meetforever/GRPO_CL huggingface dataset:
user: Let $S$ be the set of all natural numbers with the property: the sum of the largest three divisors of a number $n$, excluding $n$, is greater than $n$. Determine the largest natural number $k$ that divides every number in $S$.
Answer: 6
user: Determine the least positive integer $n$ for which the following statement is true: the product of any $n$ odd consecutive positive integers is divisible by $45$.
Answer: 6
user: We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is **kawaii** if \( T \) has the following properties:

1. \( T \) has at least two distinct elements;
2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \).

For example, the subset \( T = \{31, 71, 2024\}\) is **kawaii**, but \( T = \{5, 15, 75\}\) is not **kawaii** because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \).

What is the largest possible number of elements that a **kawaii** subset can have?
Answer: 675
user: The NIMO problem writers have invented a new chess piece called the *Oriented Knight*. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16 \times 16$ chess board?
Answer: 252
user: Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt{b}+c$, where $b$ is square-free. Find $a+b+c$.
Answer: 66
user: If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, find \( m + n \).
Answer: 97
user: An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the value of $n$?
Answer: 9
user: Triangle $ABC$ is isosceles with $AB = AC$. The bisectors of angles $\angle ABC$ and $\angle ACB$ meet at point $I$. If the measure of $\angle CIA$ is $130^\circ$, compute the measure of $\angle CAB$.
Answer: 20
user: Lines parallel to the sides of an equilateral triangle are drawn such that they divide each side into $10$ equal segments, resulting in the triangle being divided into $100$ congruent smaller triangles. Each of these $100$ triangles is referred to as a "cell." Additionally, lines parallel to each side of the original triangle are drawn through each vertex of the original triangle. The cells located between any two adjacent parallel lines form a "stripe." Determine the maximum number of cells that can be selected such that no two selected cells belong to the same stripe.
Answer: 10
user: A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 6425
user: The product 
$$\frac{1}{2} \cdot \frac{2}{4} \cdot \frac{3}{8} \cdot \frac{4}{16} \cdot \ldots \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}$$
is written in its most simplified form. What is the last digit of the denominator?
Answer: 2
user: Edward had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow. 

The guard fired an arrow with an initial velocity of $100 \frac{\text{ft}}{\text{s}}$. At the same time, Edward started running away with an acceleration of $1 \frac{\text{ft}}{\text{s}^2}$. Assuming that air resistance causes the arrow to decelerate at $1 \frac{\text{ft}}{\text{s}^2}$, and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\frac{\text{ft}}{\text{s}}$)?
Answer: 75
user: Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$.
Answer: 783
user: Say that a polynomial with real coefficients in two variables, $x, y$, is balanced if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
Answer: 2020050
user: If $x + y^{-99} = 3$ and $x + y = -2$, find the sum of all possible values of $x$.
Answer: -195
user: 480 one-centimeter unit cubes are used to build a block measuring 6 cm by 8 cm by 10 cm. A tiny ant then chews its way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that it does not "pass through" cubes if it is merely passing through where their edges or vertices meet.
Answer: 20
user: Let $\phi = \frac{1+\sqrt{5}}{2}$. A base-$\phi$ number $(a_n a_{n-1} \ldots a_1 a_0)_{\phi}$, where $0 \le a_n, a_{n-1}, \ldots, a_0 \le 1$ are integers, is defined by 
\[
(a_n a_{n-1} \ldots a_1 a_0)_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + \ldots + a_1 \cdot \phi^1 + a_0.
\]
Compute the number of base-$\phi$ numbers $(b_j b_{j-1} \ldots b_1 b_0)_\phi$ which satisfy $b_j \ne 0$ and
\[
(b_j b_{j-1} \ldots b_1 b_0)_\phi = \underbrace{(100 \ldots 100)_\phi}_{\text{Twenty} \ 100's}.
\]
Answer: 1048576
user: The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?
Answer: 9
user: Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Answer: 716
user: There are relatively prime positive integers $p$ and $q$ such that $\frac{p}{q} = \sum_{n=3}^{\infty} \frac{1}{n^5 - 5n^3 + 4n}$. Find $p+q$.
Answer: 97
user: Juan wrote a natural number. Maria then added a digit $1$ to the left and a digit $1$ to the right of Juan's number. As a result, Maria's number exceeds Juan's number by $14789$. Find the original number that Juan wrote.
Answer: 532
user: In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player had the same number of points. Determine the minimal total number of draws.
Answer: 4
user: Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}$.
Answer: 1983
user: A *form* is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, \ldots, m_n$ satisfy $m_1 \geq \ldots \geq m_n$. An *angle* in a *form* consists of a box $v$ and all the boxes to the right of $v$ and all the boxes above $v$. The size of a *form* of an *angle* is the number of boxes it contains. Find the maximum number of *angles* of size $11$ in a form of size $400$.
Answer: 36
user: Alex launches his boat into a river and heads upstream at a constant speed. At the same time, at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is flowing downstream at $2\frac{3}{10}$ miles per hour. Alex and Alice will meet at a point that is $\frac{m}{n}$ miles from Alex's starting point, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 52
user: Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
Answer: 35
user: If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
Answer: 27
user: How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$?
Answer: 99
user: For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$. If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{M}{N} \] for relatively prime positive integers $M$ and $N$, compute $100M+N$.
Answer: 611
user: Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9) > f(16)$. The probability that $f(1) > f(16) > f(25)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1, 2, \ldots, 100\}$.
Answer: 730
user: Siva has the following expression, which is missing operations:
$$\frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32}.$$ 

For each blank, he flips a fair coin: 
- If it comes up heads, he fills it with a plus.
- If it comes up tails, he fills it with a minus.

Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coin flips and operations. 

If the probability that the positive difference between his computed values is greater than $\frac{1}{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
Answer: 39
user: Determine the smallest positive integer $n$ such that there exist positive integers $x_1, x_2, \ldots, x_n$ so that each natural number from 1001 to 2021 inclusive can be expressed as the sum of one or more distinct terms $x_i$ (where $i = 1, 2, \ldots, n$).
Answer: 11
user: Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$.
Answer: 1923
user: We say that an integer $x \in \{1, \dots, 102\}$ is \textit{square-ish} if there exists some integer $n$ such that $x \equiv n^2 + n \pmod{103}$. Compute the product of all \textit{square-ish} integers modulo $103$.
Answer: 52
user: A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers, whereas $69$ and $157$ are not. 

Let $A$ and $B$ be $4$-digit palindrome numbers, and $C$ be a $3$-digit palindrome number. Given that $A - B = C$, what is the value of $C$?
Answer: 121
user: Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc = m$.
Answer: 11
user: The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120$. What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?
Answer: 142
user: How many ways are there to create two 3-digit numbers $m$ and $n$ such that $n = 3m$? Each of the digits $1$, $2$, $3$, $6$, $7$, and $8$ must be used exactly once.
Answer: 2
user: What is the integer closest to $\pi^{\pi}$?
Answer: 36
user: Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\frac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Answer: 123
user: A square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second quadrilateral.
Answer: 50
user: The largest prime factor of $199^4 + 4$ has four digits. Compute the second largest prime factor.
Answer: 181
user: How many permutations $p(n)$ of the set $\{1, 2, \dots, 35\}$ satisfy the condition that if $a \mid b$ (where $a$ divides $b$), then $p(a) \mid p(b)$?
Answer: 48
user: How many ways can you arrange $3$ Alice's, $1$ Bob, $3$ Chad's, and $1$ David in a line if the Alice's are all indistinguishable, the Chad's are all indistinguishable, and Bob and David want to be adjacent to each other? In other words, how many ways can you arrange $3$ A's, $1$ B, $3$ C's, and $1$ D in a row where the B and D are adjacent?
Answer: 280
user: Let $a, b, c, d$ be real numbers satisfying the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 2, \\
(a+c)(b+d) &= 3, \\
(a+d)(b+c) &= 4.
\end{align*}
\]
Find the minimum value of $a^2 + b^2 + c^2 + d^2$.
Answer: 7
user: Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
Answer: 13
user: There are $n$ players in a round-robin ping-pong tournament, meaning every two persons 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
user: Find all four-digit natural numbers $\overline{xyzw}$ such that the sum of the number and the sum of its digits equals $2003$.
Answer: 1978
user: Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only condition is that if the union of two different subsets $A$ and $B$ equals $M$, then $A$ and $B$ must be given different colors. What is the least possible value of $k$?
Answer: 2014
user: Let $\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a $1$. For example, $123$ is in $\mathcal{S}$ and $450$ is not. The probability that a randomly chosen $3$-digit positive integer is in $\mathcal{S}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 197
user: Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters that her garden will take up?
Answer: 40
user: Let $ABC$ be a triangle with angles $A$, $B$, and $C$. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1 \le \tan A + \tan B + \tan C \le 2015$, find the number of possible integer values for $\tan B$.
Answer: 11
user: Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?
Answer: 5052
user: In the parallelogram $ABCD$, we are given that $AC = 10$ and $BD = 28$. The points $K$ and $L$ are such that they move in the plane of $ABCD$ satisfying $AK = BD$ and $BL = AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. Determine the maximum value of the expression $\cot^2 \left(\frac{\angle BMD}{2}\right) + \tan^2 \left(\frac{\angle ANC}{2}\right)$.
Answer: 2
user: A circle is inscribed in a regular octagon with area $2024$. A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers and $c$ is square-free. Compute $a + b + c$.
Answer: 1520
user: Let \( a\uparrow\uparrow b = a^{a^{\cdots^{a}}} \), where there are \( b \) instances of \( a \). The operation \( a\uparrow\uparrow b \) is defined recursively as follows:  
\[  a\uparrow\uparrow b = \begin{cases} a & \text{if } b=1 \\ a^{a\uparrow\uparrow (b-1)} & \text{if } b\ge2 \end{cases} \]  
Determine the remainder of \( 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) \) when divided by \( 60 \).
Answer: 27
user: Call a date \( mm/dd/yy \) \textit{multiplicative} if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, \( 01/21/21 \), \( 03/07/21 \), and \( 07/03/21 \) are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
Answer: 29
user: In the plane, there are $2020$ points, some of which are black and the rest are green. For every black point, the following condition holds: there are exactly two green points that are at a distance of $2020$ from that black point. Find the smallest possible number of green dots.
Answer: 45
user: Farmer John has $5$ cows, $4$ pigs, and $7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other.
Answer: 100800
user: Eva draws an equilateral triangle and its altitudes. In the first step, she draws the center triangle of the equilateral triangle. In the second step, she draws the center triangle of this center triangle, and so on. After each step, Eva counts all triangles whose sides lie completely on the drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?
Answer: 7
user: There are $24$ participants who attended a meeting. Each pair of them either shook hands once or did not. A total of $216$ handshakes occurred during the meeting. For any two participants who have shaken hands, at most $10$ among the remaining $22$ participants have shaken hands with exactly one of these two persons. Define a *friend circle* as a group of $3$ participants where each person has shaken hands with the other two. Find the minimum possible number of friend circles.
Answer: 864
user: Find the value of $2(\sin 2^{\circ} \tan 1^{\circ} + \sin 4^{\circ} \tan 1^{\circ} + \cdots + \sin 178^{\circ} \tan 1^{\circ})$.
Answer: 2
user: A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.
Answer: 6
user: A sequence with the first term $a_0$ is defined such that $a_{n+1} = 2a_n^2 - 1$ for $n \geq 0$. Let $N$ denote the number of possible values of $a_0$ such that $a_0 = a_{2020}$. Find the number of factors of $N$.
Answer: 4
user: Ben flips a coin 10 times and records the absolute difference between the total number of heads and tails. He then flips the coin one more time and records the absolute difference again. If the probability that the second recorded number is greater than the first can be expressed as \( \frac{a}{b} \) for positive integers \( a, b \) with \( \gcd(a, b) = 1 \), find \( a + b \).
Answer: 831
user: Find the number of positive integers less than $2018$ that are divisible by $6$ but are not divisible by at least one of the numbers $4$ or $9$.
Answer: 280
user: In how many ways can you divide the set of eight numbers $\{2, 3, \ldots, 9\}$ into $4$ pairs such that no pair of numbers has a greatest common divisor (gcd) equal to $2$?
Answer: 36
user: Find $x$ such that 
\[2010^{\log_{10}x} = 11^{\log_{10}(1+3+5+\cdots +4019)}.\]
Answer: 121
user: Find the number of positive $6$-digit integers such that the sum of their digits is $9$, and four of its digits are $2, 0, 0, 4$.
Answer: 180
user: Let $n$ be a natural number. A cube of edge $n$ may be divided into 1996 smaller cubes, each having edge lengths that are also natural numbers. Find the minimum possible value for $n$.
Answer: 13
user: Let \((x_n)\) be a sequence of positive integers defined as follows: \(x_1\) is a fixed six-digit number and for any \(n \geq 1\), \(x_{n+1}\) is a prime divisor of \(x_n + 1\). Find \(x_{19} + x_{20}\).
Answer: 5
user: Find the sum of the real roots of the polynomial:
\[
\prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right).
\]
Answer: 330
user: Compute how many permutations of the numbers $1, 2, \dots, 8$ have no adjacent numbers that sum to $9$.
Answer: 13824
user: A nonempty word is called pronounceable if it alternates between vowels (A, E, I, O, U) and consonants (all other letters) and contains at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
Answer: 68
user: Let $B$, $M$, and $T$ be the three roots of the equation $x^3 + 20x^2 - 18x - 19 = 0$. What is the value of $|(B + 1)(M + 1)(T + 1)|$?
Answer: 18
user: What is the smallest perfect square that ends in \( 9009 \)?
Answer: 1503
user: On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, we have $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
Answer: 494
user: William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hour during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it takes William to get to school and back.
Answer: 31
user: How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
Answer: 120
user: At the beginning of a game, the numbers $1, 2, \ldots, 2004$ are written on a desk. A move consists of:

1. Selecting some numbers from those on the desk.
2. Calculating the remainder of the sum of these numbers when divided by $11$ and writing this remainder on the desk.
3. Deleting the selected numbers.

In this game, after several moves, only two numbers remained on the desk. One of them was $1000$. What was the other number?
Answer: 4
user: Triangle $ABC$ has side lengths $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with side lengths $AG$, $BG$, and $CG$?
Answer: 28