sungyub/dapo-math-17k-verl huggingface dataset:
user: Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$.

Answer: 468
user: Let $f(N) = N \left( \frac{9}{10} \right)^N$, and let $\frac{m}{n}$ denote the maximum value of $f(N)$, as $N$ ranges over the positive integers. If $m$ and $n$ are relatively prime positive integers, find the remainder when $m + n$ is divided by $1000$.

Answer: 401
user: Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k, m \in \mathbb{N} = \{0, 1, 2, \ldots\}$.

Answer: 79
user: Find the sum of all integers $0 \le a \le 124$ such that $a^3 - 2$ is a multiple of $125$.

Answer: 53
user: Find the least possible value of $\frac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positive real numbers.

Answer: 8
user: I have $8$ unit cubes of different colors, which I want to glue together into a $2 \times 2 \times 2$ cube. How many distinct $2 \times 2 \times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

Answer: 1680
user: There are $2016$ customers who entered a shop on a particular day. Each customer entered the shop exactly once (i.e., the customer entered the shop, stayed there for some time, and then left the shop without returning).

Find the maximal $k$ such that the following condition holds:

There are $k$ customers such that either all of them were in the shop at a specific time instance, or no two of them were both in the shop at any time instance.

Answer: 45
user: Let $x$ be a number such that $x + \frac{1}{x} = -1$. Determine the value of $x^{1994} + \frac{1}{x^{1994}}$.

Answer: -1
user: Let $r_1, r_2, \ldots, r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute \( \sum_{i=1}^{47} r_i^{2020} \).

Answer: 0
user: To each pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:

$a*(b*c) = (a*b)*c$ for all $a, b, c$.
$a*a = 1$ for all $a$.

Solve the equation $x*36 = 216$.

Answer: 7776
user: While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the "blue screen of death" (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk.

He had computed three positive integers $a$, $b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.

Answer: 10
user: Let $\Gamma$ be a circle with center $A$, radius $1$, and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$, and diameter $DY$, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.

Answer: 16
user: Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition $\{A, B\}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number among these numbers. Find the sum of the digits of $N$.

Answer: 17
user: How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?

Answer: 1920
user: Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2 \parallel \ell_1 \parallel \ell_2$, $F_1$ lies on $\mathcal{P}_2$, and $F_2$ lies on $\mathcal{P}_1$. The two parabolas intersect at distinct points $A$ and $B$. Given that $F_1F_2=1$, the value of $AB^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.

Answer: 1504
user: Compute the sum of all positive integers $n$ for which the expression \[ \frac{n+7}{\sqrt{n-1}} \] is an integer.

Answer: 89
user: Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have?

Answer: 4374
user: Let $g(x) = 1 + 2x + 3x^2 + 4x^3 + \ldots$. Find the coefficient of $x^{2015}$ in $f(x) = \frac{g(x)}{1-x}$.

Answer: 2033136
user: Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$.

Answer: 1009
user: How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation:
\[
 a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2
\]

Answer: 408
user: In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$.

Answer: 25
user: In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$.

Answer: 8960
user: Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college, one-third of the freshman class drops out of school, 40 students in the sophomore class drop out of school, and one-tenth of the junior class drops out of school. Given that the college only admits new freshman students and wants to begin each school year with 3400 students enrolled, how many students does it need to admit into the freshman class each year?

Answer: 1185
user: Given that
\[\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!}\] 
for some positive integer $m$, find $m \pmod{71}$.

Answer: 12
user: On a $3 \times 3$ board, the numbers from $1$ to $9$ are written in some order without repeating. We say that the arrangement obtained is Isthmian if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.

Two arrangements are considered equal if one can be obtained from the other by rotating the board.

Answer: 720
user: 39 students participated in a math competition. The exam consisted of 6 problems, and each problem was worth 1 point for a correct solution and 0 points for an incorrect solution. For any 3 students, there is at most 1 problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the 39 students. Find the smallest possible value of $B$.

Answer: 165
user: Suppose $N$ is a $6$-digit number with base-$10$ representation $\underline{a}\,\underline{b}\,\underline{c}\,\underline{d}\,\underline{e}\,\underline{f}$. If $N$ is $\frac{6}{7}$ of the number with base-$10$ representation $\underline{d}\,\underline{e}\,\underline{f}\,\underline{a}\,\underline{b}\,\underline{c}$, find $N$.

Answer: 461538
user: Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$?

Answer: 25
user: Let $S$ be a subset of the natural numbers such that $0 \in S$, and for all $n \in \mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?

Answer: 47
user: Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)

Answer: 17
user: The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.

Answer: 15
user: For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$.

Answer: 40320
user: One hundred billion light years from Earth is planet Glorp. The inhabitants of Glorp are intelligent, uniform, amorphous beings with constant density which can modify their shape in any way, and reproduce by splitting. Suppose a Glorpian has somehow formed itself into a spinning cylinder in a frictionless environment. It then splits itself into two Glorpians of equal mass, which proceed to mold themselves into cylinders of the same height, but not the same radius, as the original Glorpian. If the new Glorpians' angular velocities after this are equal and the angular velocity of the original Glorpian is $\omega$, let the angular velocity of each of the new Glorpians be $\omega'$. Then, find $ \left( \frac {\omega'}{\omega} \right)^{10} $.

Answer: 1024
user: There is a rectangle $ABCD$ such that $AB = 12$ and $BC = 7$. Points $E$ and $F$ lie on sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Let $X$ be the intersection of lines $AF$ and $DE$. What is the area of pentagon $BCFXE$?

Answer: 47
user: How many ways are there to choose (not necessarily distinct) integers $a, b, c$ from the set $\{1, 2, 3, 4\}$ such that $a^{(b^c)}$ is divisible by $4$?

Answer: 28
user: Find $a + b + c$, where $a$, $b$, and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$.

Answer: 12
user: David, when submitting a problem for CMIMC, wrote his answer as $100\frac{x}{y}$, where $x$ and $y$ are two positive integers with $x < y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's. What is the smallest possible value of $x + y$?

Answer: 299
user: The sweet shop named "Olympiad" sells boxes containing either 6, 9, or 20 chocolates. Groups of students from a nearby school collect money to buy chocolates so that each student receives one. They buy a box and distribute the chocolates accordingly. For example, students can form groups of 15 (using one box of 6 and one box of 9 chocolates) or 38 (using two boxes of 9 and one box of 20 chocolates).

The shopkeeper has assured the students that he can fulfill the chocolate needs of any group. If a new box needs to be opened for any group size that doesn't fit perfectly (such as groups of 4, 7, or 10 students), then the shopkeeper will give all the chocolates for free to that group.

Is it possible to construct the largest group that receives free chocolates, and if so, how many students are there in this group?

Answer: 43
user: Let $a > b$ be positive integers. Compute the smallest possible integer value of \( \frac{a! + 1}{b! + 1} \).

Answer: 103
user: Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9.

For example, if \( n = 2024 \) and we choose \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 3 \), then \( m = 4177 \) is an alero of \( n \), but if we choose the digits \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 6 \), then we don't obtain an alero of \( n \), because \( 4 + 6 \) exceeds 9.

Find the smallest \( n \) which is a multiple of \( 2024 \) that has an alero which is also a multiple of \( 2024 \).

Answer: 10120
user: Consider a right-angled triangle $ABC$ with $\angle C = 90^\circ$. Suppose that the hypotenuse $AB$ is divided into four equal parts by the points $D, E, F$, such that $AD = DE = EF = FB$. If $CD^2 + CE^2 + CF^2 = 350$, find the length of $AB$.

Answer: 20
user: In a tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded 1 point, the loser got 0 points, and each player earned \(\frac{1}{2}\) point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. In particular, each of the ten lowest-scoring players earned half of their points against the other nine of the ten. What was the total number of players in the tournament?

Answer: 25
user: This sequence lists the perfect squares in increasing order: $0, 1, 4, 9, 16, \cdots, a, 10^8, b, \cdots$. Determine the value of $b-a$.

Answer: 40000
user: In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows: 

The first contestant takes $\frac{1}{n}$ of the cake.
The second contestant takes $\frac{2}{n}$ of the remaining cake.
The third contestant takes $\frac{3}{n}$ of what remains after the first and second contestants have taken their portions.
This process continues until the last contestant, who takes all of the remaining cake.

Determine which competitor takes the largest piece of cake.

Answer: 100
user: An ant starts at the origin of the Cartesian coordinate plane. Each minute, it moves randomly one unit in one of the directions: up, down, left, or right, with all four directions being equally likely. Its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

Answer: 3907
user: A 10-digit arrangement $0,1,2,3,4,5,6,7,8,9$ is called beautiful if:
When read from left to right, $0,1,2,3,4$ form an increasing sequence.
$5,6,7,8,9$ form a decreasing sequence.
$0$ is not the leftmost digit.

For example, $9807123654$ is a beautiful arrangement.

Determine the number of beautiful arrangements.

Answer: 126
user: How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a > b > c > d > e > f$ and $a + f = b + e = c + d = 30$ are simultaneously true?

Answer: 364
user: $A_1, A_2, \cdots, A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.

Answer: 9
user: In a country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by some air company. Find the minimal possible number of air companies required if direct flights between any three cities are operated by three different air companies.

Answer: 2015
user: A positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k > 1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. For example, 6 is beautiful since $6 = 3 \cdot 2 \cdot 1 = 3 + 2 + 1$, and this is unique. But 8 is not beautiful since $8 = 4 + 2 + 1 + 1 = 4 \cdot 2 \cdot 1 \cdot 1$ as well as $8 = 2 + 2 + 2 + 1 + 1 = 2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost. Find the largest beautiful number less than 100.

Answer: 95
user: In a drawer, there are at most $2009$ balls. Some of these balls are white, and the rest are blue, randomly distributed. If two balls are taken at the same time, the probability that both balls are either blue or both white is $\frac{1}{2}$. Determine the maximum number of white balls in the drawer such that the probability statement holds true.

Answer: 990
user: $f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$.

Answer: 424
user: In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that:
Contains more than two cities,
Does not repeat any city,
Ends where it begins,

It must include all types of airports. What is the maximum number of flights in this country?

Answer: 53
user: Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write and submit before he has written half of the total problems?

Answer: 65
user: Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a \cdot 21^b}$ is a positive integer.

Answer: 837
user: Compute the number of ordered triples \((x, y, z)\) of integers satisfying 
\[x^2 + y^2 + z^2 = 9.\]

Answer: 30
user: Evaluate $\lim_{n \to \infty} n^2 \int_{-\frac{1}{n}}^{\frac{1}{n}} (2005\sin x + 2006\cos x)|x| \, dx.$

Answer: 2006
user: How many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimeter less than 20?

Answer: 41
user: Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

Answer: 32
user: In triangle $ABC$, $AB = 52$, $BC = 34$, and $CA = 50$. The side $BC$ is divided into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. What is the smallest possible value of $n$?

Answer: 102
user: Find the numerator of \(\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\) when reduced.

Answer: 9091
user: Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.

Answer: 7114
user: Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$.

Answer: 4
user: Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac{1}{2}$, then compute $a^2 + b^2$.

Answer: 3
user: Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. For integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Words cannot cross between lines but may be repeated. Compute the number of distinct haikus Ted can make, and find the remainder when this number is divided by $100$.

Answer: 28
user: Which is the smallest natural number $n$ such that $n!$ has at least 2013 divisors?

Answer: 14
user: Amandine and Brennon play a turn-based game, with Amandine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked; if only $3$ has been selected so far, all numbers not divisible by three are eligible. A player loses immediately if they select the integer $1$. 

Call a number $n$ \textit{feminist} if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the \textit{feminist} numbers less than $40$.

Answer: 192
user: There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighboring numbers in the sequence is $0$, and that the sum of every $12$ neighboring numbers in the sequence is not zero. What is the maximal value of $n$?

Answer: 15
user: A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime, while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

Answer: 60
user: Let $ABCD$ be a square with $AB=6$. A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD$. What is the distance from $P$ to $A$?

Answer: 5
user: Let $\alpha ,\ \beta$ be the distinct positive roots of the equation $2x=\tan x$. Evaluate the following definite integral: 
\[ \int_{0}^{1}\sin(\alpha x)\sin(\beta x)\, dx \]

Answer: 0
user: The sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by the initial conditions $a_{0}=1$ and $a_{1}=-4$, with the recurrence relation $a_{n+2}=-4a_{n+1}-7a_{n}$ for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50} - a_{49}a_{51}$.

Answer: 51
user: Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?

Answer: 3
user: 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
user: Given a rectangle $ABCD$, where the segment $MA$ is perpendicular to the plane $ABC$. The lengths are as follows: $MB = 15$, $MC = 24$, and $MD = 20$. Determine the length of $MA$.

Answer: 7
user: For $a > 0$, find the minimum value of $\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.$

Answer: 4
user: Let $(x, y, z)$ be an ordered triplet of real numbers that satisfies the following system of equations: 
\[ \begin{align*} x + y^2 + z^4 &= 0, \\ y + z^2 + x^4 &= 0, \\ z + x^2 + y^4 &= 0. \end{align*} \] 
If $m$ is the minimum possible value of $\lfloor x^3 + y^3 + z^3 \rfloor$, find the modulo $2007$ residue of $m$.

Answer: 2006
user: Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \).

Answer: 105
user: A $20 \times 20 \times 20$ block is cut up into 8000 non-overlapping unit cubes, and a number is assigned to each cube. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Find the sum of all numbers of the cubes outside these slices.

Answer: 333
user: For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8$?

Answer: 1
user: A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find $\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3$.

Answer: 8
user: Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number $n$ such that $\sigma_1(n) = 1854$?

Answer: 1234
user: Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that
\[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\]
Find $a^3+b^3+c^3+d^3$.

Answer: 43