ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: The value of
\[
\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)
\]
can be written as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(2m - n\).
Answer: 1
Problem: Find all $x \in \mathbb{R}$ such that $x - \left\lfloor \frac{x}{2016} \right\rfloor = 2016$, where $\lfloor k \rfloor$ represents the largest integer less than or equal to $k$.
Answer: 2017
Problem: Sally's salary in 2006 was $\$37,500$. For 2007, she received a salary increase of $x$ percent. In 2008, she received another salary increase of $x$ percent. In 2009, she experienced a salary decrease of $2x$ percent, resulting in a salary of $\$34,825$.

Suppose instead, Sally had received a $2x$ percent salary decrease in 2007, followed by an $x$ percent salary increase in 2008, and another $x$ percent salary increase in 2009. What would her salary be in 2009 then?
Answer: 34825
Problem: Let $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is located on the segment $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at point $F$, find the area of quadrilateral $BFPC$.
Answer: 62
Problem: Alex writes down some distinct integers on a blackboard. For each pair of integers, he writes the positive difference of those on a piece of paper. Find the sum of all $n \leq 2022$ such that it is possible for the numbers on the paper to contain only the positive integers between $1$ and $n$, inclusive, exactly once.
Answer: 10
Problem: Starting with a positive integer $M$ written on the board, Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x + 2$. Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x + 27$. Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M + N$.
Answer: 10
Problem: Compute the \( \textit{number} \) of ordered quadruples \((w,x,y,z)\) of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\[
\begin{align*}
    wxyz &= 1 \\
    wxy^2 + wx^2z + w^2yz + xyz^2 &= 2 \\
    wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
    w^2xy + x^2yz + wy^2z + wxz^2 &= -1
\end{align*}
\]
Answer: 16
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: Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$, $BC = 21$, and $CA = 17$. Points $D$ and $E$ lie on side $BC$ such that $BD = 4$, $DE = 6$, and $EC = 11$. The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$, and line $IP$ meets the altitude from $A$ to $BC$ at $X$. Find $(DX \cdot EX)^2$.
Answer: 85
Problem: Find $$\sum_{i=1}^{100} i \cdot \gcd(i, 100).$$
Answer: 31000
Problem: Compute the largest prime factor of $111^2 + 11^3 + 1^1$.
Answer: 41
Problem: A jeweler can purchase different gold alloys: one that is $40\%$ gold for $200$ dollars per ounce, another that is $60\%$ gold for $300$ dollars per ounce, and a third that is $90\%$ gold for $400$ dollars per ounce. The jeweler plans to melt and mix these alloys to create a new alloy that is $50\%$ gold. Determine the minimum cost per ounce for creating this $50\%$ gold alloy.
Answer: 240
Problem: Given a prime $p$ and a positive integer $k$, an integer $n$ with $0 \le n < p$ is called a $(p, k)$-Hofstadterian residue if there exists an infinite sequence of integers $n_0, n_1, n_2, \ldots$ such that $n_0 \equiv n$ and $n_{i + 1}^k \equiv n_i \pmod{p}$ for all integers $i \ge 0$. If $f(p, k)$ is the number of $(p, k)$-Hofstadterian residues, then compute $\displaystyle \sum_{k = 1}^{2016} f(2017, k)$.
Answer: 1296144
Problem: A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an L-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible locations, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?
Answer: 10080
Problem: Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.

Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c} - \sqrt{d})$ for positive integers $a, b, c, d$, where $a, b$ are relatively prime, and the sum of $a, b, c, d$ is minimal.

Find $a + b + c + d$.
Answer: 23
Problem: The 40 unit squares of the $9 \times 9$ table are labeled. A horizontal or vertical row of 9 unit squares is considered "good" if it contains more labeled unit squares than unlabeled ones. How many "good" rows (both horizontal and vertical) can the table have in total?
Answer: 16
Problem: The positive reals $x$, $y$, $z$ satisfy the relations
\begin{align*}
x^2 + xy + y^2 &= 1 \\
y^2 + yz + z^2 &= 2 \\
z^2 + zx + x^2 &= 3.
\end{align*}
The value of $y^2$ can be expressed uniquely as $\frac{m-n\sqrt{p}}{q}$, where $m$, $n$, $p$, $q$ are positive integers such that $p$ is not divisible by the square of any prime and no prime dividing $q$ divides both $m$ and $n$. Compute $m+n+p+q$.
Answer: 23
Problem: Determine the smallest $n$ such that $n \equiv (a - 1) \pmod{a}$ for all $a \in \{2,3,..., 10\}$.
Answer: 2519
Problem: Two ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c,d)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.
Answer: 66
Problem: Determine all positive numbers $x$ such that \[ \frac{16}{x+2} + \frac{4}{x} + \frac{9}{x+1} + \frac{100}{x+8} = 19. \]
Answer: 2
Problem: A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways to select two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes forms a $45^\circ$ angle with the horizontal plane.
Answer: 60
Problem: A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We say a pair of players $A$ and $B$ is a dominant pair if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
Answer: 2015
Problem: Let $a_n$ be the product of the complex roots of $x^{2n} = 1$ that are in the first quadrant of the complex plane. That is, roots of the form $a + bi$ where $a, b > 0$. Let $r = a_1 \cdot a_2 \cdot \ldots \cdot a_{10}$. Find the smallest integer $k$ such that $r$ is a root of $x^k = 1$.
Answer: 1260
Problem: In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$.
Answer: 23
Problem: For a positive integer $n$, let $f(n)$ be equal to $n$ if there exists an integer $x$ such that $x^2 - n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when \( \sum_{n=0}^{2^{12}-1} f(n) \) is divided by $2^{12}$.
Answer: 0
Problem: Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every $i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No. $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series in one row on his shelf in such a way that there do not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition.
Answer: 2500
Problem: Let $ABC$ be a right triangle with $AB=3$, $BC=4$, and $\angle B = 90^\circ$. Points $P$, $Q$, and $R$ are chosen on segments $AB$, $BC$, and $CA$, respectively, such that $PQR$ is an equilateral triangle, and $BP=BQ$. Given that $BP$ can be written as $\frac{\sqrt{a}-b}{c}$, where $a,b,c$ are positive integers and $\gcd(b,c)=1$, what is $a+b+c$?
Answer: 451
Problem: John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains $k$ ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of flips it takes for John’s bottle to land on its bottom?
Answer: 6
Problem: The teacher secretly thought of a three-digit number $S$. Students $A$, $B$, $C$, and $D$ tried to guess the number by suggesting $541$, $837$, $291$, and $846$ respectively. The teacher informed them that each of them got exactly one digit correct, and it was in the correct position. What is the number $S$?
Answer: 247
Problem: Compute \[\left\lfloor \frac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
Answer: 2006
Problem: Let $ABC$ be a right triangle with a right angle at $A$ and side lengths $AC=8$ and $BC=16$. The lines tangent to the circumcircle of $\triangle ABC$ at points $A$ and $B$ intersect at $D$. Let $E$ be the point on side $\overline{AB}$ such that $\overline{AD} \parallel \overline{CE}$. Then $DE^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 451
Problem: Find the least three-digit number that is equal to the sum of its digits plus twice the product of its digits.
Answer: 397
Problem: Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} - \frac{2547^3}{2548\cdot 2549}$$.
Answer: 8
Problem: An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1 \leq x < y \leq 26$, there is exactly one passenger that goes from the $x$-th stop to the $y$-th one. If every passenger wants to take a seat during his journey, find the minimum number of seats that must be available on the train.
Answer: 169
Problem: Nineteen weights of mass $1$ gm, $2$ gm, $3$ gm, \ldots, $19$ gm are given. Nine are made of iron, nine are of bronze, and one is pure gold. It is known that the total mass of all the iron weights is $90$ gm more than the total mass of all the bronze ones. Find the mass of the gold weight.
Answer: 10
Problem: 2020 magicians are divided into groups of 2 for the Lexington Magic Tournament. After every 5 days, which is the duration of one match, teams are rearranged so no 2 people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n$?
Answer: 10095
Problem: Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
Answer: 182
Problem: Find the sum of all integers $n \geq 3$ such that the measure, in degrees, of an interior angle of a regular $n$-gon is an integer.
Answer: 1167
Problem: How many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? Here, $A$, $B$, $C$, and $D$ are digits, possibly equal. 

**Clarification:** $A$ and $D$ cannot be zero because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit integer.
Answer: 161
Problem: Let $p=101$. The sum \[\sum_{k=1}^{10}\frac{1}{\binom{p}{k}}\] can be written as a fraction of the form $\dfrac{a}{p!}$, where $a$ is a positive integer. Compute $a \pmod{p}$.
Answer: 5
Problem: Find the last digit of $7^1 \times 7^2 \times 7^3 \times \cdots \times 7^{2009} \times 7^{2010}$.
Answer: 3
Problem: In how many ways can Alice, Bob, Charlie, David, and Eve split $18$ marbles among themselves so that no two of them have the same number of marbles?
Answer: 2160
Problem: José has the following list of numbers: $100, 101, 102, \ldots, 118, 119, 120$. He calculates the sum of each pair of different numbers that can be formed from this list. How many different prime numbers can be obtained from these sums?
Answer: 6
Problem: Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots, 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Answer: 15
Problem: Evin and Jerry are playing a game with a pile of marbles. On each player's turn, they can remove $2$, $3$, $7$, or $8$ marbles. If they can't make a move because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the game?
Answer: 573
Problem: Assume the polynomial $p(x) = x^8 - 12x^6 + 49x^4 - 78x^2 + 42$ has no complex roots. How many negative roots does it have?
Answer: 4
Problem: A positive integer $N$ greater than $1$ is described as special if, in its base-$8$ and base-$9$ representations, both the leading and ending digit of $N$ are equal to $1$. What is the smallest special integer in decimal representation?
Answer: 793
Problem: In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} - d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
Answer: 31
Problem: Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10x - 22.$
Answer: 12
Problem: Determine for how many positive integers $n \in \{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[ n^2 - 1, \quad n^3 - 1, \quad n^4 - 1 \]
Answer: 31
Problem: How many positive integers $a$ less than $100$ satisfy that $4a^2 + 3a + 5$ is divisible by $6$?
Answer: 33
Problem: Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in \mathbb{N}$.
Answer: 16
Problem: Andrew has a four-digit number whose last digit is $2$. Given that this number is divisible by $9$, determine the number of possible values for this number that Andrew could have. Note that leading zeros are not allowed.
Answer: 100
Problem: Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition until you get a single-digit number. Find the number of positive integers $N \leq 1000$, such that the final single-digit number $n$ is equal to $5$. 

Example: $N = 563 \rightarrow (5 + 6 + 3) = 14 \rightarrow (1 + 4) = 5$ will be counted as one such integer.
Answer: 111
Problem: Let $a_0, a_1, \ldots$ be a sequence of positive integers such that $a_0 = 1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0, a_1, \ldots, a_{n-1}$. Compute $a_{10}$.
Answer: 841
Problem: Let $f(x)$ be a real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in the range $[0,5)$, find the number of zeros that $g(x)$ has in the range $[1,4)$.
Answer: 11
Problem: In a basketball tournament, every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points, and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?
Answer: 12
Problem: The four points $A(-1,2)$, $B(3,-4)$, $C(5,-6)$, and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points $P$ in the plane.
Answer: 23
Problem: Let $I, T, E, S$ be distinct positive integers such that the product $ITEST = 2006$. What is the largest possible value of the sum $I + T + E + S + T + 2006$?

\[ \textbf{(A) } 2086 \quad \textbf{(B) } 4012 \quad \textbf{(C) } 2144 \]
Answer: 2086
Problem: If $x$, $y$, and $z$ are distinct positive integers such that $x^2 + y^2 = z^3$, what is the smallest possible value of $x + y + z$?
Answer: 18
Problem: Evaluate the sum \[1^2 + 2^2 - 3^2 - 4^2 + 5^2 + 6^2 - 7^2 - 8^2 + \cdots - 1000^2 + 1001^2\]
Answer: 1001
Problem: At a certain grocery store, cookies may be bought in boxes of $10$ or $21$. What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?
Answer: 52
Problem: How many numbers greater than $1000$ but less than $10000$ have a product of their digits equal to $256$?
Answer: 31
Problem: The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42$, and another is a multiple of $72$. What is the minimum possible length of the third side?
Answer: 7
Problem: Determine the greatest common divisor (gcd) of all numbers of the form $p^8 - 1$, where $p$ is a prime number greater than 5.
Answer: 480
Problem: Let $f(x)$ be a polynomial of degree 3 with real coefficients, satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.
Answer: 72
Problem: What is the least positive integer $x$ for which the expression $x^2 + 3x + 9$ has 3 distinct prime divisors?
Answer: 27
Problem: The following diagram shows four adjacent $2 \times 2$ squares labeled $1, 2, 3$, and $4$. A line passing through the lower left vertex of square $1$ divides the combined areas of squares $1, 3$, and $4$ in half so that the shaded region has area $6$. The difference between the areas of the shaded region within square $4$ and the shaded region within square $1$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 49
Problem: The diagram below shows a circle with center $F$. The angles are related by the following equations:

- $\angle BFC = 2\angle AFB$
- $\angle CFD = 3\angle AFB$
- $\angle DFE = 4\angle AFB$
- $\angle EFA = 5\angle AFB$

Find the degree measure of $\angle BFC$.
Answer: 48
Problem: In each cell of a chessboard with $2$ rows and $2019$ columns, a real number is written such that:
1. There are no two numbers written in the first row that are equal to each other.
2. The numbers written in the second row coincide with (in some other order) the numbers written in the first row.
3. The two numbers written in each column are different, and they add up to a rational number.

Determine the maximum quantity of irrational numbers that can be in the chessboard.
Answer: 4032
Problem: In the convex quadrilateral $ABCD$, we have the following side lengths: $AB = 15$, $BC = 16$, $CD = 12$, $DA = 25$, and the diagonal $BD = 20$. Let $M$ be the circumcenter and $\gamma$ the circumcircle of $\triangle ABD$. Line $CB$ intersects $\gamma$ again at point $F$. Line $AF$ intersects $MC$ at point $G$, and line $GD$ intersects $\gamma$ again at point $E$. Determine the area of the pentagon $ABCDE$.
Answer: 396
Problem: Three builders are scheduled to build a house in 60 days. However, they procrastinate and do nothing for the first 50 days. To complete the house on time, they decide to hire more workers and work at twice their original speed. If the new workers also work at this doubled rate, how many new workers are needed? Assume each builder works at the same rate and does not interfere with the others.
Answer: 6
Problem: A square measuring $15 \times 15$ is partitioned into five rows of five congruent squares. The small squares are alternately colored black and white. Find the total area of the part colored black.
Answer: 117
Problem: Consider the set of all ordered $6$-tuples of nonnegative integers $(a, b, c, d, e, f)$ such that:
\[ a + 2b + 6c + 30d + 210e + 2310f = 2^{15}. \]
In the tuple with the property that $a + b + c + d + e + f$ is minimized, what is the value of $c$?
Answer: 1
Problem: There are $n$ integers $a$ such that $0 \leq a < 91$ and $a$ is a solution to the congruence $x^3 + 8x^2 - x + 83 \equiv 0 \pmod{91}$. What is $n$?
Answer: 6
Problem: Determine the greatest common divisor of the numbers:
\[ 5^5 - 5, \ 7^7 - 7, \ 9^9 - 9, \ \ldots, \ 2017^{2017} - 2017 \]
Answer: 24
Problem: Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ written on the board with $x > y$, and writes the number $x^2 - y^2$ on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of $N$.
Answer: 252
Problem: You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m + n$.
Answer: 1007
Problem: Real numbers $x, y > 1$ are chosen such that the three numbers \(\log_4 x\), \(2\log_x y\), \(\log_y 2\) form a geometric progression in that order. If \(x + y = 90\), then find the value of \(xy\).
Answer: 729
Problem: The integer $111111$ is the product of five prime numbers. Determine the sum of these prime numbers.
Answer: 71
Problem: Let us consider all rectangles with sides of length $a$ and $b$, both of which are whole numbers. Do more of these rectangles have a perimeter of $2000$ or a perimeter of $2002$?
Answer: 2002
Problem: There are $n \ge 3$ puddings in a room. If a pudding $A$ hates a pudding $B$, then $B$ hates $A$ as well. Suppose the following two conditions hold:

1. Given any four puddings, there are two puddings who like each other.
2. For any positive integer $m$, if there are $m$ puddings who like each other, then there exist 3 puddings (from the other $n-m$ puddings) that hate each other.

Find the smallest possible value of $n$.
Answer: 7
Problem: Let $ABCD$ be a rhombus with sides $AB = BC = CD = DA = 13$. Construct another rhombus $BAFE$ on the side $AB$ such that it is outside $ABCD$ and the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.
Answer: 120
Problem: Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.
Answer: 170