inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the minimal number of parties, $z$, that must be thrown to find out whether each pair of people from different groups like each other, given $n$ groups of $k$ people each, with no person in more than one group. Express $z$ in terms of $k$ and $n$. For example, if $n=k=2$, then $z=4$. Note that $k^2 \leq z \leq k^n$. Provide your answer as a function of $k$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^2}
Find the largest integer \( m \) such that \( n^{12} \equiv 1 \pmod{m} \) for all integers \( n \) that are relatively prime to \( m \), given that the Carmichael function \( \lambda(m) = 12 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65520}
Two lines pass through point \(N\) and are tangent to a circle with center \(O\). Point \(A\) is on one line, and point \(B\) is on the other such that \(OA = OB\) and \(OA > ON\). Given \(NA = a\), \(NB = b\), and \(OA = c\) where \(a \neq b\), find the length of \(ON\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{c^2 + ab}}
Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \[ g(g(y) - x) + 2x = g(g(x) + y) \] for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = x + c}
A point is randomly selected from the interval [6, 11]. Let $k$ be the selected value. Determine the probability that the roots of the quadratic equation $(k^2 - 2k - 24)x^2 + (3k - 8)x + 2 = 0$ satisfy the condition $x_1 \leq 2x_2$. Express your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Evaluate the expression \( \frac{bc}{a^{5}} + \frac{ac}{b^{5}} + \frac{ab}{c^{5}} \) in terms of \( p \), where \( a, b, c \) are the roots of the polynomial \( x^3 - px + 1 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-p^6 + 6p^3 - 3}
Evaluate the surface integral \( \iint_S (x^2 y^2 + y^2 z^2 + x^2 z^2)\, dS \) using the divergence theorem, where \( S \) is the entire surface of the sphere of unit radius centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{5}}
Calculate the volume of the larger region cut from the solid sphere defined by \(x^2 + y^2 + z^2 = 4\) by the plane \(z = 1\). Use double integrals in spherical coordinates to find the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\pi}
Compute the line integral \( \oint_C (y^2 \, dx - xy \, dy) \) where \( C \) is the boundary of the triangle with vertices \((1,0)\), \((4,0)\), and \((1,10)\), oriented counterclockwise. Use Green's Theorem to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-150}
Evaluate the contour integral \( \oint_{|z|=3} \frac{1}{2-z} e^{\frac{1}{z}} \,dz \) by finding the residue at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Let \( f(x) \) be a polynomial with integer coefficients such that \( f(15) f(21) f(35) - 10 \) is divisible by 105. Given \( f(-34) = 2014 \) and \( f(0) \ge 0 \), find the smallest possible value of \( f(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{620}
Determine the number of permutations \(\sigma\) in \(S_6\) such that \(\gamma ' = \sigma \gamma \sigma ^{-1}\), where \(\gamma = (123)(45)(6)\) and \(\gamma ' = (1)(23)(456)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Using ropes that burn irregularly, determine the least expensive way to measure exactly 1 minute. A rope that burns in 16 minutes costs 32 rupees, and a rope that burns in 7 minutes costs 14 rupees. You can light the ropes at one end, extinguish them, and relight them as needed. What is the minimum cost to measure exactly 1 minute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Is there a base in which all prime numbers end with 5 different symbols? If such a base exists, identify it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given quadrilateral $ACDB$ and parallelogram $AEDF$, the following vector equations are satisfied: \[ 6 \overrightarrow{A E}+3 \overrightarrow{D E}=2 \overrightarrow{A C}, \] \[ 4 \overrightarrow{F B}+5 \overrightarrow{F D}=2 \overrightarrow{A D}. \] Determine the ratio of the area of quadrilateral $ACDB$ to the area of parallelogram $AEDF$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{8}}
Eight identical dice are rolled simultaneously. How many possible outcomes are there in which each of the six numbers appears at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{191520}
Determine the number of 1-dimensional characters for a nonabelian group of order 57.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given 11 chess players and 5 distinct tables, in how many ways can the players be paired to play, considering that the order and position (color) of the players matter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39916800}
In triangle $ABC$ with circumcircle $\omega$, the bisector of $\angle ABC$ meets segment $AC$ at $D$ and circle $\omega$ at $M \neq B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E \neq B$, and $CE$ meets $\omega$ at $P \neq C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q \neq C$. Given that $PQ = MC$, find the degree measure of $\angle ABC$. \n\n\textit{(Ray Li)}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the range of values of $p > 0$ for which the improper integral \( \int_0^{\infty} x^p e^{-x^8 \sin^2 x} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 3)}
Let $X_1$ and $X_2$ be independent and identically distributed random variables with an exponential distribution, EXP(1). The joint PDF of $X_1$ and $X_2$ is given by $f(x_1, x_2) = e^{-x_1-x_2}$ for $x_1, x_2 \geq 0$. Find the joint PDF of $T_1 = \min(X_1, X_2)$ and $T_2 = X_1 + X_2$. Express your answer in terms of $t_1$ and $t_2$. Assume $t_1 \leq t_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2e^{-t_2}}
Evaluate the integral $$\int_{-\infty}^{+\infty} \frac{b}{1+2ax+(cx)^2} \cdot \frac{dx}{1-2ax+(cx)^2},$$ given that $a^2+b^2=c^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{2}$}
In a 110x110 array of guinea pigs, with 110 guinea pigs for each of the 110 species, determine the maximum integer n such that there is always a row or column containing at least n different species.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the sum of all values of \(a\) that satisfy the equation:
\[ \int_{0}^{\pi/2} (\sin x + a \cos x)^3 \, dx - \frac{4a}{\pi-2} \int_{0}^{\pi/2} x \cos x \, dx = 2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{2}}
Evaluate the line integral \( \int \pmb{f} \cdot d\pmb{\alpha} \) for the path \( \pmb{\alpha}(\theta) = R(\sin{\theta}\hat{i} + \cos{\theta}\hat{k}) \) where \( \theta \in [0, \pi] \), and the vector field \( \pmb{f}(r, \theta, \phi) = r^2(\cos{\theta}\hat{r} + \cos{\phi}\hat{\theta} - \cos{\theta}\sin{\phi}\hat{\phi}) \). Express \( \pmb{f}[\pmb{\alpha}(\theta)] \) when \( \pmb{\alpha} \) is in Cartesian coordinates and \( \pmb{f} \) is in spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi R^3}
Given that \((AA^t)^r = I\) for an \(m \times n\) real matrix \(A\) and a natural number \(r\), determine the possible values of \(\det(AA^t)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number of zeros of the polynomial \( h(z) = z^4 - z^3 + 1 \) in the region \( \{ z \in \mathbb{C} : \text{Re}(z) < 0 < \text{Im}(z) \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the lambda expression: $[[\lambda f.\lambda m. f(m + m^2))]([\lambda n.2n])](3)$. Perform the function evaluation correctly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the limit: \[ \lim_{x \to 0^+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Find the coefficient of the term $x^3y^4z^5$ in the expansion of the polynomial $(x + y + z)^8(x + y + z + 1)^8$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1940400}
Simplify the vector equation:
\[ (p^2 + q^2 |\mathbf{P}|^2 + (\mathbf{P} \bullet \mathbf{Q})^2)\mathbf{X} + (\mathbf{Q} \bullet \mathbf{X})(|\mathbf{P}|^2 \mathbf{Q} - (\mathbf{P} \bullet \mathbf{Q})\mathbf{P}) + (\mathbf{P} \bullet \mathbf{X})(|\mathbf{Q}|^2 \mathbf{P} - (\mathbf{P} \bullet \mathbf{Q})\mathbf{Q}) + (\mathbf{X} \bullet (\mathbf{P} \times \mathbf{Q}))(\mathbf{P} \times \mathbf{Q}) \]
where \(p\) and \(q\) are scalars, \(\mathbf{P}, \mathbf{Q}, \mathbf{X}\) are 3D vectors, and \(|\mathbf{P}|^2 \equiv (\mathbf{P} \bullet \mathbf{P})\). Given that \(p^2 + |\mathbf{P}|^2 = q^2 + |\mathbf{Q}|^2 = 1\), express the equation in terms of \(\mathbf{X}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{X}}
Calculate the moment of inertia around the x-axis for the solid bounded by the planes $x=0$, $y=0$, $z=0$, and $x+y+z=1$, assuming a constant density.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\rho}{30}}
Determine the number of real numbers \(x\) that satisfy the equation:
\[
\left(|x^2-12x+20|^{\log{x^2}}\right)^{-1+\log x}=|x^2-12x+20|^{1+\log{\frac{1}{x}}}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the length of the longest simple circuit in a wheel graph $W_n$ for $n > 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the number of invertible $2\times2$ matrices over the field $\mathbb{Z}/p\mathbb{Z}$ that are unimodular, meaning their determinant is either $1$ or $-1$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2p(p^2 - 1)}
Determine the ideal generated by \((2+3i, 4+5i)\) in the ring of Gaussian integers \(\mathbb{Z}[i]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1)}
Evaluate the line integral \(\oint_{\partial T} \vec{F} \cdot \hat{a} \, ds\) for the vector field \(\vec{F}(x,y) = (2x^2 + 3x -2 \cos^4(y) \sin^3(y) , 4 e^{2x} \sinh(x) - 3y)\) and the triangle \(T\) with vertices \((-1,0)\), \((1,1)\), and \((0,2)\), traversed counterclockwise. The outward normal to \(T\) in the \(xy\) plane is \(\hat{a}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the area of the region that lies between the polar curves \( r = 1 + \sin\theta \) and \( r = 1 + \cos\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{2}}
Given the function $f(x)=\frac{1}{2}x^{2}+(2a^{3}-a^{2})\ln x-(a^{2}+2a-1)x$, and $x=1$ is its extreme point, find the real number $a=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the field to which the quotient \( \frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle} \) is isomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_5}
Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monogrammed on their towels. Different letters may cost different amounts to monogram. If it costs \$$21$to monogram BRYAN, \$$25$ to monogram BARRY and \$$18$ to monogram SARAH, how much does it cost to monogram SHANA?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In a certain foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies within the government. Friendship and enmity are symmetric relations. A triplet of ministers is called **uniform** if all three are either mutual friends or mutual enemies. How many uniform triplets are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In a one-dimensional lattice $\mathbb{Z}^1$, a thief starts at the origin (0) and $N$ policemen are initially positioned at 2. All participants perform independent random walks, where each step to the left or right occurs with equal probability (1/2). Let $\tau_N$ denote the first encounter time between the thief and a policeman. Given that $E\tau_1=\infty$, find the smallest $N$ for which $E\tau_N<\infty$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the mass of the tetrahedron bounded by the planes \(x=0\), \(y=0\), \(z=0\), and \(\frac{x}{10} + \frac{y}{8} + \frac{z}{3} = 1\), given that the mass density at each point is \(\rho = \left(1 + \frac{x}{10} + \frac{y}{8} + \frac{z}{3}\right)^{-6}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A regular triangular prism \(ABC A_1 B_1 C_1\) is inscribed in a sphere, with \(ABC\) as its base and lateral edges \(A A_1, B B_1, C C_1\). The segment \(CD\) is the diameter of this sphere. Point \(K\) is the midpoint of edge \(A A_1\), and point \(L\) is the midpoint of edge \(A B\). Given that \(D L = \sqrt{6}\) and \(D K = 3\), find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\sqrt{3}}
Find the supremum of \( \left| z + e^{i\theta} w \right| \) for any complex numbers \( z \) and \( w \), where \( \theta \in [0, 2\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| + |w|}
Find the second derivative $F''(\pi)$ of the function $$ F(x) = \int_x^{\sin x} \left( \int_0^{\sin t} \sqrt{1+u^4} \, du \right) dt $$ using the fundamental theorem of calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
An ant is sitting at the center of a circle with a radius of 3 meters. Every minute, the ant randomly chooses a direction and moves 1 meter in that direction. On average, how many minutes does it take for the ant to leave the circle? Assume that if the ant reaches the edge of the circle within a minute, it is considered to have left the circle at the end of that minute.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
If the complex number \( z \) satisfies \( |z| = 2 \), find the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine an upper bound for \(|\cot z|\) on the boundary of the square with vertices at \(k\pi(1+i)\), \(k\pi(-1+i)\), \(k\pi(-1-i)\), and \(k\pi(1-i)\), where \(k\) is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\coth(k\pi)}
Let X be a random variable uniformly distributed over the interval [0,2]. Find the expected value of \( \max(X, X^3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{8}}
Determine the index \([B : A]\) of the subgroup \(A\) in the subgroup \(B\) of the group \(\mathbb{Q}[x]\) of one variable rational polynomials under addition, where:

- \(A = \{p(x) \in \mathbb{Z}[x]\mid p\) has degree at most \(2\}\), and
- \(B = \{p(x) \in \mathbb{Q}[x] \mid p\) has degree at most \(2\), and \(p(\mathbb{Z}) \subseteq \mathbb{Z}\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In preparation for the World Cup, Coach F plans to evaluate seven players, labeled as \( A_{1}, A_{2}, \cdots, A_{7} \), over three training matches, each lasting 90 minutes. At any given time during a match, only one player is on the field. The total playing time for players \( A_{1}, A_{2}, A_{3}, A_{4} \) must be divisible by 7, and the total playing time for players \( A_{5}, A_{6}, A_{7} \) must be divisible by 13. With unlimited substitutions allowed, determine the number of different scenarios for the total playing time of each player.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208806}
Given that $a^x = b^y = c^z$ and $a^3 = b^2c$, find the value of $\dfrac{3}{x} - \dfrac{2}{y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{z}}
Find the number of distinct elements in the smallest subgroup of \(GL(2,\mathbb{R})\) that contains the matrices \(A = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the complex number $z$ that satisfies the equation $z^{1+i} = 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1 - i}}
Determine the pointwise limit of the sequence of functions \( f_n(x) = x^2 - \frac{x^n}{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2}
Evaluate the triple integral \( \int_A |z| \,dx \, dy \, dz \) where \( A = \{(x,y,z): x^2+y^2+z^2\le4, x^2+y^2-2y\le0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5\pi}{2}}
Gary purchased a large beverage, but drank only $m/n$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37
}
Let $z_1$ and $z_2$ be two nth roots of unity that form a right angle at the origin. If the roots are represented as $z_k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right)$ for $k = 1, 2$, determine the condition on $n$ such that the angle between $z_1$ and $z_2$ is $\frac{\pi}{2}$ radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the values of $p$ for which a function in the Sobolev space $W^{1,p}(\Omega;\mathbb{R}^n)$ has a continuous representative, given that $\Omega$ is a bounded and smooth domain in $\mathbb{R}^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > n}
A crate contains blocks of four different dimensions: $13 \times 17 \times 21$, $13 \times 17 \times 37$, $13 \times 21 \times 37$, and $17 \times 21 \times 37$, with $33$ blocks of each kind. A tower is constructed by stacking $100$ of these blocks, one from each cross-section, to form a tower of height $h$. Determine the number of unique possible values for $h$. \(\boxed{\text{?}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{596}
In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament? 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
Evaluate the integral: $$\int_0^\infty \frac{\cos{kt}}{\pi}\,\mathrm{d}k.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(t)}
Calculate the integral \( \int_{0}^{1}\int_{0}^{2-2x}\frac{\left(2x-y\right)^{2}}{2x+y}dydx \) using the substitution \( u=2x-y \) and \( v=2x+y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{9}}
In an equilateral triangle $ABC$, a point $P$ is such that $|AP| = 2$ and $|BP| = 3$. Determine the maximum possible value of $|CP|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{19}}
Given a random sample \( X_1, X_2, \ldots, X_n \) from a population with density function \( f(x) = \begin{cases} e^{\theta-x}, & x \geq \theta \\ 0, & \text{otherwise} \end{cases} \), find the maximum likelihood estimator for \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min\{X_1, X_2, \ldots, X_n\}}
Determine the values of $z$ for which the function $\exp(-2 z\sin^2 t)$ is the characteristic function of a discrete random variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z \geq 0}
A company had a total sales revenue of 10 million yuan last year, with annual costs consisting of 5 million yuan in production costs and 2 million yuan in advertising costs. If the annual profit is taxed at a rate of p% and any annual advertising expenses exceeding 2% of the annual sales revenue are also taxed at p%, with no other taxes applied, and the company paid a total tax of 1.2 million yuan last year, calculate the tax rate p%. Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25%}
Find all polynomials \( P(x) \) that satisfy
$$
(x-1)P(x+1) - (x+2)P(x) \equiv 0, \quad x \in \mathbb{R}.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a\left( x^3 - x \right)}
Consider a finite field \( F \) with cardinality \( q \). Let \( F_k[x] \) be the set of polynomials in \( F[x] \) with degree less than or equal to \( k \). Given distinct elements \( x_1, \ldots, x_n \) in \( F \), define a code \( C \) as in the original question. Determine the minimum distance of this code.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - k}
30 students participated in the mathematical Olympiad. Each of them was given 8
 problems to solve. Jury estimated their work with the following rule:
 1）Each problem was worth \( k \) points,if it wasn’t solved by exactly \( k \) students;
 2) Each student received the maximum possible points in each problem or got 0 in it
 Lasha got the least number of points. What's the maximal number of points he could
 have?
 Remark: 1) means that if the problem was solved by exactly \( k \) students,than each of
 them got \( {30} - k \) points in it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
60
}
Given a power function $y=f(x)$ that passes through the point $(4, \frac{1}{2})$, find the value of $f\left( \frac{1}{4} \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
8-8. A chess tournament is held according to the following system: each of the 15 students from the "White Rook" school must play one game with each of the 20 students from the "Black Bishop" school, i.e., a total of 300 games should be played. At any given time, no more than one game is played.

After $n$ games, a spectator named Sasha, who has watched all the games and knows all the participants, exclaims: "I can definitely name one of the participants of the next game!" What is the smallest $n$ for which this could happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}