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	$A$	$B$	$C$	$D$	$E$
$A$	0	$y$	52	78	99
$B$	$y$	0	64	$x$	75
$C$	52	64	0	25	31
$D$	78	$x$	25	0	84
$E$	99	75	31	84	0

The table above shows the distance, in miles, by the most direct route, between any two of the five cities $A, B, C, D$ and $E$ . For example, the distance between City $B$ and City $C$ is $64$ 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 $D$ is four times the distance between $A$ and $B$ .

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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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	$A$	$B$	$C$	$D$
$A$	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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A square patch from a quilt has a square interior surrounded by a border of uniform width. If the ratio of the area of the interior of the patch to the area of its border is 100 to 44, which of the following could be the width, in centimeters, of the border?

I. $2$

II. $4$

III. $11$

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A square table has a square glass inlay in the center, and a wooden strip of uniform width surrounding the glass. If the ratio of the glass inlay to the wooden strip is 4 to 5, which of the following could be the width, in feet, of the wooden strip?

I. $\tfrac{1}{4}$

II. $\tfrac{1}{2}$

III. 1

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$y-4x=8$

$|y| \leq 15$

For how many ordered pairs $(x,y)$ that are solutions of the system above are $x$ and $y$ both integers?

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At a seminar, Workshops A, B, and C take place throughout the day. Workshop A takes place every hour, beginning at 8:00 a.m., and lasts $\frac{3}{4}$ hour. Workshop B takes place every $\frac{1}{2}$ hour, beginning at 8:30 a.m., and lasts 1 hour. Workshop C takes place every hour, beginning at 8:45 a.m., and lasts $1\frac{1}{2}$ hours. What is the least total amount of time a person planning to attend all 3 workshops, in order, must spend waiting between workshops if all workshops keep to their schedules?

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A passenger is planning a trip that involves three connecting trains that leave from Stations A, B, and C, respectively. The first train leaves Station A every hour, beginning at 6:30 a.m., and arrives at Station B $1\frac{1}{2}$ hours later. The second train leaves Station B every hour, beginning at 7:30 a.m., and arrives at Station C $2\frac{3}{4}$ hours later. The third train leaves Station C every 30 minutes, beginning at 9:00 a.m. What is the least total amount of time the passenger must spend waiting between trains if all trains keep to their schedules?

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On a certain day, a movie theater is playing all 3 parts of a trilogy. Part 1 plays every hour, beginning at 10:00 a.m., and lasts $2\frac{1}{4}$ hours. Part 2 plays every $1\frac{1}{2}$ hours, beginning at 10:00 a.m., and lasts 2 hours. Part 3 plays every $1\frac{1}{2}$ hours beginning at 12:15 p.m. What is the least total amount of time a person planning to watch all 3 parts of the trilogy in order must spend waiting between movies if all movies keep to their schedules?

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A couple decides to have four children. If they succeed in having four children and each child is equally likely to be a boy or a girl, what is the probability that they will have at least two girls?

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A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have 1 boy and 3 girls?

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Sasha flips a coin 4 times. What is the probability that she will flip exactly 2 heads and 2 tails?

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A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?

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A building has $48$ one- and two-bedroom apartments to rent, some with balconies and others without. If one of these apartments is to be selected at random, is the probability greater than $\frac{1}{2}$ that the apartment has two bedrooms and a balcony?

(1) Of the apartments, $40$ have two bedrooms.

(2) Of the apartments, $30$ have a balcony.

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A store received a shipment of $150$ coats. Each coat was either red or black, and there was at least one of each color coat. These coats have either buttons or a zipper. If one of these coats is to be selected at random, is the probability less than $\frac{1}{3}$ that a red coat with buttons is selected?

(1) Of the coats, $90$ are black.

(2) Of the coats, twice as many have a zipper than buttons.

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On a certain day, each of $120$ students chose either pizza or chicken to eat and either milk or juice to drink for lunch. If one of these students is to be selected at random, is the probability less than $\frac{1}{2}$ that the student chose pizza and milk for lunch?

(1) Of the students, $48$ chose pizza.

(2) Of the students, $75$ chose milk.

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In the sequence $N$ of numbers, each term after the first two terms is the product of the two immediately preceding terms and no term is equal to 0. What is the 4th term of $N$ ?