base, in that order, are in arithmetic progression. Then:

If

$$
\begin{aligned} &\frac{a}{10^x-1}+\frac{b}{10^x+2}\\ &\qquad=\frac{2\cdot10^x+3} {(10^x-1)(10^x+2)} \end{aligned}
$$

is an identity for positive rational values of $x,$ then the value of $a-b$ is:

If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $AC,$ in inches, is:

If the (convex) area bounded by the $x$-axis and the lines $y=mx+4,$ $x=1,$ and $x=4$ is $7,$ then $m$ equals:

A triangle $ABC$ is to be constructed given side $a$ (opposite angle $A$), angle $B,$ and $h_c,$ the altitude from $C.$ If $N$ is the number of noncongruent solutions, then $N$

Let $f(t)=\dfrac{t}{1-t},$ $t\ne1.$ If $y=f(x),$ then $x$ can be expressed as:

The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle, in feet, is:

Let the product $(12)(15)(16),$ each factor written in base $b,$ equal $3146$ in base $b.$ Let $s=12+15+16,$ each term expressed in base $b.$ Then $s,$ in base $b,$ is:

If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0,$ then it must follow that:

If $x^2-5x+6\lt0$ and $P=x^2+5x+6,$ then

The area of a rectangle remains unchanged when it is made $2\dfrac12$ inches longer and $\dfrac23$ inch narrower, or when it is made $2\dfrac12$ inches shorter and $\dfrac43$ inch wider. Its area, in square inches, is:

A circle is inscribed in a square of side $m,$ then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches: