inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a boundary value problem with the operator $\mathscr{L}:=-\Delta$ and the boundary condition $u=0$. If $u$ satisfies the inequality $u^5-u \le 0$ in the domain $\Omega$, determine an upper bound for $\sup_{\Omega} |u|$. Provide your answer as a specific numerical value or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $(b_i)_{i\in \Bbb{N}}$ be a sequence of positive integers such that $b_3 = 7$, $b_{2020} = 2021$, and $b_n=b_{b_{n-1}}$ for all other positive $n$. Determine the possible values of $b_{2021}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
In triangle \(ABC\), the angle bisector \(AL\) (where \(L \in BC\)) is drawn. Points \(M\) and \(N\) lie on the other two angle bisectors (or their extensions) such that \(MA = ML\) and \(NA = NL\). Given that \(\angle BAC = 50^\circ\), find the measure of \(\angle MAN\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Determine the minimum distance $d(C)$ of the ternary linear code $C$ with the following parity check matrix:
$$H=\begin{pmatrix} 2 & 0 & 1 & 1 & 1 \\ 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 1 \\ 1 & 1 & 2 & 2 & 0 \end{pmatrix}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given $n \equiv 0 \,(\mathrm{mod}\,4)$, determine the minimum value of $m$ such that there exists an $m \times n$ partial Hadamard matrix that cannot be extended to an $(m+1) \times n$ partial Hadamard matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{2}}
In a regular triangle \( ABC \), three nails are driven into the floor, one near each side, preventing the triangle from rotating without lifting. The first nail divides side \( AB \) in the ratio \( 1:3 \) from vertex \( A \), and the second nail divides side \( BC \) in the ratio \( 2:1 \) from vertex \( B \). Determine the ratio in which the third nail divides side \( AC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3}
Evaluate the surface integral \( \iint_S \vec{A} \cdot \hat{n} \, dS \), where \( \vec{A} = 18z\hat{i} - 12\hat{j} + 3y\hat{k} \) and \( S \) is the part of the plane \( 2x + 3y + 6z = 12 \) located in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given 365 unique days in a year, how many friends would you need on average to ensure that at least one friend's birthday falls on each day? Assume that birthdays are equally distributed across all days of the year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2365}
Compute the integral \( \int_{C} \log(z-1) \, dz \) over a circle centered at the origin with a radius of 3, oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i}
In an $n \times n$ grid, each cell contains a student facing one of the four cardinal directions: up, down, left, or right. The students on the grid's borders do not face each other directly, meaning no student looks directly at another student's face; they can only see another student's ear or back. Determine the minimum number of students who can see the ear of another student. For example, in a $5 \times 5$ grid, the minimum number of students with this view is $n+4$. What is the minimum number of students who see the ear of another student for a general $n \times n$ grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 4}
Determine the number of equivalence relations \(S\) over the power set of natural numbers \(\mathcal P(\mathbb N)\) such that the equivalence class of \(\{8\}\) under \(S\) is \(\{A \in \mathcal P(\mathbb N) \mid A \neq \{1\} \wedge A \neq \{2\}\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the order of the splitting field of the polynomial \(x^8 - 1\) over the field with 4 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a 2015x2015 square table filled with positive numbers, the product of the numbers in each row and each column is 2, and the product of the numbers in any 3x3 sub-square is 1. Determine the number located at the center of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate $ \int_0^1 \frac{(1-2x)e^{x}+(1+2x)e^{-x}}{(e^x+e^{-x})^3}\ dx.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{e^{2}}{\left(1 + e^{2}\right)^{2}} \]}
Evaluate the limit: $$\lim_{x\to 0}\left\{\frac{e^{-x^2}\sin|x|-|x|}{|x^3|}\right\}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{6}}
Find the smallest positive number \( \lambda \) such that for any positive numbers \( a_{1}, a_{2}, \ldots, a_{n} \) and any \( n \) numbers \( b_{1}, b_{2}, \ldots, b_{n} \) from the interval \( \left[0, \frac{1}{2}\right] \), if \( a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+\cdots+b_{n}=1 \), then \( a_{1} a_{2} \cdots a_{n} \leq \lambda\left(a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2(n-1)^{n-1}}}
Determine the number of distinct values that the integral \( \oint_\gamma \frac{dz}{(z-a_1)(z-a_2)...(z-a_n)} \) can take, where \( \gamma \) is a simple closed curve not passing through any of the points \( a_1, a_2, \ldots, a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
We write the digits of $1995$ in the following way:
$199511999955111999999555......$
Which digit is in position number $1995$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the largest positive integer $r$ such that in any 500-element subset of the set $\{1, 2, \ldots, 1000\}$, there exist two subsets whose intersection contains at least $r$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Find the minimum value of the integral \( \int_{0}^{1}|t^3-a-bt-ct^2|^2dt \) by using the projection of \( t^3 \) onto the subspace of polynomials of degree at most 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2800}}
Determine the number of conjugacy classes of the group \( G = \langle s,t \mid s^8=t^2=1, st=ts^3 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the common ring to which the quotient ring \( \mathbb{Z}_6[x]/\langle 2x-3 \rangle \) is isomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_3}
Suppose $A$ and $B$ are matrices with dimensions $n \times K$ and $n \times M$ respectively, where $n > K$ and $\text{rank}(A) = K$. If $\text{rank}(A + B) = K - s$ for some $s > 0$, determine the minimum possible dimension of the column space of $B$. \(\text{Express your answer as an integer.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s}
Find a particular solution to the differential equation \( y(x) = xy' + \ln y' \) given the initial condition \( y(-1) = -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = -1 - \ln(-x)}
Determine the group of grouplike elements in the Hopf algebra $U(L)$, where $U(L)$ is the universal enveloping algebra of a Lie algebra $L$. The algebra $U(L)$ is constructed as $U(L) = T(L) / I(L)$, with $I(L)$ being the two-sided ideal generated by elements of the form $xy - yx - [x,y]$, and $T(L)$ is the tensor algebra of $L$. A grouplike element in a coalgebra is an element $x \neq 0$ with $\Delta(x) = x \otimes x$. The set of grouplike elements forms a group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1\}}
Evaluate the triple summation \( S = \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{1}{3^{i+j+k}} \) where \( i \neq j \neq k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{208}}
As $n_1 \to \infty$ and $n_2 \to \infty$, does the statistic \( T = \dfrac{\bar{X}_{1, n_1} - \bar{X}_{2, n_2}}{\sqrt{\dfrac{S_1^2}{n_1} + \dfrac{S_2^2}{n_2}}} \) converge in distribution to a random variable? If so, identify the distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N(0,1)}
Calculate the Laplacian of the function \( u(x,y,z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the largest possible number of inversions in a sequence of positive integers whose sum is 2014?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{507024}
Determine the maximum percentage increase in the number of basic solutions for a bounded linear programming problem when an additional constraint is added.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100\%}
Let $x_1, x_2, \ldots, x_{2560}$ be the roots of the polynomial $x^{2560} + 2x^{2559} + 3x^{2558} + \ldots + 2561$. Determine the value of the expression:
$$(x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1)\ldots(x_{2560}^2 + 1)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3279361}
Provide an example of a bipartite connected graph with an even number of vertices that has an Eulerian circuit but does not have a perfect matching.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K_{2,4}}
Calculate the limit \( \lim_{n\to \infty}\left(\dfrac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^{1/n} \) given that \( f:\mathbb{R}\to \mathbb{R} \) is differentiable at \( a \) and \( f(a) > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the limit superior of the sequence \( a_j = |\sin j|^{\sin j} \) for all \( j \in \mathbb{N} \). Determine \( \limsup_{j \to \infty} \{ a_j \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{1/e}}
Find the number of permutations \( \pi \) of \([n]\) such that the numbers 1, 2, and 3 appear in the same cycle of \( \pi \), while the number 4 appears in a different cycle. Provide a general formula for any \( n \geq 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n!}{12}}
Calculate the expected number of collisions when hashing 80,000 unique entries using a 36-bit hash function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0466}
Determine the number of distinct sets of eight numbers such that each number in the set is the product of two other numbers from the same set. Sets that differ only in the order of the numbers are considered identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the points of continuity for the function \( f:(0,1)\rightarrow \mathbb{R} \) defined by \( f(x)= x^{2} \) if \( x \) is rational and \( f(x) = 2-x^{2} \) if \( x \) is irrational.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$25^{\circ} $}
In a grand hall, there are 1000 people having dinner. One person is initially sick, while the other 999 are healthy. Each minute, every person talks to one other person in the room at random, ensuring they do not talk to someone they have previously talked to. If a sick person talks to a healthy person, the healthy person becomes sick. Once a person becomes sick, they remain sick. Determine the maximum number of minutes until all 1000 people in the hall are sick.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
Evaluate the integral \( \int^{+\infty}_0 \frac{e^{-x^2}}{(x^2+\frac{1}{2})^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
Determine the number of distinct ways to select four 3-combinations from a set of six elements, such that each pair of combinations shares exactly one common element. Consider two sets of combinations equivalent if one can be transformed into the other by an element of the dihedral group D6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Each cell of a 2 × 5 grid of unit squares is to be colored white or black. Compute the number of such colorings for which no 2 × 2 square is a single color.
  
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
634
}
Evaluate the limit: \( \lim_{n\to\infty}\sqrt[n]{c_n^n+d_n^n+2023} \), given that \( c_n \rightarrow c > 1 \) and \( d_n \rightarrow d > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{c, d\}}
Find the value of \( \Phi \), the infinite product of successive approximations to the golden ratio \( \phi \), defined as \[ \Phi = \frac{\sqrt{1}}{\phi} \times \frac{\sqrt{1+\sqrt{1}}}{\phi} \times \frac{\sqrt{1+\sqrt{1+\sqrt{1}}}}{\phi} \times \dots \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the smallest natural number $b$ such that the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots is 20.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
Find an expression for the function \( h(z) = \sum_{n=1}^{\infty}\frac{(z-2)^n}{n} \) as an elementary function. Consider using techniques similar to those used for the related function \( f(z) = \sum_{n=1}^{\infty} n(z-2)^n = \frac{z+3}{(z+2)^2} \). Knowledge of logarithms in the complex plane may be required.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln(3 - z)}
Evaluate the integral \( \int_{|z|=3\pi/2} \frac{2\cosh(z)}{1-\sinh(z)} \, dz \) by considering the singularities of the integrand within the contour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12\pi i}
Compute the minimum value of the integral \(\displaystyle\int_{0}^{\infty} \left | x^3-a-bx-cx^2 \right |^2e^{-x}\, dx\) subject to the constraints \(a,b,c \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Consider a sequence of independent and identically distributed (i.i.d.) Gaussian matrices $A_i \in \mathbb{R}^{p \times n}$ with $p < n$. Each matrix $A_i$ can be decomposed into its reduced singular value decomposition (SVD) as $A_i = V_i \Sigma_i U_i^T$. When we concatenate the column spaces of these matrices by forming $\sum_{i=1}^m U_i U_i^T$, what is the minimum value of $m$ for this sum to be a full-rank matrix, assuming that the matrices $U_i$ follow the Haar measure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lceil \dfrac{n}{p} \right\rceil}
Find the greatest constant \( k > 0 \) such that \[ \max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right) \geq k\cdot\frac{2+3b}{5-c-2d} \] for all \( 0 \leq b \leq 1 \) and \( 0 \leq c \leq d \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{16}}
Find the term with the highest power of \(x\) in the expression \((x^3-2)^{16}-(x^4 + 3)^{12}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-32x^{45}}
Determine all values of the parameter $m$ such that the system of equations \( x^2 = 2^{|x|} + |x| - y - m \) and \( 1 - y^2 = 0 \) has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the condition under which equality holds for the inequality \(|x+y| \leq |x| + |y|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{xy \geq 0}
64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players \( P \) is it possible that \( P \) wins the first four rounds that he plays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Determine the value of \( x \) for which all eigenvalues of the matrix \( A \) are real, where:
\[ A=\begin{pmatrix} 10 & 5+i & 4 \\ x & 20 & 2 \\ 4 & 2 & -10 \end{pmatrix} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 - i}
Find \( \lim_{n \to \infty} \int_{1}^{n}\frac{nx^{1/2}}{1+nx^2}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle $ABC$ with sides $AB = 30$, $BC = 51$, and $CA = 63$, points $P$, $Q$, $R$, and $S$ are on sides $BC$, $CA$, and $AB$, respectively, such that $PQRS$ forms a parallelogram with its center at the centroid of triangle $ABC$. Determine the area of parallelogram $PQRS$ using an analytic approach.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
Suppose $X_1, X_2, \ldots, X_n$ are independent random variables, each uniformly distributed over the interval $[0, 1]$. Let $X_{(n)}$ be the largest and $X_{(n-1)}$ be the second largest of these variables. Calculate the expected value of the ratio $\frac{X_{(n-1)}}{X_{(n)}}$. In other words, find $\mathbb{E}\left[\frac{X_{(n-1)}}{X_{(n)}}\right]$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n-1}{n}}
Find the value of \( r \), where \( r > \frac{1}{2} \), such that two non-vertical normal lines from the point \((r, 0)\) to the parabola \(x = y^2\) intersect perpendicularly. One of these normal lines is the x-axis. The derivative of the parabola is \(\frac{dy}{dx} = \frac{1}{2\sqrt{x}}\), so the gradient of the normal is \(-2\sqrt{x}\). The equation of the normal can be expressed as \(y = -2\sqrt{x}(x - r)\). Determine the correct value of \( r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Assume a problem $R \in \mathit{NP}$ can be solved with a verifier $M(x,y)$ that runs in $O(n^3)$ time, where $y$ is additional information no longer than $3\log n$ bits. Determine the function $f(n)$ such that $R \in \mathit{TIME}(f(n))$. Provide your answer as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^6}
Let $X_1, \ldots, X_{735}$ and $Y_1, \ldots, Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$, and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Using the Central Limit Theorem, find $P\left(\sum_{i=1}^{735} X_i < \sum_{i=1}^{880} Y_i\right)$. Express your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8413}
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that\[(\cos(a\pi)+i\sin(b\pi))^4\]is a real number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{25}}
Evaluate the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \ln(2468^{2} \cos^2x + 990^2 \sin^2x) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \ln 1729}
Calculate the expectation value for the function: $$ f(x)=\begin{cases} \frac{x^2}{2\beta^{3}}e^{-\frac{x}{\beta}} & x>0\\ 0 & \text{otherwise} \end{cases}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\beta}
Determine the number of non-isomorphic groups of order 30, given that every Sylow 3 and Sylow 5 subgroup is normal in the group, and the group has a normal subgroup of order 15.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the constant $p$ such that the product of any positive number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
What is the smallest number of vertices, $n$, in a polygon such that there exists an interior point $p$ where the distance from any vertex $b$ to $p$ is greater than the distance from $b$ to either of its adjacent vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the minimum number of combination attempts required to guarantee opening a safe with 3 wheels, each having 8 positions, if the safe opens when any two of the three wheels are set correctly? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} = \frac 1m + \frac 1n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
Find the degree of the differential equation \((y''')^\frac{4}{3} + (y')^\frac{1}{5} + 4=0\) after rationalizing the indices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Evaluate the limit \( \lim_{h \to 0} \frac{2\sin\left(\frac{1}{2}\ln\left(1+\frac{h}{x}\right)\right)}{h} \) without using L'Hopital's rule, given that \( \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{x}}
Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]

Determine the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1511}
Determine the order of the group $G$ given by the presentation $G = \langle a, b \mid a^3 = b^2 = (ab)^3 = e \rangle$, where $e$ is the identity element.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the radius of convergence of the power series \( \sum_{k=0}^{\infty} a_k x^{k^2} \), where \( a_k = (2k)! \) if \( k \) is a perfect square, and \( a_k = 0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the minimum value of the function \( f(x,y,z) = \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| \) given that \( x, y, z \) are non-negative real numbers satisfying \( x+y+z=1 \) and \( y^2 \le 3xz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52 - 28\sqrt{3}}
A battery has a lifetime of 24 hours and is used for a maximum of three days. Each day, the battery is used for a random number of hours, denoted by \( K \), where \( K \) is uniformly distributed over \([0, 24]\) and independent of usage on other days. Calculate the expected remaining lifetime of the battery at the end of the third day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum value of \((f^{-1})'f(x)\) for an invertible increasing function \(f(x)\) that satisfies \(f''(x)+2f'(x)\geq 2\) and \(f'(0)=2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}