1967 AMC 12

Question 1

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:

$1$
$2xy$
$2x^2y^2+2$
$2xy+\dfrac{2}{xy}$
$\dfrac{2x}{y}+\dfrac{2y}{x}$

Question 3

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:

$\dfrac{s^2}{24}$
$\dfrac{s^2}{6}$
$\dfrac{s^2\sqrt2}{6}$
$\dfrac{s^2\sqrt3}{6}$
$\dfrac{s^2}{3}$

Question 4

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:

$\dfrac{q^2}{p+r}$
$\dfrac{p+r}{2q}$
$2q-p-r$
$2q-pr$
$q^2-pr$

Question 5

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:

independent of the value of $r$
$\dfrac{\sqrt2}{r}$
$\dfrac{2}{\sqrt r}$
$\dfrac2r$
$\dfrac r2$

Question 6

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

$4$
$f(x)$
$2f(x)$
$3f(x)$
$4f(x)$

Question 7

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

$a$ must be negative
$a$ must be positive
$a$ must not be zero
$a$ can be negative or zero, but not positive
$a$ can be positive, negative, or zero

Question 8

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:

$\dfrac{10m}{m-10}$
$\dfrac{5m}{m-10}$
$\dfrac{m}{m-10}$
$\dfrac{5m}{m-20}$
not determined by the given information

Question 9

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:

$K$ must be an integer
$K$ must be a rational fraction
$K$ must be an irrational number
$K$ must be an integer or a rational fraction
taken alone neither $(A)$ nor $(B)$ nor
$(C)$ nor $(D)$ is true

Question 10

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:

$\dfrac43$
$\dfrac53$
$2$
$\dfrac{11}{4}$
$3$

Question 11

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

$0$
$\sqrt{50}$
$10$
$\sqrt{200}$
none of these

Question 12

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:

$-\dfrac12$
$-\dfrac23$
$-\dfrac32$
$-2$
none of these

Question 13

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 $N$

is $1$
is $2$
must be zero
must be infinite
must be zero or infinite

Question 14

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

$f\left(\dfrac1y\right)$
$-f(y)$
$-f(-y)$
$f(-y)$
$f(y)$

Question 15

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:

$12$
$9$
$6\sqrt2$
$6$
$3\sqrt2$

Question 16

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:

$43$
$44$
$45$
$46$
$47$

Question 17

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

$\lvert r_1+r_2\rvert\gt4\sqrt2$
$\lvert r_1\rvert\gt3$ or $\lvert r_2\rvert\gt3$
$\lvert r_1\rvert\gt2$ and $\lvert r_2\rvert\gt2$
$r_1\lt0$ and $r_2\lt0$
$\lvert r_1+r_2\rvert\lt4\sqrt2$

Question 18

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

$P$ can take any real value
$20\lt P\lt30$
$0\lt P\lt20$
$P\lt0$
$P\gt30$

Question 19

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:

$30$
$\dfrac{80}{3}$
$24$
$\dfrac{45}{2}$
$20$

Question 20

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:

$\dfrac{\pi m^2}{2}$
$\dfrac{3\pi m^2}{8}$
$\dfrac{\pi m^2}{3}$
$\dfrac{\pi m^2}{4}$
$\dfrac{\pi m^2}{8}$