A rectangle measuring 6 inches by 7 inches is centered inside a larger rectangle such that the perimeters of the two rectangles are separated by a uniform distance of <i>x i>inches. If the difference between the areas of the two rectangles is 168 square inches, what is <i>xi>?

In a certain business, a marketing measure <i>mi> is directly proportional to the sales measure <i>si>, which is in turn directly proportional to the profit measure <i>pi>. What is <i>mi> if $p=100$?

1. $s=40$ whenever $p=80$.
2. $m=80$ whenever $p=70$.

For any integers <i>ni> and <i>mi>, $\ddot{m}\left(n,m\right)$ and $\dot{m}\left(n,m\right)$ denote the minimum and the maximum of <i>ni> and <i>mi>, respectively. For example, $\ddot{m}\left(4,3\right)=3$ and $\dot{m}\left(4,3\right)=4$. For the integer <i>mi>, what is the value of $\ddot{m}\left(100,m\right)$?

1. $m=\dot{m}\left(100,n\right)$ for some integer <i>ni>.
2. $m=\dot{m}\left(100,m\right)$

If ^ represents one of the operations $-$, $÷$, and $\times$, is $a^\left(b^c\right)=c^\left(a^b\right)$for all numbers <i>ai>, <i>bi>, and <i>ci>?

1. $a^\left(b+c\right)$ is not equal to $a^b+a^c$ for some values of <i>a, bi>, and <i>ci>.
2. ^ represents division or subtraction.

If <i>ki>, <i>ni>, and <i>mi> are positive integers, is $kn$ even?

1. $k=m^3$
2. $n={\left(m-1\right)}^3$

If <i>mi> is an integer, is <i>mi> odd?

1. $1+m^2$ is an odd integer.
2. $5m-2$ is an even integer.

If <i>ui> and <i>vi> are positive, is $\frac{u}{v}$ less than 1?

1. ${\left(uv\right)}^2>1$
2. $u-v>0$

A stack of colored cards is sorted such that <i>x i>blue cards, <i>y i>red cards, and <i>z i>green cards are placed on top of each of a number of yellow cards, and there are no cards left over. How many yellow cards are there?

1. The numbers of blue, red, and green cards placed on a given yellow card are in the ratio 3:4:7, respectively.
2. A total of 90 blue cards, 120 red cards, and 210 green cards are placed.

If <i>ei>, <i>fi>, <i>gi>, and <i>hi> are positive numbers, and $f\ne h$, is $\frac{e}{f}<\frac{g}{h}$?

1. $\frac{e-g}{h-f}$ is negative.
2. $fg{\left(\frac{eh}{fg}\right)}^2<eh$

A box contains <i>bi> blue balls and <i>ri> red balls. If 4 blue balls and 3 red balls are added to the box, and if one ball is obtained at random from the box, then the probability that a blue ball is obtained can be represented by

If <i>i> is an integer greater than 6, then all of the following must be divisible by 3 EXCEPT

If <i>ki> is an integer, is <i>ki> even?

1. $\frac{k}{2}$ is not an even integer.
2. $5-k$ is an even integer.

Pat bought a new car for <i>Pi> dollars and sold it later for <i>Mi> dollars. The difference between the two sale prices was what percent of the initial price?

1. $M-P=10,000$
2. $\frac{P}{M}=\frac{3}{2}$

In the figure above, <i>EF i>is a diameter of the circle with center <i>O i>and is perpendicular at each end to the trapezoid <i>ABCDi>. If the area of the circle is $225\pi$, the length of <i>AD i>is 20 and the length of <i>BC i>is 10, what is the area of the <i>ABCDi>?