meetforever/GRPO_CL huggingface dataset:
user: In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.
Answer: 222
user: Determine all $n \geq 3$ for which there exist $n$ positive integers $a_1, a_2, \ldots, a_n$ such that any two of them have a common divisor greater than 1, but any three of them are coprime. Furthermore, assuming that the numbers $a_i$ are less than 5000, find the greatest possible $n$.
Answer: 4
user: Given that $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer, solve the following:

- $b \leq d$
- $c \geq a$
- $a \leq e$
- $b \geq e$
- $d \neq 5$

Determine the value of $a^b + c^d + e$.
Answer: 628
user: Find the remainder when $19^{92}$ is divided by 92.
Answer: 49
user: The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. 

Find the smallest possible number of independent deputies after the vacation.
Answer: 50
user: Navi and Ozna are playing a game where Ozna starts first and the two take turns making moves. A positive integer is written on the board. A move consists of either:

1. Subtracting any positive integer at most $2015$ from it, or
2. If the integer on the board is divisible by $2014$, dividing by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \geq 2015$, ensuring that all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.
Answer: 2013
user: Suppose $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB = 10$ and $CD = 15$, compute the maximum possible value of $XY$.
Answer: 6
user: In $\triangle ABC$, $\angle A = 30^{\circ}$, $BC = 13$. Given two circles $\gamma_1$ and $\gamma_2$ with radii $r_1$ and $r_2$, respectively, contain point $A$ and touch line $BC$ at points $B$ and $C$. Find the product $r_1r_2$.
Answer: 169
user: What is the least positive integer such that the product of its digits is $9!$?
Answer: 2578899
user: A week ago, Sandy's seasonal Little League batting average was $0.360$. After five more at bats this week, Sandy's batting average is now $0.400$. What is the smallest number of hits that Sandy could have had this season?
Answer: 12
user: Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$.
Answer: 630
user: Given a square $ABCD$. Let $P \in AB$, $Q \in BC$, $R \in CD$, $S \in DA$ such that $PR \parallel BC$ and $SQ \parallel AB$. Let $Z = PR \cap SQ$. If $BP = 7$, $BQ = 6$, and $DZ = 5$, find the side length of the square.
Answer: 10
user: Ann and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens.

Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least $N$ total tokens from the slot machine before Drew receives at least $M=2^{2018}$ total tokens, and Drew wins if he receives $M$ tokens before Ann receives $N$ tokens.

If each person has the same probability of winning, compute the remainder when $N$ is divided by $2018$.
Answer: 5
user: Construction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, with each project taking exactly one more day than the previous one to complete. The first project takes 3 days, the second takes 4 days, and so on.

Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
Answer: 51
user: Suppose $x, y \in \mathbb{Z}$ satisfy the equation:
\[
y^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 + 24).
\]
Find the sum of all possible values of $|xy|$.
Answer: 18
user: How many positive integers less than $200$ are relatively prime to either $15$ or $24$?
Answer: 120
user: Four people were guessing the number, $N$, of jellybeans in a jar. No two guesses were equally close to $N$. The closest guess was 80 jellybeans, the next closest guess was 60 jellybeans, followed by 49 jellybeans, and the furthest guess was 125 jellybeans. Find the sum of all possible values for $N$.
Answer: 1256
user: Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$, each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\frac{a\sqrt{b}-c}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$.
Answer: 46
user: On March $6$, $2002$, during the celebrations of the $500$th anniversary of the birth of mathematician Pedro Nunes, ten people entered the Viva bookstore for science. Each person bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. "The Adventures of Mathematics" by Pedro Nunes was one of the books that achieved the highest number of sales that morning. What is the smallest value this number could have taken?
Answer: 5
user: Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, \ldots). What is the largest integer $j$ such that $P$ is divisible by $101^j$?
Answer: 1980
user: How many sequences of $5$ positive integers $(a, b, c, d, e)$ satisfy the inequality $abcde \leq a + b + c + d + e \leq 10$?
Answer: 116
user: Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that \(\text{area} (\triangle ADE) = 16\), \(\text{area} (\triangle CEF) = 9\) and \(\text{area} (\triangle ABF) = 25\). What is the area of \(\triangle AEF\)?
Answer: 30
user: Natural number $M$ has 6 divisors such that the sum of them is equal to 3500. Find all values of $M$.
Answer: 1996
user: Let $\ell$ be a real number satisfying the equation $\frac{(1+\ell)^2}{1+\ell^2}=\frac{13}{37}$. Then
\[\frac{(1+\ell)^3}{1+\ell^3}=\frac{m}{n},\] where $m$ and $n$ are positive coprime integers. Find $m+n$.
Answer: 62
user: In a party of $2020$ people, some pairs of people are friends. We define a person's popularity as the size of the largest group of people in the party containing them such that every pair of people in that group are friends. A person has a popularity number of $1$ if they have no friends. What is the largest possible number of distinct popularities in the party?
Answer: 1010
user: Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\frac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
Answer: 446
user: In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.
Answer: 793
user: Let $\triangle ABC$ be a triangle with points $D$, $M$, and $N$ on sides $BC$, $AB$, and $AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. Given that the area of $\triangle BMD$ is $9\, \text{cm}^2$ and the area of $\triangle DNC$ is $25\, \text{cm}^2$, compute the area of $\triangle AMN$.
Answer: 15
user: Suppose that $x_1 + 1 = x_2 + 2 = x_3 + 3 = \cdots = x_{2008} + 2008 = x_1 + x_2 + x_3 + \cdots + x_{2008} + 2009$. 

Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \displaystyle\sum_{n=1}^{2008} x_n$.
Answer: 1005
user: Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \neq 0$. What is the four-digit number $\overline{abcd}$?
Answer: 9376
user: Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times does $111$ appear in this sequence?
Answer: 14
user: Let $S$ be the set of ordered pairs $(x, y)$ of positive integers for which $x + y \le 20$. Evaluate \[ \sum_{(x, y) \in S} (-1)^{x+y}xy. \]
Answer: 715
user: Let
\[ S = \sum_{i = 1}^{2012} i! \].
The tens and units digits of \( S \) (in decimal notation) are \( a \) and \( b \), respectively. Compute \( 10a + b \).
Answer: 13
user: What is the largest positive integer $n$ less than $10,000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$.
Answer: 4369
user: In the diagram, congruent rectangles $ABCD$ and $DEFG$ share a common vertex $D$. The sides $BC$ and $EF$ intersect at point $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$, find the area of the region $ABHED$.
Answer: 54
user: For which natural number $k$ does \( \frac{k^2}{1.001^k} \) attain its maximum value?
Answer: 2001
user: Find the number of ordered pairs \((a,b)\) such that \(a,b \in \{10,11,\ldots,29,30\}\) and \(\text{GCD}(a,b)+\text{LCM}(a,b)=a+b\).
Answer: 35
user: The sequence $a_0$, $a_1$, $a_2$, $\ldots$ satisfies the recurrence equation:
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
Answer: 181
user: An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?
Answer: 4
user: Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even.
Answer: 2017
user: How many ordered pairs of real numbers $(x, y)$ are there such that $x^2 + y^2 = 200$ and \[ \sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \] is an integer?
Answer: 12
user: Bored in an infinitely long class, Evan jots down a fraction whose numerator and denominator are both $70$-character strings, as follows:
\[ r = \frac{\text{loooloolloolloololllloloollollolllloollloloolooololooolololooooollllol}}
{\text{lolooloolollollolloooooloooloololloolllooollololoooollllooolollloloool}}. \]
If $o=2013$ and $l=\frac{1}{50}$, find $\lceil roll \rceil$.
Answer: 1
user: Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $(2016^{2016}, N)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016} - N$.
Answer: 10087
user: Compute the number of ordered pairs of non-negative integers \((x, y)\) which satisfy \(x^2 + y^2 = 32045.\)
Answer: 4
user: Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\triangle ABC$ out of paper and then folding $\triangle AFD$ up along $FD$, folding $\triangle BED$ up along $DE$, and folding $\triangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.
Answer: 80
user: Evaluate the integral: $$\int_0^5 \max(2x, x^2) \, dx$$
Answer: 43
user: Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers $\underline{a}\, \underline{b} \, \underline{c}$, if $\underline{a} \, \underline{b} \, \underline{c}$ is a multiple of $x$, then the three-digit (base 10) number $\underline{b} \, \underline{c} \, \underline{a}$ is also a multiple of $x$.
Answer: 64
user: Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying the following equations:
\[
\begin{align*}
&a + b - c - d &= 3, \\
&ab - 3bc + cd - 3da &= 4, \\
&3ab - bc + 3cd - da &= 5.
\end{align*}
\]
Find the value of $11(a - c)^2 + 17(b - d)^2$.
Answer: 63
user: Find the smallest positive integer $n$ such that $n^2 + 4$ has at least four distinct prime factors.
Answer: 94
user: A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For every distinct pair of lights $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 < x_2$ and $y_1 > y_2$, all lights strictly between them (i.e., $x_1 < x < x_2$ and $y_2 < y < y_1$) are toggled. After this procedure is done, how many lights are on?
Answer: 1022121
user: The **Collatz function** is defined as: 
$$C(n) = \begin{cases} 
3n - 1 & \text{if } n \text{ is odd}, \\
\frac{n}{2} & \text{if } n \text{ is even}.
\end{cases}$$

We obtain the **Collatz sequence** of a number by repeatedly applying the Collatz function to that number. For example, the Collatz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. 

Find the sum of the three smallest positive integers $n$ whose Collatz sequences do not contain $1$, or in other words, do not **collatzse**.
Answer: 21
user: Let $ABC$ be a triangle, and let $M$ be the midpoint of side $AB$. If $AB$ is $17$ units long and $CM$ is $8$ units long, find the maximum possible value of the area of $\triangle ABC$.
Answer: 68
user: Let $S=\left\{ \frac{p}{q} \mid q \leq 2009, \frac{p}{q} < \frac{1257}{2009}, p,q \in \mathbb{N} \right\}$. If the maximum element of $S$ is $\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$.
Answer: 595
user: Alexander has chosen a natural number $N > 1$ and has written down, in increasing order, all its positive divisors $d_1 < d_2 < \ldots < d_s$ (where $d_1 = 1$ and $d_s = N$). For each pair of neighboring numbers, he has found their greatest common divisor. The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N - 2$. Find all possible values of $N$.
Answer: 3
user: Which positive integers satisfy that the sum of the number's last three digits added to the number itself yields $2029$?
Answer: 2019
user: Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle FPQ$.
Answer: 210
user: A circle is inscribed in the square $ABCD$, where each side measures $10$ cm. The circle intersects sides $BC$ and $AD$ at points $M$ and $N$, respectively. Let $I$ be the intersection point of line $AM$ with the circle, distinct from $M$. Point $P$ is the orthogonal projection of $I$ onto line $MN$. Determine the length of segment $PI$.
Answer: 4
user: You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Suppose that $F(8) = \frac{43}{256}$, and $F(10)$ can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.
Answer: 1195
user: Consider the sequence of six real numbers: 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? 

Note: The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.
Answer: 135
user: Let $x_1, x_2, x_3$ be the solutions to the equation $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of
\[
\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].
\]
Answer: 12033
user: For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2 + 4px - p + 1$ an integer?
Answer: 4
user: The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
Answer: 21
user: In a certain country, there are $2023$ islands and $2022$ bridges. Each bridge connects two different islands, and any two islands share at most one bridge. It is possible to travel from any island to any other island using these bridges. If, among any three islands, there exists an island connected by bridges to each of the other two islands, these three islands form an "island group." It is known that any two "island groups" share at least one common island. What is the minimum number of islands with only one bridge connected to it?
Answer: 1011
user: There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats such that neither little boy sits next to either little girl. Determine the number of different ways these six people can be seated.
Answer: 176
user: Let two positive integers $x, y$ satisfy the condition $44 \mid (x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$.
Answer: 21296
user: Michael is jogging along the beach when he observes a group of girls playing a game around an ellipse drawn in the sand. The ellipse has two foci, represented by two softball bats that are $24$ feet apart. Two children stand at the points where the ellipse intersects the line containing the bats, and these points are $40$ feet apart. Five other children stand at various points on the ellipse.

The objective of the game is for each child to run from her starting point on the ellipse to one bat, then to the other bat, and finally return to her starting point. The child who completes this circuit in the shortest distance wins.

Michael calculates the shortest possible distance a child could run from her starting point on the ellipse, touching both bats and returning to her starting point. If Michael's calculations are correct, what is the shortest distance one of the girls could run during the game?
Answer: 64
user: A triangle $ABC$ is given in which $\angle BAC = 40^\circ$ and $\angle ABC = 20^\circ$. Find the length of the angle bisector drawn from the vertex $C$, given that the sides $AB$ and $BC$ differ by $4$ centimeters.
Answer: 4
user: The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns:

1. A digit that is repeated 3 or more times in a row. For example, $0001$ and $5555$ are not PINs, but $0010$ is a PIN.
2. A pair of digits that is duplicated. For example, $1212$ and $6363$ are not PINs, but $1221$ and $6633$ are PINs.

How many distinct possible PINs are there?
Answer: 9720
user: Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
Answer: 31
user: The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
Answer: 683
user: Alice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants to book two flights: one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways can she plan her flight itinerary?
Answer: 45
user: Including the original, how many ways are there to rearrange the letters in "PRINCETON" so that no two vowels \((I, E, O)\) are consecutive and no three consonants \((P, R, N, C, T, N)\) are consecutive?
Answer: 17280
user: Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, \ldots, r_{10}$, and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i} + \frac{1}{r_j} - \frac{1}{r_i r_j}$ where $i < j$ and $i, j \in \{1, \ldots, 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $\frac{a}{b}$ for relatively prime integers $a, b$. Find $a + b$.
Answer: 19
user: The number of positive integer pairs \((a,b)\) such that \(a\) divides \(b\) and \(b\) divides \(2013^{2014}\) can be expressed as \(2013n+k\), where \(n\) and \(k\) are integers with \(0 \leq k < 2013\). What is \(k\)? Recall \(2013=3\cdot 11\cdot 61\).
Answer: 27
user: Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BPD$?
Answer: 60
user: Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?
Answer: 200
user: Let $f(n)=\left|\binom{n}{0}+\binom{n}{3}+\dots +\binom{n}{3\left[\frac{n}{3}\right]}-\frac{2^n}{3}\right|$, where $[x]$ is the greatest integer not exceeding $x$. Find $f(1)+f(2)+\dots+f(2021)$.
Answer: 898
user: Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose $$(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.$$ Compute the product of all possible values of $a$.
Answer: 2009
user: Let $P_1, P_2, \dots, P_n$ be points on the plane. There is an edge between distinct points $P_x, P_y$ if and only if $x \mid y$. Find the largest $n$, such that the graph can be drawn with no crossing edges.
Answer: 15
user: Calculate the single-digit result obtained from the repeated sum of the digits of the number $2^{2004}$. Follow these steps:
1. Calculate the sum $S$ of all digits of $2^{2004}$.
2. Calculate the sum of the digits of $S$ to obtain $R$.
3. Continue calculating the sum of the digits until a single-digit number is reached.

For example, for $2^7 = 128$, we have:
- $S = 1 + 2 + 8 = 11$
- $R = 1 + 1 = 2$
Thus, the single-digit result for $2^7$ is $2$.
Answer: 1
user: Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups.
Answer: 16