inclusionAI/Ring-lite-rl-data huggingface dataset:
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?

Clarification: If  $A$  is a friend of  $B$  then  $B$  is a friend of  $A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the number $ABCC$ (written in decimal system) given that

$$
ABCC = (DD - E) \cdot 100 + DD \cdot E
$$

where $A, B, C, D,$ and $E$ are distinct digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1966}
If $x + y^{-99} = 3$ and $x + y = -2$, find the sum of all possible values of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-195}
在四棱锥 E-ABCD 中，一只蚂蚁在 0 时刻从顶点 A 出发，在 t 时刻时，蚂蚁会等概率地从 t 时刻所在顶点移向与之相邻的某一顶点，并且在 t+1 时刻到达该顶点. 记 P_E(t) 为 t 时刻蚂蚁恰好在 E 点处的概率，I=\displaystyle\lim_{t\rightarrow+\infty}P_E(t)，则 [100I]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
There are two spheres on a table, touching each other externally. A cone touches the table with its lateral surface and also touches both spheres externally. The apex of the cone lies on the segment connecting the points where the spheres touch the table. It is known that the rays connecting the apex of the cone with the centers of the spheres form equal angles with the table. Find the maximum possible angle at the apex of the cone. (The angle at the apex of the cone is defined as the angle between its generatrices in the axial section.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \operatorname{arcctg} 2}
# Task 7. (14 points)

In a company, there are 168 employees. Among any four people, at least one can be chosen who is acquainted with the other three. What is the minimum possible number of people who are acquainted with everyone? #
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
设 $a, b, c$ 在同一平面内的三个单位向量，且 $\boldsymbol{a} \perp \boldsymbol{b}$，求 $(c-a) \cdot(c-b)$ 的最大值。原始答案的形式为 $k + \sqrt{m}$，请给出 $k + m$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
3. Let $E$ be a point on the side $AB$ of square $ABCD$ such that $|AE|=3|EB|$, and let $F$ be a point on the side $DA$ such that $|AF|=5|FD|$. Denote the intersection of segments $DE$ and $FC$ by $K$, the intersection of $DE$ and $BF$ by $L$, and the intersection of $FB$ and $EC$ by $M$. Let $p_{1}$ be the sum of the areas of triangles $EML$ and $DKC$, and $p_{2}$ be the sum of the areas of triangles $FLK$ and $MBC$. Determine the ratio $p_{1}: p_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
给定整数 $n \geq 2$. 在 $n \times n$ 方格表的每个小方格中各填入一个不超过 $n$ 的正整数, 使得每一行填入的数从左至右不减, 每一列填入的数从上至下不减。若两个有公共边的小方格填入的数相同，则称这两个（无序的）小方格为一个"好对"。求好对个数的最小可能值。当 $n = 4$ 时，求最小可能的好对个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If the function $f(x)=x^{3}+mx^{2}+x+2023$ has no extreme points on $R$, then the range of real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\sqrt{3}, \sqrt{3}]}
Define the two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations:
- $a_{n+1} = 3a_n + b_n$
- $b_{n+1} = 3b_n - a_n$

for all nonnegative integers $n$. Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31$, respectively. Compute $100r + s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3010}
Palm oil has increased in price by 10%. As a result, the cheese produced by one of the manufacturers increased in price by 3%. What percentage of palm oil is in this manufacturer's cheese?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
2.2. Given a convex pentagon $A B C D E$, such that

$$
A B=A E=D C=B C+D E=1 \text { and } \angle A B C=D E A=90^{\circ} .
$$

What is the area of this pentagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given $n$ points on a plane ($n>2$), no three of which are collinear, in how many distinct ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} n(n-1)}
Under the call for the development of the western region by the country, a certain western enterprise received a $4$ million yuan interest-free loan for equipment renewal. It is predicted that after the equipment renewal, the income of the enterprise in the first month is $200,000$. In the following $5$ months, the income of each month increases by $20\%$ compared to the previous month. Starting from the $7$th month, the income of each month increases by $20,000$ more than the previous month. Then, starting from the use of the new equipment, the enterprise needs ______ months to repay the $4$ million interest-free loan with the income obtained. (Round the result to the nearest whole number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
An insect moves on the ellipse $2x^2 + y^2 = 3$ in the clockwise direction at a constant speed of 3 cm/s on the $xy$-plane. The temperature function experienced by the insect is $T(x, y) = 3x^2 - 2xy$ (in °C), with $x$ and $y$ measured in cm. Determine the rate of change of temperature (in °C/s) when the insect is at the point $(1, 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{24}{\sqrt{5}}\)}
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn  on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3030}
Let $ABCD$ be a trapezoid with $AB$ and $CD$ parallel, $\angle D = 2 \angle B$, $AD = 5$, and $CD = 2$. Find the length of $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
With one mighty blow, Maria cracked open the pinata, and countless candies spilled all over the floor. There were 60 red candies, 30 less than three times as many yellow candies as red candies, and a quarter as many blue candies as twice the number of yellow candies. Additionally, there were 40 green candies, which was thrice the number of purple candies, and 1.5 times the number of red candies. Finally, 15 silver candies rolled to the side, and there were half as many gold candies as silver candies.

If Carlos ate all of the yellow candies, 75% of the green candies, and a third of the blue candies, how many candies remained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Find the smallest number \( n > 1980 \) such that the number

$$
\frac{x_{1} + x_{2} + x_{3} + \ldots + x_{n}}{5}
$$

is an integer for any given integer values \( x_{1}, x_{2}, x_{3}, \ldots, x_{n} \), none of which is divisible by 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1985}
Given the function $f(x) = 2\sin(\omega x + \phi) + 1$ with $\omega > 0$ and $|\phi| \leq \frac{\pi}{2}$, the shortest distance between the two nearest intersection points of its graph and the line $y=2$ is $\frac{\pi}{3}$. If $f(x) > 1$ holds for all $x \in \left(-\frac{\pi}{8}, \frac{\pi}{3}\right)$, determine the range of $\phi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{\pi}{4}, \frac{\pi}{3}]}
$n$ teams are to hold a double round robin tournament with home and away games (each pair of teams plays 2 games, with each team having one home game). Each team can play multiple away games in a week (from Sunday to Saturday). However, if a team has a home game in a week, it cannot have any away games in that same week. If all games can be completed within four weeks, find the maximum value of $n$.

Note: A game between teams $A$ and $B$ held at $A$'s home venue is called a home game for $A$ and an away game for $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the maximum possible value of $(a+b+c)$ for an ordered triple $(a, b, c)$ of positive integers that satisfy the following conditions:

* $c \ge b\geq a$
* gcd$(a, b, c) = 1$
* $(a^n + b^n + c^n)$ is divisible by $(a +b+ c)$ for all positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $S$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Let us call two subgroups $H$ and $K$ belonging to $S$ as equivalent if they are conjugate (that is, there exists $g \in G$ such that $gHg^{-1}=K$). How many equivalence classes are there in $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{30}}
If $a$ and $b$ are digits for which
$\begin{array}{ccc}& 2 & a\ \times & b & 3\ \hline & 6 & 9\ 9 & 2 & \ \hline 9 & 8 & 9\end{array}$
then $a+b =$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A "Rock, Paper, Scissors" tournament is being held in a single-elimination format with 16 players. The organizers have determined the matchups for the first round, and the winners of each match will play against one another in the following rounds until a champion is determined. Each player will make the same move (rock, paper, or scissors) throughout all their games, which they choose randomly before the first game. How many choices do the players have so that the tournament concludes with a winner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \times 2^{15}}
Vasya needs to write one digit on each face of several cubes in such a way that any ordered combination of three digits from 000 to 999 inclusive can be obtained by selecting some three different cubes and placing them with suitable faces up in the correct order. Note that the digits 6 and 9 are not considered interchangeable when rotated by 180 degrees. What is the minimum number of cubes that Vasya must use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $BC = 6$, $BX = 3$, $CX = 5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX} \perp \overline{BC}$ and $AF = \sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c + 3b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1288}
Given a triangle \( ABC \) where \( |AB| = |AC| \) and \( \angle BAC = 80^\circ \). Inside the triangle, there is a point \( M \) such that \( \angle MBC = 30^\circ \) and \( \angle MCB = 10^\circ \). Find \( \angle AMC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Let $x$ be a positive integer. Consider the following iterated sums: $$f(x)=\sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1$$ What is the asymptotic behavior of the total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^u/u!}
Given the function $g(x)=e^{1+x^{2}}- \dfrac {1}{1+x^{2}}+|x|$, find the range of $x$ for which $g(x-1) > g(3x+1)$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,0)}
A product is initially purchased for 80 yuan and sold for 90 yuan, with a sales volume of 400 units. For every 1 yuan increase in price, the sales volume decreases by 20 units. Find the selling price per unit that maximizes the profit. Express your answer in yuan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Given that the solution set of the system of inequalities $2x-a<1$ and $x-2b>3$ is $-1<x<1$, calculate the value of $(a+1)(b-1)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
In the convex pentagon \(ABCDE\), \(\angle A = \angle B = \angle D = 90^\circ\). Find the angle \(ADB\) if it is known that a circle can be inscribed in this pentagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Towers grow at points along a line. All towers start with height $0$ and grow at the rate of $1$ meter per second. As soon as any two adjacent towers are each at least $1$ meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time $0$, there are no towers, but at time $0$, the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time $10$ seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1033}
Given a differentiable function $f(x)$ such that $f(0)=f(1)=0$ and $\int_0^1 f(x) \, dx = 1$, find the maximum value of $c$ such that $\sup_{t \in [0,1]} |f'(t)| \geq c$ is guaranteed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Inside the triangle \(ABC\), a point \(D\) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). It is known that \(AB = 15\) and \(CD = 8\). Find the length of segment \(AD\). If necessary, round the answer to 0.01 or write the answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
A major convention took place at the city's convention center with 5,000 people attending. Of the attendees, 60% decided to have a snack at the snack bar. After an hour, 25 new convention-goers joined in to have a snack. Subsequently, 70% of these snack eaters got full and left. After another hour, a group of 50 newcomers joined to have a snack. Then, 800 of the snack eaters decided that they had their fill and left. Afterward, two-thirds of the remaining snack eaters left. Lastly, a group of 100 latecomers arrived and decided to have snacks. How many snack eaters are left at the snack bar?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
What is the smallest number $k$ greater than $6$ for which:
\[1 + 2 + \ldots + k = k + (k+1) + \ldots + n\]
where $n$ is an integer greater than $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Find the number of sets $A$ that satisfy the following conditions:

1. $A$ is a set of two positive integers.
2. Each number in $A$ is at least $22\%$ the size of the other number.
3. $A$ contains the number $30$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
A certain Xinhua bookstore has recently acquired a new best-selling book. On the first day, they sold half of the total stock, and on the second day, they sold a quarter more than that, which was 50 books extra. As a result, the bookstore still has 200 books of this kind left. How many of these new best-selling books did the bookstore acquire?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Two people travel toward each other from points \(A\) and \(B\) with speeds \(v_{1} = 6 \, \mathrm{m/s}\) and \(v_{2} = 4 \, \mathrm{m/s}\). At the moment they meet, one of them turns around and walks in the opposite direction, while the other does not change direction. The person who did not change direction reached their final destination \(t_{2} = 10\) minutes earlier than the person who turned around. Determine the time \(t_{1}\) that passed from the start of the journey until the meeting. (15 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30 \text{ minutes}}
A prime number $p$ is called a \textbf{moderate} number if for every $2$ positive integers $k > 1$ and $m$, there exist $k$ positive integers $n_1, n_2, \ldots, n_k$ such that:
\[ n_1^2 + n_2^2 + \ldots + n_k^2 = p^{k+m} \]

Determine the smallest prime $r$ which is not moderate, given that $q$ is the smallest \textbf{moderate} number and $q < r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the minimum value of $\displaystyle\int_{-1}^1|g(x)|^2dx$ given that $\displaystyle\int_{-1}^1x^3g(x)dx=\int_{-1}^1x^2g(x)dx=\int_{-1}^1xg(x)dx=0$ and $\displaystyle\int_{-1}^1|x^3-a-bx-cx^2|dx=2/15$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/15}
Given the proposition ∀x > 1, x² + x + 1 > 0, what is the negation of this statement? Express your answer as a logical statement, using appropriate quantifiers and mathematical expressions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃x > 1, x² + x + 1 ≤ 0}
Let $O$ be the centre of the square $ABCD$. Let $P,Q,R$ be respectively on the segments $OA,OB,OC$ such that $OP=3,OQ=5,OR=4$. Suppose $S$ is on $OD$ such that $X=AB\cap PQ,Y=BC\cap QR$ and $Z=CD\cap RS$ are collinear. Find $OS$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{60}{23}}
Consider the sequence \( u_{n} \) defined as follows: \( u_{0} = 2010^{2010} \), and \( u_{n+1} = u_{n} + 7 \) if \( u_{n} \) is odd, and \( \frac{u_{n}}{2} \) if it is even. What is the smallest integer that this sequence will reach?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle.

Find the area of the quadrilateral \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
设$H = \{ h \in C \vert h ^ { n } = - 1 \}$，n是正整数， n≥2. 求$\sum _ { h \in H } ( 1 - h ) ^ { - 2 }$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ $\frac { n ( 2 - n ) } { 4 }$}
An empty swimming pool was filled with water by two taps \(A\) and \(B\), both with constant flow rates. For 4 hours, both taps were open and filled 50% of the pool. Then, tap \(B\) was closed and for the next 2 hours, tap \(A\) filled 15% of the pool's volume. After this period, tap \(A\) was closed and tap \(B\) was opened. How long did tap \(B\) need to remain open to fill the pool by itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
10. A positive integer $n$ is called "flippant" if $n$ does not end in 0 (when written in decimal notation) and, moreover, $n$ and the number obtained by reversing the digits of $n$ are both divisible by 7 . How many flippant integers are there between 10 and 1000 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Determine the number of zeros of the function $f(z)=5z^3+15z^2+7z+10$ inside the annulus $1<|z|<2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In triangle \( \triangle ABC \), the three interior angles \( \angle A, \angle B, \angle C \) satisfy \( \angle A = 3 \angle B = 9 \angle C \). Find the value of 
\[ \cos A \cdot \cos B + \cos B \cdot \cos C + \cos C \cdot \cos A = \quad . \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1/4}
In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take?

Recall that among 99 numbers, the median is such a number that is greater than 49 others and less than 49 others.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Given that $a, b, c,$ and $d$ are positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$, find the prime number that must be a divisor of $a$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
25. Detective Sherlock Holmes found a password safe, and the password is a three-digit number $A$. Currently, there is a clue: among the four three-digit numbers $123,931,297,419$, each number contains exactly one digit from the three-digit number $A$, and the position of the digit is different from that in $A$, then the three-digit number $A$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{374}
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$63$}
$908 \times 501 - [731 \times 1389 - (547 \times 236 + 842 \times 731 - 495 \times 361)] =$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5448}
Mom decides to take Xiaohua on a road trip to 10 cities during the vacation. After checking the map, Xiaohua is surprised to find that for any three cities among these 10, either all three pairs of cities are connected by highways, or exactly two pairs of cities among the three are not connected by highways. What is the minimum number of highways that must be opened among these 10 cities? (Note: There can be at most one highway between any two cities.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find all three-digit integers \( abc = n \) such that \( \frac{2n}{3} = a! \cdot b! \cdot c! \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
Find the smallest positive integer  $n$  for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than  $1.999$ .

[i]Proposed by James Lin
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
## Problem Statement

Calculate the definite integral:

$$
\int_{2 \operatorname{arctg} \frac{1}{3}}^{2 \operatorname{arctg} \frac{1}{2}} \frac{d x}{\sin x(1-\sin x)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln3-\ln2+1}