inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.

[i]Proposed by Huawei Zhu[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
Given the bivariate normal distribution $\mathbf{X} = (X_1, X_2)' \sim N(\mu, \Lambda)$, where $\mu = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $\Lambda = \begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix}$, calculate the conditional probability $P(X_1 \geq 2 \mid X_2 + 3X_1 = 3)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0003}
A builder has two identical bricks. She places them side by side in three different ways, resulting in shapes with surface areas of 72, 96, and 102. Calculate the surface area of one original brick. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
How many unordered triples  $A,B,C$  of distinct lattice points in  $0\leq x,y\leq4$  have the property that  $2[ABC]$  is an integer divisible by  $5$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Determine the fundamental group of a sphere with two points removed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the least possible value of $f(1998)$, where $f: \mathbb{N} \to \mathbb{N}$ is a function such that for all $m, n \in \mathbb{N}$,  

$$f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares \(\square_{1}, \square_{2}, \square_{3}\) on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set \(\{1,2,3,4,5\}\) such that no integer is used more than once. Ana's goal is to minimize the minimum value \(M\) that the polynomial \(a_{1} x^{2} + a_{2} x + a_{3}\) attains over all real \(x\), where \(a_{1}, a_{2}, a_{3}\) are the integers written in \(\square_{1}, \square_{2}, \square_{3}\) respectively. Banana aims to maximize \(M\). Assuming both play optimally, compute the final value of \(100 a_{1} + 10 a_{2} + a_{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{451}
Let $f (x) = 2x -1$. Find the value of $x$ that minimizes $| f ( f ( f ( f ( f (x)))))-2020|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2051}{32}}
Let $X$ be a random variable following a geometric distribution with parameter $p = \frac{2}{3}$, i.e., $X \sim \text{Geo}(\frac{2}{3})$. Define $Y = \left\lfloor \frac{X}{2} \right\rfloor + 1$. Find the expected value $\mathbb{E}[Y]$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{8}}
Given a positive integer \( n \) and an integer \( k \) where \( 2 \leq k \leq 9 \), determine the \( n \)-th number that either contains the digit \( k \) or is divisible by \( k \). For example, if \( n = 15 \) and \( k = 3 \), the answer is \( 33 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Calculate \( \int \int_X f(x, y) \, dx \, dy \) where \( f(x, y) = x \) and the region \( X \) is defined by the inequalities \( 2rx \leq x^2 + y^2 \leq R^2 \) with \( 0 < 2r < R \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi r^3}
Provide an example of an open set in the domain of the function \( f(x) = x^2 \) whose image under \( f \) is not open in the codomain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Determine the number of non-isomorphic graphs with 30 vertices and 432 edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In Papa Carlo's room, there is a clock on each wall, and they all show incorrect times: the first clock is off by 2 minutes, the second by 3 minutes, the third by 4 minutes, and the fourth by 5 minutes. One day, Papa Carlo decided to find out the exact time before leaving the house, and he saw the following times on the clocks: 14:54, 14:57, 15:02, and 15:03. Help Papa Carlo determine the exact time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14:58}
What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Find a function \( f(x) \) such that the sequence \( f(n) = a_n \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(\pi x)}
A matrix $H$ over $\Bbb F_2$ has as its first seven rows the identity matrix $I_7$, and the other rows are all vectors of $\{0,1\}^7$ with exactly three $1$'s per vector (weight-3 vectors). This matrix $H$ serves as a parity-check matrix for a linear code $C$. Determine the distance $d(C)$ of this linear code.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( b_n \) be the sequence of all positive integers such that the decimal expression for \( \frac{1}{b_n} \) terminates in an odd digit: 1, 2, 4, 8, 10, \ldots (For instance, 3 is not included because \( \frac{1}{3} = 0.33\ldots \) does not terminate, 4 is included because \( \frac{1}{4} = 0.25 \) terminates in 5, which is odd; 5 is not included because \( \frac{1}{5} = 0.2 \) terminates in 2, which is even.) Find the sum to infinity of \( \frac{1}{b_n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{9}}
Let $f(x) \geq 0$ be a Riemann integrable function such that \( \int_0^1 f(x) \, \mathrm{d}x = 1 \) and \( \int_0^1 x f(x) \, \mathrm{d}x = \frac{1}{6} \). Find the minimum value of \( \int_0^1 f^2(x) \, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{170}
A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. Find the number of checkerboard squares which are completely covered by the disc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
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 \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the integral \( \iint_D \left(\sqrt{x^2+y^2}-\sqrt{a^2-x^2-y^2}~\right)dxdy \) where \(D_{xy}\) is the disk \(x^2+y^2\le a^2\), using a change of variables different from polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{441}
Determine the set of values of $m\in\mathbb{C}$ such that the equation $(1+iz)^n=m(1-iz)^n$ has only real roots, where $n\in\mathbb{N}$ and $z$ is a complex variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{1}}
Find the square root of the real number 4. Express your answer using the ± symbol to denote both positive and negative roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2}
Evaluate the limit: $$\lim_{x \to 0} \left(\ln\left(\frac{\cos x}{1+\sin x}\right)\right)^{\frac{1}{x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the function $f\left(x\right)=2\sin x+\sin 2x$, the minimum value of $f\left(x\right)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3\sqrt{3}}{2}}
Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all $a + p$ for the solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
James is standing at the point $(0,1)$ on the coordinate plane and wants to eat a hamburger. For each integer $n \geq 0$, the point $(n, 0)$ has a hamburger with $n$ patties. There is also a wall at $y=2.1$ which James cannot cross. In each move, James can go either up, right, or down 1 unit as long as he does not cross the wall or visit a point he has already visited. Every second, James chooses a valid move uniformly at random, until he reaches a point with a hamburger. Then he eats the hamburger and stops moving. Find the expected number of patties that James eats on his burger.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{3}}
A diagonal of a regular 2006-gon \( P \) is called good if its endpoints divide the boundary of \( P \) into two parts, each containing an odd number of sides. The sides of \( P \) are also called good. Let \( P \) be divided into triangles by 2003 diagonals, none of which have common interior points. What is the greatest number of isosceles triangles, each of which has two good sides, that such a division can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
Let \( S \) be the set of points \(\left(x_{1}, x_{2}, \ldots, x_{2012}\right)\) in 2012-dimensional space such that \(\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{2012}\right| \leq 1\). Let \( T \) be the set of points in 2012-dimensional space such that \(\max _{i=1}^{2012}\left|x_{i}\right|=2\). Let \( p \) be a randomly chosen point on \( T \). What is the probability that the closest point in \( S \) to \( p \) is a vertex of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2^{2011}}}
The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \( O \) of one rectangle and the midpoint \( M \) of one edge of the other. What is the distance \( OM \)? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the integral \( \int_0^n \frac{(1-\frac{x}{n})^n}{ne^{-x}}dx \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the Krull dimension of the trivial ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given the probability density function \( f(x, \theta) = \frac{\theta}{x^{2}} \) for \( 0 < \theta \le x < \infty \), and an i.i.d. sample \( (X_i)_{i=1}^{n} \), find the sufficient statistic for \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_{(1)}}
Use Taylor's inequality to estimate the accuracy of the approximation \( f(x) \approx T_3(x) \) for \( f(x) = e^{-x^2} \cos(x) \) at \( a = 0 \), where \( T_3(x) = 1 - \frac{3x^2}{2} \). Estimate the error for \( |x| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{24}}
In triangle \(ABC\), where \(AB = AC\), the distances from vertex \(C\) to the trisection points of side \(AB\) are 17 cm and 20 cm, respectively. Determine the distances from vertex \(A\) to the trisection points of side \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{26}}
Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{n+1} }
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all  integers $n \ge 1$ are defined:
$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
$O_n$ is the center of $\odot (PA_nB_n)$
$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$

If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$ 

[hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2^{1007}}}
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\frac{2}{3}$ probability of voting for Alice and a $\frac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1513}{2017}}
The radius of the circle circumscribed around a triangle is equal to half of one of the sides of the triangle. What can be said about the angle opposite this side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11621}
Given $f\left(x\right)=ax\cdot e^{x}(a \gt 0)$, if the range of $y=f\left(f\left(x\right)\right)$ is the same as the range of $y=f\left(x\right)$, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[e,+\infty \right)}
For which natural numbers $n$ does the inequality $2^{n} \geq n^{3}$ hold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 10}
Lolita can drink 3 boxes of milk during weekdays. During Saturdays, she can drink twice the number of boxes of milk than on the weekdays, and during Sundays, she can drink some boxes of milk. She drinks 30 boxes of milk per week. What is the ratio of the number of boxes of milk she drinks on Sundays to the number of boxes she drinks on weekdays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:5}
Calculate: ${(\frac{81}{16})}^{-\frac{1}{4}}+\frac{1}{4}\log_{\sqrt{2}}3\times\log_{3}4\times{[(-\frac{1}{3})}^{2]}^{\frac{1}{2}}+{7}^{\log_{7}\frac{1}{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Let $\Omega = D(0,2)/\{\frac{1}{2}\}$, where $D(0,2)$ is a disc, $f$ holomorphic in $\Omega$. $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate $$ lim_{n \to \infty} \frac{f^{n}(0)}{2^{n}n!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
At a football game, the total initial attendance was 1300 people comprising boys, girls, and adults. At the start of the game, there were 350 boys, 450 girls, and the rest were adults. After the first quarter, one-fourth of the boys, one-eighth of the girls, and one-fifth of the adults left the stadium. During halftime, 5% of the remaining boys, 7% of the remaining girls, and 50 more adults entered the stadium. At the beginning of the third quarter, 3% of the remaining boys and 4% of the remaining girls left the stadium, and 2% of the remaining adults entered the stadium. Lastly, 15 boys and 25 girls entered the stadium before the start of the fourth quarter and 1% of the remaining adults left the stadium. Find the total number of spectators in the stadium at the end of these changes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1166}
Consider the ring $R=\mathbb{Z}_p[x]/(x^p-x)$. Find the number of elements in $R$ that satisfy $r^3=r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^p}
Tom has a list of nine integers: 1, 2, 3, 4, 5, 6, 7, 8, and 9. He creates a second list by adding 2 to some of the integers in the first list and by adding 5 to all of the other integers in the first list. What is the smallest number of different integers he can obtain in the second list? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
What is the difference between a number and the local value of 3 in 28943712, if the local value of 7 in the difference is 7000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7000}
Given that the area of the parallelogram $ABCD$ is $240$, $E$ and $H$ are the midpoints of sides $AD$ and $AB$ respectively. Point $G$ is on side $BC$ such that $BG = 2GC$, and point $F$ is on side $CD$ such that $DF = 3FC$. Point $K$ is on side $AC$ such that the area of $\triangle EKF$ is 33. Find the area of $\triangle HKG$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
On a map, the distance between two mountains is 312 inches. The actual distance between the mountains is 136 km. Ram is camped at a location that on the map is 34 inches from the base of the mountain. How many km is he from the base of the mountain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.82}
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, and it is given that $S_n=(-1)^n a_n-\frac{1}{2^n}$. Find the value of $S_1+S_3+S_5+\cdots +S_{2017}=\_\_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}\left(\frac{1}{2^{2018}}-1\right)}
A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent cuts, forming a tray with taped edges. Determine the height of this tray, which is the perpendicular distance from the base of the tray to the plane formed by the upper edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2}}
Determine the minimum number of shifts required for a group of 6 people on a ship, such that for any two distinct persons $A$ and $B$, there exists a shift where $A$ is awake while $B$ is asleep.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in \mathbf{N} \) such that \( a_{1} + a_{2} + a_{3} = b_{1} + b_{2} + b_{3} \),
\[ 
a_{1}a_{2}a_{3} = b_{1}b_{2}b_{3} 
\]
\[ 
a_{1}a_{2} + a_{1}a_{3} + a_{2}a_{3} \neq b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3} 
\]

Then find the minimal value of the largest number among \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In how many different ways can the digits $0,1,2,3,4,5,6$ form a seven-digit number that is divisible by 4? (The number cannot start with 0.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1248}
The coefficients of the polynomial \(P(x)\) are nonnegative integers, each less than 100. Given that \(P(10)=331633\) and \(P(-10)=273373\), compute \(P(1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Calculate the following integral:

$$I_p = \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr$$

where $p$ is a real number. Determine the values of $p$ for which the integral converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 < p < 3}
A circle \( O \) with radius \( R \) is given. There is a line \( l \) outside the circle. Let the projection of \( O \) on \( l \) be \( H \). The distance \( OH = d \). \( OH \) intersects the circle \( O \) at points \( C \) and \( D \) with \( CH > DH \). Let \( P \) and \( Q \) be points on \( l \), on the same side of \( H \), such that \( PH = a \) and \( QH = b \) with \( a > b \). There are 2008 chords \( A_i B_i \) \((i=1,2,\ldots, 2008)\) in the circle parallel to \( l \), and these 2008 chords divide \( CD \) into equal segments. Find the value of \(\frac{1}{2008} \sum_{i=1}^{2008} (PA_i^2 + PB_i^2 + QA_i^2 + QB_i^2)\) in terms of \( a, b, d, \) and \( R \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a^2 + 2b^2 + 4d^2 + 4R^2}
What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
24
}
$\mathrm{Find}\lim\limits_{a\to\infty}\frac{1}{a^2}\int_0^a\log(1+e^x)dx.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}$.

}
In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). If the area of triangle \( BQC \) is 1, find the area of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{4}}
Evaluate

\[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{4}{3} - \frac{2\sqrt{3}}{3} \]}
Given the function \( f(x) = \frac{1+10x}{10-100x} \), and the composition \( f^n = \underbrace{f\circ f\circ f \circ\cdots \circ f}_{n} \), find the sum \( f(1/2) + f^2(1/2) + f^3(1/2) + \cdots + f^{6000}(1/2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{595}
Calculate the expected exit time of a random walk on the interval $(-a, b)$ starting at position $x$ using a recurrence relation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a + x)(b - x)}
集合$\{ x \vert - 1 \leq \log _ { \frac { 1 } { x } } 1 0 < - \frac { 1 } { 2 } , x \in N _ { + } \}$的真子集个数是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 ^ { 9 0 } - 1$}
Determine the rate at which \( \mathrm{E} \min \left(\sigma_{2n}, 2n\right) \) approaches infinity as \( n \to \infty \), where \( \sigma_{2n} = \min \{ 1 \leq k \leq 2n : S_k = 0 \} \) and \( \sigma_{2n} = 2n \) if \( S_k \neq 0 \) for all \( 1 \leq k \leq 2n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n}}