Gary purchased a large beverage, but drank only $\frac{m}{n}$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m + n.$
On square $ABCD,$ point $E$ lies on side $\overline{AD}$ and point $F$ lies on side $\overline{BC},$ so that $BE = EF = FD = 30.$ Find the area of square $ABCD.$
The degree measures of the angles of a convex $18$-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
In triangle $ABC,$ $AB = \frac{20}{11} AC.$ The angle bisector of angle $A$ intersects $\overline{BC}$ at point $D,$ and point $M$ is the midpoint of $\overline{AD}.$ Let $P$ be the point of intersection of $\overline{AC}$ and line $BM.$ The ratio of $CP$ to $PA$ can be expressed in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
The sum of the first $2011$ terms of a geometric series is $200.$ The sum of the first $4022$ terms of the same series is $380.$ Find the sum of the first $6033$ terms of the series.
Define an ordered quadruple of integers $(a, b, c, d)$ to be *interesting* if $1 \le a \lt b \lt c \lt d \le 10$ and $a + d \gt b + c.$ How many interesting ordered quadruples are there?
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways in which Ed can arrange the $m + 5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000.$
Let $z_1,$ $z_2,$ $z_3,$ $\ldots,$ $z_{12}$ be the $12$ zeroes of the polynomial $z^{12} - 2^{36}.$ For each $j,$ let $w_j$ be one of $z_j$ or $\mathrm{i}z_j.$ Then the maximum possible value of the real part of $\sum_{j=1}^{12} w_j$ can be written as $m + \sqrt{n},$ where $m$ and $n$ are positive integers. Find $m + n.$
Let $x_1,$ $x_2,$ $\ldots,$ $x_6$ be nonnegative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 =$ $1,$ and $x_1x_3x_5 + x_2x_4x_6 \ge \frac{1}{540}.$ Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $$
\begin{aligned} &x_1x_2x_3 + x_2x_3x_4 \\ &\quad {}+ x_3x_4x_5 + x_4x_5x_6 \\ &\quad {}+ x_5x_6x_1 + x_6x_1x_2. \end{aligned}
$$ Find $p + q.$
A circle with center $O$ has radius $25.$ Chord $\overline{AB}$ of length $30$ and chord $\overline{CD}$ of length $14$ intersect at point $P.$ The distance between the midpoints of the two chords is $12.$ The quantity $OP^2$ can be represented as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by $1000.$
Let $M_n$ be the $n \times n$ matrix with entries as follows: for $1 \le i \le n,$ $m_{i,i} = 10;$ for $1 \le i \le n - 1,$ $m_{i+1,i} = m_{i,i+1} = 3;$ all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n.$ Then $\sum_{n=1}^{\infty} \frac{1}{8D_n + 1}$ can be represented as $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$
Note: The determinant of the $1 \times 1$ matrix $[a]$ is $a,$ and the determinant of the $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $ad - bc;$ for $n \ge 2,$ the determinant of an $n \times n$ matrix with first row or first column $a_1$ $a_2$ $a_3$ $\ldots$ $a_n$ is equal to $a_1C_1 - a_2C_2$ $+ a_3C_3 - \cdots$ $+ (-1)^{n+1}a_nC_n,$ where $C_i$ is the determinant of the $(n - 1) \times (n - 1)$ matrix formed by eliminating the row and column containing $a_i.$
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP \gt CP.$ Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP,$ respectively. Given that $AB = 12$ and $\angle O_1PO_2 = 120^\circ,$ then $AP = \sqrt{a} + \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a + b.$
There are $N$ permutations $(a_1, a_2, \ldots, a_{30})$ of $1,$ $2,$ $\ldots,$ $30$ such that for $m \in \{2, 3, 5\},$ $m$ divides $a_{n+m} - a_n$ for all integers $n$ with $1 \le n \lt n + m \le 30.$ Find the remainder when $N$ is divided by $1000.$
Let $P(x) = x^2 - 3x - 9.$ A real number $x$ is chosen at random from the interval $5 \le x \le 15.$ The probability that $\left\lfloor \sqrt{P(x)} \right\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e},$ where $a,$ $b,$ $c,$ $d,$ and $e$ are positive integers. Find $a + b + c + d + e.$
Jar $A$ contains four liters of a solution that is $45$% acid. Jar $B$ contains five liters of a solution that is $48$% acid. Jar $C$ contains one liter of a solution that is $k$% acid. From jar $C$, $\frac{m}{n}$ liters of the solution is added to jar $A$, and the remainder of the solution in jar $C$ is added to jar $B$. At the end both jar $A$ and jar $B$ contain solutions that are $50$% acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n.$
In rectangle $ABCD,$ $AB = 12$ and $BC = 10.$ Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9,$ $DF = 8,$ $\overline{BE} \parallel \overline{DF},$ $\overline{EF} \parallel \overline{AB},$ and line $BE$ intersects segment $\overline{AD}.$ The length $EF$ can be expressed in the form $m\sqrt{n} - p,$ where $m,$ $n,$ and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n + p.$
Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A = (24, -1),$ and let $M$ be the line perpendicular to line $L$ that contains the point $B = (5, 6).$ The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14, 27)$ in the original system has coordinates $(\alpha, \beta)$ in the new coordinate system. Find $\alpha + \beta.$
In triangle $ABC,$ $AB = 125,$ $AC = 117,$ and $BC = 120.$ The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L,$ and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K.$ Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL},$ respectively. Find $MN.$
The vertices of a regular nonagon ($9$-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3.$ Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.