sungyub/dapo-math-17k-verl huggingface dataset:
user: On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds?

Answer: 150
user: A sequence consists of the digits $122333444455555\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501^{\text{st}}$ and $4052^{\text{nd}}$ digits of this sequence.

Answer: 13
user: Evan has $66000$ omons, particles that can cluster into groups of a perfect square number of omons. An omon in a cluster of $n^2$ omons has a potential energy of $\frac{1}{n}$. Evan accurately computes the sum of the potential energies of all the omons. Compute the smallest possible value of his result.

Answer: 284
user: 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
user: Find the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \dots, a_n$, the product of all differences $a_i - a_j$ for $i < j$ is divisible by $1991$.

Answer: 182
user: Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x, y)$ such that $1 \leq x, y \leq p$ and $x^3 - 3xy + y^3 + 1$ is divisible by $p$.

Answer: 30024
user: Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$.

Answer: 25
user: Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he have drawn from the box?

Answer: 990
user: Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF$. The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4}{5}$ the area of $ABC$. The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Answer: 7
user: How many ways are there to express $91$ as the sum of at least two consecutive positive integers?

Answer: 3
user: Define a common chord between two intersecting circles as the line segment connecting their two intersection points. Let $\omega_1, \omega_2, \omega_3$ be three circles with radii $3, 5,$ and $7$, respectively. Suppose they are arranged such that:

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

Compute the square of the area of the triangle formed by the centers of the three circles.

Answer: 96
user: Merlin summons the $n$ knights of Camelot for a conference. Each day, he assigns them to the $n$ seats at the Round Table. From the second day on, any two neighbors may interchange their seats if they were not neighbors on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?

Answer: 2
user: The positive integer $8833$ has the property that $8833 = 88^2 + 33^2$. Find the unique other four-digit positive integer $\overline{abcd}$ such that $\overline{abcd} = (\overline{ab})^2 + (\overline{cd})^2$.

Answer: 1233
user: Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruits is the banana!" The last fruit responds "No, one of the \emph{last} forty fruits is the banana!" The fruit in the middle yells "I'm the banana!" In how many positions could the banana be?

Answer: 21
user: Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.

Answer: 72
user: Three students $A$, $B$, and $C$ are traveling from a location on the National Highway No. $5$ towards Hanoi to participate in the HOMC 2018. Initially, $A$ takes $B$ on the motorcycle, while $C$ rides the bicycle. After one and a half hours, $B$ switches to a bicycle and immediately continues the trip to Hanoi, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on the motorcycle to Hanoi with $A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motorcycle is $50$ km per hour and that of both bicycles is $10$ km per hour. Find the distance from the starting point to Hanoi.

Answer: 100
user: A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.

Answer: 203
user: How many natural numbers $n$ less than $2015$ are divisible by $\lfloor\sqrt[3]{n}\rfloor$?

Answer: 266
user: Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab} > \underline{bc} > \underline{ca}$?

Answer: 120
user: Marcus and four of his relatives are at a party. Each pair of the five people are either $\textit{friends}$ or $\textit{enemies}$. For any two enemies, there is no person that they are both friends with. In how many ways is this possible?

Answer: 52
user: A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?

Answer: 439
user: A car has a 4-digit integer price, displayed in a digital format (similar to digital numbers on a watch). While the salesman isn't looking, the buyer flips the price upside down and pays $1626$ less than the original price. Determine the initial cost of the car.

Answer: 6815
user: Find the smallest integer $k > 1$ for which $n^k - n$ is a multiple of $2010$ for every positive integer $n$.

Answer: 133
user: Find the maximum number of different integers that can be selected from the set $\{1, 2, \ldots, 2013\}$ such that no two integers have a difference of $17$.

Answer: 1010
user: A positive integer $n$ is considered lucky if the numbers $2n + 1$, $3n + 1$, and $4n + 1$ are all composite. Determine the smallest lucky number.

Answer: 16
user: Let $\alpha$ and $\beta$ be real numbers. Find the least possible value of:
\[
(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.
\]

Answer: 100
user: The number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by the two-digit numbers formed by both their first two digits and their last two digits?

Answer: 155
user: Let $x_1, x_2, \ldots, x_6$ be non-negative reals such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ and $x_1x_3x_5 + x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of the expression:

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

Find $p+q$.

Answer: 559
user: Find the area of the region in the coordinate plane satisfying the following conditions:

- $x \le 2y$
- $y \le 2x$
- $x + y \le 60$

Answer: 600
user: In each square of an $11 \times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

Answer: 11
user: Suppose $x$ and $y$ are real numbers such that 
\[x^2+xy+y^2=2\] and \[x^2-y^2=\sqrt{5}.\] 
The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers, what is $100m+n$?

Answer: 803
user: A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We define a pair of players $A$ and $B$ as a \textit{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
user: Find the largest prime factor of \( \frac{1}{\sum_{n=1}^\infty \frac{2012}{n(n+1)(n+2)\cdots(n+2012)}} \).

Answer: 2011
user: If $xy:yz:zx = 6:8:12$, and the ratio $\frac{x^3+y^3+z^3}{xyz}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Answer: 41
user: Calvin makes a number. He starts with $1$, and on each move, he multiplies his current number by $3$, then adds $5$. After $10$ moves, find the sum of the digits (in base $10$) when Calvin's resulting number is expressed in base $9$.

Answer: 21
user: Consider all cubic polynomials $f(x)$ such that:
$f(2018) = 2018$,
The graph of $f$ intersects the $y$-axis at height 2018,
The coefficients of $f$ sum to 2018,
$f(2019) > 2018$.

We define the infimum of a set $S$ as follows: Let $L$ be the set of lower bounds of $S$. That is, $\ell \in L$ if and only if for all $s \in S$, $\ell \leq s$. Then the infimum of $S$ is $\max(L)$.

Of all such $f(x)$, what is the infimum of the leading coefficient (the coefficient of the $x^3$ term)?

Answer: 0
user: 2021 people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up 1000 positions to the left of their original position.

Answer: 1021000
user: Let $p$ be a prime and $a$ and $n$ be natural numbers such that \( \frac{p^a - 1}{p-1} = 2^n \). Find the number of natural divisors of $na$.

Answer: 4
user: Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$.

Answer: 287
user: Determine \[ \left\lfloor \prod_{n=2}^{2022} \frac{2n+2}{2n+1} \right\rfloor, \] given that the answer is relatively prime to 2022.

Answer: 29
user: Let $n$ be the largest number such that 
\[ \frac{2014^{100!} - 2011^{100!}}{3^n} \]
is still an integer. Compute the remainder when $3^n$ is divided by $1000$.

Answer: 83
user: We call a number greater than $25$ a \textit{semi-prime} if it is the sum of two different prime numbers. What is the greatest number of consecutive natural numbers that can be \textit{semi-prime}?

Answer: 5
user: Find the smallest prime number that cannot be written in the form $|2^a - 3^b|$ with non-negative integers $a, b$.

Answer: 41
user: Let $x_0, x_1, x_2, \ldots$ be the sequence defined by: 
\[x_i = 2^i \text{ if } 0 \leq i \leq 2003\] 
\[x_i = \sum_{j=1}^{2004} x_{i-j} \text{ if } i \geq 2004\] 
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.

Answer: 2003
user: A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the midpoint of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ is the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$?

Answer: 18
user: Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

Answer: 18
user: Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?

Answer: 150
user: Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that 
\[ 20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2. \]
If 
\[ \tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019, \]
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

Answer: 201925
user: Triangle $ABC$ has $AB = 4$, $BC = 5$, and $CA = 6$. Points $A'$, $B'$, $C'$ are such that $B'C'$ is tangent to the circumcircle of $\triangle ABC$ at $A$, $C'A'$ is tangent to the circumcircle at $B$, and $A'B'$ is tangent to the circumcircle at $C$. Find the length $B'C'$.

Answer: 5
user: Let $x, y, z > 0$ such that \((x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\frac{91}{10}\). Compute \(\left[(x^3+y^3+z^3)\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)\right]\) where \([.]\) represents the integer part.

Answer: 9
user: An alien script has $n$ letters $b_1, b_2, \dots, b_n$. For some $k < \frac{n}{2}$, assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. A $k$-word is considered **sacred** if:

No letter appears twice, and
If a letter $b_i$ appears in the word, then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$).

For example, if $n = 7$ and $k = 3$, then $b_1b_3b_6$, $b_3b_1b_6$, $b_2b_4b_6$ are sacred 3-words. On the other hand, $b_1b_7b_4$, $b_2b_2b_6$ are not sacred.

What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.

Answer: 600
user: If there is only \(1\) complex solution to the equation \(8x^3 + 12x^2 + kx + 1 = 0\), what is \(k\)?

Answer: 6
user: Let $\triangle ABC$ be an isosceles triangle with $AB = AC$ and $BC = 14$. This triangle is inscribed in a circle $\omega$. Let $D$ be a point on the ray $BC$ such that $CD = 6$. The intersection of line $AD$ with the circle $\omega$ is point $E$. Given that $AE = 7$, find $AC^2$.

Answer: 105
user: Compute the number of nonempty subsets $S$ of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max S + \min S}{2}$ is an element of $S$.

Answer: 234
user: How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points?

Answer: 36
user: Compute the sum of $x^2 + y^2$ over all four ordered pairs $(x, y)$ of real numbers satisfying $x = y^2 - 20$ and $y = x^2 + x - 21$.

Answer: 164
user: A circle of radius 1 has four circles $\omega_1, \omega_2, \omega_3$, and $\omega_4$ of equal radius internally tangent to it, such that $\omega_1$ is tangent to $\omega_2$, which is tangent to $\omega_3$, which is tangent to $\omega_4$, which is tangent to $\omega_1$. The radius of the circle externally tangent to $\omega_1, \omega_2, \omega_3$, and $\omega_4$ is $r$. If $r = a - \sqrt{b}$ for positive integers $a$ and $b$, compute $a + b$.

Answer: 11
user: A set $S$ of positive integers is \textit{sum-complete} if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.

Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$.

Answer: 223
user: Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as \(\frac{m}{n}\) for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Answer: 12
user: Determine the value of the series $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \ldots + 10000$, where the signs change after each perfect square.

Answer: 1000000
user: We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relatively prime. Find the maximum value of $n$.

Answer: 25
user: A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. 

Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

Answer: 7
user: In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$?

$$
\begin{array}{ccccccc}
A & B & C & D & E & D & B \\
& B & C & D & E & D & B \\
&& C & D & E & D & B \\
&&& D & E & D & B \\
&&&& E & D & B \\
&&&&& D & B \\
+&&&&&& B \\
\hline
A & A & A & A & A & A & A
\end{array}
$$

Answer: 84269
user: Find the largest positive integer $n$ such that $n$ is 167 times the number of its positive divisors.

Answer: 2004
user: Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value of $n$.

Answer: 50308
user: Find the smallest perfect square whose last digit is not $0$, such that when its last two digits are removed, the resulting number is also a perfect square.

Answer: 121
user: Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the Primle. For each guess:

- A digit is highlighted blue if it is in the Primle but not in the correct place.
- A digit is highlighted orange if it is in the Primle and is in the correct place.
- A digit is left unhighlighted if it is not in the Primle.

If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the Primle?

$$
\begin{array}{c} 
 \boxed{1} \quad \boxed{3} \\
 \boxed{4} \quad \fcolorbox{black}{blue}{\color{white}7}
\end{array}
$$

Answer: 79
user: An $8 \times 8 \times 8$ cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?

Answer: 56
user: Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_5)$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, allowing Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$ and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes:
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]
Out of all $\binom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?

Answer: 125
user: A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division?

Answer: 2892
user: Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} = \frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.

Answer: 14
user: Find $n$ such that $\frac{1}{2!9!} + \frac{1}{3!8!} + \frac{1}{4!7!} + \frac{1}{5!6!} = \frac{n}{10!}$.

Answer: 92
user: A total fixed amount of $N$ thousand rupees is given to $A$, $B$, and $C$ every year, each receiving an amount proportional to their age. In the first year, $A$ received half of the total amount. When the sixth payment was made, $A$ received six-sevenths of the amount she received in the first year; $B$ received 1000 Rs less than she received in the first year, and $C$ received twice the amount she received in the first year. Find $N$.

Answer: 35
user: Square $ABCD$ has side length $7$. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. Similarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares.

Answer: 98
user: Let $x$ and $y$ be two real numbers such that \( 2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7 \). Find \( \tan^2 x + 2 \tan^2 y \).

Answer: 9
user: How many integer values of $k$, with $1 \leq k \leq 70$, are such that $x^{k}-1 \equiv 0 \pmod{71}$ has at least $\sqrt{k}$ solutions?

Answer: 21
user: $M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC$. The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1$. On the line passing through $C$ and parallel to $MD$, there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150^\circ$. Find the $\angle MTD$.

Answer: 120
user: A good approximation of $\pi$ is $3.14.$ Find the least positive integer $d$ such that if the area of a circle with diameter $d$ is calculated using the approximation $3.14,$ the error will exceed $1.$

Answer: 51
user: Triangle $ABC$ has side lengths $AB=20$, $AC=14$, and $BC=22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let \( \frac{p}{q} = \frac{[AMPN]}{[ABC]} \) for positive integers \( p \) and \( q \) that are coprime. Note that \([ABC]\) denotes the area of triangle $ABC$. Find \( p+q \).

Answer: 331
user: Evaluate the sum of all positive irreducible fractions less than $1$ with the denominator $1991$.

Answer: 900
user: A positive integer $n$ is called \textit{un-two} if there does not exist an ordered triple of integers $(a,b,c)$ such that exactly two of \[ \frac{7a+b}{n}, \frac{7b+c}{n}, \frac{7c+a}{n} \] are integers. Find the sum of all un-two positive integers.

Answer: 660
user: Let $N_6$ be the answer to problem 6.

Given positive integers $n$ and $a$, the $n$th tetration of $a$ is defined as:
\[ ^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}. \]

For example, $^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536$.

Compute the units digit of $^{2024}N_6$.

Answer: 6