meetforever/GRPO_CL huggingface dataset:
user: A circle passes through the vertices of a triangle with side-lengths of \(7\frac{1}{2}, 10, 12\frac{1}{2}\). The radius of the circle can be expressed in the form \(\frac{k}{m}\). Find the value of \(k + m\).
Answer: 29
user: If \(V = gt + V_0\) and \(S = \frac{1}{2}gt^2 + V_0t\), find the expression for \(t\) in terms of \(S\), \(V\), and \(V_0\). The original answer is in the form \(\frac{kS}{m(V + V_0)}\), please give the value of k + m.
Answer: 3
user: If the decimal representation of $2^n$ starts with $7$, what is the first digit in the decimal representation of $5^n$? Provide your answer as a single digit.
Answer: 1
user: We draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of the circle is placed in the middle of one of the squares. Find the number of squares through which the circle passes. Please provide your answer as an integer.
Answer: 40
user: If $a$, $b$, and $d$ are the lengths of a side, the shortest diagonal, and the longest diagonal, respectively, of a regular nonagon (see adjoining figure), determine the relationship between these variables in the form of an equation. The original answer is in the form $d = m \cdot a + n \cdot b + p$. Please give the value of $m + n + p$.
Answer: 2
user: How many pairs of integers \((x, y)\) are there such that \(2x + 5y = xy - 1\)?
Answer: 4
user: Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. Find the probability that Flora will eventually land at $10$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 3
user: A right circular cone and a right cylinder with the same height $20$ do not have the same circular base, but the circles are coplanar, and their centers are the same. If the cone and the cylinder are on the same side of the plane and their base radii are $20$ and $10$, respectively, find the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder.
Answer: 1
user: Triangle \(ABC\) is a right triangle with \(\angle ACB\) as its right angle, \(m\angle ABC = 60^\circ\), and \(AB = 10\). Let \(P\) be randomly chosen inside \(\triangle ABC\), and extend \(\overline{BP}\) to meet \(\overline{AC}\) at \(D\). Find the probability that \(BD > 5\sqrt{2}\). The original answer is in the form \(\frac{k - \sqrt{m}}{n}\), please provide the value of k + m + n.
Answer: 9
user: The integer $k$ is a \textit{good number}, if we can divide a square into $k$ squares. Find how many good numbers not greater than $2006$ there are.
Answer: 2003
user: For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities \[ \begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases} \]?
Answer: 3
user: Side \(AB\) of triangle \(ABC\) has length \(8\) inches. Line \(DEF\) is drawn parallel to \(AB\) so that \(D\) is on segment \(AC\), and \(E\) is on segment \(BC\). Line \(AE\) extended bisects angle \(FEC\). If \(DE\) has length \(5\) inches, then find the length of \(CE\) in inches. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 43
user: Acute-angled triangle \(ABC\) is inscribed in a circle with center at \(O\). The arcs are given as follows: \(\overarc{AB} = 120^\circ\) and \(\overarc{BC} = 72^\circ\). A point \(E\) is located on the minor arc \(AC\) such that \(OE\) is perpendicular to \(AC\). Determine the ratio of the magnitudes of angles \(\angle OBE\) and \(\angle BAC\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 4
user: If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6}$, find the value of the middle part in its simplest fractional form. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 55
user: The point $(-3, 2)$ is rotated $90^\circ$ clockwise around the origin to point $B$. Point $B$ is then reflected over the line $y = x$ to point $C$. What are the coordinates of $C$? Provide your answer as an ordered pair $(k, m)$, and then find the sum of k and m.
Answer: 5
user: What is the least possible value of \((xy-1)^2+(x+y)^2\) for real numbers \(x\) and \(y\)?
Answer: 1
user: A permutation $ (a_1, a_2, a_3, a_4, a_5)$ of $ (1, 2, 3, 4, 5)$ is heavy-tailed if $ a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
Answer: 48
user: The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt{3}i$, where $i=\sqrt{-1}$, form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt{q}-r\sqrt{s}$, where $p$, $q$, $r$, and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. Find the value of $p+q+r+s$.
Answer: 20
user: The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Find an equation for $G'$. The original answer is in the form $y = a^{-bx}$, please give the value of a + b.
Answer: 11
user: In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? Provide your answer as an integer.
Answer: 5720
user: How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?
Answer: 6
user: For all positive integers $n$, the function $f$ satisfies the following conditions:
- $f(1) = 1$
- $f(2n + 1) = 2f(n)$
- $f(2n) = 3f(n) + 2$

Determine how many positive integers $x \leq 100$ make the value of $f(x)$ odd.
Answer: 7
user: The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \, \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \, \text{cm}$. The square has a perimeter greater than 0. How many positive integers are NOT possible values for $d$?
Answer: 663
user: A quadrilateral has consecutive sides of lengths $70$, $90$, $130$, and $110$. It is both inscribed in a circle and has a circle inscribed within it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find the value of $|x-y|$.
Answer: 13
user: In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark.
```latex
\begin{asy}
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$2$",(0,0));
draw(shift(1,0)*box); label("$?$",(1,0));
draw(shift(2,0)*box); label("$?$",(2,0));
draw(shift(3,0)*box); label("$?$",(3,0));
draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4));
draw(shift(1,0.8)*box); label("$5$",(1,0.8));
draw(shift(2,0.8)*box); label("$?$",(2,0.8));
draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2));
\end{asy}
```
Answer: 13
user: What is the perimeter of the right triangle whose exradius of the hypotenuse is $30$?
Answer: 60
user: How many pairs of positive integers $(x, y)$ satisfy the equation $2^x - y^2 = 4$?
Answer: 1
user: $X$, $Y$, and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X$, $Y$, $Z$, $X \cup Y$, $X \cup Z$, and $Y \cup Z$ are given in the table below.

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Set} & X & Y & Z & X \cup Y & X \cup Z & Y \cup Z \\
\hline
\text{Average age of people in the set} & 37 & 23 & 41 & 29 & 39.5 & 33 \\
\hline
\end{array}
\]

Find the average age of the people in set $X \cup Y \cup Z$.
Answer: 34
user: Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$, $m(\widehat{ACB})=90^\circ$. Find the area of $\triangle DAC$.
Answer: 84
user: If \( y = x + \frac{1}{x} \), then transform the equation \( x^4 + x^3 - 4x^2 + x + 1 = 0 \) into the form \( x^2(y^2 + y - k) = 0 \). What is the value of \( k \)?
Answer: 6
user: $ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Find the value of $\sin \theta$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

```asy
\draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
\draw((0,0)--(2,1));
\draw((0,0)--(1,2));
\label("A", (0,0), SW);
\label("B", (0,2), NW);
\label("C", (2,2), NE);
\label("D", (2,0), SE);
\label("M", (1,2), N);
\label("N", (2,1), E);
\label("$\theta$", (.5,.5), SW);
```
Answer: 8
user: Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both $x^{4} + 6x^{2} + 25$ and $3x^{4} + 4x^{2} + 28x + 5$, find the value of $p(1)$.
Answer: 4
user: Let \(ABCD\) be a rectangle and let \(\overline{DM}\) be a segment perpendicular to the plane of \(ABCD\). Suppose that \(\overline{DM}\) has integer length, and the lengths of \(\overline{MA}, \overline{MC},\) and \(\overline{MB}\) are consecutive odd positive integers (in this order). Find the volume of pyramid \(MABCD\). The original answer is in the form \(k\sqrt{m}\), please provide the value of \(k + m\).
Answer: 29
user: Let \(ABCD\) be a unit square. Let \(Q_1\) be the midpoint of \(\overline{CD}\). For \(i=1,2,\dots,\) let \(P_i\) be the intersection of \(\overline{AQ_i}\) and \(\overline{BD}\), and let \(Q_{i+1}\) be the foot of the perpendicular from \(P_i\) to \(\overline{CD}\). The sum of the areas of \(\triangle DQ_i P_i\) for \(i=1\) to \(\infty\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 5
user: A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) The original answer is in the form \( k\sqrt{m} - n \). Please give the value of \( k + m + n \).
Answer: 35
user: Given the sets of consecutive integers $\{1\}$, $\{2, 3\}$, $\{4, 5, 6\}$, $\{7, 8, 9, 10\}$, $\cdots$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the $N$th set. Find the value of $S_{21}$.
Answer: 4641
user: Several sets of prime numbers, such as $\{7, 83, 421, 659\}$, use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Answer: 207
user: What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ Provide your answer as an integer.
Answer: 1
user: In $\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of $BD$?
Answer: 3
user: Let \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find the value of \(m + n + p\).
Answer: 16
user: Diagonal $DB$ of rectangle $ABCD$ is divided into $3$ segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. If the area of $ABCD$, rounded to the nearest tenth, is $x$, what is the value of $10x$?
Answer: 42
user: Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Express $f(n+1)-f(n-1)$ in terms of $f(n)$. The original answer is in the form of a single variable k, please provide the value of k.
Answer: 1
user: If the five-digit number $3AB76$ is divisible by $9$ and $A < B < 6$, find the value of $B - A$.
Answer: 2
user: Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. The probability of obtaining a six on any toss is \(\frac{1}{6}\), independent of the outcome of any other toss. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 116
user: Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$, and $p(4)=4$. Find the value of $p(5)$.
Answer: 29
user: At Rachelle's school, an A counts as 4 points, a B as 3 points, a C as 2 points, and a D as 1 point. Her GPA for the four classes she is taking is calculated as the total sum of points divided by 4. She is certain to get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a \frac{1}{6} chance of getting an A in English, and a \frac{1}{4} chance of getting a B. In History, she has a \frac{1}{4} chance of getting an A, and a \frac{1}{3} chance of getting a B, independently of her grades in English. What is the probability that Rachelle will achieve a GPA of at least 3.5? Provide your answer in the form of \frac{k}{m} and give the value of k + m.
Answer: 35
user: In right triangle \(ABC\), \(BC = 5\), \(AC = 12\), and \(AM = x\); \(\overline{MN} \perp \overline{AC}\), \(\overline{NP} \perp \overline{BC}\); \(N\) is on \(AB\). If \(y = MN + NP\), one-half the perimeter of rectangle \(MCPN\), find the value of \(y\). The original answer is in the form \(\frac{k - mx}{n}\). Please provide the value of \(k + m + n\).
Answer: 163
user: A fair die is rolled six times. Find the probability of rolling at least a five at least five times. Express your answer as a fraction in simplest form. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 742
user: If the function \( f \) satisfies the equation \( f(x) + f\left( \frac{1}{\sqrt[3]{1-x^3}}\right) = x^3 \) for every real \( x \neq 1 \), find the value of \( f(-1) \). The original answer is in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please provide the value of \( k + m \).
Answer: 5
user: A square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in the form \(a + b\pi\), where \(a\) and \(b\) are integers. Please give the value of \(a + b\).
Answer: 175
user: For what value of $n$ is the equation $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$ satisfied? Note: here $i = \sqrt{-1}$.
Answer: 97
user: How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.) Provide your answer as a number.
Answer: 24
user: Compute the remainder when $20^{(13^{14})}$ is divided by $11$.
Answer: 5
user: The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is the value of $b-a$?
Answer: 7
user: An ice ballerina rotates at a constant angular velocity at one particular point, meaning she does not translationally move. Her arms are fully extended as she rotates, and her moment of inertia is \(I\). Now, she pulls her arms in, and her moment of inertia becomes \(\frac{7}{10}I\). What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)? The original answer is in the format \(\frac{k}{m}\), please give the value of \(k + m\).
Answer: 17
user: Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping squares, respectively. If the pattern were continued, find the number of nonoverlapping squares in figure $100$.
Answer: 20201
user: Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? Provide your answer in the form of \frac{k}{m}, and calculate the value of k + m.
Answer: 5
user: By inserting parentheses, it is possible to give the expression $2 \times 3 + 4 \times 5$ several values. How many different values can be obtained?
Answer: 4
user: At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. How many pairs are there in which both students are wearing yellow shirts?
Answer: 32
user: A coin is altered so that the probability that it lands on heads is less than \(\frac{1}{2}\). When the coin is flipped four times, the probability of getting an equal number of heads and tails is \(\frac{1}{6}\). What is the probability that the coin lands on heads? Express your answer in the form \(\frac{k - \sqrt{m}}{n}\), where the fraction is in its simplest form. Please calculate the value of k + m + n.
Answer: 12
user: Hull Street is nine miles long. The first two miles of Hull Street are in Richmond, and the last two miles of the street are in Midlothian. If Kathy starts driving along Hull Street from a random point in Richmond and stops on the street at a random point in Midlothian, what is the probability that Kathy drove less than six miles? The original answer is in the form of \(\frac{k}{m}\), where both k and m are integers, and the fraction is in its simplest form. Please find the value of k + m.
Answer: 9
user: Adams plans a profit of $10\%$ on the selling price of an article and his expenses are $15\%$ of sales. If the rate of markup on an article that sells for $\$5.00$ is $x\%$, find the value of $3x$. Provide your answer as an integer without the percentage symbol.
Answer: 100
user: If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, find the probability that this number is divisible by $11$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 6
user: \(\overline{AB}\) is the hypotenuse of a right triangle \(ABC\). Median \(\overline{AD}\) has length \(7\) and median \(\overline{BE}\) has length \(4\). The length of \(\overline{AB}\) is in the form of a\sqrt{b}. Please give the value of a + b.
Answer: 15
user: A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. Find the number of chords of the circle that pass through $P$ and have integer length.
Answer: 12
user: In the sequence $2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the value of the $2004^\text{th}$ term in this sequence?
Answer: 0
user: Consider the graphs $y = Ax^2$ and $y^2 + 3 = x^2 + 4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. Determine the number of points at which the two graphs intersect.
Answer: 4
user: At a party with $n \geq 4$ people, if every 3 people have exactly 1 common friend, find the number of different values that $n$ can take.
Answer: 4
user: Given \( \frac{\log{a}}{p} = \frac{\log{b}}{q} = \frac{\log{c}}{r} = \log{x} \), all logarithms to the same base and \( x \neq 1 \). If \( \frac{b^2}{ac} = x^y \), find the value of \( y \). The original answer is in the form \(aq - bp - cr\), please provide the value of a + b + c.
Answer: 4
user: If $y = 2x$ and $z = 2y$, then find the expression for $x + y + z$ in terms of $x$. The answer should be in the form $ax$, where $a$ is a constant. Please determine the value of $a$.
Answer: 7
user: How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$? Find the number of such coefficients.
Answer: 16
user: In triangle \(ABC\), \(\angle BAC = 60^\circ\), \(AB = 3a\), and \(AC = 4a\), where \(a > 0\). Let \(M\) be a point on segment \(AB\) such that \(AM = \frac{1}{3} AB\), and \(N\) be a point on side \(AC\) such that \(AN = \frac{1}{2} AC\). Let \(I\) be the midpoint of \(MN\). Determine the length of \(BI\). The original answer is in the form \(\frac{ka\sqrt{m}}{n}\), please give the value of \(k + m + n\).
Answer: 22
user: Find the integer that does not divide $n^{2225} - n^{2005}$ for every integer value of $n$.
Answer: 7
user: Let \(ABC\) be a triangle with incenter \(I\), centroid \(G\), and \(|AC| > |AB|\). If \(IG \parallel BC\), \(|BC| = 2\), and \(\text{Area}(ABC) = \frac{3\sqrt{5}}{8}\), then find the value of \(|AB|\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Answer: 17
user: Find the number of primes $p$, such that $x^{3} - 5x^{2} - 22x + 56 \equiv 0 \pmod{p}$ has no three distinct integer roots in $[0, p)$.
Answer: 4
user: Point \(P\) is taken interior to a square with side-length \(a\) such that it is equally distant from two consecutive vertices and from the side opposite these vertices. If \(d\) represents the common distance, find the value of \(d\) in terms of \(a\). The original answer is in the form \(\frac{ka}{m}\), where the fraction is in its simplest form. Please find the value of \(k + m\).
Answer: 13
user: Let $E$ be a point inside the rhombus $ABCD$ such that $|AE| = |EB|$, $m(\widehat{EAB}) = 12^\circ$, and $m(\widehat{DAE}) = 72^\circ$. What is the value of $m(\widehat{CDE})$ in degrees?
Answer: 66
user: A man with mass $m$ jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the bungee cord is four times the weight of the man.

Find the maximum extension of the bungee cord $h$ in terms of $H$. The original answer is in the format \frac{k}{m}, please give the value of k + m.
Answer: 3
user: For how many integers \( n \) between 1 and 100 does \( x^2 + x - n \) factor into the product of two linear factors with integer coefficients?
Answer: 9
user: What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set \( \{1,2,\dots, 9\} \), there exist $k$ of them having at least one common element." true?
Answer: 4
user: Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. Find the number of different possible denominators.
Answer: 5
user: The geometric series \(a + ar + ar^{2} + \ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in the form \(\frac{k}{m}\). Find the value of \(a + r\) and give the sum of \(k + m\).
Answer: 7
user: Find the value of \(i + 2i^2 + 3i^3 + \ldots + 2002i^{2002}\). The original answer is in the form of \(k + mi\), please give the value of \(k + m\).
Answer: -1
user: Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $4$ replaced by $3$ and the other die has the $3$ replaced by $4$. When these dice are rolled, find the probability that the sum is an odd number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 14