1973 AMC 12

A chord which is the perpendicular bisector of a radius of length $12$ in a circle has length

One thousand unit cubes are fastened together to form a large cube with edge length $10$ units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is

The stronger Goldbach conjecture states that any even integer greater than $7$ can be written as the sum of two different prime numbers. For such representations of the even number $126,$ the largest possible difference between the two primes is

*Note:* The regular Goldbach conjecture states that any even integer greater than $3$ is expressible as a sum of two primes. Neither this conjecture nor the stronger version has been settled.

Two congruent $30^\circ$-$60^\circ$-$90^\circ$ triangles are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is $12,$ the area common to both triangles is

Of the following five statements, $\mathrm{I}$ to $\mathrm{V},$ about the binary operation of averaging (arithmetic mean),

$\mathrm{I}.$ Averaging is associative

$\mathrm{II}.$ Averaging is commutative

$\mathrm{III}.$ Averaging distributes over addition

$\mathrm{IV}.$ Addition distributes over averaging

$\mathrm{V}.$ Averaging has an identity element

those which are always true are

If $554$ is the base $b$ representation of the square of the number whose base $b$ representation is $24,$ then $b,$ when written in base $10,$ equals

The sum of all the integers between $50$ and $350$ which end in $1$ is

If $1$ pint of paint is needed to paint a statue $6$ ft. high, then the number of pints it will take to paint (to the same thickness) $540$ statues similar to the original but only $1$ ft. high is

In $\triangle ABC$ with right angle at $C,$ altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K,$ then the area of $\triangle ABC$ is

If $n$ is a real number, then the simultaneous system $$
\begin{cases} nx+y=1,\\ ny+z=1,\\ x+nz=1 \end{cases}
$$ has no solution if and only if $n$ is equal to

A circle with a circumscribed and an inscribed square centered at the origin $O$ of a rectangular coordinate system with positive $x$- and $y$-axes $OX$ and $OY$ is shown in each figure $\mathrm{I}$ to $\mathrm{IV}$ below.



The inequalities $$
\begin{aligned} |x|+|y| &\le \sqrt{2(x^2+y^2)}\\ &\le 2\operatorname{Max}(|x|,|y|). \end{aligned}
$$ are represented geometrically by the figure numbered

*Geometric representation:* An inequality $f(x,y)\le g(x,y)$ for all $x,$ $y$ is represented by a figure showing, for a typical real number $a,$ the containment $$
\begin{aligned} &\{(x,y):g(x,y)\le a\}\\ &\quad\subset \{(x,y):f(x,y)\le a\}. \end{aligned}
$$

The average (arithmetic mean) age of a group consisting of doctors and lawyers is $40.$ If the doctors average $35$ and the lawyers $50$ years old, then the ratio of the number of doctors to the number of lawyers is

The fraction $$
\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}}
$$ is equal to

Each valve $A,$ $B,$ and $C,$ when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in $1$ hour, with only valves $A$ and $C$ open it takes $1.5$ hours, and with only valves $B$ and $C$ open it takes $2$ hours. The number of hours required with only valves $A$ and $B$ open is

A sector with acute central angle $\theta$ is cut from a circle of radius $6.$ The radius of the circle circumscribed about the sector is

If the sum of all the angles except one of a convex polygon is $2190^\circ,$ then the number of sides of the polygon must be

If $\theta$ is an acute angle and $$
\sin\frac12\theta=\sqrt{\frac{x-1}{2x}},
$$ then $\tan\theta$ equals

If $p\ge5$ is a prime number, then $24$ divides $p^2-1$ without remainder

Define $n_a!$ for positive $n$ and $a$ to be $$
\begin{aligned} n_a!={}&n(n-a)(n-2a)\\ &\cdot(n-3a)\cdots(n-ka), \end{aligned}
$$ where $k$ is the greatest integer for which $n\gt ka.$ Then the quotient $$
\frac{72_8!}{18_2!}
$$ is equal to

A cowboy is $4$ miles south of a stream which flows due east. He is also $8$ miles west and $7$ miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is