inclusionAI/Ring-lite-rl-data huggingface dataset:
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A convex quadrilateral is drawn on a board. Alyosha claims that it can be divided by a diagonal into two acute triangles. Borya claims that it can be divided into two right triangles, and Vasya claims that it can be divided into two obtuse triangles. It turns out that exactly one of them is wrong. About whom can we be certain that he is correct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
In $\triangle ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a, b$ are coprime positive integers, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Find all the positive integers  $n$  so that  $n$  has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 48, 72}
Given the function $f(x)=3\sin(2x-\frac{π}{3})-2\cos^{2}(x-\frac{π}{6})+1$, the graph of function $f(x)$ is shifted to the left by $\frac{π}{6}$ units, resulting in the graph of function $g(x)$. Find $\sin (2x_{1}+2x_{2})$, where $x_{1}$ and $x_{2}$ are the two roots of the equation $g(x)=a$ in the interval $[0,\frac{π}{2}]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{5}}
During a two-week sale, the price of a pair of shoes is marked down by 13% for the first week and then further marked down by 8% for the second week from the new discounted price of the first week. The original tax rate on the shoes is 7% before the sale begins. After the sale ends, the price goes back to the original price, but with a new increased tax rate of 9%. Calculate the overall percentage increase from the final sale price (including discounts and original tax rate) back to the post-sale regular price (including the new tax rate) to the nearest percent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27\%}
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square  $\sqrt{5}$  away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability  $p$ . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of  $p$  that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Let \( a \) and \( b \) be positive integers. The quotient of \( a^{2} + b^{2} \) divided by \( a + b \) is \( q \), and the remainder is \( r \), such that \( q^{2} + r = 2010 \). Find the value of \( ab \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1643}
Find the least degree of a polynomial $p(x)$, monic with integer coefficients, such that $p(n)$ is a multiple of 2021 for every positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
In the diagram, \(PRTY\) and \(WRSU\) are squares. Point \(Q\) is on \(PR\) and point \(X\) is on \(TY\) so that \(PQXY\) is a rectangle. Also, point \(T\) is on \(SU\), point \(W\) is on \(QX\), and point \(V\) is the point of intersection of \(UW\) and \(TY\), as shown. If the area of rectangle \(PQXY\) is 30, calculate the length of \(ST\). Express your answer as a single number, rounded to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.48}
For  $n \geq 1$ , let  $a_n$  be the number beginning with  $n$   $9$ 's followed by  $744$ ; eg.,  $a_4=9999744$ . Define   $$ f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \} $$  , for  $n\geq 1$ . Find  $f(1)+f(2)+f(3)+ \cdots + f(10)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \mathrm{~cm}$, and the son's step length is $60 \mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What distance did they travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \, \text{km} \, 440 \, \text{m}}
How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the length of the real axis of the hyperbola. Find the eccentricity of this hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
已知$a<0,b\leq 0,c>0$，且$\sqrt { b ^ { 2 } - 4 a c } = b - 2 a c$，求$b ^ { 2 } - 4 a c$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In $\triangle ABC$, $AB=10$, $AC=8$, and $BC=6$. Circle $P$ passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection of circle $P$ with sides $AC$ and $BC$ (excluding $C$). What is the length of segment $QR$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.8}
Compute the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_{15} \le 3005$ such that $b_i-i$ is odd for $1\le i \le 15$. Express your answer as ${p \choose q}$ for some integers $p > q$ and find the remainder when $p$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{509}
Among all the five-digit numbers formed by digits 1, 2, 3, 4, 5 without repetition, count how many numbers satisfy the "wave number" pattern, where the digits follow the alternating sequence "less-greater-less-greater...less-greater". Provide your answer as a single integer representing the total count of such "wave numbers".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
An entry in a grid is called a saddle point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $3 \times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that this grid has at least one saddle point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{10}}
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 $BE+DE+CD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Calculate the number of ways of choosing $4$ numbers from the set ${1,2,\cdots ,11}$ such that at least $2$ of the numbers are consecutive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{260}
Given the line $l: x-y+2=0$ intersects the $x$-axis at point $A$, and point $P$ is on line $l$. There is exactly one point $B$ on the circle $C: (x-2)^{2}+y^{2}=2$ such that $AB \perp BP$. The set of possible values for the $x$-coordinate of point $P$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{ \dfrac {1}{3},5\right\}}
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4027}
今有官司依立方招兵，初招方面三尺，次招方面转多一尺，每人日支钱二百五十文.已招二万三千四百人，支钱二万三千四百六十二贯.问招来几日？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given  $w$  and  $z$  are complex numbers such that  $|w+z|=1$  and  $|w^2+z^2|=14$ , find the smallest possible value of  $|w^3+z^3|$ . Here  $| \cdot |$  denotes the absolute value of a complex number, given by  $|a+bi|=\sqrt{a^2+b^2}$  whenever  $a$  and  $b$  are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41}{2}}
In the village of Bol'shie Vasyuki, there is a 50-meter long embankment along the river. A boat, which is 10 meters long, passes by it in 5 seconds when going downstream and in 4 seconds when going upstream. How many seconds will it take for a paper boat to travel from one end of the embankment to the other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 \, \text{s}}
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Let  $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$  

For a real number  $a$ , find  $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
You are given a $10 \times 2$ grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
A square is inscribed in a unit circle, then a circle is inscribed in this square, followed by inscribing another square in this circle. This process is repeated infinitely. Let \( C \) represent the total area of all circles and \( S \) represent the total area of all squares. Calculate the ratio \( \frac{C}{S} \). Express your answer as a mathematical expression or simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
Rachel has the number 1000 in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1$. When she puts the number $x$ in her right pocket, the number changes to $x^{-1}$. Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E$, then compute $\left\lfloor\frac{E}{10}\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet. Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs,  $T$ . Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then,  $T$  can be expressed in the form  $m\sqrt{n}$ , where  $n$  is not divisible by the square of a prime. Find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider the polynomial  $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$ . Find a nonzero polynomial whose roots are the reciprocal values of the roots of  $p(x)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2x + 3x^2 + \ldots + 2000x^{1999}}
Let \( Z \) be as in problem 15. Let \( X \) be the greatest integer such that \( |X Z| \leq 5 \). Find \( X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\circ$. What is the sum of the four angles shaded in gray, in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{146}
Given that $f(x)$ is an even function on $\mathbb{R}$, and for any $x_1, x_2 \in [0, +\infty)$ with $x_1 \neq x_2$, it holds that $(x_1 - x_2)[f(x_1)-f(x_2)] < 0$, find the relationship in magnitude among $f(-2)$, $f(0)$, and $f(3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(0) > f(-2) > f(3)}
For how many positive integers $k$ do the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at a point whose coordinates are positive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Alice and Bob play the following game: starting with the number $2$ written on a blackboard, each player in turn changes the current number $n$ to a number $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{22...2}_{2020}$. Assuming perfect play, who will win the game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Alice}}
Let $x, y$, and $N$ be real numbers, with $y$ nonzero, such that the sets $\left\{(x+y)^{2},(x-y)^{2}, x y, x / y\right\}$ and $\{4,12.8,28.8, N\}$ are equal. Compute the sum of the possible values of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85.2}
Vasya had 101 real coins. One of them was replaced with a fake one (it differs in weight, but it's unknown whether it is lighter or heavier than a real one). Vasya wants to find as many real coins as possible in one weighing using a balance scale without weights. What should he do and how many real coins can he find?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50 \text{ genuine coins}}
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f(x)$ be an odd function defined on $\mathbb{R}$ such that $f(x+2)=\frac{1}{f(x)}$ for any $x\in\mathbb{R}$. When $x\in[-2,0)$, $f(x)=\log_2(x+3)$. Find the value of $f(2017)-f(2015)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
把编号为1~20的20张卡片，按小号在上，大号在下的顺序叠放在一起，然后将1号卡片扔掉，2号卡片放到最后，3号卡片扔掉，4号卡片放到最后，依次下去，当手中最后只剩下一张卡片时，这张卡片编号是______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{752}
Consider a trapezoid field with base lengths of 120 meters and 180 meters and non-parallel sides each measuring 130 meters. The angles adjacent to the longer base are $60^\circ$. At harvest, the crops at any point in the field are brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let  $a_1, a_2, \ldots$  and  $b_1, b_2, \ldots$  be sequences such that  $a_ib_i - a_i - b_i = 0$  and  $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$  for all  $i \ge 1$ .  If  $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$ , then what is  $b_{6}$ ?

*Proposed by Andrew Wu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{257}
Find all integer values can take  $n$  such that   $$ \cos(2x)=\cos^nx - \sin^nx $$   for every real number  $x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \in \{2, 4\}}
Number $125$ is written as the sum of several pairwise distinct and relatively prime numbers, greater than $1$. What is the maximal possible number of terms in this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find all odd natural numbers greater than 500 but less than 1000, for which the sum of the last digits of all divisors (including 1 and the number itself) is equal to 33.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{729}
1. Find the number of natural solutions to the equation

$$
\left[\frac{x}{2010}\right]=\left[\frac{x}{2011}\right]+1
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010\cdot2011}
$\triangle ABC$ ,  $\angle A=23^{\circ},\angle B=46^{\circ}$ . Let  $\Gamma$  be a circle with center  $C$ , radius  $AC$ . Let the external angle bisector of  $\angle B$  intersects  $\Gamma$  at  $M,N$ . Find  $\angle MAN$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
The cells of a  $100 \times 100$  table are colored white. In one move, it is allowed to select some  $99$  cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
In a far-off land, three fish can be traded for two loaves of bread, and one loaf of bread can be traded for four bags of rice. Determine the value of one fish in terms of bags of rice. Express your answer as a mixed number or an improper fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \frac{2}{3}}
Given that the inequality \(\left[\left(\log _{3} m\right)^{2}-\log _{3}\left(27 m^{2}\right)\right] x^{2}-\left(\log _{3} m-3\right) x-1<0\) holds for all real numbers \(x\), find the range of the real number \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(3^{-\frac{1}{5}}, 27\right]}
Mona plays a multiplayer video game on the weekends. She joins groups with four other players to complete objectives in the game. The past weekend, she joined 25 groups. In 8 of those groups, she encountered two players she had grouped with before that weekend, and in 5 different groups, she encountered one player she had grouped with before. How many unique players did Mona group with on the video game that weekend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Given that \(a, b, c\) are non-zero rational numbers and satisfy \(a b^{2}=\frac{c}{a}-b\), then \[\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{202}}
An electrician was called to repair a string of four serially connected light bulbs, one of which burnt out. It takes 10 seconds to unscrew any bulb from the string and 10 seconds to screw one in. The time spent on other actions is negligible. What is the minimum amount of time the electrician needs to find the burnt-out bulb, if he has one spare bulb?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60 \text{ seconds}}
6. A natural number $k$ is said to be $n$-squared if, by coloring the cells of a $2 n \times k$ chessboard with $n$ different colors in any way, there exist 4 distinct cells of the same color whose centers are the vertices of a rectangle with sides parallel to the sides of the chessboard.

Determine, as a function of $n$, the smallest natural number $k$ that is $n$-squared.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^{2}-n+1}
Determine the ramification index of the prime ideal $(p)$ in the cyclotomic field $\mathbb{Q}(\zeta_{p^k})$, where $p$ is a prime in $\mathbb{Z}$ and $k \geq 1$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^{k-1}(p-1)}
In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$ given the following conditions:
- $AB \cdot \tan(\angle BCD) = 20$
- $CD = 13$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{260}
3.30. The lateral edge of a regular triangular prism is equal to the side of the base. Find the angle between the side of the base and the non-intersecting diagonal of the lateral face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos\frac{\sqrt{2}}{4}}
Calculate the limit

$$
\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+\operatorname{arctg} 4 x}-\sqrt[3]{1-\operatorname{arctg} 4 x}}{\sqrt{1-\arcsin 3 x}-\sqrt{1+\operatorname{arctg} 3 x}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{8}{9}}