meetforever/GRPO_CL huggingface dataset:
user: An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(0.8)$. Find the area of the trapezoid.
Answer: 80
user: $N$ is a $50$-digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the value of the $26^{\text{th}}$ digit?
Answer: 3
user: If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, find all possible values for $K$. The original answer is in the form of $k_1, k_2, k_3$. Please give the sum of $k_1, k_2, k_3$.
Answer: 58
user: A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a, b)$?
Answer: 2
user: Two circles with radii $15$ and $95$ have their centers $d$ units apart. The external tangents to the circles intersect at an angle of $60$ degrees. What is the distance $d$ between the centers of the circles?
Answer: 160
user: Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. The length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, the length $s$ is found to be $28$ inches. How high is the table in inches?
Answer: 30
user: Bernardo randomly picks $3$ distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $3$-digit number. Silvia randomly picks $3$ distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $3$-digit number. Find the probability that Bernardo's number is larger than Silvia's number. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 93
user: Let \(H\) be the intersection of altitudes in triangle \(ABC\). If \(\angle B = \angle C = \alpha\) and \(O\) is the center of the circle passing through \(A\), \(H\), and \(C\), then find \(\angle HOC\) in terms of \(\alpha\) and express your answer in degrees. The original answer is in the form \(k^\circ - m\alpha\), please give the value of \(k + m\).
Answer: 182
user: What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? Provide your answer as an integer.
Answer: 8
user: The graphs of $2x + 3y - 6 = 0$, $4x - 3y - 6 = 0$, $x = 2$, and $y = \frac{2}{3}$ intersect in how many points?
Answer: 1
user: Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? The original answer is in the form of $k + \sqrt{m}$, where $k$ and $m$ are integers. Please find the value of $k + m$.
Answer: 4
user: Find the integer $n$ that satisfies $n^{29} \equiv 7 \pmod{65}$.
Answer: 37
user: How many pairs of integers $(m,n)$ are there such that $mn + n + 14 = (m-1)^2$? Provide the number of such pairs.
Answer: 8
user: If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\stackrel{\frown}{AC}$ and $\stackrel{\frown}{BC}$, and to $\overline{AB}$. If the length of $\stackrel{\frown}{BC}$ is $12$, find the circumference of the circle.
Answer: 27
user: Let $p$ and $q$ be two consecutive terms of the sequence of odd primes. Find the minimum number of positive divisors of $p + q$.
Answer: 4
user: $\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. If the answer is $x^\circ$, find the value of $x$ for the angle $\angle EDF$.
Answer: 66
user: There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $n$ between 1 and 25. Betül draws $n$ balls from the bag one by one such that no ball is put back after it is drawn. If exactly 2 of the $n$ balls are blue and the second blue ball is drawn at the $n^{th}$ order, Ahmet wins; otherwise, Betül wins. To increase the possibility of winning, what integer should Ahmet choose?
Answer: 13
user: What are the last three digits of $49^{303} \cdot 3993^{202} \cdot 39^{606}$?
Answer: 561
user: A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
Answer: 26
user: All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?

\[
\begin{array}{c c c c} 
{} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams \text{ and } Baker}  \\
\textbf{Boys:} & 71 & 81 & 79   \\
\textbf{Girls:} & 76 & 90 & ?   \\
\textbf{Boys \text{ and } Girls:} & 74 & 84 &   \\
\end{array}
\]

Find the average score for the girls at the two schools combined.
Answer: 84
user: In the following base-10 equation, each letter represents a unique digit: $AM \cdot PM = ZZZ$. Find the sum of $A + M + P + Z$.
Answer: 21
user: In how many ways can $10$ distinct books be placed onto a $3$-shelf bookcase such that no shelf is empty? Express your answer in the form \( n \cdot 10! \), where \( n \) is an integer.
Answer: 36
user: What is the product of all odd positive integers less than 10000? Express your answer in the form of a factorial expression divided by powers of 2 and factorials. The original answer is in the format \(\frac{x!}{2^y \cdot z!}\). Please find the value of \(x + y + z\).
Answer: 20000
user: What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x - 3$? If the area is $A\pi$, what is the value of $A$?
Answer: 8
user: Given the line $y = \frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. Find a possible equation for line $L$. The original answer is in the form $y = \frac{3}{4}x + c$, please give the value of c.
Answer: 1
user: What is the smallest possible area, in square centimeters, of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below?

\[ \begin{asy} 
size(5cm,0);
draw((0,2)--(0,3));
draw((1,1)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,2));
draw((4,0)--(4,1));
draw((2,0)--(4,0));
draw((1,1)--(4,1));
draw((0,2)--(3,2));
draw((0,3)--(2,3));
\end{asy} \]
Answer: 768
user: In unit square \(ABCD\), the inscribed circle \(\omega\) intersects \(\overline{CD}\) at \(M\), and \(\overline{AM}\) intersects \(\omega\) at a point \(P\) different from \(M\). Find the length of \(AP\). The original answer is 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: 16
user: If $w$ is one of the imaginary roots of the equation $x^3=1$, find the value of the product $(1-w+w^2)(1+w-w^2)$.
Answer: 4
user: A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find the ratio of the total kinetic energy of the system of two masses after the perfectly elastic collision and the cord has broken to the initial kinetic energy of the smaller mass prior to the collision. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 7
user: In a restaurant, a meal consists of one sandwich and one optional drink. In other words, a sandwich is necessary for a meal but a drink is not necessary. There are two types of sandwiches and two types of drinks. How many possible meals can be purchased?
Answer: 6
user: Yan is somewhere between his home and the stadium. To get to the stadium, he can either walk directly to the stadium or walk home first and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form, and provide the value of k + m.
Answer: 7
user: The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position? If the new position is $(x, y)$, please give the coordinates in the form of $(a, b)$ and find the value of $a + b$.
Answer: 5
user: What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is a positive integer?
Answer: 1514
user: Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. Find the probability that the product is divisible by $4$. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please calculate the value of k + m.
Answer: 123
user: My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, find the expected score of the losing team. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
user: Real numbers \(a\) and \(b\) are chosen with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a, b\) or \(\frac{1}{b}, \frac{1}{a}, 1\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Find the value of \(k + m + n\).
Answer: 10
user: Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)?
Answer: 2
user: $AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $\angle BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, find the length of chord $AC$, expressed in inches. The answer should be in the form \(\frac{k\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. Please calculate the value of \(k + m + n\).
Answer: 10
user: Find the sum of the roots of $\tan^2 x - 9\tan x + 1 = 0$ that are between $x = 0$ and $x = 2\pi$ radians. If the answer is $k\pi$, what is the value of $k$?
Answer: 3
user: For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? Provide the number of such integers.
Answer: 49
user: Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k$.
Answer: 1
user: $A(-1,1)$, $B$, and $C$ are points on the hyperbola $x^2 - y^2 = 1$. If $\triangle ABC$ is a regular triangle, find the area of $\triangle ABC$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Answer: 6
user: On a large piece of paper, Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana's pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws, in cm?
Answer: 54
user: Older television screens have an aspect ratio of $4:3$. That is, the ratio of the width to the height is $4:3$. The aspect ratio of many movies is not $4:3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $2:1$ and is shown on an older television screen with a $27$-inch diagonal. If the height of each darkened strip is $x$ inches, what is the value of $10x$?
Answer: 27
user: How many pairs of positive integers $(x,y)$ are there such that $\sqrt{xy} - 71\sqrt{x} + 30 = 0$?
Answer: 8
user: Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to the conversation below:

$B$: I'm a truth teller.

$C$: $D$ is a truth teller.

$D$: $B$ and $E$ are not both truth tellers.

$E$: $A$ and $B$ are truth tellers.

How many truth tellers are there?
Answer: 3
user: Arithmetic sequences $(a_n)$ and $(b_n)$ have integer terms with $a_1 = b_1 = 1 < a_2 \le b_2$ and $a_nb_n = 2010$ for some $n$. What is the largest possible value of $n$?
Answer: 8
user: Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawing the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies between the parallel lines (excluding the intersection points on the lines)?
Answer: 6600
user: How many different integers can be expressed as the sum of three distinct members of the set $\{1, 4, 7, 10, 13, 16, 19\}$?
Answer: 13
user: Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval \([0,1]\). Two random numbers \(x\) and \(y\) are chosen independently in this manner. Find the probability that \(|x-y| > \frac{1}{2}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 23
user: Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that
\[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \]
for every $x,y \in (0,\infty)$. Find the value of $f(\frac{1}{11})$.
Answer: 21
user: Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. Find the area of triangle $H_1H_2H_3$.
Answer: 96
user: Let \(D(n)\) denote the number of ways of writing the positive integer \(n\) as a product \(n = f_1 \cdot f_2 \cdots f_k,\) where \(k \ge 1\), the \(f_i\) are integers strictly greater than 1, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as 6, \(2 \cdot 3\), and \(3 \cdot 2\), so \(D(6) = 3\). What is \(D(96)\)?
Answer: 112
user: In triangle \(ABC\), \(\measuredangle CBA=72^\circ\), \(E\) is the midpoint of side \(AC\), and \(D\) is a point on side \(BC\) such that \(2BD=DC\); \(AD\) and \(BE\) intersect at \(F\). Find the ratio of the area of triangle \(BDF\) to the area of quadrilateral \(FDCE\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 6
user: A *triangular number* is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 + \cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. Find the sum of the digits of the fourth smallest triangular number that is also a perfect square.
Answer: 18
user: Triangles $ABC$ and $ADE$ have areas $2007$ and $7002$, respectively, with $B = (0,0)$, $C = (223,0)$, $D = (680,380)$, and $E = (689,389)$. Find the sum of all possible x-coordinates of $A$.
Answer: 1200
user: Find the sum of distinct residues of the number $2012^n + m^2 \pmod{11}$ where $m$ and $n$ are positive integers.
Answer: 39
user: Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i$, $-2+i$, and $-1-2i$. Find the fourth number. The original answer is in the form x + yi, where x and y are integers. Please give the value of x + y.
Answer: 1
user: Toothpicks of equal length are used to build a rectangular grid. If the grid is 20 toothpicks high and 10 toothpicks wide, find the total number of toothpicks used.
Answer: 430
user: Let $c = \frac{2\pi}{11}.$ What is the value of \[ \frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}? \]
Answer: 1
user: Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB| + |BD| = |AC|$ and $m(\widehat{BAD}) = m(\widehat{DAC}) = 30^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{ACB})$? Please provide the value of $x$ as an integer.
Answer: 40
user: My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages?
Answer: 2
user: The sides of a triangle have lengths $11$, $15$, and $k$, where $k$ is an integer. Find the number of integer values of $k$ for which the triangle is obtuse.
Answer: 13
user: A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex \(W\)? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 13
user: Given $a_1 = 1$, $a_{n+1} = \frac{a_n}{\sqrt{1 + 4a_n^2}}$ for $n \ge 1$, find the least $k$ such that $a_k < 10^{-2}$.
Answer: 2501
user: How many solutions does the equation $\sin \left( \frac{\pi}{2} \cos x\right)=\cos \left( \frac{\pi}{2} \sin x\right)$ have in the closed interval $[0,\pi]$? Provide your answer as an integer.
Answer: 2
user: How many different solutions does the congruence $x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}$ have? Provide your answer as an integer.
Answer: 6
user: Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus, the numbers $29$, $23$, $21$, and $17$ are obtained. Find one of the original integers.
Answer: 21
user: The minimum value of the quotient of a (base ten) number of three different nonzero digits divided by the sum of its digits is x. If the answer is x, what is the value of 10x?
Answer: 105
user: The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

\[\text{[asy]}\]
\text{size(300);}
\text{defaultpen(linewidth(0.8));}
\text{real r = 0.35;}
\text{path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);}
\text{path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r);}
\text{for(int i=0;i <= 4;i=i+1)}
\text{{}}
\text{draw(shift((4*i,0)) * P);}
\text{draw(shift((4*i,0)) * Q);}
\text{{}}
\text{for(int i=1;i <= 4;i=i+1)}
\text{{}}
\text{draw(shift((4*i-2,0)) * Pp);}
\text{draw(shift((4*i-1,0)) * Qp);}
\text{{}}
\text{draw((-1,0)--(18.5,0),Arrows(TeXHead));}
\[\text{[/asy]}\]

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

- some rotation around a point of line $\ell$
- some translation in the direction parallel to line $\ell$
- the reflection across line $\ell$
- some reflection across a line perpendicular to line $\ell$

Fill in the blank with the number of transformations: \_\_
Answer: 2
user: Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8:00 \text{ AM}$, and all three always take the same amount of time to eat lunch. On Monday, the three of them painted $50\%$ of a house, quitting at $4:00 \text{ PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $2:12 \text{ PM}$. On Wednesday, Paula worked by herself and finished the house by working until $7:12 \text{ PM}$. How long, in minutes, was each day's lunch break?
Answer: 48
user: The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
Answer: 10
user: Given the linear fractional transformation of \(x\) into \(f_1(x) = \frac{2x-1}{x+1}\). Define \(f_{n+1}(x) = f_1(f_n(x))\) for \(n=1,2,3,\ldots\). Assuming that \(f_{35}(x) = f_5(x)\), find the expression for \(f_{28}(x)\). The original answer is in the form \(\frac{k}{m-nx}\). Please find the value of k + m + n.
Answer: 3
user: The expression \((x + y + z)^{2006} + (x - y - z)^{2006}\) is simplified by expanding it and combining like terms. How many terms are in the simplified expression? Please provide your answer as an integer.
Answer: 1008016
user: For a geometric series $(a_n)$ with all terms real, if $S_{10} = 10$ and $S_{30} = 70$, find the value of $S_{40}$.
Answer: 150
user: The base-nine representation of the number $N$ is $27,006,000,052_{\text{nine}}$. Find the remainder when $N$ is divided by $5$.
Answer: 3
user: A simplified model of a bicycle of mass \(M\) has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is \(w\), and the center of mass of the bicycle is located midway between the tires and a height \(h\) above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude \(a\). Air resistance may be ignored.

Case 1: Assume that the coefficient of sliding friction between each tire and the ground is \(\mu\), and that both tires are skidding: sliding without rotating. Express your answers in terms of \(w\), \(h\), \(M\), and \(g\).

What is the maximum value of \(a\) so that both tires remain in contact with the ground? The original answer is in the form \(\frac{kwg}{mh}\), please find the value of \(k + m\).
Answer: 2
user: A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find \(\frac{v_f}{v_0}\). The answer should be in the form \(\frac{k}{\sqrt{m}}\). Please calculate the value of k + m.
Answer: 7
user: Let the bisectors of the exterior angles at \(B\) and \(C\) of triangle \(ABC\) meet at \(D\). Then, if all measurements are in degrees, find the value of angle \(BDC\). The original answer is in the form \(\frac{k}{m}(n - A)\). Please provide the value of k + m + n.
Answer: 183
user: The parabola $P$ has its focus at $(0,0)$ and passes through the points $(4,3)$ and $(-4,-3)$. Determine the number of points $(x,y) \in P$ with integer coefficients such that $|4x + 3y| \leq 1000$.
Answer: 40