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Matt has a twenty dollar bill and buys two items worth $7.99 each. How much change does he receive, in dollars?

The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers?

Evaluate $\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}$.

A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$.

Yannick ran 100 meters in 14.22 seconds. Compute his average speed in meters per second, rounded to the nearest integer.

The mean of the numbers 2, 0, 1, 5, and $x$ is an integer. Find the smallest possible positive integer value for $x$.

Let $f(x) = \sqrt{2^2-x^2}$. Find the value of $f(f(f(f(f(-1)))))$.

Find the smallest positive integer $n$ such that 20 divides $15n$ and 15 divides $20n$.

A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ .

Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl?

Let $k$ be a positive integer. After making $k$ consecutive shots successfully, Andy's overall shooting accuracy increased from 65% to 70%. Determine the minimum possible value of $k$.

In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $\angle AMN = \angle MNP = 90^{\circ}$. Compute the ratio $\frac{AP}{PB}$ .

Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?

Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube.

A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons?

All positive integers relatively prime to 2015 are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}\u2212 1$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $a + b$.

Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $(y - x)$.

In triangle $ABC$, where $AC$ > $AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are 5 and 6, respectively, compute the area of triangle $ABC$.

For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?

A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game.

Find the least composite positive integer that is not divisible by any of 3, 4, and 5.

Five checkers are on the squares of an $8 \times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers?

Let the operation $x@y$ be $y - x$. Compute $((\cdots((1@2)@3)@ \cdots @2013)@2014)@2015$.

In a town, each family has either one or two children. According to a recent survey, 40% of the children in the town have a sibling. What fraction of the families in the town have two children?

Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand?