inclusionAI/Ring-lite-rl-data huggingface dataset:
A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first round. How many people were sitting at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A bicycle wheel has a certain diameter. It makes 482.5323296660876 complete revolutions in 1 km. What is the diameter of the wheel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.66}
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and its partial sum \(S_{n}\) has a maximum value, then when \(S_{n}\) attains its smallest positive value, \(n = \) \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Determine the number of ways to select a positive number of squares on an $8 \times 8$ chessboard such that no two lie in the same row or the same column and no chosen square lies to the left of and below another chosen square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["12869"]}
Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffrey replaces them with \[ \frac{x + y}{2} \text{ and } 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}. \]

After $2017$ minutes, the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
In a four-level symbol pyramid, the symbols in the two bottom cells determine the cell above them: if they are the same, the cell above is "+", otherwise, it is "-". How many possible ways are there to fill the bottom row of four cells such that the top of the pyramid results in a "+"? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
What is the cardinality of the class of all rational Cauchy sequences that converge to $0$ in $\mathbb{R}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a holomorphic function such that for \( x, y \in \mathbb{R} \), \( \text{Im}f(x + iy) = x^2 - y^2. \) Find the function \( f. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i z^2}
Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial with $4$ non-necessarily distinct real roots $x_1, x_2, x_3, x_4$. If $b \geq d + 5$, what is the minimum possible value of the expression $(x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)(x_4^2 + 1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Ada has a set of identical cubes. She makes solids by gluing together 4 of these cubes. When cube faces are glued together, they must coincide. Each of the 4 cubes must have a face that coincides with a face of at least one of the other 3 cubes. Find the number of unique solids that Ada can make using 4 cubes, considering all possible configurations, rotations, and reflections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given \( \theta = \frac{\pi}{6} \), evaluate the expression:
\[
\frac{\sin(9\theta)}{\left(1 + \sum_{k=1}^{8}\frac{\cos(k\theta)}{\cos^k(\theta)}\right) \sin\left(\frac{\pi}{6}\right) \left(\cos\left(\frac{\pi}{6}\right)\right)^8}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the value of \( \sup_{f \in \mathscr{H}} \inf_{z \in D} [f_x^2(z) + f_y^2(z)] \), where \( f: \mathbb{R}^2 \to \mathbb{R} \) is continuous, has partial derivatives in \( D = \{(x,y) : x^2 + y^2 \leq 1\} \), and satisfies \( \sup_D |f| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \(ABC\) be an equilateral triangle of side length 15. Let \(A_b\) and \(B_a\) be points on side \(AB\), \(A_c\) and \(C_a\) be points on side \(AC\), and \(B_c\) and \(C_b\) be points on side \(BC\) such that \(\triangle AA_bA_c\), \(\triangle BB_cB_a\), and \(\triangle CC_aC_b\) are equilateral triangles with side lengths 3, 4, and 5, respectively. Compute the radius of the circle tangent to segments \(\overline{A_bA_c}\), \(\overline{B_aB_c}\), and \(\overline{C_aC_b}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3}}
Determine the spectrum of the operator $M_b$ on $L_2 (\mathbb{R}^n)$, where $M_b$ is defined by multiplication by the function $b(x) = e^{|x|^2}$ and $|x|^2 = x_1^2 + \ldots + x_n^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
Given the recurrence relation \(x_{n+1} = 4x_n - x_{n-1}\) for \(n \geq 1\), with initial conditions \(x_0 = 1\) and \(x_1 = 2\), find an odd prime factor of \(x_{2015}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{181}
Identify the unique degree 2 subfield of the field extension $k(x)$ over the field $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k(x^2)}
Find the equation for the trajectory of the dynamical system defined by \(\dot x = \frac{(x+iy)^3 + (x-iy)^3}{2}\) and \(\dot y = \frac{(x+iy)^3 - (x-iy)^3}{2i}\) in polar coordinates, given that the trajectory passes through the point \((x,y) = (1,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\theta = 0}
At a party, there are 20 girls and 20 boys. How many dance pairs can be formed if boys may dance with boys and girls with girls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{780}
In triangle ABC, let H, G, and O be the orthocenter, centroid, and circumcenter, respectively. Given that the Euler line is parallel to AC and \( \angle HBC = 2 \angle OCA \), calculate the distance GO if AH = a. Express your answer in terms of a.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a}{3}}
Determine the range of \( \beta \) values for which the random walk \( S_n = \varepsilon_1 + \cdots + \varepsilon_n \), with i.i.d. increments \( \{\varepsilon_n\} \) and distribution:

\[ P(\varepsilon_1 = -1) = 1 - C_{\beta} \quad \text{and} \quad P(\varepsilon_1 > t) = C_{\beta}e^{-t^{\beta}}, \quad t \geq 0, \]

exhibits heavy-tailed behavior, given that \( C_{\beta} \in (0,1) \) and \( \mathbb{E}\varepsilon_1 = -\frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Find the sum of the 11th powers of all the roots of the polynomial equation \(x^5 + 5x + 1 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A circular disk is divided by $2n$ equally spaced radii ($n > 0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided? The original answer is in the form $kn + m$. Please find the value of $k + m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of equally spaced data points required to uniquely determine a three-harmonic Fourier series representation of a periodic function $f(\theta)$ with a period of $\frac{\pi}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that the function $f(x)$ has a domain of $(0, +\infty)$, $f(1) = e$, and for any $x_{1}$, $x_{2} \in (0, +\infty)$, when $x_{2} > x_{1}$, it holds that $\frac{{f(x_{1}) - f(x_{2})}}{{x_{1}x_{2}}} > \frac{{e^{x_{2}}}}{{x_{1}}} - \frac{{e^{x_{1}}}{{x_{2}}}}$ (where $e$ is the base of the natural logarithm). If $f(\ln a) > 2e - a\ln a$, then the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, e)}
The numbers from 1 to 9 are placed in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in five shaded cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and above $M$ and $B$ is in the same column and below $M$. It is possible for $M$ to form a floating plus with unit squares that are not next to it. Find the largest positive integer $k$ (depending on $n$) such that we can color some $k$ squares black in such a way that there is no black colored floating plus.

[i]Proposed by Nikola Velov[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 4n-4 }
Given the function $f(x)=2\sin \omega x \sin(\omega x+\frac{\pi}{3})+\cos 2\omega x-\frac{1}{2}$, find the range of real number $\omega$ such that the function $f(x)$ has exactly two zeros in the interval $\left[0,\pi \right]$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{11}{12}, \frac{17}{12})}
How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{233}
What is the largest integer \( n \) such that 

$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$

is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343}
Evaluate the integral \( \int_{|x| \geq 1} \frac{1}{|x|^n} \, dx \) where \( x \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \( g_{1}(x) = \frac{1}{3} \left(1 + x + x^{2} + \cdots \right) \) for all values of \( x \) for which the right-hand side converges. Let \( g_{n}(x) = g_{1}(g_{n-1}(x)) \) for all integers \( n \geq 2 \). What is the largest integer \( r \) such that \( g_{r}(x) \) is defined for some real number \( x \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find all integer/s \( n \) such that \( \frac{{5}^{n} - 1}{3}\) is a prime or a perfect
 square of an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n = 0
}
Consider a disk with an even number of points $n$ randomly distributed uniformly within it. These points are paired to form $n/2$ line segments, creating a perfect matching. Construct the intersection graph $G$ where each node represents a segment, and an edge exists between two nodes if their corresponding segments intersect properly. Determine the diameter of the graph $G$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the dual one-form field of the vector field \( V = x\, \frac{\partial}{\partial z} + y \frac{\partial}{\partial x} \) in \( (x,y,z) \in \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y \, dx + x \, dz}
Given propositions $P$ and $Q$, where $P$ states that the equation $\frac{x^{2}}{2m} + \frac{y^{2}}{9-m} = 1$ represents an ellipse with its foci on the $y$-axis, and $Q$ states that the eccentricity $e$ of the hyperbola $\frac{y^{2}}{5} - \frac{x^{2}}{m} = 1$ is within the range $(\frac{\sqrt{6}}{2}, \sqrt{2})$. If only one of the propositions $P$ and $Q$ is true, find the range of values for the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac{5}{2}] \cup [3, 5)}
31. [23] Let
$$
P=\prod_{i=0}^{2016}\left(i^{3}-i-1\right)^{2} .
$$

The remainder when $P$ is divided by the prime 2017 is not zero. Compute this remainder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1994}
Find the total length of the intervals on the number line where the inequalities \( |x| < 1 \) and \( \operatorname{tg} \log_{5} |x| < 0 \) hold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \cdot 5^{\pi/2}}{1+5^{\pi/2}}}
Three natural numbers are written on the board: two ten-digit numbers \( a \) and \( b \), and their sum \( a + b \). What is the maximum number of odd digits that could be written on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The lengths of the sides of triangle \(ABC\) are \(a\), \(b\), and \(c\) (with \(AB = c\), \(BC = a\), \(CA = b\) and \(a < b < c\)). Points \(B_1\) and \(A_1\) are marked on rays \(BC\) and \(AC\) respectively such that \(BB_1 = AA_1 = c\). Points \(C_2\) and \(B_2\) are marked on rays \(CA\) and \(BA\) respectively such that \(CC_2 = BB_2 = a\). Find the ratio \(A_1B_1 : C_2B_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{c}{a}}
Find the integral \(\int \operatorname{ch}^{3} x \operatorname{sh} x \, dx\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\cosh^4(x)}{4} + C}
A fly trapped inside a cubical box with side length \(1\) meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. The maximum possible length of its path is in the form of \(k\sqrt{2} + m\sqrt{3}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that tetrahedron $ABCD$ is inscribed in sphere $O$, and $AD$ is the diameter of sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, what is the volume of tetrahedron $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{2}}{12}$}
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom{5}{2} = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Four new students are to be assigned to three classes: A, B, and C, with at least one student in each class. Student A cannot be assigned to class A. How many different assignment plans are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let P be a polynomial with each root real and each coefficient either 1 or -1. What is the maximum possible degree of P? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
一个含有底面的半球形容器内放置有三个两两外切的小球。若这三个小球的半径均为 1 ，且每个小球都与半球的底面和球面相切，则该半球的半径 $R=$ 的原始答案是 $\frac{m+\sqrt{n}}{k}$，请给出m + n + k的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Define a sequence \( a_{i, j} \) of integers such that \( a_{1, n} = n^n \) for \( n \geq 1 \) and \( a_{i, j} = a_{i-1, j} + a_{i-1, j+1} \) for all \( i, j \geq 1 \). Find the last (decimal) digit of \( a_{128, 1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The monotonic decreasing interval of the function $y = -x^2 + 2x + 1$ is __ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2093124}
Evaluate the integral \( K = \iiint_{P}(x^2+\sin(xy^2)+ y^2) \,dx\,dy\,dz \) where \( P = \{(x,y,z) : 0 \leq z \leq 2 -|x| -|y|\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{64}{15}}
There are 150 'zeroes' and some 'ones' written on a blackboard. A step involves choosing 3 digits and replacing them with a single digit. If all the 3 digits are identical, they are replaced with the same digit. Otherwise, the digit that appears twice replaces the 3 chosen digits. The minimum number of steps after which there will be no zero written on the blackboard is 76. How many 'ones' are initially written on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1236}
Given the functions $f(x, y)$ and $g(x, y)$ where $f(1, 2) = 2$, $g(1, 2) = 1$, and their differentials at $(1, 2)$ are $df = -2dx + 5dy$ and $dg = 4dx - 3dy$, respectively. An implicitly defined function $z(x, y)$ satisfies the equation $z^2f + zg + x - y = 0$ with $z(1, 2) = -1$. Determine $\frac{\partial z}{\partial x}(1, 2)$. \(\boxed{\text{Answer}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{3}}
8、A digital clock uses two digits to display hours, minutes, and seconds (such as $10: 09: 18$). Between $05: 00: 00 \sim 22: 59: 59$ on the same day, the probability that all six digits on the clock are different is $\qquad$ -
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{135}}
Find the invariant factor decomposition of the tensor product \( \mathbb{Z}_6 \otimes_{\mathbb{Z}} \mathbb{Z}_9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_3}
Consider the set \(X=\{2^m3^n|0 \le m, \ n \le 9 \}\). Determine the number of quadratic equations of the form \(ax^2+2bx+c\) that have equal roots, where \(a, b, c\) are distinct elements of \(X\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2400}
Calculate the maximum probability that the random variable \( \xi \) exceeds 0.5, where \( \xi \) is defined as follows:

\[\xi\overset{\circ}{=}\begin{cases} \frac{p_{1}\cdot0.3}{p_{1}\cdot0.3+p_{2}\cdot0.7} & \text{with}\;p=p_{1}\cdot0.3+p_{2}\cdot0.7\\ \frac{\left(1-p_{1}\right)\cdot0.3}{\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7} & \text{with}\;1-p=\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7 \end{cases}\]

where \( p_{1},p_{2}\in\left[0,1\right] \). Find \( \max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Determine the cardinality of the set of functions \( F: \mathbb{N} \longrightarrow \{0,1,2\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Using the formula for relativistic mass, \( m = m_0 \left(1 - \frac{v^2}{c^2}\right)^{-\frac{1}{2}} \), estimate the percentage increase in mass when the velocity of an object increases from \( v = 0.90c \) to \( v = 0.92c \) using differentials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.5\%}
What is the minimum number of cells that need to be colored on a $6 \times 6$ board so that, no matter how a 4-cell L-shaped figure is placed on the board (it can be rotated or flipped), there is at least one colored cell in that figure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given a machine that transforms a positive integer \( N \) based on the rule: if \( N = 7 \), the machine outputs \( 3 \times 7 + 1 = 22 \).

By inputting the result back into the machine and repeating five times, the output sequence is:
\[ 7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26 \]

If there exists some positive integer \( N \) whereby six iterations of the process results in the final output being 1,
\[ N \rightarrow_{-} \rightarrow_{-} \rightarrow_{-} \rightarrow_{-} \rightarrow 1 \]

What is the sum of all such positive integers \( N \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
8. Two Go teams, Team A and Team B, each have 5 players who compete in a relay match according to a predetermined order. The No. 1 players of both teams start the match, and the loser is eliminated; then the No. 2 player of the losing team competes against the winner, and the loser is eliminated. This continues until all players of one team are eliminated, at which point the other team is declared the winner. Assuming each player has equal strength, the expected number of Team A's players who have not played when the match ends, $E(X)=$ $\qquad$

---

I hope this translation meets your needs. If you have any further requests or need additional assistance, feel free to let me know!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{187}{256}}
Given the equations:
\[ xw^3 + yz^2 + z^3 = -1 \]
\[ zw^3 - xz^3 + y^2w = 1 \]
find the partial derivative \( \frac{\partial w}{\partial y} \) at the point \((x, y, z, w) = (1, -1, -1, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Let $S_n$ be the sum of the first $n$ terms of a geometric series $\{a_n\}$ with positive terms. If $S_4 - 2S_2 = 2$, find the minimum value of $S_6 - S_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. Find the weight of the basket in kilograms. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Compute the expected value \( \mathbb{E}[ \tau C^2_{\tau}] \) where \( \tau = \inf \{t \geq 0 : |C_t| = 1 \} \) and \( C \) is a Standard Brownian Motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
King Arthur has two advisors, Merlin and Percival, each providing correct answers with probability \( p \). If both advisors agree, the king follows their advice. If they disagree, the king flips a coin to decide. Arthur wonders if he should dismiss one advisor, believing the probability of making correct decisions won't decrease. If true, he will dismiss one advisor by coin flip. Percival answers truthfully, while Merlin answers to minimize his dismissal probability. What is the probability that Merlin will be dismissed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}