inclusionAI/Ring-lite-rl-data huggingface dataset:
In a chess tournament with 12 participants, each participant creates 12 lists after the tournament. The first list includes only themselves, the second list includes themselves and those they defeated, the third list includes everyone from the second list and those they defeated, and so on. The 12th list includes everyone from the 11th list and those they defeated. It is known that for any participant, there is a person in their 12th list who was not in their 11th list. Determine the number of drawn games played in the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Determine the number of continuous functions \( f: [0,1] \rightarrow \mathbb{R} \) that satisfy the equation \[ \int_{0}^{1} f(x) \, dx = \frac{1}{3} + \int_{0}^{1} f^2(x^2) \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $A$ and $B$ be $3 \times 3$ matrices such that $A^3 + 5AB = I$ and $A^3 - 5BA = 2I$. Find $\det(A)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine if there exists a function $f$ such that for all $a, b, c, d$, the following equations hold: \[ f(a)f(b)-f(c)f(d) = (f(a)-f(c))(f(b)-f(d)) \] and \[ f(a)f(b)+f(c)f(d) = (f(a)+f(c))(f(b)+f(d)). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = 0}
Solve the recurrence relation \( A(n) = 2A(n/2) + O(1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n)}
Determine the minimum number of elements that the subset \(X\) of the set \(\{00, 01, \ldots, 98, 99\}\) must contain so that in any infinite sequence of numbers, there are always two adjacent numbers that form an element of \(X\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Determine the minimal polynomial of \( \overline{x+1} \) over the field \( \mathbb{F}_2 \) in the field \( K = \mathbb{F}_2[x] / (x^3+x+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^3 + t^2 + 1}
Let \( A = (a_{ij}) \) be the \( n \times n \) matrix, where \( a_{ij} \) is the remainder of the division of \( i^j + j^i \) by \( 3 \) for \( i, j = 1, 2, \ldots, n \). Find the greatest \( n \) for which \( \det A \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit \( \lim_{x\rightarrow 0} \frac{\sin x^2}{\ln(\cos x^2 \cos x + \sin x^2 \sin x)} \) without using L'Hôpital's rule, by applying trigonometric identities and known limits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
A cylindrical fuel tank with hemispherical ends must hold 300 liters of jet fuel. The side wall of the cylinder costs $0.1 per cm² to manufacture, while the hemispherical ends cost $0.3 per cm². Determine the optimal radius of the tank that minimizes the total manufacturing cost.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.7}
Determine the number of equivalence relations on a set with 7 elements such that the equivalence relation contains exactly 29 ordered pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum
\[ \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024

}
A total fixed amount of $N$ thousand rupees is given to $A,B,C$ every year, each being given an amount proportional to her age. In the first year, A got half the total amount. When the sixth payment was made, A got six-seventh of the amount that she had in the first year; B got 1000 Rs less than that she had in the first year, and C got twice of that she had in the first year. Find N.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
35
}
Find a function $g(x)$ such that $\lim_{x\to\infty}\frac{g(x^2)}{g(x)}=1$ and $\lim_{x\to\infty}g(x)=\infty$. Provide your answer as a function expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log \log x}
Determine the minimum number of equally spaced subintervals, N, required such that the difference between the Upper and Lower Riemann sums for the integral \( \int_{1}^{17} \frac{1}{x^3} \, dx \) is less than \( \frac{1}{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1600}
Given that $T_0$ is the minimum variance unbiased estimator (MVUE) for $\gamma(\theta)$ and $T_1$ is another unbiased estimator for $\gamma(\theta)$ with an efficiency of 0.0169, find the correlation coefficient between $T_0$ and $T_1$. \( \text{corr}(T_0, T_1) = ? \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.13}
Determine the maximum possible price difference between two American-style call options on a stock with different strike prices, assuming no dividends are paid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K_2 - K_1}
Given the matrix \( A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} \), where \( A = B + C \), \( B \) is symmetric \( (B = B^T) \), and \( C \) is skew-symmetric \( (C = -C^T) \). If \( (CB)^T = D \) and \( D = [ d_{ij} ]_{3 \times 3} \), find the value of \( d_{12} + d_{21} + d_{13} + d_{31} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
84}
Let \( P \) and \( Q \) be two points on the curve \( y = \log_{\frac{1}{2}}\left(x - \frac{1}{2}\right) + \log_2\left(\sqrt{4x^2 - 4x + 1}\right) \). Point \( P \) is also on the circle \( x^2 + y^2 = 10 \). Point \( Q \) lies inside the circle such that its abscissa is an integer. Find the smallest possible value of \( \overrightarrow{OP} \cdot \overrightarrow{OQ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{(b-a)^2}{8a}$}
Given two positive integers $a$ and $b$, where $a$ has 20,041 '1's and $b$ has 20,051 '1's in their binary representations, determine the smallest possible number of '1's in the binary representation of $a + b$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the cubic equation \(x^3 + 2x^2 + 3x + 1 = 0\) with roots \(a, b, c\), it is known that \(\frac{1}{a^3} + \frac{1}{b^3} - \frac{1}{c^3}\), \(\frac{1}{a^3} + \frac{1}{c^3} - \frac{1}{b^3}\), and \(\frac{1}{c^3} + \frac{1}{b^3} - \frac{1}{a^3}\) are the roots of another cubic equation \(px^3 + qx^2 + rx + s = 0\). Find the value of \(p + q - 12r + s\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Géza starts with the sequence \((1, 2, \dotsc, 9)\) and extends it 2017 times. After the final extension, he randomly selects one element from the resulting sequence. What is the probability that the chosen element is 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{9}}
Consider a Markov chain with state space $S=\{0, 1, 2, \ldots \}$ and transition probabilities $p(0,0)=3/4$, $p(0,1)=1/4$, $p(x, x-1)=3/4$, and $p(x,x+1)=1/4$ for $x \geq 1$. Let $e(x)$ denote the expected number of steps to reach the origin starting from state $x$. Find $e(1)$. (Hint: First consider the expected return time starting at the origin and express $e(1)$ in terms of this.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the number of connected graphs that can be formed with 6 labeled vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26704}
Consider the infinite series \(S\) represented by \(2 - 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} - \frac{1}{81} + \frac{1}{243} - \cdots\). Find the sum \(S\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
For the four-digit number \(\overline{abcd}\) where \(1 \leqslant a \leqslant 9\) and \(0 \leqslant b, c, d \leqslant 9\), if \(a > b, b < c, c > d\), then \(\overline{abcd}\) is called a \(P\)-type number. If \(a < b, b > c, c < d\), then \(\overline{abcd}\) is called a \(Q\)-type number. Let \(N(P)\) and \(N(Q)\) represent the number of \(P\)-type and \(Q\)-type numbers respectively. Find the value of \(N(P) - N(Q)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285}
The library initially has 250 books. Over the course of three weeks, the library experiences the following transactions:

Week 1:
- Tuesday: 120 books are taken out to be read by children.
- Wednesday: 35 books are returned.
- Thursday: 15 books are withdrawn from the library.
- Friday: 42 books are taken out by adults.

Week 2:
- Monday: 60% of the books taken out by children last week are returned.
- Tuesday: 80% of the books taken out by adults last week are returned.
- Wednesday: 75 books are withdrawn from the library.
- Thursday: 40 books are taken out by teenagers.
- Friday: An inventory is conducted.

Week 3:
- Monday: 50% of the books taken out by teenagers last week are returned.
- Tuesday: 20 books are taken out by senior citizens.
- Wednesday: 95% of the books taken out by children in week 1 that had not yet returned are returned.
- Thursday: 10 more books are withdrawn from the library.
- Friday: 55 books are taken out by a local book club.

Taking into account all these transactions, how many books are in the library by the end of the third week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$. Let $D$ be the point such that $AB \parallel CD$ and $CA \parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively. If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Calculate the line integral

$$
\int_{L} \operatorname{arctg} \frac{y}{x} d l
$$

where \( L \) is part of the Archimedean spiral \( \varrho = \varphi \) (for \( 0 \leq \varphi \leq \pi / 2 \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\left(\pi^2 + 4\right)^{3/2} - 8}{24}}
If for any $x\in R$, $2x+2\leqslant ax^{2}+bx+c\leqslant 2x^{2}-2x+4$ always holds, then the maximum value of $ab$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{2}"]}
How many ways are there for Nick to travel from \((0,0)\) to \((16,16)\) in the coordinate plane by moving one unit in the positive \(x\) or \(y\) direction at a time, such that Nick changes direction an odd number of times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{310235040}
Determine the radius of convergence for the series \( \sum_{n=0}^{\infty} a_n (x-3)^n \), where the series is given by \( 1 - x^2 + x^4 - x^6 + \cdots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Provide an example of a function $f(x)$ whose derivative $\frac{df}{dx}$ is periodic, but the function $f(x)$ itself is not periodic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x + \sin(x)}
Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following :
(a)
$x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$
(b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.
How many polynomials are in $\mathbf{W}?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$|W|=p!$

}
Calculate the conditional Shannon entropy $H[R_{513}, C_{513} | R_1, R_2, \ldots, R_{512}]$ for a scenario where 1024 rooks are randomly and uniquely placed on a 1024 × 1024 chessboard without any rook attacking another. Here, $R_i$ represents the row number and $C_i$ the column number of the i-th rook.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Compute the derivative of the expression \( \operatorname{trace}(XWW^T X^T) \) with respect to the matrix \( W \), where \( X \) and \( W \) are \( n \times n \) real matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 X^\top X W}
Calculate the area of the quadrilateral formed by the intersection of two circles with radii \(a\) and \(b\), intersecting at an angle \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab \sin \theta}
Evaluate the limit: $$\lim_{x\to0^-} (x^{-x^x+1})$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the limit: $$\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Determine the Galois group of the splitting field of the polynomial $t^4 - 3t^2 + 4$ over $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V_4}
In the frequency distribution histogram of a sample, there are a total of $m(m\geqslant 3)$ rectangles, and the sum of the areas of the first $3$ groups of rectangles is equal to $\frac{1}{4}$ of the sum of the areas of the remaining $m-3$ rectangles. The sample size is $120$. If the areas of the first $3$ groups of rectangles, $S_1, S_2, S_3$, form an arithmetic sequence and $S_1=\frac{1}{20}$, then the frequency of the third group is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is 2018 cm. The sum of all the distances between the second point and the other points, including the first one, is 2000 cm. Find the distance between the first and second points. Express your answer in centimeters (cm) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the cofinality of the ordinal $\omega+1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the convergence and determine the value, if it exists, of the integral \( \int_{0}^{+\infty }\frac{x \log(x)}{(x^{2}+1)^{2}}dx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a complex circular zero-mean Gaussian random process (or vector) \( \textbf{x} \) where \( \textbf{x} \sim \mathcal{CN}(0,\sigma^2) \), find the expectation \( \mathbb{E}\{\textbf{x}^n\} \) for \( n \in \mathbb{N}^{*+} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the value of \( \alpha \in \mathbb{R} \) such that the integral \( \int_{\mathbb{S}^{n}} \frac{d\theta}{|\theta-\theta_{0}|^{\alpha}} \) converges, where \( \theta_{0} \in \mathbb{S}^{n} \) is fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < n}
Evaluate the integral \( \int_0^{2\pi}\frac{\cos 3x}{5-4\cos x}dx \) using complex integration techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Determine the domain of convergence for the series \( \sum_{n = 1}^{\infty} \frac{\cos(nz)}{n^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt \) given that the Fourier transform of \( f \) is \( \hat{f}(w)=\frac{w}{1+w^4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the values of $x$ for which $\lim_{n \to \infty}\cos(nx)=1$. Provide your answer in terms of $x$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = 2\pi k}
Find the surface area of the cone defined by $x^2 + y^2 = z^2$ that lies above the $xy$-plane and beneath the cylinder $y^2 + z^2 = 16$. Provide your answer in terms of exact values or expressions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
Evaluate the limit: $$\lim\limits_{n\to\infty} \int_0^n \left( 1- \frac{x}{n}\right)^n \log \left(2+\cos\left( \frac{x}{n}\right)\right)\, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
Evaluate the integral \( \int_0^{\pi} [\cot(x)] \, dx \), where \([\cdot]\) denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Since counting the numbers from 1 to 100 wasn‘t enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$, the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k＜ 15$ on the board. Then, he told Gauss:

On every turn, erase two random numbers $a, b$, chosen uniformly randomly, from the board and then write the term $2ab - a - b + 1$ on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
16384
}
For what values of \( \gamma \in \mathbb{C} \) do there exist non-singular matrices \( A, B \in \mathbb{C}^{n \times n} \) such that \( AB = \gamma BA \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma^n = 1}
Given a Poisson random variable \( Z \) with parameter \( \mathbb{E}Z \) and another random variable \( G \) that converges in distribution to \( Z \) with an error bound \( d_{TV}(G,Z) \leq c \), where \( c \) is a constant and \( d_{TV} \) is the total variation distance, find an upper bound for the probability \( P( |G-Z| > t) \) for a fixed real \( t > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c}
Determine the points on the helix parameterized by $x=\cos(t)$, $y=\sin(t)$, $z=t$ where the function $f(x,y,z)$, with partial derivatives $f_x=\cos(t)$, $f_y=\sin(t)$, and $f_z=t^2$, can take an extreme value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 0, 0)}
For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
149
}
Find the number of subgroups isomorphic to $\mathbb{Z}_{10}\oplus\mathbb{Z}_{10}$ in the group $A=\mathbb{Z}_{600}\oplus\mathbb{Z}_{200}\oplus\mathbb{Z}_{100}\oplus\mathbb{Z}_{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1085}
Find a quadratic polynomial \( B(x) \in (\mathbb{Z}/3\mathbb{Z})[x] \) such that \( \bar{B}(x) = x^3 + 2x^2 + x + 2 \in (\mathbb{Z}/3\mathbb{Z})[x]/(x^2 + 2x + 2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + x + 1}
Calculate the flux of the vector field \( \nabla f \) across the boundary of the region \( S \) bounded between the spheres centered at the origin with radii 1 and 2, and in the first octant, using Green's theorem. Here, \( f(x,y,z) = \frac{1}{x^2+y^2+z^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Calculate the limit \( \lim\limits_{n \to \infty} \int_0^2 \frac{nx^{\frac{n}{2}}}{(1+x^n)} \, dx \) using the convergence theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the limit of the difference between the prime zeta function $P(s)$ and the natural logarithm of the Riemann zeta function $\ln(\zeta(s))$ as $s$ approaches 1. Given that $M = \gamma + \sum_{n=2}^{\infty} \mu(n) \frac{\ln(\zeta(n))}{n}$, $P(s) = \sum_{n=1}^{\infty} \mu(n) \frac{\ln(\zeta(sn))}{n}$, where $M$ is Mertens' constant, $\gamma$ is the Euler-Mascheroni constant, $\mu(s)$ is the Möbius function, and $\zeta(s)$ is the Riemann zeta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M - \gamma}
Find \( \frac{dy}{dz} \) when \( y = \cot^{-1}(f(x)) \) and \( z = \cos^{-1}(\sqrt{1-x^4}) \), where \( f(x) = \frac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Determine the median area of a random n-gon formed inside a unit square. The square is bounded by the coordinates (0,0), (0,1), (1,0), and (1,1). Random x and y coordinates are chosen in the interval [0,1] for each of the n points, and these points are randomly connected to form a cycle. What is the median area of this polygon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the cumulative distribution function \( F(x,y) = \frac{(x \cdot y)^2}{4} \) for \( 0 < x < 2 \) and \( 0 < y < 1 \), find the probability \( P(X+Y < 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{24}}
In the ring \( \mathbb{Z}_n = \{0, 1, \ldots, n-1\} \) under addition and multiplication modulo \( n \), suppose \( a, b, c, x \in \mathbb{Z}_n \) are nonzero and the cyclic order \( R(a, b, c) \) holds. Under what conditions does \( R(ax, bx, cx) \) hold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gcd(x, n) = 1}
Determine the dimension of the vector space of multilinear forms \( \Delta: V^n \rightarrow K \), where \( V \) is a finite-dimensional vector space over the field \( K \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\dim V)^n}
Consider a system of polynomials $f_1, \ldots, f_m \in \mathbb{F}[x_1, \ldots, x_n]$ each of degree $d$, where $n = O(1)$. If this system has a solution in the algebraic closure $\bar{\mathbb{F}}$, what is the minimum degree of extension of $\mathbb{F}$ required to guarantee a solution? Express your answer in terms of $m$ and $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d^m}
Determine if there exists a function \( f(x) \) such that \( f(f(x)\cos x) = f(x)\sin x \) for all \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = 0}
Evaluate the integral: \[ \int_0^{ + \infty } \frac{x \ln x}{(x^2 + 1)^2} \, \mathrm{d}x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}