sungyub/dapo-math-17k-verl huggingface dataset:
user: In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. Find the value of $A + H$.

Answer: 25
user: Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. The value of \frac{r_a}{r_b} is given in the form \frac{k}{m}(n-\sqrt{p}). Please calculate the value of k+m+n+p.

Answer: 43
user: Given a rectangle \(R_1\) with one side measuring \(2\) inches and an area of \(12\) square inches. Rectangle \(R_2\) has a diagonal of \(15\) inches and is similar to \(R_1\). What is the area of \(R_2\), expressed in the form \(\frac{k}{m}\) square inches? Please provide the value of \(k + m\).

Answer: 137
user: In $\triangle ABC$, $\angle ABC = 45^\circ$. Point $D$ is on $\overline{BC}$ such that $2 \cdot BD = CD$ and $\angle DAB = 15^\circ$. Find the value of $\angle ACB$ in degrees. If the answer is $x^\circ$, what is the value of $x$?

Answer: 75
user: If \(i^2 = -1\), then find the sum 
\[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \]
\[ + \cdots + i^{40}\cos{3645^\circ} \]
The original answer is in the form \(\frac{k}{m}(\sqrt{2})(a - bi)\), please calculate the value of a + b + k + m.

Answer: 44
user: An unfair coin lands on heads with a probability of $\frac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$?

Answer: 11
user: A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. The rotational inertia of the rod about its center is $md^2$.

Find the ratio $\frac{L}{d}$. The original answer is in the form k\sqrt{m}, please provide the value of k + m.

Answer: 5
user: For any positive integer \(n\), let \[ f(n) = \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases} \] Find the value of \(\sum_{n = 1}^{1997}{f(n)}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 58
user: Convex polygons \(P_1\) and \(P_2\) are drawn in the same plane with \(n_1\) and \(n_2\) sides, respectively, where \(n_1 \le n_2\). If \(P_1\) and \(P_2\) do not share any line segment, what is the maximum number of intersections between \(P_1\) and \(P_2\)? The answer is in the form \(kn_1 + m\). Please provide the value of \(k + m\).

Answer: 2
user: Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?

Answer: 11
user: Points \(A(6,13)\) and \(B(12,11)\) lie on circle \(\omega\) in the plane. Suppose that the tangent lines to \(\omega\) at \(A\) and \(B\) intersect at a point on the \(x\)-axis. Find the value of \(\frac{\text{Area of }\omega}{\pi}\) in the format \(\frac{k}{m}\), and give the sum of k and m.

Answer: 93
user: Find the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$. The answer is an integer. Please provide the integer value.

Answer: -2
user: A particle is placed on the parabola $y = x^2 - x - 6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. Find the horizontal distance traveled by the particle, which is the numerical value of the difference in the $x$-coordinates of $P$ and $Q$.

Answer: 3
user: How many ordered pairs of real numbers \((x,y)\) satisfy the following system of equations?
\[
\begin{align*}
x+3y &= 3 \\
\big||x|-|y|\big| &= 1
\end{align*}
\]

Answer: 3
user: A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning 8 times, came home by bus in the afternoon 15 times, and commuted by train (either morning or afternoon) 9 times. Find the total number of working days $x$.

Answer: 16
user: How many positive integers $n < 10^6$ are there such that $n$ is equal to twice the square of an integer and is equal to three times the cube of an integer?

Answer: 3
user: Let $C_{1}$ and $C_{2}$ be two externally tangent circles with diameters $[AB]$ and $[BC]$, and centers $D$ and $E$, respectively. Let $F$ be the intersection point of the tangent line from $A$ to $C_{2}$ and the tangent line from $C$ to $C_{1}$ (both tangent lines are on the same side of $AC$). If $|DB| = |BE| = \sqrt{2}$, find the area of triangle $AFC$. The answer is in the form k\sqrt{m}, where m is the smallest possible integer. Please provide the value of k + m.

Answer: 4
user: Tetrahedron $ABCD$ has the following side lengths: $AB = 5$, $AC = 3$, $BC = 4$, $BD = 4$, $AD = 3$, and $CD = \frac{12}{5}\sqrt{2}$. Find the volume of the tetrahedron. The answer is in the form \(\frac{m}{n}\), please calculate the value of m + n.

Answer: 29
user: In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC = DC = 3$, $OD = 2$, and $AB = 6$, find the radius of the circle. If the radius is \(\sqrt{x}\), what is the value of \(x\)?

Answer: 22
user: Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D \in [AB]$ and $[DC] \perp [BC]$. If $m(\widehat{ABC}) = \alpha$, $m(\widehat{BCA}) = 3\alpha$, and $|AC| - |AD| = 10$, find the length of $|BD|$.

Answer: 20
user: If \(x \ge 0\), then find the value of \(\sqrt{x \sqrt{x \sqrt{x}}}\). The original answer is in the format \(\sqrt[m]{x^{k}}\). Please give the value of \(k + m\).

Answer: 15
user: A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$, she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.

Answer: 2024
user: The adjacent sides of the decagon shown meet at right angles. What is the perimeter of the decagon?

\[ \begin{asy} \defaultpen(linewidth(.8pt)); \dotfactor=4; \dot(origin); \dot((12,0)); \dot((12,1)); \dot((9,1)); \dot((9,7)); \dot((7,7)); \dot((7,10)); \dot((3,10)); \dot((3,8)); \dot((0,8)); \draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8)--(0,8)--cycle); \label("$8$",midpoint(origin--(0,8)),W); \label("$2$",midpoint((3,8)--(3,10)),W); \label("$12$",midpoint(origin--(12,0)),S); \end{asy} \]

Answer: 44
user: For the infinite series \(1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} - \frac{1}{32} + \frac{1}{64} - \frac{1}{128} - \cdots\), let \(S\) be the (limiting) sum. The original answer is in \(\frac{k}{m}\) format, where \(k\) and \(m\) are integers. Please find the value of \(k + m\).

Answer: 9
user: The sequence $a_0, a_1, a_2, \ldots$ is a strictly increasing arithmetic sequence of positive integers such that:
\[
2^{a_7} = 2^{27} \cdot a_7.
\]
What is the minimum possible value of $a_2$?

Answer: 12
user: For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer? Provide your answer as an integer.

Answer: 9
user: A large rectangle is partitioned into four smaller rectangles by two segments parallel to its sides. The areas of three of these rectangles are given. Determine the area of the fourth rectangle.

```
\begin{asy}
draw((0,0)--(10,0)--(10,7)--(0,7)--cycle);
draw((0,5)--(10,5));
draw((3,0)--(3,7));
label("6", (1.5,6));
label("?", (1.5,2.5));
label("14", (6.5,6));
label("35", (6.5,2.5));
\end{asy}
```

Find the area of the fourth rectangle.

Answer: 15
user: What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$? Provide your answer as an integer.

Answer: 20
user: Find the largest number among the following expressions: \(30^{30}\), \(50^{10}\), \(40^{20}\), \(45^{15}\), and \(5^{60}\). The original answer is in the form a^b, where a is the base and b is the exponent. Please provide the sum of a and b.

Answer: 60
user: Let $T$ be the triangle in the coordinate plane with vertices $(0,0)$, $(4,0)$, and $(0,3)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find how many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position.

Answer: 12
user: Find the sum of distinct real roots of the polynomial $x^5 + x^4 - 4x^3 - 7x^2 - 7x - 2$.

Answer: 0
user: A semi-palindrome is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. Find the difference between the largest semi-palindrome and the smallest semi-palindrome.

Answer: 8989
user: What is the least integer $n \geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$? Find the value of $n$.

Answer: 119
user: Triangle $BAD$ is right-angled at $B$. On $AD$, there is a point $C$ such that $AC = CD$ and $AB = BC$. Determine the magnitude of angle $DAB$ in degrees. If the answer is $x^\circ$, what is the value of $x$?

Answer: 60
user: When $\left(1 - \frac{1}{a}\right)^6$ is expanded, find the sum of the last three coefficients.

Answer: 10
user: Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a path length of $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.) If the answer is in the form of \(\arccos\frac{k}{m}\), provide the value of k + m.

Answer: 11
user: Find the number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$.

Answer: 2
user: Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{\left(\frac{n}{2}\right)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the digits of $n$.

Answer: 15
user: A circle centered at $O$ has radius $1$ and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, find the value of $OC$. The original answer is in the form \frac{k}{1+m\sin(\theta)}, please give the value of k + m.

Answer: 2
user: How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3? Fill in the number of such integers.

Answer: 96
user: A house and a store were each sold for $\$12000$. The house was sold at a loss of $20\%$ of its cost, and the store at a gain of $20\%$ of its cost. Find the entire transaction's net loss or gain in dollars.

Answer: 1000
user: Point $F$ is taken on side $AD$ of square $ABCD$. At $C$, a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches, and the area of triangle $CEF$ is $200$ square inches. Determine the number of inches in $BE$.

Answer: 12
user: Let $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1) = 0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. Find the integer $k$.

Answer: 3
user: Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. The ratio of the area of the smaller circle to that of the larger circle is in the form \frac{k}{m}. Please find the value of k + m.

Answer: 10
user: Delete all perfect squares from the sequence $1, 2, 3, \ldots$. Determine the 2003rd number in the remaining sequence.

Answer: 2048
user: In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(BC\) and the median from \(A\) to \(BC\) have the same length. The original answer is in the form \(\sqrt{k}\). Please determine the value of \(k\).

Answer: 2
user: Given a square $ABCD$ with side length $8$ feet, a circle is drawn through vertices $A$ and $D$ and is tangent to side $BC$. Determine the radius of the circle, in feet. What is the value of the radius?

Answer: 5
user: Tom, Dick, and Harry started out on a $100$-mile journey. Tom and Harry traveled by automobile at a speed of $25$ mph, while Dick walked at a speed of $5$ mph. After a certain distance, Harry got off and continued walking at $5$ mph, while Tom went back for Dick and brought him to the destination at the same time that Harry arrived. Find the number of hours required for the trip.

Answer: 8
user: If \(b > 1\), \(\sin x > 0\), \(\cos x > 0\), and \(\log_b \sin x = a\), find the value of \(\log_b \cos x\). The original answer is in the form \(\frac{m}{n} \log_b(1-b^{p})\). Please provide the value of \(m + n + p\).

Answer: 5
user: In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?

Answer: 14
user: Given that $a_{i} \in \{0,1,2,3,4\}$ for every $0 \le i \le 9$, if $6\sum_{i=0}^{9}a_{i} 5^{i} \equiv 1 \pmod{5^{10}}$, find the value of $a_{9}$.

Answer: 4
user: On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?

Answer: 3
user: How many ordered pairs $(p,n)$ are there such that $(1+p)^n = 1+pn + n^p$ where $p$ is a prime and $n$ is a positive integer?

Answer: 1
user: On Halloween, 31 children walked into the principal's office asking for candy. They can be classified into three types: some always lie, some always tell the truth, and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order:

"Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

Answer: 7
user: 在\(\triangle ABC\)中，已知边长\(|AB|=3\)，且从点\(C\)作出的高的长度为\(2\)。求另外两条高的长度乘积的最大值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

Answer: 169
user: If the Highest Common Divisor of $6432$ and $132$ is diminished by $8$, what is the resulting value?

Answer: 4
user: How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5? Provide your answer as an integer.

Answer: 801
user: At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First, bowler #5 competes against bowler #4. The loser receives the 5th prize, and the winner competes against bowler #3 in another game. The loser of this game receives the 4th prize, and the winner competes against bowler #2. The loser of this game receives the 3rd prize, and the winner competes against bowler #1. The winner of this game gets the 1st prize, and the loser gets the 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes? Provide your answer as an integer.

Answer: 16
user: The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how many lockers are there at the school? Provide your answer as an integer.

Answer: 2001
user: What is the median of the following list of $4040$ numbers?

$$1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$$

If the answer is $x$, what is the value of $10x$?

Answer: 19765
user: In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D, E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. Find the perimeter of parallelogram $ADEF$.

Answer: 56
user: Consider two solid spherical balls:

**First Ball**:
 - Centered at $(0, 0, \frac{21}{2})$
 - Radius $6$

**Second Ball**:
 - Centered at $(0, 0, 1)$
 - Radius $\frac{9}{2}$

Determine the number of lattice points (points with integer coordinates $(x, y, z)$) that lie in the intersection of these two spheres.

Answer: 13
user: We define a positive integer as "special" if it is divisible by all of its non-zero digits. Determine the maximum number of consecutive special numbers that can exist.

Answer: 12
user: Find the largest integer not exceeding $\lfloor (n+1)a \rfloor - \lfloor na \rfloor$, where $n$ is a natural number and $a=\frac{\sqrt{2013}}{\sqrt{2014}}$.

Answer: 1
user: The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. Find the net change over the four years, to the nearest percent. If the net change is $x\%$, what is the value of $x$?

Answer: -12
user: A man makes a trip by automobile at an average speed of $50$ mph. He returns over the same route at an average speed of $45$ mph. What is his average speed for the entire trip in miles per hour? Express your answer as a fraction in the form $\frac{k}{m}$ where $k$ and $m$ are integers. Please provide the value of k + m.

Answer: 919
user: If the remainder is $2013$ when a polynomial with coefficients from the set $\{0,1,2,3,4,5\}$ is divided by $x-6$, find the least possible value of the coefficient of $x$ in this polynomial.

Answer: 5
user: Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? The original answer is in the form k\sqrt{m}, please provide the value of k + m.

Answer: 8
user: What is the sum of the digits of the first $2010$ positive integers?

Answer: 28068
user: A circular disk is divided by $2n$ equally spaced radii ($n > 0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided? The original answer is in the form $kn + m$. Please find the value of $k + m$.

Answer: 4
user: A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? Provide your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and find the value of \(k + m\).

Answer: 8
user: A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$?

Answer: 36
user: The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. What is the value of $n$?

Answer: 112
user: The lines \(x = \frac{1}{4}y + a\) and \(y = \frac{1}{4}x + b\) intersect at the point \((1,2)\). The original answer is in \(\frac{k}{m}\) format, please find the value of \(k + m\).

Answer: 13
user: The number of distinct pairs \((x,y)\) of real numbers satisfying both of the following equations: 
\[ \begin{align*} x &= x^2 + y^2, \\ y &= 2xy \end{align*} \]
is \(\underline{\phantom{4}}\).

Answer: 4
user: How many numbers \( (x_1 x_2 \ldots x_{1998}) \) are there such that:

\[
(x_1 x_2 \ldots x_{1998}) = 7 \cdot 10^{1996} (x_1 + x_2 + \ldots + x_{1998})
\]

where \( (x_1 x_2 \ldots x_{1998}) \) shows a number with 1998 digits in the decimal system? Provide your answer as an integer.

Answer: 4
user: Juan and Mary play a two-person game where the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, find the total number of games they played.

Answer: 7
user: Regular octagon \(ABCDEFGH\) has area \(n\). Let \(m\) be the area of quadrilateral \(ACEG\). The answer is in the form \(\frac{\sqrt{k}}{l}\). Find the value of \(k + l\).

Answer: 4
user: Find the number of scalene triangles having all sides of integral lengths, and a perimeter less than $13$.

Answer: 3
user: If \(10^{2y} = 25\), then find the value of \(10^{-y}\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 6
user: $ABCDE$ is a convex pentagon. Given the following measurements: \( m(\widehat{B}) = m(\widehat{D}) = 90^\circ \), \( m(\widehat{C}) = 120^\circ \), \( |AB| = 2 \), \( |BC| = |CD| = \sqrt{3} \), and \( |ED| = 1 \). Find the length of \( |AE| \). The original answer is in the form \(\sqrt{k}\), please give the value of \(k\).

Answer: 3
user: Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt{5}$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt{5}$. The original answer for the length of $|AD|$ is in the form $k\sqrt{m}$. Please find the value of k + m.

Answer: 19
user: $A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, find the circumference of the track in yards.

Answer: 480
user: Let $r$ be the speed in miles per hour at which a wheel, 11 feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second, the speed $r$ is increased by 5 miles per hour. What is the value of $r$?

Answer: 10
user: Let $ABC$ be a triangle such that $|BC| = 7$ and $|AB| = 9$. If $m(\widehat{ABC}) = 2m(\widehat{BCA})$, then find the area of the triangle. The original answer is in the form of $m\sqrt{n}$, please give the value of m + n.

Answer: 19
user: For how many different values of integer $n$, can one find $n$ different lines in the plane such that each line intersects with exactly $2004$ of the other lines? Provide your answer as an integer.

Answer: 12
user: In Cartesian coordinates, two areas \(M\) and \(N\) are defined as follows: 
- \(M: y \geq 0, y \leq x, y \leq 2-x\); 
- \(N: t \leq x \leq t+1\). 
Here, \(t\) is a real number such that \(t \in [0,1]\). 

Determine the area of \(M \cap N\). The original answer is in the form \(-at^2 + bt + \frac{c}{d}\), where \(\frac{c}{d}\) is a simplified fraction. Please give the value of \(a + b + c + d\).

Answer: 5
user: The graph of \( y = x^2 + 2x - 15 \) intersects the \( x \)-axis at points \( A \) and \( C \) and the \( y \)-axis at point \( B \). Find the value of \( \tan(\angle ABC) \). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

Answer: 11
user: How many ordered integer pairs \((a, b)\) satisfy the equation \(a^2b + ab^2 = 2009201020092010\)?

Answer: 0