ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Consider a $1$-indexed array that initially contains the integers $1$ to $10$ in increasing order.

The following action is performed repeatedly (any number of times):

\[\text{def action():} \]
\[\text{Choose an integer } n \text{ between } 1 \text{ and } 10 \text{ inclusive} \]
\[\text{Reverse the array between indices } 1 \text{ and } n \text{ inclusive} \]
\[\text{Reverse the array between indices } n+1 \text{ and } 10 \text{ inclusive (If } n = 10, \text{ we do nothing)} \]

How many possible orders can the array have after we are done with this process?
Answer: 20
Problem: Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.
Answer: 217
Problem: Andrew and Beatrice practice their free throws in basketball. One day, they attempted a total of $105$ free throws between them, with each person taking at least one free throw. If Andrew made exactly $\frac{1}{3}$ of his free throw attempts and Beatrice made exactly $\frac{3}{5}$ of her free throw attempts, what is the highest number of successful free throws they could have made between them?
Answer: 59
Problem: Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? 

(The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)
Answer: 40
Problem: $F(0) = 3$ and $F(n) = F(n-1) + 4$ when $n$ is positive. Find $F(F(F(5)))$.
Answer: 383
Problem: Let \( \{x\} \) denote the fractional part of \( x \). For example, \( \{5.5\} = 0.5 \). Find the smallest prime \( p \) such that the inequality \[ \sum_{n=1}^{p^2}\left\{\frac{n^p}{p^2}\right\} > 2016 \] holds.
Answer: 67
Problem: Define the minimum real $C$ such that for any real numbers $0 = a_0 < a_1 < \dots < a_{1000}$, the inequality $$\min_{0 \le k \le 1000} (a_k^2 + (1000-k)^2) \le C(a_1 + \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor$.
Answer: 127
Problem: There are $2022$ equally spaced points on a circular track $\gamma$ with a circumference of $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.

Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labelings of the $2022$ points.
Answer: 2042222
Problem: A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?
Answer: 4
Problem: The diagram below shows an equilateral triangle and a square with a side length of $2$ joined along an edge. What is the area of the shaded triangle?

```latex
\begin{asy}
fill((2,0) -- (2,2) -- (1, 2 + \sqrt{3}) -- cycle, gray);
draw((0,0) -- (2,0) -- (2,2) -- (1, 2 + \sqrt{3}) -- (0,2) -- (0,0));
draw((0,2) -- (2,2));
\end{asy}
```
Answer: 1
Problem: How many six-letter words formed from the letters of "AMC" do not contain the substring "AMC"? For example, "AMAMMC" has this property, but "AAMCCC" does not.
Answer: 622
Problem: The sides and vertices of a pentagon are labeled with the numbers $1$ through $10$. Each side of the pentagon must have the same sum of numbers. What is the smallest possible value of this sum?
Answer: 14
Problem: The fraction $\frac{1}{2015}$ has a unique "(restricted) partial fraction decomposition" of the form \[\frac{1}{2015} = \frac{a}{5} + \frac{b}{13} + \frac{c}{31},\] where $a$, $b$, and $c$ are integers with $0 \leq a < 5$ and $0 \leq b < 13$. Find $a + b$.
Answer: 14
Problem: For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2 y}$. The sum of the solutions to the equation  
\[4096f(f(x, x), x) = x^{13}\]  
can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.
Answer: 177
Problem: Two circles have radii of 5 and 26, respectively. The smaller circle passes through the center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?
Answer: 4
Problem: Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$.
Answer: 7294927
Problem: On a large wooden block, there are four twelve-hour analog clocks of varying accuracy. At 7 PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7 PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.)
Answer: 12
Problem: For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree. For each sapling, she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. Once a sapling is turned into a perfectly-shaped apple tree, it will remain so. 

The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Answer: 789953
Problem: Let $a$, $b$, $c$, $d$, $e$, $f$, and $g$ be seven distinct positive integers not bigger than 7. Find all primes that can be expressed as $abcd + efg$.
Answer: 179
Problem: Find the sum of the four least positive integers each of whose digits add to $12$.
Answer: 210
Problem: A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2 - 4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
Answer: 6900
Problem: Find all integer solutions of the equation:

\[
\left\lfloor \frac{x}{1!} \right\rfloor + \left\lfloor \frac{x}{2!} \right\rfloor + \cdots + \left\lfloor \frac{x}{10!} \right\rfloor = 1001.
\]
Answer: 584
Problem: Let $ABCD$ be a rhombus with sides $AB = BC = CD = DA = 13$. Construct another rhombus $BAFE$ on the side $AB$ outside $ABCD$, such that 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 $x$ and $y$ be positive real numbers. Define $a = 1 + \frac{x}{y}$ and $b = 1 + \frac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.
Answer: 50
Problem: A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 11
Problem: The positive integer $m$ is a multiple of $111$, and the positive integer $n$ is a multiple of $31$. Their sum is $2017$. Find $n - m$.
Answer: 463
Problem: Consider a cyclic quadrilateral with side lengths $1$, $4$, $8$, and $7$ in that order. What is its circumdiameter? Let the answer be in the form $a\sqrt{b} + c$, where $b$ is squarefree. Find the value of $a + b + c$.
Answer: 66
Problem: Let $f(x) = x^3 + ax + b$, where $a \neq b$. Suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. Find $f(1)$.
Answer: 1
Problem: On a natural number $n$, you are allowed two operations: 
1. Multiply $n$ by $2$, or 
2. Subtract $3$ from $n$. 

For example, starting with $8$, you can reach $13$ as follows: $8 \longrightarrow 16 \longrightarrow 13$. You need two steps, and you cannot do it in less than two steps. 

Starting from $11$, what is the least number of steps required to reach $121$?
Answer: 10
Problem: Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \frac{1}{2}| + |y - \frac{1}{2}| < \frac{1}{2}$ is $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$
Answer: 3
Problem: Let $f(x) = x^2 + x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y) = f(z) = m + \sqrt{n}$ and $f\left(\frac{1}{y}\right) + f\left(\frac{1}{z}\right) = \frac{1}{10}$. Compute $100m + n$.
Answer: 1735
Problem: Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\triangle MNP$ can be written in simplest form as $\frac{a}{b}$. Find $a + b$.
Answer: 457
Problem: Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e., it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
Answer: 70
Problem: Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.
Answer: 1481480
Problem: Find the remainder when $2020!$ is divided by $2020^2$.
Answer: 0
Problem: Let $H$ be a regular hexagon with area 360. Three distinct vertices $X$, $Y$, and $Z$ are picked randomly, with all possible triples of distinct vertices equally likely. Let $A$, $B$, and $C$ be the unpicked vertices. What is the expected value (average value) of the area of the intersection of $\triangle ABC$ and $\triangle XYZ$?
Answer: 24
Problem: What is the sum of all positive integer values of $n$ that satisfy the equation:
\[
  \cos \left( \frac{\pi}{n} \right) \cos \left( \frac{2\pi}{n} \right) \cos \left( \frac{4\pi}{n} \right) \cos \left( \frac{8\pi}{n} \right) \cos \left( \frac{16\pi}{n} \right) = \frac{1}{32}?
\]
Answer: 47
Problem: Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630,000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd(c, d) = 70$ and $\operatorname{lcm}(c, d) = 630,000$ is $\frac{m}{n}$ for some relatively prime integers $m$ and $n$, compute $100m + n$.
Answer: 106
Problem: Let $a, b, c, d$ be distinct digits such that the product of the two-digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$. Find all possible values of $a + b + c + d$.
Answer: 21
Problem: In a room, there are 2019 aligned switches, each connected to a light bulb. Initially, all the light bulbs are switched on. Then, 2019 people enter the room one by one and perform the following operation: 

- The first person toggles every switch.
- The second person toggles every second switch.
- The third person toggles every third switch.
- This continues until the 2019th person toggles every 2019th switch.

Determine how many light bulbs remain switched on after all 2019 people have entered the room and completed their operations.
Answer: 1975
Problem: A football tournament is played between 5 teams, each pair playing exactly one match. Points are awarded as follows:
- 5 points for a victory
- 0 points for a loss
- 1 point each for a draw with no goals scored
- 2 points each for a draw with goals scored

In the final ranking, the five teams have points that are 5 consecutive numbers. Determine the least number of goals that could be scored in the tournament.
Answer: 6
Problem: Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are at a distance of $2$ apart such that $AC < AD$. We can write $\tan \angle BOD = \frac{a}{b}$, where $a$ and $b$ are positive integers such that $\gcd(a, b) = 1$. Compute $a + b$.
Answer: 113
Problem: Given triangle $ABC$ and a point $P$ inside it, $\angle BAP=18^\circ$, $\angle CAP=30^\circ$, $\angle ACP=48^\circ$, and $AP=BC$. If $\angle BCP=x^\circ$, find $x$.
Answer: 6
Problem: Let $P_i(x) = x^2 + b_i x + c_i$, where $i = 1, 2, \ldots, n$, be pairwise distinct polynomials of degree $2$ with real coefficients. It is given that for any $0 \le i < j \le n$, $i, j \in \mathbb{N}$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.
Answer: 3
Problem: Let $I$ be the incenter of a triangle $\triangle ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $\triangle ABC$ at $D \neq C$. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $\triangle AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.
Answer: 13
Problem: Jon wrote the $n$ smallest perfect squares on one sheet of paper, and the $n$ smallest triangular numbers on another sheet (note that $0$ is both square and triangular). Jon notices that there are the same number of triangular numbers on the first paper as there are squares on the second paper. However, if $n$ had been one smaller, this would not have been true. If $n < 2008$, let $m$ be the greatest number Jon could have written on either paper. Find the remainder when $m$ is divided by $2008$.
Answer: 89
Problem: How many multiples of $2005$ are factors of $(2005)^2$?
Answer: 4
Problem: A quadruple $(a, b, c, d)$ of distinct integers is said to be $\text{balanced}$ if $a + c = b + d$. Let $\mathcal{S}$ be any set of quadruples $(a, b, c, d)$ where $1 \leq a < b < d < c \leq 20$ and where the cardinality of $\mathcal{S}$ is $4411$. Find the least number of balanced quadruples in $\mathcal{S}$.
Answer: 91
Problem: Let $S(n)$ be the sum of the squares of the positive integers less than and coprime to $n$. For example, $S(5) = 1^2 + 2^2 + 3^2 + 4^2$, but $S(4) = 1^2 + 3^2$. Let $p = 2^7 - 1 = 127$ and $q = 2^5 - 1 = 31$ be primes. The quantity $S(pq)$ can be written in the form \( \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) \) where $a$, $b$, and $c$ are positive integers, with $b$ and $c$ coprime and $b < c$. Find $a$.
Answer: 7561
Problem: In baseball, a player's batting average is calculated as the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $0.399$. What is the fewest number of at bats that Danielle could have?
Answer: 138
Problem: Given that for reals $a_1, a_2, \ldots, a_{2004}$, the equation

$$x^{2006} - 2006x^{2005} + a_{2004}x^{2004} + \cdots + a_2x^2 + a_1x + 1 = 0$$

has 2006 positive real solutions, find the maximum possible value of $a_1$.
Answer: -2006
Problem: Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is \( \frac{7}{2} \) times the original one.
Answer: 153846
Problem: Find the smallest prime number that does not divide \(9 + 9^2 + 9^3 + \cdots + 9^{2010}\).
Answer: 17
Problem: Determine the smallest prime number that does not divide any five-digit number whose digits are in strictly increasing order.
Answer: 11
Problem: Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.
Answer: 1500
Problem: Find the maximum possible value of $8 \cdot 27^{\log_6 x} + 27 \cdot 8^{\log_6 x} - x^3$ as $x$ varies over the positive real numbers.
Answer: 216
Problem: Evaluate \( \int_0^{2008} \left(3x^2 - 8028x + 2007^2 + \frac{1}{2008}\right)\, dx \).
Answer: 2009
Problem: For $n \geq 1$, let $a_n$ be the number beginning with $n$ 9's followed by 744; for example, $a_4 = 9999744$. Define \[ f(n) = \max \{ m \in \mathbb{N} \mid 2^m \text{ divides } a_n \}, \] for $n \geq 1$. Find $f(1) + f(2) + f(3) + \cdots + f(10)$.
Answer: 75
Problem: The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Find the positive integer.
Answer: 89
Problem: A crazy physicist has discovered a new particle called an omon. He has a machine that takes two omons of mass $a$ and $b$ and entangles them. This process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac{1}{2}(a+b)$. The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$. What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
Answer: 9
Problem: Observing the temperatures recorded in Cesenatico during December and January, Stefano noticed an interesting coincidence: for each day in this period, the low temperature is equal to the sum of the low temperatures of the preceding day and the succeeding day.

Given that the low temperatures on December $3$ and January $31$ were $5^\circ \text{C}$ and $2^\circ \text{C}$ respectively, find the low temperature on December $25$.
Answer: -3
Problem: 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
Problem: $ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$. $F$ lies on $AD$ such that $EF$ is perpendicular to $FD$. Given $EF = 2$ and $FD = 1$. Find the area of the rhombus $ABCD$.
Answer: 20
Problem: According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatched pairs of shoes? Note that left and right shoes are distinct, and Moor must always wear one of each.
Answer: 2018
Problem: In the diagram below, the larger square has a side length of $6$. Find the area of the smaller square. 

![Diagram](https://cdn.artofproblemsolving.com/attachments/f/3/f4479ee5fa3ee0998a8e0e9ca0e1ef8bc04807.jpg)
Answer: 18
Problem: Let $\frac{a}{b}$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a + b$ is divided by $2007$.
Answer: 79
Problem: The positive integer $m$ is a multiple of $101$, and the positive integer $n$ is a multiple of $63$. Their sum is $2018$. Find $m - n$.
Answer: 2
Problem: Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC$. Suppose that $BC = 27$, $CD = 25$, and $AP = 10$. If $MP = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
Answer: 2705
Problem: We glue together $990$ one-inch cubes into a $9 \times 10 \times 11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?
Answer: 382
Problem: Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1, z_2$ with $|z_1| = |z_2| = 1$ and $\{z_1, z_2\} \neq \{-i, i\}$ such that:
\[
z_1(z_1^3 + z_1^2 + rz_1 + 1) = z_2(z_2^3 + z_2^2 + rz_2 + 1).
\]
There exist real numbers $a, b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r \le b$. If the value of $|a| + |b|$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $100p + q$.
Answer: 2504
Problem: Suppose that a planet contains $(CCAMATHBONANZA_{71})^{100}$ people ($100$ in decimal), where in base $71$ the digits $A, B, C, \ldots, Z$ represent the decimal numbers $10, 11, 12, \ldots, 35$, respectively. Suppose that one person on this planet is snapping, and each time they snap, at least half of the current population disappears. Estimate the largest number of times that this person can snap without disappearing. An estimate of $E$ earns $2^{1-\frac{1}{200}\left|A-E\right|}$ points, where $A$ is the actual answer.
Answer: 8355
Problem: A positive integer with at least two digits written in base $10$ is called 'ascending' if its digits increase in value from left to right. For example, $123$ is 'ascending', but $132$ and $122$ are not. How many 'ascending' numbers are there?
Answer: 502
Problem: Find the number of integers $n$ for which $n^2 + 10n < 2008$.
Answer: 91
Problem: Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations.
Answer: 9720
Problem: What is the greatest number of balls of radius $\frac{1}{2}$ that can be placed within a rectangular box of size $10 \times 10 \times 1$?
Answer: 106
Problem: There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is, each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?
Answer: 48
Problem: Amanda has the list of even numbers $2, 4, 6, \dots, 100$ and Billy has the list of odd numbers $1, 3, 5, \dots, 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list?
Answer: 50
Problem: Find the least positive integer $N$ such that $N$ is 50 times the number of positive integer divisors of $N$.
Answer: 1800
Problem: In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD = DP = 3$, and $GC = 4$, find $AB^2$.
Answer: 168
Problem: One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle?
Answer: 1875
Problem: A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of $STUV$?
Answer: 36
Problem: The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0$. What is $a + b + c + d$?

$\textbf{(A)}\ 0 \ \textbf{(B)}\ 1 \ \textbf{(C)}\ 4 \ \textbf{(D)}\ 9 \ \textbf{(E)}\ 16$
Answer: 9
Problem: Six integers $a, b, c, d, e, f$ satisfy the following system of equations:
\[
\begin{cases}
ace + 3ebd - 3bcf + 3adf = 5 \\
bce + acf - ade + 3bdf = 2
\end{cases}
\]
Find all possible values of $abcde$.
Answer: 0
Problem: Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$.
Answer: 1023