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A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square. If the ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width, in inches, of the wooden strip?

I. 1

II. 3

III. 4

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A square mirror has a square wooden frame of uniform width surrounding it. If the ratio of the area of the mirror to the area of the wooden frame is 25 to 24, which of the following could be the width, in inches, of the wooden frame?

I. 1

II. 2

III. 5

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	$A$	$B$	$C$	$D$
$A$	0	81	$x$	$y$
$B$	81	0	58	43
$C$	$x$	58	0	25
$D$	$y$	43	25	0

The table above shows the distance, in miles, by the most direct route, between any two of the four cities $A, B, C$ and $D$ . For example, the distance between City $B$ and City $D$ is $43$ miles. What is the value of $x$ ?

(1) By the most direct route, the distance between $A$ and $B$ is three times the distance between $A$ and $D$ .

(2) By the most direct route, the distance between $B$ and $C$ is half the distance between $A$ and $C$ .

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	W	X	Y	Z
W	0	83	45	$y$
X	83	0	$b$	43
Y	45	$b$	0	$a$
Z	$y$	43	$a$	0

The table above shows the distance, in miles, by the most direct route between any two of the four cities W, X, Y, and Z. For example, the distance between City W and City X is 83 miles. What is the value of $a$ ?

(1) By the most direct route, the distance between W and X is one-third of the distance between Y and Z.

(2) By the most direct route, the distance between W and Y is four times the distance between X and Y.

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Each hour, a radio station plays 600 seconds worth of both 30-second and 60-second commercials. At least one commercial of each length is played. How many 60-second commercials are played each hour?

(1) Fewer than 5 commercials that are 30 seconds are played each hour.

(2) More than 8 commercials that are 60 seconds are played each hour.

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Enough small and large pizzas are ordered for $30$ party guests that are expected to eat $3$ slices each. Each small pizza has $6$ slices and each large pizza has $8$ slices. How many small pizzas are ordered?

(1) Fewer than $5$ small pizzas are ordered.

(2) More than $5$ large pizzas are ordered.

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Does the integer $n$ have at least three different positive prime factors?

(1) $\frac{n}{8}$ is an integer

(2) $\frac{n}{6}$ is an integer

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Does the positive integer $x$ have at least two different positive prime factors?

(1) $\frac{x}{4}$ is an integer.

(2) $\frac{x}{25}$ is an integer.

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Running at their respective constant rates, Machine A takes 4 days longer to produce $n$ widgets than Machine B. At these rates, if the two machines together produce $\frac{10}{7}n$ widgets in 3 days, how many days would it take Machine A alone to produce $3n$ widgets?

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In the figure shown, angles $n$ , $q$ , and $r$ make up three of the interior angles of quadrilateral $ABCD$ . Is $n=r$ ?

(1) Lines $h$ and $k$ are parallel.

(2) $p+q=180$

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In the rectangular coordinate system above, if $\Delta HIJ$ and $\Delta JKL$ have equal area, what are the coordinates of point $L$ ?

(1) The coordinates of point $J$ are $(6,0)$ .

(2) $\overline{JK}=2\overline{LK}$

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The sequence $a_1, a_2, a_3, ..., a_n, ...$ is defined such that for all integers $n \geq 1$ :

$a_n = \frac{1}{n+1}-\frac{1}{n+2}$

If $k$ is a positive integer, is the sum of the first $k$ terms of the sequence greater than $\frac{5}{12}$ ?

(1) $k < 17$

(2) $k > 11$

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If two different students are to be selected at random from a group of $10$ male and female students, and if $p$ is the probability that one male and one female student will be selected, is $p>\frac{1}{3}$ ?

(1) More than $\frac{1}{2}$ of the $10$ students are female.

(2) The probability that both students selected will be female is greater than $\frac{1}{3}$ and less than $\frac{1}{2}$ .

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If arc $XYZ$ above is a semicircle, what is the length of diameter $\overline{XZ}$ ?

(1) $m=16$

(2) $\overline{YZ}=4\sqrt{5}$

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If $x$ and $y$ are positive integers, is $\frac{x}{y} < \frac{5}{7}$ ?

(1) $\frac{x}{y} < 0.714$

(2) $\frac{y}{x} > 1.414$

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$A=\{2,3,x,y\}$

$B=\{x,y,5,6,7\}$

Two integers will be randomly selected, one integer from set $A$ and one integer from set $B$ . What is the probability that the sum of the two integers will equal $7$ ?

(1) $x=3$

(2) $y=4$

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The inside of a rectangular carton is $36$ inches long, $30$ inches wide, and $6$ inches high. The carton is filled to capacity with $x$ identical cylindrical cans of soda that stand upright in rows and columns, as indicated in the figure above. If the cans are $6$ inches high, what is the value of $x$ ?

(1) Each of the cans has a radius of $3$ inches.

(2) Six of the cans fit exactly along the length of the carton.

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The inside of a rectangular carton is $44$ centimeters long, $28$ centimeters wide, and $5$ centimeters high. The carton is filled to capacity with $y$ identical cylindrical cans of tomato paste that stand upright in rows and columns, as indicated in the figure above. If the cans are $5$ centimeters high, what is the value of $y$ ?