ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: You have two blackboards, $A$ and $B$. You need to write some integers on them, where each integer is greater than or equal to $2$ and less than or equal to $20$. The condition is that each number on blackboard $A$ must be co-prime with each number on blackboard $B$. Determine the maximum possible value of the product of the number of integers written on $A$ and the number of integers written on $B$.
Answer: 65
Problem: We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$?
Answer: 8
Problem: The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$.
Answer: 32
Problem: Calculate \(1^2 - 2^2 + 3^2 - 4^2 + \ldots - 2018^2 + 2019^2\).
Answer: 2039190
Problem: Let $\triangle ABC$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle ABC$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.
Answer: 622
Problem: Determine the maximum integer $n$ such that for each positive integer $k \le \frac{n}{2}$, there exist two positive divisors of $n$ with a difference of $k$.
Answer: 24
Problem: Three students write on the blackboard three two-digit square numbers next to each other. In the end, they observe that the resulting 6-digit number is also a perfect square. Find this number!
Answer: 166464
Problem: Find $x$ if \[ \frac{1}{\frac{1}{\frac{1}{\frac{1}{x} + \frac{1}{2}} + \frac{1}{\frac{1}{x} + \frac{1}{2}}} + \frac{1}{\frac{1}{\frac{1}{x} + \frac{1}{2}} + \frac{1}{\frac{1}{x} + \frac{1}{2}}}} = \frac{x}{36}. \]
Answer: 70
Problem: Let $m$ and $n$ be positive integers such that $x = m + \sqrt{n}$ is a solution to the equation $x^2 - 10x + 1 = \sqrt{x}(x + 1)$. Find $m + n$.
Answer: 55
Problem: Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a$, $b$, and $c$ that satisfy the equation: $$a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.$$
Answer: 5
Problem: In a circle, 15 equally spaced points are drawn, and arbitrary triangles are formed by connecting 3 of these points. How many non-congruent triangles can be drawn?
Answer: 19
Problem: The following figure shows a walk of length 6:

```asy
unitsize(20);
for (int x = -5; x <= 5; ++x)
  for (int y = 0; y <= 5; ++y)
    dot((x, y));
label("$O$", (0, 0), S);
draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3));
```

This walk has three interesting properties:

- It starts at the origin, labeled $O$.
- Each step is 1 unit north, east, or west. There are no south steps.
- The walk never comes back to a point it has been to.

Let's call a walk with these three properties a northern walk. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?
Answer: 239
Problem: Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged, it is possible to travel between any two cities in Graphistan using only GA flights, provided that the absolute value of the difference between the number of flights originating from and terminating at any city is not more than $k$.
Answer: 1005
Problem: Evaluate $2023 \cdot \frac{2023^6 + 27}{(2023^2 + 3)(2024^3 - 1)} - 2023^2$.
Answer: -6066
Problem: Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\frac{1}{\sqrt[4]{3}}$. What is the value of $900p$, rounded to the nearest integer?
Answer: 628
Problem: Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m+n$.
Answer: 4306
Problem: Find the number of pairs \((n, C)\) of positive integers such that \(C \leq 100\) and \(n^2 + n + C\) is a perfect square.
Answer: 180
Problem: Let $a_1 < a_2 < a_3 < \ldots < a_n < \ldots$ be positive integers such that, for $n = 1, 2, 3, \ldots,$  
\[ a_{2n} = a_n + n. \]  
Given that if $a_n$ is prime, then $n$ is also, find $a_{2014}$.
Answer: 2014
Problem: At Ignus School, there are $425$ students. Of these students, $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects.
Answer: 9
Problem: Four rectangular strips, each measuring $4$ by $16$ inches, are laid out with two vertical strips crossing two horizontal strips, forming a single polygon resembling a tic-tac-toe pattern. What is the perimeter of this polygon?
Answer: 96
Problem: In each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted?
Answer: 9
Problem: When flipped, coin A shows heads $\frac{1}{3}$ of the time, coin B shows heads $\frac{1}{2}$ of the time, and coin C shows heads $\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$.
Answer: 273
Problem: We wish to find the sum of $40$ given numbers using $40$ processors. Initially, each processor displays the number $0$ on its screen. Each processor can add the number on its screen with a number entered directly (only the given numbers can be entered directly into the processors) or transferred from another processor in a unit time. Whenever a number is transferred from one processor to another, the former processor resets to zero. Determine the minimum time required to compute the desired sum.
Answer: 7
Problem: On the planet Mars, there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions:
1. Each block must have at most $50$ states.
2. Every pair of states must be together in at least one block.

Find the minimum number of blocks that must be formed.
Answer: 6
Problem: If
\[
\sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n}
\]
for relatively prime positive integers $m,n$, compute $m+n$.
Answer: 2004001
Problem: If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?
Answer: 27
Problem: Let $a_1, a_2, \ldots, a_{18}$ be a list of prime numbers such that $\frac{1}{64} \times a_1 \times a_2 \times \cdots \times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18} \frac{1}{\log_{a_i} n}$$ is a positive integer.
Answer: 64008420
Problem: Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$. For each $i$, where $1 \leq i \leq 200$, calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. Determine the sum of the distinct values of $g_i$.
Answer: 25
Problem: A circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with diameter $1700$. Two runners start at point $A$ at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point $A$. At that time, each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 2128
Problem: Find the decimal form of the largest prime divisor of \(100111011_6\).
Answer: 181
Problem: Find the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that consists of only $5$'s.
Answer: 1010
Problem: Calculate $\sum_{i=2}^{\infty} \frac{i^2-2}{i!}$.
Answer: 3
Problem: For any interval $\mathcal{A}$ on the real number line not containing zero, define its reciprocal to be the set of numbers of the form $\frac{1}{x}$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m, n)$ with $m < n$ such that the length of the interval $[m, n]$ is $10^{10}$ times the length of its reciprocal.
Answer: 60
Problem: Let $n$ be an arbitrary positive integer. Calculate \( S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n} \).
Answer: 1
Problem: The triangle $ABC$ is isosceles with $AB = BC$. The point $F$ on the side $[BC]$ and the point $D$ on the side $AC$ are the feet of the internal bisector drawn from $A$ and the altitude drawn from $B$, respectively, such that $AF = 2BD$. Find the measure of the angle $\angle ABC$.
Answer: 108
Problem: Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$, there exist at least $k$ consecutive composite integers in the set \{$n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$\}.
Answer: 7
Problem: For what single digit $n$ does $91$ divide the 9-digit number $12345n789$?
Answer: 7
Problem: In trapezoid $ABCD$, $AB < CD$, $AB \perp BC$, $AB \parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD \perp DC$. If
\[\frac{QP}{AP} + \frac{AP}{QP} = \left( \frac{51}{14} \right)^4 - 2,\]
then $\frac{BP}{AP} - \frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$.
Answer: 61
Problem: A $24$-hour digital clock shows times $h : m : s$, where $h$, $m$, and $s$ are integers with $0 \leq h \leq 23$, $0 \leq m \leq 59$, and $0 \leq s \leq 59$. How many times $h : m : s$ satisfy $h + m = s$?
Answer: 1164
Problem: Let $P$ be the product of the first $50$ nonzero square numbers. Find the largest integer $k$ such that $7^k$ divides $P$.
Answer: 16
Problem: Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 95
Problem: Find the sum of all positive integers $n$ where the mean and median of \( \{20, 42, 69, n\} \) are both integers.
Answer: 45
Problem: Compute the number of positive integers $n$ satisfying the inequalities
\[
2^{n-1} < 5^{n-3} < 3^n.
\]
Answer: 5
Problem: Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x + y) = f(x) + f(y) + f(xy)$ for all $x, y \in \mathbb{R}$.
Answer: 0
Problem: The letters $R$, $M$, and $O$ represent whole numbers. If the following conditions are satisfied: 
1. $R \times M \times O = 240$, 
2. $R \times O + M = 46$, 
3. $R + M \times O = 64$, 

what is the value of $R + M + O$?
Answer: 20
Problem: Call a positive integer $N \ge 2$ "special" if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?
Answer: 50
Problem: Compute the smallest positive integer $x$ which satisfies the following congruences:

1. $x^2 - 8x + 1 \equiv 0 \pmod{22}$
2. $x^2 - 22x + 1 \equiv 0 \pmod{8}$
Answer: 35
Problem: How many subsets with $3$ elements of the set $S = \{1, 2, 3, \ldots, 19, 20\}$ exist, such that their product is divisible by $4$?
Answer: 795
Problem: Let $\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Assume that $B$ and $C$ lie on the same side of $l$, and that the distance from $B$ to $l$ is twice the distance from $C$ to $l$. Suppose also that the length $BA$ is twice that of $CA$. If $\triangle ABC$ has integer side lengths and is as small as possible, what is $AB^2+BC^2+CA^2$?
Answer: 61
Problem: Two distinct similar rhombi share a diagonal. The smaller rhombus has an area of $1$, and the larger rhombus has an area of $9$. Compute the side length of the larger rhombus.
Answer: 3
Problem: In rectangle $ABCD$ with center $O$, where $AB=10$ and $BC=8$, circle $\gamma$ is centered at $O$ and is tangent to both $\overline{AB}$ and $\overline{CD}$. Points $M$ and $N$ are located on $\overline{AD}$ and $\overline{BC}$, respectively. The segment $MN$ intersects $\gamma$ at two distinct points $P$ and $Q$, with $P$ between $M$ and $Q$. Given that the ratio $MP : PQ : QN = 3 : 5 : 2$, the length $MN$ can be expressed as $\sqrt{a} - \sqrt{b}$, where $a$ and $b$ are positive integers. Determine the value of $100a + b$.
Answer: 16808
Problem: Find the number of prime numbers $p$ between $100$ and $200$ for which the congruence equation $x^{11} + y^{16} \equiv 2013 \pmod{p}$ has a solution in integers $x$ and $y$.
Answer: 21
Problem: On a circle, $2018$ points are marked. Each of these points is labeled with an integer. Let each number be larger than the sum of the preceding two numbers in clockwise order. Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers.
Answer: 1008
Problem: Stacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First, she goes to the lottery, and her money is doubled. Then, she goes into the first shop and spends $1024$ dollars. After that, she alternates between playing the lottery, where her money is doubled (Stacy always wins), and going into a new shop, spending $1024$ each time. When she comes out of the last shop, she has no money left. What is the minimum possible value of $d$?
Answer: 1023
Problem: Let $a$ be a positive number. Find the number of integer solutions $x$ of the inequality $2 < xa < 3$, given that the inequality $1 < xa < 2$ has exactly 3 integer solutions. Consider all possible cases.
Answer: 3
Problem: A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.
Answer: 15
Problem: There is a square of checkered paper measuring $102 \times 102$ squares and a connected figure of unknown shape, consisting of 101 cells. What is the largest number of such figures that can be cut from this square with a guarantee? A figure made up of cells is called connected if any two of its cells can be connected by a chain of its cells in which any two adjacent cells have a common side.
Answer: 102
Problem: How many four-digit multiples of $8$ are greater than $2008$?
Answer: 998
Problem: You roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a + b$.
Answer: 706
Problem: Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$.
Answer: 17
Problem: If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$, then what is the value of $a+b+c$?
Answer: 16
Problem: Find the last three digits in the product $1 \cdot 3 \cdot 5 \cdot 7 \cdot \ldots \cdot 2009 \cdot 2011$.
Answer: 875
Problem: Determine the smallest positive integer $m$ such that $529^n + m \cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.
Answer: 1984
Problem: William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder when $M$ is divided by $1000$.
Answer: 173
Problem: Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} $, where $ \# A $ is the cardinality of $ A $.
Answer: 82
Problem: The area of the region in the $xy$-plane satisfying the inequality 
\[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\]
is $k\pi$, for some integer $k$. Find $k$.
Answer: 210
Problem: The integers from $1$ through $9$ inclusive are placed in the squares of a $3 \times 3$ grid. Each square contains a different integer. The product of the integers in the first and second rows are $60$ and $96$ respectively. Find the sum of the integers in the third row.
Answer: 17
Problem: Suppose $a, b$ are positive real numbers such that $a + a^2 = 1$ and $b^2 + b^4 = 1$. Compute $a^2 + b^2$.
Answer: 1
Problem: Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?
Answer: 273
Problem: There are three bags:
1. Bag A contains 3 green candies and 1 red candy.
2. Bag B contains 2 green candies and 2 red candies.
3. Bag C contains 1 green candy and 3 red candies.

A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. 

- If the first candy is green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag.  
- If the first candy is red, the child randomly selects a second candy from the same bag as the first candy.  

If the probability that the second candy is green is given by the fraction $\frac{m}{n}$ in lowest terms, find $m + n$.
Answer: 217
Problem: In the star-shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.
Answer: 14
Problem: Given that $x_{n+2}=\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\sum_{n=0}^\infty\dfrac{x_{3n}}{2^n}=\dfrac{p}{q}$ for some positive integers $p$, $q$ with $\text{GCD}(p,q)=1$. Find $p+q$.
Answer: 1685
Problem: Camilla drove $20$ miles in the city at a constant speed and $40$ miles in the country at a constant speed that was $20$ miles per hour greater than her speed in the city. Her entire trip took one hour. Find the number of minutes that Camilla drove in the country, rounded to the nearest minute.
Answer: 35
Problem: Determine the number of three-digit numbers such that:
- The number formed by the first two digits is prime.
- The number formed by the last two digits is prime.
Answer: 52
Problem: Let the sequence $\{a_n\}$ for $n \ge 0$ be defined as $a_0 = c$, and for $n \ge 0$,  
$$a_n = \frac{2a_{n-1}}{4a^2_{n-1} - 1}.$$  
Compute the sum of all values of $c$ such that $a_{2020}$ exists but $a_{2021}$ does not exist.
Answer: 0
Problem: Shrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After Shrek throws the balls, the probability that there is exactly one empty bin can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 173
Problem: Let $x$ be a real number such that $3 \sin^4 x - 2 \cos^6 x = -\frac{17}{25}$. Then, $3 \cos^4 x - 2 \sin^6 x = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $10m + n$.
Answer: 445
Problem: Let $m$ be the answer to this question. What is the value of $2m - 5$?
Answer: 5
Problem: Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?
Answer: 12
Problem: A student walks down a hallway with a row of closed lockers numbered from $1$ to $1024$. He opens locker No. $1$, skips the next one, opens the following one, and continues this pattern. Upon reaching the end of the row, he turns around and opens the first closed locker he encounters, skips the next closed one, and continues this pattern back to the start of the hallway. This process is repeated until all lockers are open. What is the number of the last locker he opens?
Answer: 1024
Problem: Real numbers $x$, $y$, and $z$ satisfy the following equality: 
\[4(x+y+z) = x^2 + y^2 + z^2\] 
Let $M$ be the maximum of $xy + yz + zx$, and let $m$ be the minimum of $xy + yz + zx$. Find $M + 10m$.
Answer: 28
Problem: The equation $ax^3 + bx^2 + cx + d = 0$ has three distinct solutions. How many distinct solutions does the following equation have:

$4(ax^3 + bx^2 + cx + d)(3ax + b) = (3ax^2 + 2bx + c)^2$?
Answer: 2
Problem: There are $n$ players in a round-robin ping-pong tournament, meaning every two players will play exactly one game. After some matches have been played, it is known that the total number of matches that have been played among any $n-2$ people is equal to $3^k$, where $k$ is a fixed integer. Find the sum of all possible values of $n$.
Answer: 9
Problem: How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1 \pmod{720}$?
Answer: 16
Problem: 12. In a rectangle $ABCD$ with $AB = 8$ and $BC = 20$, let $P$ be a point on $AD$ such that $\angle BPC = 90^\circ$. If $r_1, r_2, r_3$ are the radii of the incircles of triangles $\triangle APB$, $\triangle BPC$, and $\triangle CPD$, what is the value of $r_1 + r_2 + r_3$?
Answer: 8
Problem: Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

$x\,\,\,\, z = 15$

$x\,\,\,\, y = 12$

$x\,\,\,\, x = 36$

If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x + 10y + z$.
Answer: 2037
Problem: A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$.
Answer: 769
Problem: A number is called *purple* if it can be expressed in the form $\frac{1}{2^a 5^b}$ for positive integers $a > b$. The sum of all purple numbers can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
Answer: 109
Problem: Let $n$ be a positive integer. Determine the smallest possible value of $1 - n + n^2 - n^3 + \dots + n^{1000}$.
Answer: 1
Problem: Let $x_1, x_2, \dots, x_n$ be a sequence of integers such that $-1 \leq x_i \leq 2$ for $i = 1, 2, \dots, n$. It is given that:
\[ x_1 + x_2 + \dots + x_n = 7 \]
and
\[ x_1^8 + x_2^8 + \dots + x_n^8 = 2009. \]
Let $m$ and $M$ be the minimal and maximal possible values of $x_1^9 + x_2^9 + \dots + x_n^9$, respectively. Find $\frac{M}{m}$. Round your answer to the nearest integer, if necessary.
Answer: 511
Problem: There is a row of $100$ cells, each containing a token. For $1$ dollar, it is allowed to interchange two neighboring tokens. Additionally, it is allowed to interchange, with no charge, any two tokens such that there are exactly $3$ tokens between them. What is the minimum price for arranging all the tokens in the reverse order?
Answer: 50
Problem: Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$ Find $n$ such that the first $n$ terms sum up to $2010$.
Answer: 1027