sungyub/dapo-math-17k-verl huggingface dataset:
user: Evaluate the following sum: 
\[ \binom{n}{1} \sin(a) + \binom{n}{2} \sin(2a) + \ldots + \binom{n}{n} \sin(na) \]
Find the value of the expression in terms of \( n \) and \( a \). The answer is in the form \( k \cdot m^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{n \cdot a}{2}\right) \). Please provide the sum of k and m.

Answer: 3
user: Let $K$ be the number of sequences $A_1$, $A_2$, \dots, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of \{1, 2, 3, \dots, 10\}, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, \{\}, \{5, 7\}, \{2, 5, 7\}, \{2, 5, 7\}, \{2, 5, 6, 7, 9\} is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?

Answer: 5
user: On a certain math exam, $10\%$ of the students got 70 points, $25\%$ got 80 points, $20\%$ got 85 points, $15\%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?

Answer: 1
user: A softball team played ten games, scoring $1, 2, 3, 4, 5, 6, 7, 8, 9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

Answer: 45
user: How many pairs of positive integers $(a,b)$ are there such that the roots of the polynomial $x^2-ax-b$ are not greater than $5$? Provide the number of such pairs.

Answer: 50
user: Find the sum of the digits of the largest even three-digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged.

Answer: 25
user: Regular hexagon \(ABCDEF\) has an area of \(1\). Starting with edge \(AB\) and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side \(AB\), the new point, \(G\), is drawn so that \(AG = \frac{1}{2} AB\). This forms hexagon \(GHIJKL\), as shown. What is the area of this new hexagon? The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\).

Answer: 7
user: In triangle $ABC$, lines $CE$ and $AD$ are drawn such that \( \frac{CD}{DB} = \frac{3}{1} \) and \( \frac{AE}{EB} = \frac{3}{2} \). Let \( r = \frac{CP}{PE} \), where $P$ is the intersection point of $CE$ and $AD$. Find the value of \( r \).

Answer: 5
user: If $a, b,$ and $c$ are digits for which \[ \begin{array}{cccc} & 7 & a & 2 \\ - & 4 & 8 & b \\ \hline & c & 7 & 3 \end{array} \] then find the value of $a + b + c$.

Answer: 17
user: How many ordered pairs of integers $(x,y)$ are there such that $2011y^2 = 2010x + 3$?

Answer: 0
user: In base $R_1$, the expanded fraction $F_1$ becomes $0.373737\ldots$, and the expanded fraction $F_2$ becomes $0.737373\ldots$. In base $R_2$, fraction $F_1$, when expanded, becomes $0.252525\ldots$, while fraction $F_2$ becomes $0.525252\ldots$. Find the sum of $R_1$ and $R_2$, each written in base ten.

Answer: 19
user: Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$?

Answer: 68
user: Let's write \(p\), \(q\), and \(r\) as three distinct prime numbers, where 1 is not a prime. Find the smallest positive perfect cube that leaves \(n = pq^2r^4\) as a divisor. The original answer is in the form \((p^a q^b r^c)^3\), please give the value of a + b + c.

Answer: 4
user: How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order? Provide your answer as an integer.

Answer: 34
user: In the adjoining figure, $AB$ and $BC$ are adjacent sides of square $ABCD$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $BC$. The lines $AN$ and $CM$ intersect at $O$. Determine the ratio of the area of quadrilateral $AOCD$ to the area of square $ABCD$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 5
user: What is the sum of real roots of the equation $x^4-8x^3+13x^2 -24x + 9 = 0$?

Answer: 7
user: Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is the area of the hexagon? If the area is expressed as $x\sqrt{3}$, what is the value of $x$?

Answer: 25
user: In triangle $ABC$, $AB = AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP = PC = CB$, then if the answer is $x^\circ$, what is the value of $x$?

Answer: 36
user: If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, find the value of $m$. The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 1
user: What is the perimeter of the smallest rectangle with integer side lengths that fits three non-overlapping squares with areas $4$, $9$, and $16$?

Answer: 24
user: Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day?

Answer: 100
user: Find the last 5 digits of the number $M = 5^{2010}$.

Answer: 65625
user: A mass is attached to an ideal spring. At time \(t = 0\) the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is \(T\). At what time \(t\) is the power delivered to the mass by the spring first a maximum, in terms of \(T\)? The answer is in the form \(\frac{kT}{m}\). Please find the value of \(k + m\).

Answer: 9
user: Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.

Answer: 13
user: First, $a$ is chosen at random from the set $\{1, 2, 3, \ldots, 99, 100\}$, and then $b$ is chosen at random from the same set. Find the probability that the integer $3^a + 7^b$ has a units digit of $8$. The original answer is in the form of a common fraction \(\frac{k}{m}\). Please provide the value of k + m.

Answer: 19
user: Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?

Answer: 803
user: An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? Write your answer as an integer.

Answer: 121
user: A square $ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $E, F, G$ where $E$ is the midpoint of $BC$, $F$ and $G$ are on $AB$ and $CD$, respectively. They are positioned such that $AF < FB$, $DG < GC$, and $F$ is directly opposite $G$. If $FB = x$, the length of the longer parallel side of each trapezoid, find the value of $x$. The original answer is in the form of \frac{k}{m}, please give the value of k + m.

Answer: 11
user: The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a negligible height above the equator, find the best estimate of the number of hours of flight.

Answer: 50
user: Find the number of distinct integral solutions of $x^{4} + 2x^{3} + 3x^{2} - x + 1 \equiv 0 \pmod{30}$ where $0 \leq x < 30$.

Answer: 1
user: What is the minimum value of \((x^2+2x+8-4\sqrt{3})\cdot(x^2-6x+16-4\sqrt{3})\) where \(x\) is a real number? The answer should be in the form \(a - b\sqrt{3}\). Please find the value of \(a + b\).

Answer: 176
user: A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[
a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}
\]
for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q$.

Answer: 8078
user: Circles with centers \(O\) and \(P\) have radii 2 and 4, respectively, and are externally tangent. Points \(A\) and \(B\) are on the circle centered at \(O\), and points \(C\) and \(D\) are on the circle centered at \(P\), such that \(\overline{AD}\) and \(\overline{BC}\) are common external tangents to the circles. Find the area of hexagon \(AOBCPD\). The original answer is in the form \(k\sqrt{m}\). Please give the value of \(k + m\).

Answer: 26
user: $(a_n)_{n = 1}^\infty$ is defined on real numbers with $a_n \neq 0$, $a_n a_{n + 3} = a_{n + 2} a_{n + 5}$ and $a_1 a_2 + a_3 a_4 + a_5 a_6 = 6$. Find the value of $a_1 a_2 + a_3 a_4 + \cdots + a_{41} a_{42}$.

Answer: 42
user: Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is the length of $|BC|$?

Answer: 8
user: The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a \(0.5\) probability of winning any game), find the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 21
user: Let \(f(x) = \frac{x+1}{x-1}\). Then for \(x^2 \neq 1\), find the expression for \(f(-x)\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 2
user: A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. If the measure of the angle subtended at the center of the circle by the shortest side is $x^\circ$, what is the value of $x$?

Answer: 60
user: A point \(P\) lies in the same plane as a given square with side length \(1\). The vertices of the square, taken counterclockwise, are labeled as \(A\), \(B\), \(C\), and \(D\). The distances from point \(P\) to the vertices \(A\), \(B\), and \(C\) are \(u\), \(v\), and \(w\), respectively. Determine the greatest distance that \(P\) can be from vertex \(D\), given that \(u^2 + v^2 = w^2\). The original answer is in the form \(k + \sqrt{m}\). Please provide the value of \(k + m\).

Answer: 4
user: A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? Provide your answer in the form of \(\frac{m}{n}\) and find the value of m + n.

Answer: 20922789888256
user: Suppose that \(\cos(3x) + 3\cos(x) = -2\). Find the value of \(\cos(2x)\). The original answer is in the form \(\sqrt[3]{a} - b\). Please provide the value of a + b.

Answer: 3
user: At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, find the number of campers who are in exactly two classes.

Answer: 12
user: If Scott rolls four fair six-sided dice, find the probability that he rolls more 2's than 1's. The probability is given in the form \(\frac{m}{n}\). Please calculate the value of m + n.

Answer: 1717
user: If \(\frac{x}{x-1} = \frac{y^2+2y-1}{y^2-2y-2}\), find the value of \(x\). The original answer can be expressed in the form \(ay^2 + by + c\). Please find the value of \(a + b + c\).

Answer: 2
user: How many real roots of the equation \[x^2 - 18\lfloor x \rfloor + 77 = 0\] are not integer, where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\)?

Answer: 3
user: If \( |x - \log y| = x + \log y \) where \( x \) and \( \log y \) are real, find the expression that represents the relationship between \( x \) and \( y \). The original answer is in the form \( mx(y-n)=p \). Please provide the value of \( m+n+p \).

Answer: 2
user: The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. What is the sum of $d$ and $e$?

\[
\begin{array}{ccccccc}
& 7 & 4 & 2 & 5 & 8 & 6 \\
+ & 8 & 2 & 9 & 4 & 3 & 0 \\
\hline
1 & 2 & 1 & 2 & 0 & 1 & 6 \\
\end{array}
\]

Answer: 8
user: Find the first step in finding the product \((3x + 2)(x - 5)\) by use of the distributive property in the form \(a(b + c) = ab + ac\). The original answer is in the form \((a+b)x + (a+b)(c)\). Please give the value of a + b + c.

Answer: 0
user: The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)

Answer: 100
user: Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find the value of $m + n$.

Answer: 128
user: In \(\triangle ABC\), point \(D\) is on \(AC\) and point \(F\) is on \(BC\). It is given that \(AB \perp AC\), \(AF \perp BC\), and \(BD = DC = FC = 1\). Find the length of \(AC\). The original answer is in \(\sqrt[k]{m}\) format, please give the value of k + m.

Answer: 5
user: For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3 + ax^2 + bx + 6$ have 3 distinct integer roots?

Answer: 5
user: A telephone number has the form $ABC - DEF - GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find the value of $A$.

Answer: 8
user: Halfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, with other possible scores being $8$, $4$, $2$, and $0$ points. Chelsea always scores at least $4$ points on each shot. If Chelsea's next $n$ shots are bullseyes, she will be guaranteed victory. What is the minimum value for $n$?

Answer: 42
user: Equilateral $\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $T$ and tangent to sides $AB$ and $AC$ at points $P$ and $Q$. If side $BC$ has length $12$, find the length of segment $PQ$.

Answer: 8
user: A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. Find the number of ordered triples $(a,b,c)$ that are possible.

Answer: 10
user: At the beginning of the school year, $50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$? If the answer is $y\%$, what is the value of $y$?

Answer: 60
user: Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. Find the maximum number of elements in the range set of $f$.

Answer: 7
user: After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents. If the answer is \( x \) cents, what is the value of \( x \)?

Answer: 11
user: From a two-digit number $N$, we subtract the number with the digits reversed and find that the result is a positive perfect cube. How many values of $N$ satisfy this condition?

Answer: 7
user: Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$.

Answer: 30
user: If $\frac{8}{19}$ of the product of the largest two elements of a positive integer set is not greater than the sum of the other elements, what is the minimum possible value of the largest number in the set?

Answer: 13
user: How many positive perfect squares less than $2023$ are divisible by $5$? Fill in the blank with the number of such perfect squares.

Answer: 8
user: Let \(ABCD\) be a regular tetrahedron and let \(E\) be a point inside the face \(ABC\). Denote by \(s\) the sum of the distances from \(E\) to the faces \(DAB\), \(DBC\), \(DCA\), and by \(S\) the sum of the distances from \(E\) to the edges \(AB\), \(BC\), \(CA\). The original answer is in \(\frac{k\sqrt{m}}{n}\) format. Please give the value of \(k + m + n\).

Answer: 7
user: If five boys and three girls are randomly divided into two four-person teams, find the probability that all three girls will end up on the same team. Express your answer in the form of \(\frac{k}{m}\) where the fraction is simplified, and provide the value of k + m.

Answer: 8
user: If $ab \neq 0$ and $|a| \neq |b|$, find the number of distinct values of $x$ satisfying the equation
\[ \frac{x-a}{b} + \frac{x-b}{a} = \frac{b}{x-a} + \frac{a}{x-b} \].

Answer: 3
user: Let $ABCD$ be a convex quadrilateral such that $m(\widehat{ABD})=40^\circ$, $m(\widehat{DBC})=70^\circ$, $m(\widehat{BDA})=80^\circ$, and $m(\widehat{BDC})=50^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{CAD})$?

Answer: 30
user: For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. Find how many different values of $p < 2015$ are possible.

Answer: 31
user: Circles $A$, $B$, and $C$ each have a radius of $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. Find the area inside circle $C$ but outside circle $A$ and circle $B$.

Answer: 2
user: Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor \sqrt{x} \rfloor = 12$, find the probability that $\lfloor \sqrt{100x} \rfloor = 120$. ($\lfloor v \rfloor$ means the greatest integer less than or equal to $v$.) The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 2741
user: If $x^4 + 4x^3 + 6px^2 + 4qx + r$ is exactly divisible by $x^3 + 3x^2 + 9x + 3$, find the value of $(p + q)r$.

Answer: 15
user: For how many integer values of $k$ do the graphs of $x^2 + y^2 = k^2$ and $xy = k$ \textit{not} intersect?

Answer: 2
user: Circles with centers \(P\), \(Q\), and \(R\), having radii \(1\), \(2\), and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P'\), \(Q'\), and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{a}-\sqrt{b}\), please provide the value of \(a+b\).

Answer: 8
user: Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. Find the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$.

Answer: 6
user: A 16-step path is to go from $( -4, -4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. Find the number of such paths that stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.

Answer: 1698
user: How many solutions does the equation $\tan{(2x)} = \cos{\left(\frac{x}{2}\right)}$ have on the interval $[0, 2\pi]$? Write your answer as an integer.

Answer: 5
user: In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circles $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$, and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, find the length of chord $EF$.

Answer: 24
user: A rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, find the expression for $x$ in terms of $b$ and $h$. The original answer is in the form \frac{kb}{mh+b}, please give the value of k + m.

Answer: 3
user: If \(x \in \left[-\frac{5\pi}{12}, -\frac{\pi}{3}\right]\), find the maximum value of \(y = \tan\left(x + \frac{2\pi}{3}\right) - \tan\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right)\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). Please provide the value of \(k + m + n\).

Answer: 20
user: If $n$ is any whole number, find the largest number by which $n^2(n^2 - 1)$ is always divisible.

Answer: 12
user: For every \( x, y \in \mathbb{R}^+ \), the function \( f: \mathbb{R}^+ \to \mathbb{R} \) satisfies the condition \( f(x) + f(y) = f(x)f(y) + 1 - \frac{1}{xy} \). If \( f(2) < 1 \), find the value of \( f(3) \). The original answer is in \(\frac{a}{b}\) format, please give the value of a + b.

Answer: 5
user: What is the sum of the real roots of the equation \(4x^4 - 3x^2 + 7x - 3 = 0\)? Please provide your answer in the form of an integer \(k\), where \(k\) is the sum of the real roots.

Answer: -1
user: Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing?

Answer: 7
user: For each $x$ in $[0,1]$, define
\[ f(x) = \begin{cases} 2x, & \text{if } 0 \leq x \leq \frac{1}{2}; \\
2 - 2x, & \text{if } \frac{1}{2} < x \leq 1. \end{cases} \]
Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. Find the number of values of $x$ in $[0,1]$ for which $f^{[2005]}(x) = \frac{1}{2}$. The original answer is in the form of $k^m$, please provide the value of $k + m$.

Answer: 2007
user: How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$? Find the total number of such integers.

Answer: 133
user: If $n$ is a real number, then the simultaneous system of equations:
\[
\begin{align*}
 nx + y &= 1 \\
 ny + z &= 1 \\
 x + nz &= 1
\end{align*}
\]
has no solution if and only if $n$ is equal to $x$. Please provide the absolute value of $x$ as your final answer.

Answer: 1