ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are considered the same.)
Answer: 333333
Problem: What is the minimum possible perimeter of a right triangle with integer side lengths whose perimeter is equal to its area?
Answer: 24
Problem: Let $ABC$ be a triangle such that $AB = AC$. Suppose the tangent to the circumcircle of $\triangle ABC$ at $B$ is perpendicular to $AC$. Find the measure of angle $\angle ABC$ in degrees.
Answer: 30
Problem: Compute
$$
2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111}
$$
where there are $2022$ instances of $2022$ in the tower. (Give the answer as an integer from $0$ to $110$).
Answer: 75
Problem: Suppose that $a$ and $b$ are positive real numbers such that:
\[
3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab).
\]
Find $101 - \sqrt[3]{a} - \sqrt[3]{b}$.
Answer: 0
Problem: For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\]
Answer: 499500
Problem: Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, for $i = 1, 2, \dots, m$. It is known that every two elements of $X$ are contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.
Answer: 50
Problem: Evaluate the expression:
\[2 \times (2 \times (2 \times (2 \times (2 \times (2 \times 2 - 2) - 2) - 2) - 2) - 2) - 2.\]
Answer: 2
Problem: \[\left(1 + \frac{1}{1+2^1}\right)\left(1+\frac{1}{1+2^2}\right)\left(1 + \frac{1}{1+2^3}\right)\cdots\left(1 + \frac{1}{1+2^{10}}\right)= \frac{m}{n},\] where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 3073
Problem: We have a positive integer $n$, whose sum of digits is 100. If the sum of digits of $44n$ is 800, then what is the sum of digits of $3n$?
Answer: 300
Problem: Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $\triangle ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $\triangle OMG$.
Answer: 15
Problem: Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$.
Answer: 9
Problem: Find the number of real numbers $x$ that satisfy the equation $(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1$.
Answer: 4
Problem: Let $G=10^{10^{100}}$ (also known as a googolplex). Then, \[ \log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G \] can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine the sum of the digits of $m+n$.
Answer: 18
Problem: Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14th jump lands at the origin?
Answer: 613470
Problem: Given two positive integers $m$ and $n$, we say that $m \mid\mid n$ if $m \mid n$ and $\gcd(m,\, n/m) = 1$. Compute the smallest integer greater than \[ \sum_{d \mid 2016} \sum_{m \mid\mid d} \frac{1}{m}. \]
Answer: 52
Problem: Line $\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are the orthogonal projections of points $B$ and $C$ onto line $\ell$, respectively. Given that $DE = 1$ and $2BD = CE$, determine the area of triangle $ABC$ expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the value of $m+n$.
Answer: 10
Problem: Given a triangle $ABC$ with an area of $1$. Let $A'$, $B'$, and $C'$ be the midpoints of the sides $[BC]$, $[CA]$, and $[AB]$ respectively. What is the minimal possible area of the common part of the two triangles $A'B'C'$ and $KLM$, if the points $K$, $L$, and $M$ are lying on the segments $[AB']$, $[CA']$, and $[BC']$ respectively?
Answer: 0
Problem: Compute the number of (not necessarily convex) polygons in the coordinate plane with the following properties:

- If the coordinates of a vertex are $(x,y)$, then $x,y$ are integers and $1\leq |x|+|y|\leq 3$.
- Every side of the polygon is parallel to either the x or y axis.
- The point $(0,0)$ is contained in the interior of the polygon.
Answer: 256
Problem: Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder?
Answer: 60
Problem: Triangle $\triangle ABC$ has $AB = 8$, $AC = 10$, and $AD = \sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars are drawn from $D$ to meet $AB$ and $AC$ at $E$ and $F$, respectively. The length of $EF$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.
Answer: 33
Problem: Given a natural number $n$, suppose that $N$ is the maximum number of elephants (bishops) that can be placed on a chessboard measuring $2 \times n$ such that no two elephants are mutually under attack. Determine the number of ways to place $N$ elephants on a $2 \times n$ chessboard so that no two elephants attack each other.

Alternative Formulation: Determine the number of ways to place $2015$ elephants on a chessboard measuring $2 \times 2015$ so that no two elephants attack each other.
Answer: 2
Problem: There is $1$ integer between $300$ and $400$ (inclusive) in base $10$ such that its last digit is $7$ when written in bases $8$, $10$, and $12$. Find this integer in base $10$.
Answer: 367
Problem: Evaluate \[2 \times (2 \times (2 \times (2 \times (2 \times (2 \times 2 - 2) - 2) - 2) - 2) - 2) - 2.\]
Answer: 2
Problem: For any positive integer $k$, define $f_1(k)$ as the square of the digital sum of $k$ in the decimal system. For $n > 1$, define $f_{n}(k)=f_1(f_{n-1}(k))$. Compute $f_{1992}(2^{1991})$.
Answer: 256
Problem: Points $E$ and $F$ lie inside rectangle $ABCD$ such that $AE = DE = BF = CF = EF$. Given $AB = 11$ and $BC = 8$, find the area of quadrilateral $AEFB$.
Answer: 32
Problem: How many integers $x$, from $10$ to $99$ inclusive, have the property that the remainder of $x^2$ divided by $100$ is equal to the square of the units digit of $x$?
Answer: 26
Problem: Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at their own roll and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma, and Fred each state the expected sum of the dice rolls based on the information they have. All six people say different numbers; in particular, Alice, Bob, Charlie, and Diana say $19$, $22$, $21$, and $23$, respectively. Compute the product of the dice rolls.
Answer: 1440
Problem: A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, it holds that $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
Answer: 2842
Problem: Determine the largest three-digit prime number that is a factor of \( \binom{2000}{1000} \).
Answer: 661
Problem: $A, B, C$, and $D$ are points on a circle, forming a convex quadrilateral $ABCD$. Given that $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, find the area of quadrilateral $ABCD$.
Answer: 504
Problem: Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$. Given $AB = 21$, $CD = 7$, $AD = 13$, and the area of trapezoid $[ABCD] = 168$. A line parallel to $AB$ through $P$ intersects the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at points $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.
Answer: 17
Problem: If $15$ and $9$ are the lengths of two medians of a triangle, what is the maximum possible area of the triangle, rounded to the nearest integer?
Answer: 90
Problem: What is the smallest positive integer $n$ such that $n = x^3 + y^3$ for two different positive integer tuples $(x, y)$?
Answer: 1729
Problem: The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day, they caught $3$ fewer rabbits and $2$ more squirrels than the day before. This pattern continued, with them catching $3$ fewer rabbits and $2$ more squirrels each day. Determine the total number of rabbits caught by the day when they caught more squirrels than rabbits.
Answer: 5491
Problem: In a round-robin chess tournament organized at a school, every two students played one match against each other. Determine the minimum possible number of students in the school given the following conditions:

- Each girl student has at least 21 wins in matches against boy students.
- Each boy student has at least 12 wins in matches against girl students.
Answer: 65
Problem: Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
Answer: 1944
Problem: Given the digits \( \{0, 1, 2, \ldots, 9\} \). Find the number of 6-digit numbers that contain the digit \(7\) and can be permuted. For example, 137456 and 314756 are considered the same number.
Answer: 2002
Problem: If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $|m - n|$?
Answer: 3
Problem: All possible $6$-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., $877550$), are written as a sequence in increasing order. Find the $2005^{\text{th}}$ number in this sequence.
Answer: 864100
Problem: For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$.
Answer: 15
Problem: Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$. Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$. If $BC'=30$, find the sum of all possible side lengths of $\triangle ABC$.
Answer: 130
Problem: A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?
Answer: 53
Problem: Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers. Let $f(n) = \prod_{i=1}^k e_i^{p_i}$. Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128$.
Answer: 35
Problem: How many integers $n > 1$ are there such that $n$ divides $x^{13} - x$ for every positive integer $x$?
Answer: 31
Problem: Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with a common difference of $\sqrt{2}$. Given that the product $abcd = 2021$, $d$ can be expressed as $\frac{m+\sqrt{n}}{\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers that are not divisible by the square of any prime. Find $m + n + p$.
Answer: 100
Problem: During a day, 2016 customers visited the store. Each customer visited the store only once (a customer enters the store, spends some time, and then leaves). Find the greatest integer $k$ such that the following statement is always true:

We can find $k$ customers such that either all of them have been at the store at the same time, or any two of them have not been at the store at the same time.
Answer: 45
Problem: For each positive integer $m$ and $n$, define the function $f(m, n)$ as follows:
- $f(1, 1) = 1$
- $f(m+1, n) = f(m, n) + m$
- $f(m, n+1) = f(m, n) - n$

Find the sum of all values of $p$ such that $f(p, q) = 2004$ for some $q$.
Answer: 3007
Problem: A subset $B$ of \( \{1, 2, \dots, 2017\} \) is said to have property \( T \) if any three elements of \( B \) are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property \( T \) may contain.
Answer: 1009
Problem: Meghana writes two (not necessarily distinct) prime numbers $q$ and $r$ in base 10 next to each other on a blackboard, resulting in the concatenation of $q$ and $r$. For example, if $q = 13$ and $r = 5$, the number on the blackboard is $135$. She notices that three more than the resulting number is the square of a prime $p$. Find all possible values of $p$.
Answer: 5
Problem: Consider the permutation of $1, 2, \ldots, n$, which we denote as $\{a_1, a_2, \ldots, a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions:

1. $a_1 = 1$
2. $|a_i - a_{i-1}| \le 2$, for $i = 2, 3, \ldots, n$

What is the residue when we divide $f(2015)$ by $4$?
Answer: 3
Problem: An ant is at a vertex of a cube. Every $10$ minutes, it moves to an adjacent vertex along an edge. If $N$ is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$.
Answer: 74
Problem: Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide on a specific time. Each of the four group members signs on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?
Answer: 48
Problem: A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the $x$-coordinate and $y$-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability $\frac{1}{4}$. The ant moves $3$ times and doesn't reach an anthill during this time. On average, how many additional moves will the ant need to reach an anthill? (Compute the expected number of additional moves needed.)
Answer: 3
Problem: Circle $\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\omega$ with radius $7$. Suppose that point $P$ lies on $\omega$ with $AP=4$. Determine the product of the distances from $P$ to the two intersections of $\omega$ and $\Gamma$.
Answer: 15
Problem: On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds?
Answer: 150
Problem: A sequence consists of the digits $122333444455555\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501^{\text{st}}$ and $4052^{\text{nd}}$ digits of this sequence.
Answer: 13
Problem: Evan has $66000$ omons, particles that can cluster into groups of a perfect square number of omons. An omon in a cluster of $n^2$ omons has a potential energy of $\frac{1}{n}$. Evan accurately computes the sum of the potential energies of all the omons. Compute the smallest possible value of his result.
Answer: 284
Problem: On a straight line $\ell$, there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed. Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn, which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.
Answer: 30
Problem: Find the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \dots, a_n$, the product of all differences $a_i - a_j$ for $i < j$ is divisible by $1991$.
Answer: 182
Problem: Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x, y)$ such that $1 \leq x, y \leq p$ and $x^3 - 3xy + y^3 + 1$ is divisible by $p$.
Answer: 30024
Problem: Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$.
Answer: 25
Problem: Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he have drawn from the box?
Answer: 990
Problem: Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF$. The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4}{5}$ the area of $ABC$. The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 7
Problem: How many ways are there to express $91$ as the sum of at least two consecutive positive integers?
Answer: 3
Problem: Define a common chord between two intersecting circles as the line segment connecting their two intersection points. Let $\omega_1, \omega_2, \omega_3$ be three circles with radii $3, 5,$ and $7$, respectively. Suppose they are arranged such that:

1. The common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$.
2. The common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$.
3. The common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$.

Compute the square of the area of the triangle formed by the centers of the three circles.
Answer: 96
Problem: Merlin summons the $n$ knights of Camelot for a conference. Each day, he assigns them to the $n$ seats at the Round Table. From the second day on, any two neighbors may interchange their seats if they were not neighbors on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?
Answer: 2
Problem: The positive integer $8833$ has the property that $8833 = 88^2 + 33^2$. Find the unique other four-digit positive integer $\overline{abcd}$ such that $\overline{abcd} = (\overline{ab})^2 + (\overline{cd})^2$.
Answer: 1233
Problem: Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruits is the banana!" The last fruit responds "No, one of the \emph{last} forty fruits is the banana!" The fruit in the middle yells "I'm the banana!" In how many positions could the banana be?
Answer: 21
Problem: Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.
Answer: 72
Problem: Three students $A$, $B$, and $C$ are traveling from a location on the National Highway No. $5$ towards Hanoi to participate in the HOMC 2018. Initially, $A$ takes $B$ on the motorcycle, while $C$ rides the bicycle. After one and a half hours, $B$ switches to a bicycle and immediately continues the trip to Hanoi, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on the motorcycle to Hanoi with $A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motorcycle is $50$ km per hour and that of both bicycles is $10$ km per hour. Find the distance from the starting point to Hanoi.
Answer: 100
Problem: A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
Answer: 203
Problem: How many natural numbers $n$ less than $2015$ are divisible by $\lfloor\sqrt[3]{n}\rfloor$?
Answer: 266
Problem: Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab} > \underline{bc} > \underline{ca}$?
Answer: 120
Problem: Marcus and four of his relatives are at a party. Each pair of the five people are either $\textit{friends}$ or $\textit{enemies}$. For any two enemies, there is no person that they are both friends with. In how many ways is this possible?
Answer: 52
Problem: A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?
Answer: 439
Problem: A car has a 4-digit integer price, displayed in a digital format (similar to digital numbers on a watch). While the salesman isn't looking, the buyer flips the price upside down and pays $1626$ less than the original price. Determine the initial cost of the car.
Answer: 6815
Problem: Find the smallest integer $k > 1$ for which $n^k - n$ is a multiple of $2010$ for every positive integer $n$.
Answer: 133
Problem: Find the maximum number of different integers that can be selected from the set $\{1, 2, \ldots, 2013\}$ such that no two integers have a difference of $17$.
Answer: 1010
Problem: A positive integer $n$ is considered lucky if the numbers $2n + 1$, $3n + 1$, and $4n + 1$ are all composite. Determine the smallest lucky number.
Answer: 16
Problem: Let $\alpha$ and $\beta$ be real numbers. Find the least possible value of:
\[
(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.
\]
Answer: 100
Problem: The number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by the two-digit numbers formed by both their first two digits and their last two digits?
Answer: 155
Problem: Let $x_1, x_2, \ldots, x_6$ be non-negative reals such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ and $x_1x_3x_5 + x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of the expression:

$$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2.$$ 

Find $p+q$.
Answer: 559
Problem: Find the area of the region in the coordinate plane satisfying the following conditions:

- $x \le 2y$
- $y \le 2x$
- $x + y \le 60$
Answer: 600
Problem: In each square of an $11 \times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.
Answer: 11
Problem: Suppose $x$ and $y$ are real numbers such that 
\[x^2+xy+y^2=2\] and \[x^2-y^2=\sqrt{5}.\] 
The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers, what is $100m+n$?
Answer: 803