ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2} = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
Answer: 462
Problem: Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. Note that numbers like $012$ and $007$ are not considered $3$-digit numbers.
Answer: 400185
Problem: Find the remainder when $2^{5^9} + 5^{9^2} + 9^{2^5}$ is divided by $11$.
Answer: 8
Problem: The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence.
Answer: 16384
Problem: What is the last three digits of the sum \(11! + 12! + 13! + \ldots + 2006!\)?
Answer: 400
Problem: What is the smallest number $k$ greater than $6$ for which:
\[1 + 2 + \ldots + k = k + (k+1) + \ldots + n\]
where $n$ is an integer greater than $k$?
Answer: 35
Problem: Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.
Answer: 13
Problem: Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_A$ through $O$ and $A$, and another circle $\omega_B$ through $O$ and $B$; the circles $\omega_A$ and $\omega_B$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega$, $\omega_A$, $\omega_B$ are congruent. If $CO = \sqrt{3}$, what is the perimeter of $\triangle ABC$?
Answer: 6
Problem: A company that sells keychains incurs a daily maintenance cost of $\$500$. Additionally, each worker is paid $\$15$ per hour. Each worker produces 5 keychains per hour, and each keychain is sold for $\$3.10$. What is the minimum number of workers the company must hire to make a profit during an 8-hour workday?
Answer: 126
Problem: Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make?
Answer: 112
Problem: How far is it from the point \((9, 17)\) to its reflection across the line \(3x + 4y = 15\)?
Answer: 32
Problem: Find the numerator of \[ \frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011} \] when reduced.
Answer: 9091
Problem: The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x) \cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$.
Answer: 100
Problem: Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose
\[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Answer: 702
Problem: Let $a$, $b$, and $c$ be integers simultaneously satisfying the equations:
\[
4abc + a + b + c = 2018
\]
and
\[
ab + bc + ca = -507
\]
Find \(|a| + |b| + |c|\).
Answer: 46
Problem: \( \triangle ABC \) has side lengths \( 13 \), \( 14 \), and \( 15 \). Let the feet of the altitudes from \( A \), \( B \), and \( C \) be \( D \), \( E \), and \( F \), respectively. The circumcircle of \( \triangle DEF \) intersects \( AD \), \( BE \), and \( CF \) at \( I \), \( J \), and \( K \) respectively. What is the area of \( \triangle IJK \)?
Answer: 21
Problem: For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?
Answer: 1680
Problem: An integer $n \ge 2$ is called *delicious* if there exist $n$ positive integers that sum up to 2014 and have distinct remainders when divided by $n$. Determine the smallest delicious integer.
Answer: 4
Problem: A magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \times 100$ rectangle, with one card per unit square. The cards are placed such that consecutive numbers are in neighboring squares, but the numbers are not visible. The magician then turns over $k$ cards of his choice. What is the smallest value of $k$ such that the revealed cards can determine the exact positions of all other cards?
Answer: 2
Problem: A building contractor needs to pay his $108$ workers \(\$200\) each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two \(\$100\) bills. Find the number of workers who get paid with four \(\$50\) bills.
Answer: 31
Problem: Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$
Answer: 1847
Problem: A courtyard has the shape of a parallelogram $ABCD$. At the corners of the courtyard, there stand poles $AA'$, $BB'$, $CC'$, and $DD'$, each of which is perpendicular to the ground. The heights of these poles are $AA' = 68$ centimeters, $BB' = 75$ centimeters, $CC' = 112$ centimeters, and $DD' = 133$ centimeters. Find the distance in centimeters between the midpoints of $A'C'$ and $B'D'$.
Answer: 14
Problem: Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$. Find the minimum possible value of $|b|$.
Answer: 4
Problem: Determine the number of ordered pairs of positive integers $(x, y)$ with $y < x \le 100$ such that $x^2 - y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are relatively prime if they have no common factor other than $1$.)
Answer: 99
Problem: Find all positive integers $n$ such that both $837 + n$ and $837 - n$ are cubes of positive integers.
Answer: 494
Problem: A positive integer is written on each of the six faces of a cube. For each vertex of the cube, we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?
Answer: 31
Problem: Find the sum of all positive integers $n$ such that \( \frac{2020}{n^3 + n} \) is an integer.
Answer: 13
Problem: What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
Answer: 672
Problem: Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?
Answer: 7
Problem: Compute $ \left\lfloor \frac{2005^3}{2003 \cdot 2004} - \frac{2003^3}{2004 \cdot 2005} \right\rfloor $
Answer: 8
Problem: Let $n$ be a positive integer. Define $f(n)$ as the probability that if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then \( \gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c)) \). Let $s(n)$ be the sum of the distinct prime divisors of $n$. If \( f(n) < \frac{1}{2018} \), compute the smallest possible value of $s(n)$.
Answer: 58
Problem: Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions:

1. $f(1) = 2$
2. $f(a) + f(b) \leq 2\sqrt{f(a)}$
3. For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b+100)$.
Answer: 10201
Problem: Let $a, b$ be positive real numbers with $a > b$. Compute the minimum possible value of the expression

\[
\frac{a^2b - ab^2 + 8}{ab - b^2}.
\]
Answer: 6
Problem: How many ordered triples \((a,b,c)\) of integers with \(1 \leq a \leq b \leq c \leq 60\) satisfy \(a \cdot b = c\)?
Answer: 134
Problem: An alarm clock runs 4 minutes slow every hour. It was set right $3 \frac{1}{2}$ hours ago. Now another clock, which is correct, shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
Answer: 15
Problem: Find the value of \((2^{40} + 12^{41} + 23^{42} + 67^{43} + 87^{44})^{45! + 46} \mod 11\).
Answer: 5
Problem: Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.
Answer: 29
Problem: Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b \in \{0,\ldots,2016\}$ and $(a,b) \neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$.
Answer: 121
Problem: Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$.
Answer: 2009
Problem: Let triangle $\triangle ABC$ have side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $M$ be the midpoint of $\overline{BC}$, and $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle AEM = m\angle AFM = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle AED$ and $\angle AFD$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
Answer: 28
Problem: A total of $2010$ coins are distributed in $5$ boxes. Initially, the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that, she can perform the following transformation any number of times: from a box with at least $4$ coins, she can transfer one coin to each of the other boxes.

What is the maximum number of coins that Martha can take away?
Answer: 2004
Problem: The number $27,000,001$ has exactly four prime factors. Find the sum of these prime factors.
Answer: 652
Problem: In triangle $ABC$, where $\angle ABC$ is obtuse, point $D$ is located on side $AC$ such that $\angle ABD$ is a right angle. Point $E$ is on side $AC$ between points $A$ and $D$, such that line $BD$ bisects $\angle EBC$. Given that $AC = 35$, $BC = 7$, and $BE = 5$, find the length of $CE$.
Answer: 10
Problem: What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?
Answer: 17
Problem: Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions:

1. Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime.
2. All other pairs of the five integers are relatively prime.
Answer: 47
Problem: A classroom consists of a $5 \times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?
Answer: 962
Problem: The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots and form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square.
Answer: 180
Problem: Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$, we have $a_1 + \dots + a_k \neq 0$.
Answer: 24
Problem: The four-digit base ten number $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$ has all nonzero digits and is a multiple of $99$. Additionally, the two-digit base ten number $\underline{a}\;\underline{b}$ is a divisor of $150$, and the two-digit base ten number $\underline{c}\;\underline{d}$ is a divisor of $168$. Find the remainder when the sum of all possible values of the number $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$ is divided by $1000$.
Answer: 108
Problem: We define the ridiculous numbers recursively as follows:
\begin{itemize}
\item 1 is a ridiculous number.
\item If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
\end{itemize}

A closed interval $I$ is "boring" if:
\begin{itemize}
\item $I$ contains no ridiculous numbers, and
\item There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
\end{itemize}

The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?
Answer: 9
Problem: Find all positive integers $n$ such that \( \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \), where \( \lfloor x \rfloor \) represents the largest integer less than the real number $x$.
Answer: 24
Problem: How many ordered pairs of integers \((m,n)\) are there such that \(m\) and \(n\) are the legs of a right triangle with an area equal to a prime number not exceeding 80?
Answer: 87
Problem: Let $a_1, a_2, \ldots$ be a sequence defined by $a_1 = 1$ and for $n \ge 1$, $a_{n+1} = \sqrt{a_n^2 - 2a_n + 3} + 1$. Find $a_{513}$.
Answer: 33
Problem: Let $m$ and $n$ denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum $m + n$?
Answer: 2008
Problem: Evaluate the integral: 

\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{2\sin^3 x}{\cos^5 x} \, dx. \]
Answer: 4
Problem: Let $A = \{m : m \text{ is an integer and the roots of } x^2 + mx + 2020 = 0 \text{ are positive integers}\}$ and $B= \{n : n \text{ is an integer and the roots of } x^2 + 2020x + n = 0 \text{ are negative integers}\}$. Suppose $a$ is the largest element of $A$ and $b$ is the smallest element of $B$. Find the sum of the digits of $a + b$.
Answer: 26
Problem: Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.
Answer: 0
Problem: In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.
Answer: 222
Problem: Determine all $n \geq 3$ for which there exist $n$ positive integers $a_1, a_2, \ldots, a_n$ such that any two of them have a common divisor greater than 1, but any three of them are coprime. Furthermore, assuming that the numbers $a_i$ are less than 5000, find the greatest possible $n$.
Answer: 4
Problem: Given that $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer, solve the following:

- $b \leq d$
- $c \geq a$
- $a \leq e$
- $b \geq e$
- $d \neq 5$

Determine the value of $a^b + c^d + e$.
Answer: 628
Problem: Find the remainder when $19^{92}$ is divided by 92.
Answer: 49
Problem: The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. 

Find the smallest possible number of independent deputies after the vacation.
Answer: 50
Problem: Navi and Ozna are playing a game where Ozna starts first and the two take turns making moves. A positive integer is written on the board. A move consists of either:

1. Subtracting any positive integer at most $2015$ from it, or
2. If the integer on the board is divisible by $2014$, dividing by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \geq 2015$, ensuring that all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.
Answer: 2013
Problem: Suppose $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB = 10$ and $CD = 15$, compute the maximum possible value of $XY$.
Answer: 6
Problem: In $\triangle ABC$, $\angle A = 30^{\circ}$, $BC = 13$. Given two circles $\gamma_1$ and $\gamma_2$ with radii $r_1$ and $r_2$, respectively, contain point $A$ and touch line $BC$ at points $B$ and $C$. Find the product $r_1r_2$.
Answer: 169
Problem: What is the least positive integer such that the product of its digits is $9!$?
Answer: 2578899
Problem: A week ago, Sandy's seasonal Little League batting average was $0.360$. After five more at bats this week, Sandy's batting average is now $0.400$. What is the smallest number of hits that Sandy could have had this season?
Answer: 12
Problem: Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$.
Answer: 630
Problem: Given a square $ABCD$. Let $P \in AB$, $Q \in BC$, $R \in CD$, $S \in DA$ such that $PR \parallel BC$ and $SQ \parallel AB$. Let $Z = PR \cap SQ$. If $BP = 7$, $BQ = 6$, and $DZ = 5$, find the side length of the square.
Answer: 10
Problem: Ann and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens.

Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least $N$ total tokens from the slot machine before Drew receives at least $M=2^{2018}$ total tokens, and Drew wins if he receives $M$ tokens before Ann receives $N$ tokens.

If each person has the same probability of winning, compute the remainder when $N$ is divided by $2018$.
Answer: 5
Problem: Construction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, with each project taking exactly one more day than the previous one to complete. The first project takes 3 days, the second takes 4 days, and so on.

Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
Answer: 51
Problem: Suppose $x, y \in \mathbb{Z}$ satisfy the equation:
\[
y^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 + 24).
\]
Find the sum of all possible values of $|xy|$.
Answer: 18
Problem: How many positive integers less than $200$ are relatively prime to either $15$ or $24$?
Answer: 120
Problem: Four people were guessing the number, $N$, of jellybeans in a jar. No two guesses were equally close to $N$. The closest guess was 80 jellybeans, the next closest guess was 60 jellybeans, followed by 49 jellybeans, and the furthest guess was 125 jellybeans. Find the sum of all possible values for $N$.
Answer: 1256
Problem: Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$, each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\frac{a\sqrt{b}-c}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$.
Answer: 46
Problem: On March $6$, $2002$, during the celebrations of the $500$th anniversary of the birth of mathematician Pedro Nunes, ten people entered the Viva bookstore for science. Each person bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. "The Adventures of Mathematics" by Pedro Nunes was one of the books that achieved the highest number of sales that morning. What is the smallest value this number could have taken?
Answer: 5
Problem: Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, \ldots). What is the largest integer $j$ such that $P$ is divisible by $101^j$?
Answer: 1980
Problem: How many sequences of $5$ positive integers $(a, b, c, d, e)$ satisfy the inequality $abcde \leq a + b + c + d + e \leq 10$?
Answer: 116
Problem: Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that \(\text{area} (\triangle ADE) = 16\), \(\text{area} (\triangle CEF) = 9\) and \(\text{area} (\triangle ABF) = 25\). What is the area of \(\triangle AEF\)?
Answer: 30
Problem: Natural number $M$ has 6 divisors such that the sum of them is equal to 3500. Find all values of $M$.
Answer: 1996
Problem: Let $\ell$ be a real number satisfying the equation $\frac{(1+\ell)^2}{1+\ell^2}=\frac{13}{37}$. Then
\[\frac{(1+\ell)^3}{1+\ell^3}=\frac{m}{n},\] where $m$ and $n$ are positive coprime integers. Find $m+n$.
Answer: 62
Problem: In a party of $2020$ people, some pairs of people are friends. We define a person's popularity as the size of the largest group of people in the party containing them such that every pair of people in that group are friends. A person has a popularity number of $1$ if they have no friends. What is the largest possible number of distinct popularities in the party?
Answer: 1010
Problem: Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\frac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
Answer: 446
Problem: In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.
Answer: 793
Problem: Let $\triangle ABC$ be a triangle with points $D$, $M$, and $N$ on sides $BC$, $AB$, and $AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. Given that the area of $\triangle BMD$ is $9\, \text{cm}^2$ and the area of $\triangle DNC$ is $25\, \text{cm}^2$, compute the area of $\triangle AMN$.
Answer: 15
Problem: Suppose that $x_1 + 1 = x_2 + 2 = x_3 + 3 = \cdots = x_{2008} + 2008 = x_1 + x_2 + x_3 + \cdots + x_{2008} + 2009$. 

Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \displaystyle\sum_{n=1}^{2008} x_n$.
Answer: 1005