inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the values of $x$ for which the expression $(A^TA)^x$ is well-defined, where $A$ is a real matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x > 0}
Let $x_1, x_2, x_3$ be the solutions to $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of
$$\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12033}
Compute the limit \( \lim_{n \rightarrow \infty}{ \int_0^{\infty}\left(\frac{nx^{1/n}}{ne^x+\sin(nx)}\right)dx} \) using the relation between Riemann and Lebesgue integrals and Lebesgue’s Dominated Convergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: \[ \lim_{n \to \infty} 3^{3n-1}\left(\sqrt[4]{27^n+5}-\sqrt[4]{27^n-5}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of distinct values that can be obtained by inserting the operators $+, -, \times, \div$ and parentheses into the sequence $2, 2, 2, \ldots, 2$ (with $n$ occurrences of the number 2) and evaluating the resulting expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{n-1}}
Consider a hash function \( H(x) \rightarrow \{ 0,1 \}^{160} \) that maps inputs to 160-bit strings. What is the minimum number of attempts required to find a collision, i.e., two different inputs \( x_1 \) and \( x_2 \) such that \( H(x_1) = H(x_2) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{80}}
Determine the complexity of an algorithm that tests whether a given polynomial is a square in a finite field with small characteristic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1291}
Given that the weight of each gum ball (in centigrams) follows a gamma distribution with parameters \( \alpha = 25 \) and \( \beta = 2 \), calculate the probability that the total weight of 100 gum balls exceeds 52 grams.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0228}
给定 2003 个集合,每个集合都恰含 44 个元素, 并且每两个集合恰有一个公共元. 试求这 2003 个集合的并集中元素的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86130}
Evaluate the limit: $$\lim\limits_{n \to \infty} \frac{n^{1/n^2+x}−1}{\ln(n)\cdot \frac{1}{n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the gradient with respect to the matrix $X$ of the function: $$ \log \mathrm{det}\, X^{-1} $$ where $X$ is a positive definite matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-X^{-1}}
Find the smallest integer value of \( n \) such that the remainder \( R_{n}f(x) \) of the Taylor polynomial of degree \( n \) for \( \sin(x) \) around \( a=0 \) satisfies \( R_{n}f(x) \leq 0.01 \) for all \( x \in [0, 1/3] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the positive divisors of a number $n$ are $d_1, d_2, \ldots, d_r$ such that $1 = d_1 < d_2 < \ldots < d_r = n$, and that $(d_7)^2 + (d_{15})^2 = (d_{16})^2$, find all possible values of $d_{17}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Suppose we have a Bernoulli random variable $\Theta$ representing two hypotheses, where the probability of $\Theta = 1$ is $p$. Under hypothesis $\Theta = 0$, the random variable $X$ follows a uniform distribution on $[0,1]$. For the alternative hypothesis $\Theta = 1$, the probability density function (PDF) of $X$ is $f_{X|\Theta}(x|1) = 2x$ if $0 \leq x \leq 1$, and $0$ otherwise. When deciding between the two hypotheses based on the maximum a posteriori (MAP) rule with observed data $X = x$, find the probability of error as a function of $p$. Specifically, evaluate $P(\text{error})$ when $p \geq \frac{1}{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{(1-p)(5p-1)}{4p}\)}
Calculate the probability $P(\max(X_1, X_2, \ldots, X_n) > X_{n+1})$ where $X_1, X_2, \ldots, X_n, X_{n+1}$ are independently chosen from the same distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{n+1}}
$\mathrm{Let~}S=\{1,2,...,8\}.\text{ How many ways are there to select two disjoint subsets of }S?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6561
}
Find a non-trivial lower bound for the conditional expectation \(E(X \mid X \neq 0)\), where \(X\) is the number of people who get their own hat back at a party with \(N\) people, and each person grabs a hat at random when leaving.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine the number of homomorphisms from the group \( \mathbb{Z}/3 \times \mathbb{Z}/3 \) to the group \( A_4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Let $U$ and $W$ be subspaces of polynomials over $\mathbb{R}$ defined as follows:

\[ U = \text{Span}(t^3 + 4t^2 - t + 3, t^3 + 5t^2 + 5, 3t^3 + 10t^2 -5t + 5) \]
\[ W = \text{Span}(t^3 + 4t^2 + 6, t^3 + 2t^2 - t + 5, 2t^3 + 2t^2 -3t + 9) \]

Determine the dimension of the intersection $\dim(U \cap W)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given 155 birds perched on a circle \( C \), where birds \( P_{i} \) and \( P_{j} \) are considered mutually visible if \(\overparen{P_{i} P_{j}} \leqslant 10^{\circ}\), find the minimum number of pairs of mutually visible birds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{620}
Evaluate the line integral \( \int_C F \, dr \) using Green's Theorem, where the vector field is given by \( F=\langle e^{\sin{x}}+y^2, x^2+y^2 \rangle \) and the curve \( C \) is oriented as follows: it starts at point \((1,2)\), moves along a line segment to \((1,4)\), then to \((2,4)\), and finally back to \((1,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find a sequence of positive real numbers \( \{y_n\} \) such that \( \frac{y_n}{\sqrt{n}} \to \infty \) as \( n \to \infty \), but \( \frac{y_n}{\sqrt{n^\beta}} \to 0 \) for any \( \beta > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_n = \sqrt{n} \log n}
Find all positive integer solutions to the equation \((n+1)^n = 2n^k + 3n + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 3)}
Evaluate\[  \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^9 \sum_{c=1}^9 \left( abc+ab+bc+ca+a+b+c \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{215}
Find the coefficient of $x^2$ in the expansion of $(1+2x-3x^2+4x^3)^{-1/2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the probability distribution function of the random variable \( z = |x - y| \), where \( x \) and \( y \) are independent random variables uniformly distributed between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(1 - z)}
What is the expected number of non-empty columns (or sampled dishes) $\Lambda_t$ in an Indian Buffet Process with $T$ customers and concentration parameter $\alpha$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \sum_{t=1}^T \frac{1}{t}}
Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$
Find the last seven digits of $A$, in order from millions to units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2402230}
Determine the largest possible determinant of a matrix whose elements are either $\pm1$ or $0$. Provide your answer as a single number or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Calculate the product modulo 1234 of all numbers less than 1234 that are relatively prime to 1234, excluding those numbers that are their own multiplicative inverses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a competition with 37 contestants, each contestant has a non-negative integer score. The sum of the scores of any 17 contestants is at least 45, and the total score of all 37 contestants is 150. What is the maximum score that a single contestant can achieve?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Evaluate the integral \( \int_{-\infty}^{\infty} e^{-\pi(x - i\alpha)^2}dx \) where \( \alpha \) is a real constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many permutations of the string "ABCDEF" exist such that no two adjacent characters in the original string are adjacent in the permutation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
In a convex pentagon $ABCDE$ the areas of the triangles $ABC, ABD, ACD$ and $ADE$ are all equal to the same value x. What is the area of the triangle $BCE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2x$}
In a graph where each vertex has a degree $d$, determine the minimum length of a path that can be constructed using a greedy algorithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d}
Determine the number of distinct colorings of the vertices of a cycle graph $C_{8}$ using 10 colors, where each color can be used multiple times. Two colorings are considered the same if one can be transformed into the other by an isometry of the octagon. Use Burnside's Lemma to find the number of distinct colorings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6278140}
Find a primitive root of 338.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the values of $n \geq 2$ for which there exists a permutation $\sigma$ in the alternating group $A_n$ such that the order of $\sigma$ is greater than $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 8}
Consider two Gaussian distributions, where one has a variance of zero, effectively making it a unit impulse function. Determine the Overlapping Coefficient between these two distributions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $Y$ be a connected covering space of a bouquet of 5 circles with a degree of 5. Determine the rank of the free group $\pi_{1}(Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Using the Cauchy-Schwarz inequality, determine the minimum value of \( \int_0^1 f^2(x) \, \mathrm{d}x \) for a non-negative Riemann integrable function \( f(x) \) satisfying \( \int_0^1 f(x) \, \mathrm{d}x = 1 \) and \( \int_0^1 x f(x) \, \mathrm{d}x = \frac{1}{6} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
Given the function \( f(x) = x^3 \), find the value of \( \frac{\partial^2 f}{\partial x \partial f} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On a street with 14 houses, the odd-numbered houses are on one side and the even-numbered houses are on the opposite side. Families on one side have surnames that are colors, and on the other side, surnames indicate professions. Szabó and Fazekas live opposite Zöld and Fehér, respectively, who are neighbors of Fekete. Kovács is the father-in-law of Lakatos. Lakatos lives in a higher-numbered house than Barna, and the sum of their house numbers equals the sum of Fehér and Fazekas's house numbers. Kádárné's house number is twice that of her sister, Kalaposné. Sárga lives opposite Pék. Bordóné's house number is two-digit, and she lives opposite her sister, Kádárné. What is the house number of Mr. Szürke?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find the maximum number of different permutations of the projections of 8 points onto all possible directed lines on a plane. Let this maximum number be denoted as \( N_8 \). What is the value of \( N_8 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Find the limit of the function \( \frac{\ln(\frac{1+x^2}{1-x^2}) - \arctan(2x^3) + 2x\sin^2(x)}{x\exp(-\frac{1}{x}) - 2x^2 + \ln(\frac{1+x^2}{1-x^2})} \) as \( x \) approaches 0 from the positive side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( \int\int\int_E z \, dV \), where \( E \) is the region above the paraboloid \( z = x^2 + y^2 \) and below the plane \( z = 6y \). Round your answer to the nearest hundredth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1908.52}
Evaluate the integral \( \int_0^1 \ln(\sin(2\pi z)) \, \mathrm{d}z \) using the properties of the Gamma function and the identity \( \sin(\pi z) = \pi z \prod_{k=1}^\infty \left(1 - \frac{z^2}{k^2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
Let $X$ be a random variable with probability density function \( f(x)=\frac{1}{2} e^{-|x|},\; -\infty<x<\infty. \) Calculate the expected value of \( \lfloor X \rfloor \), where \( \lfloor X \rfloor \) denotes the greatest integer less than or equal to \( X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Find the spectrum of the operator \( A: C[0,1] \rightarrow C[0,1] \) defined by \( (Af)(x) = f(x) + \int_0^x f(t)dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1\}}
Consider the set \( S = \{1, 2, \ldots, 100\} \). A partition of \( S \) is defined as \( S_1, S_2, \ldots, S_n \) for some \( n \), where each \( S_i \) is nonempty, pairwise disjoint, and \( S = \bigcup_{i=1}^n S_i \). Let \( a_i \) be the average of the elements in the set \( S_i \). The score of this partition is given by \( \frac{a_1 + a_2 + \ldots + a_n}{n} \). Determine the minimum possible score among all possible partitions of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A short-sighted rook is a rook that beats all squares in the same column and in the same row for which he can not go more than
 60-steps.
 What is the maximal amount of short-sighted rooks that don’t beat each other that can be put on a \( {100} \times  {100} \) chessboard.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
178
}
Determine the limit of the sequence \( f_n(a) = a^{f_{n-1}(a)} \) as \( n \) approaches infinity, where \( f_0(a) = a \) and \( a = 1.1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.1118}
Find an upper bound for $l_n$, the length of the curve $y=x^n$ in the interval $[0,1] \times [0,1]$, using the function $f_n(x) = \sqrt{1+n^2(1-x)^{2n-2}}$ and the approximation provided in the original answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the greatest positive real number \( M \) such that for every positive real sequence \((a_n)\) and for every real number \( m < M \), there exists an index \( n \geq 1 \) satisfying the inequality \( a_1 + a_2 + \cdots + a_n + a_{n+1} > m a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Observing the following expressions: $$1+ \frac {1}{2^{2}} < \frac {3}{2}, 1+ \frac {1}{2^{2}}+ \frac {1}{3^{2}} < \frac {5}{3}, 1+ \frac {1}{2^{2}}+ \frac {1}{3^{2}}+ \frac {1}{4^{2}} < \frac {7}{4},…$$, what could be the conjectured conclusion when $$n \in \mathbb{N}$$ and $$n \geqslant 2$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {2n-1}{n}}
If $f(x)$ is a differentiable function such that \( f(x) = \left(1 + \frac{x^3}{3}\right) + \int_0^x e^{-t} f(x-t) \, dt \), find the value of \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{5}}
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, \frac{a_{n+1}-a_{n}}{a_{n}}=\frac{a_{n+2}-a_{n+1}}{a_{n+2}}\left(n \in N^{*}\right)$, 若 $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{6} a_{7}=3$, 则 $a_{2024}=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2035}
In a Poisson process with parameter \( \lambda \), let \( S_1, S_2, \ldots \) be the arrival times. Given the time of the \( n \)-th arrival, find the expected value \( E(S_1|S_n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{S_n}{n}}
Evaluate the contour integral \( \oint_{\gamma} \frac{f'(z)}{f(z)} dz \), where \( f(z) = f(0) \prod_{\rho} \left(1 - \frac{z}{\rho}\right) \) and \( \rho \) are the roots of \( f(z) \). The contour \( \gamma \) is a rectangle with vertices at \( a \), \( a + b \), \( a + b + it \), and \( a + it \), traversed counterclockwise. Assume all roots of \( f(z) \) lie within the rectangle. Determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i N}
In a convex equilateral hexagon \(A B C D E F\), lines \(B C\), \(A D\), and \(E F\) are parallel. Let \(H\) be the orthocenter of triangle \(A B D\). Given that the smallest interior angle of the hexagon is 4 degrees, find the smallest angle of triangle \(H A D\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $k:\Bbb R\to\Bbb R$ be a twice differentiable function such that $k(a)=k(e)=0$, $k(b)=k(d)=2$, and $k(c)=1$, where $a<b<c<d<e$. Determine the minimum number of solutions to the equation $(k'(x))^2+k(x)k''(x)=0$ in the interval $[a,e]$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the average length of a cycle in a permutation of the set \( \{1, 2, 3, \dots, n\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{H_n}}
In a quadrilateral pyramid with a rhombus base $ABCD$, where $\angle BAD = 60^\circ$, it is given that $SA = SC$ and $SD = SB = AB$. A point $E$ is located on edge $DC$ such that the area of triangle $BSE$ is minimized among all sections of the pyramid containing segment $BS$ and intersecting segment $DC$. Determine the ratio $DE:EC$. \( \text{Express your answer as a simplified fraction.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the condition on $a(t)$ such that the stochastic process $X(t)$, defined by the stochastic differential equation $$ dX = a(t)Xdt + b(t)XdW(t), $$ is a $P$-martingale, where $b(t)$ is non-zero almost surely and $W(t)$ is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a(t) = 0}
Let $X$ and $Y$ be independently distributed central chi-squared random variables with degrees of freedom $m(\geq 3)$ and $n(\geq 3)$, respectively. Given that $E \left(\frac{X}{Y}\right)=3$ and $m+n=14$, find $E \left(\frac{Y}{X}\right)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{7}}
Given an $m \times n$ board where $m,n > 2$ are even integers, each cell is colored either black or white. The Guesser can ask the Oracle whether two adjacent cells (sharing an edge) have the same color. What is the minimum number of inquiries the Guesser needs to make to determine the parity of the number of adjacent cell-pairs with different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 4}
Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\ln{(2)} $}
A driver travels from point A to point B, which are 100 km apart. Initially, the driver travels at a constant speed. When the navigator indicates 30 minutes of travel remain, the driver reduces speed by 10 km/h. When the navigator indicates 20 minutes of travel remain, the driver reduces speed by another 10 km/h. The car travels 5 minutes longer at the reduced speed than it would have at the twice-reduced speed. Determine the initial speed of the car.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the range of \( t \) such that for any \( a \in (0, 4) \), there exists an \( x_0 \in [0, 2] \) satisfying \( t \leq |f(x_0)| \), where \( f(x) = x^2 - ax + a - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
Determine the smallest integer $n$ such that the approximation of $e^x$ by the series $s_n(x) = \sum_{k=0}^{n} \frac{x^k}{k!}$ satisfies the inequality \(|e^x - s_n(x)| \leq \frac{|e^x|}{10^{16}}\) for all $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$. Provide your answer as the smallest $n$ that meets this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the number of different ring homomorphisms from \(\mathbb{Z} \times \mathbb{Z}\) to \(\mathbb{Z} \times \mathbb{Z}\), given that a ring homomorphism must map identities to identities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a simple polygon with \( n \) vertices, where each vertex is exactly visible to four other vertices, determine all possible values of the positive integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral: $$\int^{1/2}_0 \int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{400}}
Let $a$ be a positive real number, $n$ a positive integer, and define the *power tower* $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Determine which is larger: $x_{42}$ or $x_{43}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_{42}}
Evaluate the integral $$\int_0^{\infty} \frac{\cos(2\alpha q)}{\sqrt{1+q^2}} \, dq.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K_0(2\alpha)}
Seven fishermen stand in a circle, each with a unique integer exaggeration factor. When asked, "How many fish did your left neighbor catch?", the responses were $12, 12, 20, 24, 32, 42,$ and $56$. When asked, "How many fish did your right neighbor catch?", six of the fishermen responded $12, 14, 18, 32, 48,$ and $70$. What did the seventh fisherman respond?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}