meetforever/GRPO_CL huggingface dataset:
user: Let $A$ be an even number but not divisible by $10$. What are the last two digits of $A^{20}$? Please provide the answer as a two-digit integer.
Answer: 76
user: Pentagon \(ABCDE\) is inscribed in a circle such that \(ACDE\) is a square with an area of \(12\). What is the largest possible area of pentagon \(ABCDE\)? The original answer is in the form of \(k + m\sqrt{2}\). Please provide the value of \(k + m\).
Answer: 12
user: The circumradius of an acute triangle \(ABC\) is twice the distance from its circumcenter to \(AB\). Given that \(|AC|=2\) and \(|BC|=3\), what is the altitude passing through \(C\)? The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please provide the value of \(k + m + n\).
Answer: 31
user: Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis.

Dennis then announces, "Aaron's number is at least three times Jerry's number."

Aaron says, "I still don't know Jerry's number."

Jerry then replies, "Oh, now I know Aaron's number."

What is the sum of their numbers?
Answer: 7
user: Five unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending from \((a,0)\) to \((3,3)\), divides the entire region into two regions of equal area. The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.
Answer: 5
user: LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid \(A\) dollars and Bernardo had paid \(B\) dollars, where \(A < B\). How many dollars must LeRoy give to Bernardo so that they share the costs equally? Provide your answer in the form of \(\frac{k(B-A)}{m}\), and then find the value of k + m.
Answer: 3
user: How many values of $\theta$ in the interval $0 < \theta \le 2\pi$ satisfy $1 - 3\sin\theta + 5\cos3\theta = 0$? Provide your answer as an integer.
Answer: 6
user: Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, find the length $B'C'$. The original answer is in the form $k+\sqrt{m(r^2-1)}$, please give the value of $k+m$.
Answer: 4
user: The length of a rectangle is \(5\) inches and its width is less than \(4\) inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is \(\sqrt{6}\), what is the width of the rectangle in the form \(\sqrt{a}\)? Please provide the value of \(a\).
Answer: 5
user: The average (arithmetic mean) age of a group consisting of doctors and lawyers is 40. If the doctors average 35 years old and the lawyers average 50 years old, what is the ratio of the number of doctors to the number of lawyers? Express your answer as a simplified ratio in the form $m: n$, and then calculate the value of $m + n$.
Answer: 3
user: If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$, then find the value of $1000 \times \left( x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} \right)$.
Answer: 51005
user: If the radius of a circle is increased $100\%$, by what percentage is the area increased?
Answer: 300
user: \(y\) varies inversely as the square of \(x\). When \(y=16\), \(x=1\). Find the value of \(y\) when \(x=8\). The original answer is in the form \(\frac{k}{m}\), please give the value of \(k + m\).
Answer: 5
user: A parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $2$ adjacent spaces. What is the probability that she is able to park? Express your answer in the form \(\frac{k}{m}\), where the fraction is simplified, and provide the value of \(k + m\).
Answer: 45
user: $ABCD$ is a cyclic quadrilateral. If $\angle B = \angle D$, $AC \cap BD = \{E\}$, $\angle BCD = 150^\circ$, $|BE| = x$, $|AC| = z$, then find $|ED|$ in terms of $x$ and $z$. The original answer is in the form \(\frac{kx + mz}{n}\), where n cannot be divisible by 2. Please provide the value of \(k + m + n\).
Answer: 1
user: The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. Find the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer.
Answer: 17
user: Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\widehat{ACB})=20^\circ$, then what is the value of $m(\widehat{BAD})$ in degrees?
Answer: 20
user: Call a positive integer *monotonous* if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. Find the total number of monotonous positive integers.
Answer: 1524
user: Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) can be expressed in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 31
user: If \(q_1(x)\) and \(r_1\) are the quotient and remainder, respectively, when the polynomial \(x^8\) is divided by \(x + \frac{1}{2}\), and if \(q_2(x)\) and \(r_2\) are the quotient and remainder, respectively, when \(q_1(x)\) is divided by \(x + \frac{1}{2}\), find the value of \(r_2\). The original answer is in the form \(-\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 17
user: The circle with radius $4$ passing through points $A$ and $B$ is tangent to another circle with radius $3$ passing through points $A$ and $C$, where the distance $|AB| = 2$. If the line $BC$ is tangent to the second circle, what is the value of $|BC|^2$?
Answer: 7
user: Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. How many distinct lines pass through at least two members of $S$?
Answer: 20
user: Find the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle. Express your answer in the form of a ratio $a: b$, and find the value of $a + b$.
Answer: 7
user: Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $AR+BR+CR+DR$.
Answer: 192
user: A circle centered at \(A\) with a radius of 1 and a circle centered at \(B\) with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is in the simplest fractional form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 13
user: Four coins are placed in a line. A passerby walks by and flips each coin, stopping if she ever obtains two adjacent heads. If the passerby manages to flip all four coins, find the number of possible head-tail combinations for her four flips.
Answer: 8
user: Let $a, b,$ and $c$ be positive integers with $a \geq b \geq c$ such that \[ \begin{align*} a^2 - b^2 - c^2 + ab &= 2011 \\ a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc &= -1997 \end{align*} \] What is the value of $a$?
Answer: 253
user: The roots of the equation \(2\sqrt{x} + 2x^{-\frac{1}{2}} = 5\) can be found by solving the quadratic equation \(kx^2 + mx + n = 0\). What are the values of \(k\), \(m\), and \(n\) in the equation? Please provide the sum of \(k\), \(m\), and \(n\).
Answer: -9
user: The circle $C_1$ with radius $6$ and the circle $C_2$ with radius $8$ are externally tangent to each other at $A$. The circle $C_3$, which is externally tangent to both $C_1$ and $C_2$, has a radius of $21$. The common tangent of $C_1$ and $C_2$ that passes through $A$ meets $C_3$ at points $B$ and $C$. The original answer is in the form $k\sqrt{m}$, please find the value of $k + m$.
Answer: 27
user: What is the area enclosed by the graph of $|3x| + |4y| = 12$?
Answer: 24
user: Integers \(a\), \(b\), \(c\), and \(d\), not necessarily distinct, are chosen independently and at random from \(0\) to \(2007\), inclusive. Find the probability that \(ad - bc\) is even. The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest terms. Please calculate the value of \(k + m\).
Answer: 13
user: A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$ correct answers, at least $2$ of any $40$ gave exactly $2$ correct answers, and at least $3$ of any $40$ gave exactly $1$ correct answer. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers?
Answer: 24
user: What is the greatest number of consecutive integers whose sum is $45$? Provide your answer as an integer.
Answer: 90
user: Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ such that the sum of the three interior angles at $A$ is $360^\circ$. Thus, the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?
Answer: 21
user: A regular polygon with $n$ sides is inscribed in a circle with radius $R$. The area of the polygon is $3R^2$. Determine the value of $n$.
Answer: 12
user: How many different sentences with two words can be written using all letters of the word $\text{YARI\c{S}MA}$?

(The Turkish word $\text{YARI\c{S}MA}$ means $\text{CONTEST}$. It will produce the same result.)

Find the total number of different sentences.
Answer: 15120
user: Let \(a_n\) denote the angle opposite to the side of length \(4n^2\) units in an integer right-angled triangle with side lengths \(4n^2\), \(4n^4+1\), and \(4n^4-1\), where \(n \in \mathbb{N}\). Find the value of:
\[
\lim_{p \to \infty} \sum_{n=1}^p a_n
\]
The original answer is in the format \(\frac{k\pi}{m}\). Please give the value of \(k + m\).
Answer: 3
user: To the nearest thousandth, $\log_{10}2$ is $0.301$ and $\log_{10}3$ is $0.477$. Find the best approximation of $\log_5 10$ in the form of a fraction \(\frac{k}{m}\). Please provide the value of k + m.
Answer: 17
user: How many pairs of positive integers \((a,b)\) with \(a \geq b\) are there such that \(a^2+b^2\) divides both \(a^3+b\) and \(a+b^3\)?
Answer: 1
user: How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?
Answer: 6
user: The sides of a triangle are \(25, 39,\) and \(40\). What is the diameter of the circumscribed circle? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 128
user: For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. Find the number of two-digit positive integers $n$ such that $R(n) = R(n+1)$.
Answer: 2
user: If the tangents of all interior angles of a triangle are integers, what is the sum of these integers?
Answer: 6
user: At a dance party, a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, with the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, find the relation between $b$ and $g$. The original answer is in the form $b = mg + n$. Please find the value of $m + n$.
Answer: -3
user: In triangle \(ABC\), the median from vertex \(A\) is perpendicular to the median from vertex \(B\). If the lengths of sides \(AC\) and \(BC\) are \(6\) and \(7\) respectively, find the length of side \(AB\). The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\).
Answer: 17
user: What is the sum of cubes of real roots of the equation $x^3-2x^2-x+1=0$?
Answer: 11
user: Let \(ABC\) be a triangle with \(m(\widehat{A})=90^\circ\), \(|AB|=4\), and \(|AC|=3\). Let \(D\) be the foot of the perpendicular from \(A\) to \([BC]\). If \(P\) is a point on \([BD]\) such that \(5|AP|=13|PD|\), find the length of \(|CP|\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 19
user: In triangle $ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Find the area of the triangle $ADE$ that is closest to an integer.
Answer: 3
user: Jenn draws a scalene triangle and measures the heights from each of the vertices to its opposite side. She discovers that the three height lengths are all roots of the polynomial \(x^3 - 3.9x^2 + 4.4x - 1.2\). Find the length of the inradius of the triangle. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 14
user: Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of $4$ to $3$. His wife got twice as much as the son. If the cook received a bequest of $500$, then what was the total value of the entire estate?
Answer: 7000
user: $ABC$ is a triangle with $|AB|=6$, $|BC|=7$, and $|AC|=8$. Let the angle bisector of $\angle A$ intersect $BC$ at $D$. If $E$ is a point on $[AC]$ such that $|CE|=2$, find the length of $|DE|$.
Answer: 3
user: All the roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$?
Answer: -88
user: Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. Find the probability that the lengths of these three segments are the three side lengths of a triangle with positive area. Provide your answer in the form \(\frac{k}{m}\) where the fraction is simplified, and give the value of \(k + m\).
Answer: 509
user: The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3 - Q_1$. Find the number of outliers in this data set.
Answer: 1
user: Suppose that \( P = 2^m \) and \( Q = 3^n \). Find the expression that is equal to \( 12^{mn} \) for every pair of integers \( (m, n) \). The original answer is in the form \( P^{a}Q^{b} \), please give the value of \( a + b + c \).
Answer: 3
user: Find the sum of integers $0 \leq x < 25$ such that $x^3 + 3x^2 - 2x + 4 \equiv 0 \pmod{25}$. What is this sum congruent to modulo 25?
Answer: 17
user: Write $2017$ numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, \ldots, -\frac{1}{1008}, 0, \frac{1}{1008}, \frac{2}{1008}, \ldots, \frac{1007}{1008}, \frac{1008}{1008}$.

One processes some steps as follows: erase two arbitrary numbers $x, y$ on the blackboard and then write the number $x + 7xy + y$. After $2016$ steps, there is only one number left. Find the last number on the blackboard in the form of a simplified fraction \(-\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 8
user: How many valence electrons are in the pyrophosphate ion, $\ce{P2O7}^{4-}$? Provide your answer as a whole number.
Answer: 56
user: The sequence $a_1$, $a_2$, $a_3$, $\dots$ satisfies $a_1 = 19$, $a_9 = 99$, and, for all $n \ge 3$, $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find the value of $a_2$.
Answer: 179
user: An engineer is given a fixed volume \(V_m\) of metal with which to construct a spherical pressure vessel. Assuming the vessel has thin walls and is always pressurized to near its bursting point, the amount of gas the vessel can contain, \(n\) (measured in moles), does not depend on the radius \(r\) of the vessel. Instead, it depends only on \(V_m\) (measured in \(\text{m}^3\)), the temperature \(T\) (measured in \(\text{K}\)), the ideal gas constant \(R\) (measured in \(\text{J/(K} \cdot \text{mol})\)), and the tensile strength of the metal \(\sigma\) (measured in \(\text{N/m}^2\)). Express \(n\) in terms of these parameters. The original answer is in the form \(\frac{k}{m} \times \frac{V_m\sigma}{RTn}\). Please give the value of \(k + m + n\).
Answer: 6
user: Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB}) = 11^{\circ}$, and $m(\widehat{EBC}) = 71^{\circ}$. If the answer is $x^{\circ}$, find the value of $x$ for $m(\widehat{DCE})$.
Answer: 68
user: In a circle with center at \(O\) and radius \(r\), chord \(AB\) is drawn with length equal to \(r\) (units). From \(O\), a perpendicular to \(AB\) meets \(AB\) at \(M\). From \(M\), a perpendicular to \(OA\) meets \(OA\) at \(D\). In terms of \(r\), find the area of triangle \(MDA\) in appropriate square units. The original answer is in the form \(\frac{k r^2 \sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. Please provide the value of \(k + m + n\).
Answer: 36
user: If the sum of digits in the decimal representation of a positive integer $n$ is $111$ and the sum of digits in the decimal representation of $7002n$ is $990$, what is the sum of digits in the decimal representation of $2003n$?
Answer: 555
user: A bug starts at a vertex of a grid made of equilateral triangles with a side length of $1$. At each step, the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. Find the probability that after $5$ moves, the bug will never have been more than $1$ unit away from the starting position. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 121
user: For how many primes $p$ less than $15$, there exist integer triples $(m,n,k)$ such that:
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod{p} \\
mn+mk+nk &\equiv& 1 \pmod{p} \\
mnk &\equiv& 2 \pmod{p}.
\end{array}
\]
Find the number of such primes $p$.
Answer: 3
user: Let $ABCD$ be a convex quadrilateral such that $m (\widehat{ADB}) = 15^{\circ}$ and $m (\widehat{BCD}) = 90^{\circ}$. The diagonals of the quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC| = 4$, $|EA| = 8$, and $|EP| = 2$. If the answer is $x^\circ$, what is the value of $x$?
Answer: 75
user: The acronym AMC is shown in the rectangular grid below with grid lines spaced \(1\) unit apart. In units, find the sum of the lengths of the line segments that form the acronym AMC. The original answer is in the form \(a + b\sqrt{c}\), please provide the value of \(a + b + c\).
Answer: 19
user: How many ordered triples \((x, y, z)\) of integers satisfy the following system of equations?

\[
\begin{align*}
x^2 - 3xy + 2yz - z^2 &= 31 \\
-x^2 + 6yz + 2z^2 &= 44 \\
x^2 + xy + 8z^2 &= 100
\end{align*}
\]

Find the number of ordered triples.
Answer: 0
user: Let $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$, where $C$ and $D$ are on different semicircles with diameter $[AB]$. Let $H$ be the foot of the perpendicular from $B$ to $[CD]$. If $|AO| = 13$, $|AC| = 24$, and $|HD| = 12$, what is $\widehat{DCB}$ in degrees? If the answer is $x^\circ$, what is the value of $x$?
Answer: 30
user: Find the number of distinct ordered pairs $(x,y)$, where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$.
Answer: 3
user: Points $A = (3,9), B = (1,1), C = (5,3),$ and $D = (a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA}$ is a square. Find the sum of the coordinates of point $D$.
Answer: 10
user: A function $f$ from the integers to the integers is defined as follows:
\[ f(n) = \begin{cases} n + 3 & \text{if } n \text{ is odd} \\
\frac{n}{2} & \text{if } n \text{ is even} \end{cases} \]
Suppose $k$ is odd and $f(f(f(k))) = 27$. Find the sum of the digits of $k$.
Answer: 6
user: Let $(a_n)_{n=1}^{\infty}$ be a real sequence such that $a_1=1$, $a_3=4$, and for every $n \geq 2$, $a_{n+1} + a_{n-1} = 2a_n + 1$. Find the value of $a_{2011}$.
Answer: 2021056
user: Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? Provide your answer as an integer.
Answer: 400
user: Find the value of \( \frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}} \). The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 9
user: What is the hundreds digit of the sum below?

$$1 + 12 + 123 + 1234 + 12345 + 123456 + 1234567 + 12345678 + 123456789$$
Answer: 2
user: Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations? Provide your answer as an integer.
Answer: 4
user: A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\frac{3}{4}$ of its former rate and arrives $3\frac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. Find the length of the trip in miles.
Answer: 600
user: A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile, the speed is constant. If the second mile is traversed in $2$ hours, find the time, in hours, needed to traverse the $n$th mile. The original answer is in the form $k(n-m)$, please give the value of $k + m$.
Answer: 3
user: Two circles with radii 1 and 2 touch each other and a line as shown in the figure. In the region between the circles and the line, there is a circle with radius $r$ which touches the two circles and the line. The original answer is in the form $m - n\sqrt{2}$, where $m$ and $n$ are integers. Please find the value of $m + n$.
Answer: 10
user: In tetrahedron $ABCD$, where $AB=1$, $CD=3$, the distance between $AB$ and $CD$ is $2$, and the intersection angle between $AB$ and $CD$ is $\frac{\pi}{3}$, find the volume of tetrahedron $ABCD$. The original answer is in \frac{\sqrt{k}}{m} format, please give the value of k + m.
Answer: 5
user: A point object of mass \(m\) is connected to a cylinder of radius \(R\) via a massless rope. At time \(t = 0\), the object is moving with an initial velocity \(v_0\) perpendicular to the rope. The rope has a length \(L_0\) and a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds \(T_{max}\). Express your answers in terms of \(T_{max}\), \(m\), \(L_0\), \(R\), and \(v_0\). The original answer is in the form \(\frac{k m^2 v_0^3}{n T_{max}}\). Please calculate the value of \(k + m + n\).
Answer: 4
user: Let \(S_1\) be the area of the regular pentagon \(ABCDE\). And let \(S_2\) be the area of the regular pentagon whose sides lie on the lines \(AC, CE, EB, BD, DA\). The original answer is in the form \(\frac{k+m\sqrt{5}}{n}\). Please find the value of \(k+m+n\).
Answer: 12