inclusionAI/Ring-lite-rl-data huggingface dataset:
123. In reverse order. What nine-digit number, when multiplied by 123456 789, gives a product where the nine least significant digits are 9,8,7,6,5,4,3, 2,1 (in that exact order)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{989010989}
Let $\mathcal{A} = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a regular 2014-simplex, meaning the 2015 vertices of $\mathcal{A}$ lie in 2014-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when $PA_0^2 + PA_1^2 + \dots + PA_{2014}^2$ is divided by $10^6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{348595}
Let \(p(x) = x^5 - 833x^4 + ax^3 + bx^2 + cx + d\) be a polynomial whose roots are in geometric progression. If the sum of the reciprocals of the roots is 17, determine all possible values of \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-16807}
Find the smallest integer $N$ such that the sequence $\{ N, N+1, N+2, \ldots, N+a\}$ consists entirely of composite numbers. Provide a general formula for $N$ in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a + 2)! + 2}
Given integers $a$, $b$, $c$, and $d$ such that $a$ and $b$ are relatively prime, $d \leq 2022$, and $a + b + c + d = ac + bd = 0$, determine the largest possible value of $d$. \( \text{(Express your answer as an integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Suppose that $f$ and $g$ are continuous functions on $(0, \\infty)$ such that $f(x \\cdot t) = f(x)g(t)$ for all $x, t > 0$. Given that $g'(1) = 2$, find $f'(1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the ring of integers of an imaginary quadratic field, determine the number of elements with norm equal to a prime number $q$, given that $q$ splits in the field and $N(q) = q^2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the degree of the splitting field of the polynomial \( p(x) = x^4 - 2x^2 - 8 \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a group \(G\), it is given that \(a^5 = e\), where \(e\) is the identity element of \(G\). If \(a, b \in G\) and \(aba^{-1} = b^2\), determine the order of the element \(b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Determine the number of real solutions $a$ to the equation:
\[ \left\lfloor \frac{1}{2}a \right\rfloor + \left\lfloor \frac{1}{3}a \right\rfloor + \left\lfloor \frac{1}{5}a \right\rfloor = a. \]
Here, if $x$ is a real number, then $\left\lfloor x \right\rfloor$ denotes the greatest integer that is less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a discrete-time Markov Chain described by the equation $X(n+1) = X(n) - 1 + A(n)$, where $A(n)$ is a random variable taking values $0, 1, 2$ with probabilities $p_0, p_1, p_2$, respectively, and $X(n) = n$ with $X(n) \geq 1$, determine the condition on $p_0, p_1, p_2$ for the stability of the system.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p_1 + 2p_2 < 1}
Find the general solution of the differential equation $y = xy' - {y'}^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = Cx - C^2}
Given a set of n natural numbers, determine the number of possible order relations among them. Express your answer in terms of Fubini numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_n}
How many trees \( T = \langle V, E \rangle \) exist such that \( V = \{ v_1, v_2, \dots, v_7 \} \), with \( \deg(v_1) = 1 \) and \( \deg(v_j) \leq 5 \) for every \( j = 2, \ldots, 7 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7770}
A square is divided by horizontal and vertical lines that form \( {n}^{2} \) squares each with side 1. What is the greatest possible value of
 \( n \) such that it is possible to select \( n \) squares such that any rectangle with area \( n \) formed by the horizontal and vertical lines
 would contain at least one of the selected \( n \) squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Calculate the limit \[ \lim_{b\to \infty}\frac{\vert \text{erf}(\sqrt{a+\mathrm{i} b})\vert^{2}}{\sqrt{a^2+b^2}} \] where \[ \text{erf}(a+\mathrm{i} b)=\frac{2}{\sqrt{\pi}}\int_{0}^{a+\mathrm{i} b}e^{-t^2}~dt \] is the error function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
## Task B-4.4.

From point $A$ outside a circle, tangents are drawn to circle $k$ with points of tangency at points $B$ and $C$. A line parallel to $A C$ passes through point $B$ and intersects circle $k$ again at point $D$. If $|A B|=49$ and $|C D|=28$, determine the length of $|A D|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Calculate the area enclosed by the curves \(y=\frac{4}{x}\), \(y=\frac{6}{x}\), \(y=4x\), and \(y=8x\) using a suitable coordinate transformation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Find the number of all seven-digit numbers that start with the digit 1 and end with the digit 9, such that the difference between any two adjacent digits is either 1 or 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Solve for \( x \) in degrees in the equation: \[ \sin 84^\circ \sin(54^\circ-x) = \sin 126^\circ \sin x. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In triangle $\triangle ABC$, $\angle B$ is obtuse. The interior bisector $BM$ is drawn from point $B$, along with the altitudes $AN$ and $CQ$. Given that $AN = 8$ cm and $CQ = 12$ cm, determine the length of the perpendicular drawn from $M$ to line $\overline{AC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{24}{5}}
Evaluate the limit: $$\lim_{n \to \infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{kn} \right].$$ Determine if the limit is 0 or another value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln k}
Evaluate the limit: $$\lim_{x\rightarrow0}\frac{1-\frac{1}{2}x^2-\cos(\frac{x}{1-x^2})}{x^4}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{23}{24}}
Find the first common value of the sequences defined by the expressions \(m^2 + 928m + 687\) and \(n^2 + 54n + 729\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11514148416}
Calculate the determinant of the matrix \(A_{2n}:=\begin{pmatrix} a &0 &\cdots & \cdots &0 &b \\ 0& a & & & b&0 \\ \vdots &\vdots &\ddots &\ddots &\vdots &\vdots \\ \vdots&\vdots & \ddots & \ddots &\vdots &\vdots \\ 0& b & & & a&0 \\ b&0 &\cdots &\cdots &0 &a \end{pmatrix}\in \mathbb{R}^{2n \times 2n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a^2 - b^2)^n}
Determine a formula to calculate the number of distinct accepted $2 \times n$ matrices, where the first row is a permutation of the numbers $1$ to $n$, and the second row is also a permutation such that no numbers are the same within rows or columns, and no column has identical numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n! \cdot !n}
Find the largest non-negative integer \(a\) such that \(\lim_{x \to 1} \left(\frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1}\right)^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Fill the integers from 1 to \(2016^2\) into a \(2016 \times 2016\) grid, ensuring each cell contains a unique number. Draw a vector from the center of one cell to the center of another if the number in the first cell is smaller than the number in the second cell. Given that the sum of the numbers in each row and each column of the grid is the same, determine the sum of all the vectors drawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a coffeehouse, the number of guests follows a Poisson distribution with parameter 10, i.e., $X \sim \text{Poi}(10)$. Each guest independently drinks a Poisson(1) number of cups of coffee and raises their hand a Poisson(1) number of times. The number of guests, the number of cups of coffee each guest drinks, and the number of hand raises each guest performs are all independent. Given that no cups of coffee have been sold, calculate the expected value of the total number of hand raises.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{e}}
Evaluate the contour integral \( \oint_C f(z) \, dz \) where \( f(z) = \pi e^{\pi \cdot \overline{z}} \) and \( C \) is the square contour with vertices at \( 0 \), \( 1 \), \( 1+i \), and \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(e^{\pi} - 1)}
Find the general solution to the ordinary differential equation \( x\frac{dy}{dx} = y - \frac{1}{y} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = \pm \sqrt{1 + C x^2}}
In how many ways can a plane intersect a regular dodecahedron in space to form a regular hexagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let \( Z = \{ (1,0), (0,1) \} \subset \mathbb{A}_k^2 \). Find the ideal \( \mathbb{I}(Z) \) in the polynomial ring \( k[x, y] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x + y - 1, xy)}
What is the maximum number of attracting and neutral orbits for a complex polynomial of degree $d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2d - 2}
Provide an example of an ideal $I$ in the polynomial ring $R[x]$ such that the set $X = \{ x \in R : f(x) = 0, \forall f \in I \}$ is empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x^2 + 1)}
Compute the infinity norm \(||B^{-1}||_{\infty}\) for the square real matrix \(B\) defined by:
\[ b_{ii}=1, \quad b_{ij}=-1 \text{ for } i<j, \quad b_{ij}=0 \text{ for } i>j. \]

Given that \(B = I + N\), where \(N\) is a strictly lower triangular matrix, find \(||B^{-1}||_{\infty}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Determine the tighter bounds for $|\sin z|$ given that $|z|=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin 1 \leq |\sin z| \leq \sinh 1}
Determine the smallest integer $a$ such that the inequality \(x^2 - ax - \ln x + e^{\sin x} - 1 < 0\) is satisfied for at least one positive value of $x$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the equation of the locus of the foot of the perpendicular drawn from the origin to a line that makes intercepts on the coordinate axes, given that the square of the length of the perpendicular is equal to the product of the lengths of the intercepts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 - xy = 0}
Find the minimum value of the expression:
\[
\left[\cos ^{2}\left(\dfrac{\theta }{2}\right) + \,\sqrt{\,{1 - \gamma}\,}\,\sin^{2}\left(\dfrac{\theta }{2}\right)\right] ^{2} + \dfrac{\gamma }{4}\,\sin^{2}\left(\theta\right)\quad \text{where}\quad 0 \le \gamma \le 1.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \gamma}
Evaluate the limit of the product as \( n \) approaches infinity: 
\[ P=\lim_{n \to \infty}\prod_{a=5}^{n}\frac{a^3-64}{a^3+64} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{259}}
A frog makes 3 jumps, each exactly 1 meter long, with the directions of the jumps chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the behavior of the iterated composition of the function \( f(x) = \sin(\cos(x)) \) as the number of iterations approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.695}
In triangle ABC, points E, F, and C are collinear, D, E, and B are collinear, and A, D, and F are collinear. Given that |CF| = |FE|, |DE| = |EB|, and |AD| = |DF|, with |AC| = |CB| = \( \sqrt{14} \) dm and the area of triangle DEF is 0.01 m², calculate the length of |AB| in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\sqrt{7}}
Determine the genus $g$ of a closed Riemann surface if its tangent bundle $T$ is trivial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $X$ and $Y$ be two random variables such that for all $1 \le m < k$, the joint probability is given by: \[ P(X=k, Y=m) = \frac{1}{16}\left(\frac{3}{4}\right)^{k-2} = \frac{1}{9}\left(\frac{3}{4}\right)^{k} \] Calculate the covariance $\text{Cov}(X,Y)$ using the formula: \[ \text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the ratio of the minimum to the maximum area of a circle that is concentric with the ellipse \(x^2 + 2y^2 = 2\), given that tangents are drawn from a point \((h, k)\) such that \(h^2 + k^2 = 3\), and the chord of contact touches the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Let \( F_1 = \mathbb{Q}(e^{\frac{2\pi i}{5}}) \) and \( F_2 = \mathbb{Q}(\cos(\frac{2\pi}{5})) \). Calculate the degree of the field extension \([F_1F_2:F_2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim _{n \rightarrow \infty}\left(\frac{\sin \left\{\frac{2}{n}\right\}}{\left[2 n \tan \frac{1}{n}\right]\left(\tan \frac{1}{n}\right)}+\frac{1}{n^{2}+\cos n}\right)^{n^{2}} \] where \(\{\}\) and \([\]\) denote the fractional part function and the greatest integer function, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( f:[0,1]\rightarrow \mathbb{R} \) be a differentiable function such that \( f(0)=f(1)=0 \) and \( f''(x)-4f'(x)+4f(x)\ge e^{2x} \) for all \( x \in [0,1] \). Determine the minimum value of \( \frac{f'(x)}{f(x)} \) on the interval \( (0,\frac{1}{2}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the range of values for the parameter $a$ such that the function \( g(x) = \frac{ax^2-4x+2}{a-4x+2x^2} \) is onto.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-6, 2)}
Find a constant \( A > 0 \) such that for all real \( x \geqslant A \), we have

\[ x^{3} + x + 1 \leq \frac{x^{4}}{1000000} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
Determine the largest constant \( C \) such that for all non-negative real numbers \( x, y, z \), the inequality \((yz + zx + xy)^{2}(x + y + z) \geq C \,xyz\,(x^{2} + y^{2} + z^{2})\) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the area of the region in the first quadrant bounded by the parabolas $y=x^2$, $y=2x^2$, $x=y^2$, and $x=2y^2$ using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Evaluate the integral \( \iint\limits_D \max\{\sin x, \sin y\} \, dx \, dy \), where \( D = \{0 \le x \le \pi, \ 0 \le y \le \pi\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many integers $n$, satisfy $|n|< 2020$ and the equation $11^3|n^3+3n^2-107n+1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
0
}
Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Player $2$ has a winning strategy.}
Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring; $180$ or $144$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game.
If after ending of tournament participant have at least $ 75 %
$ of maximum possible points he called $winner$ $of$ $tournament$.
Find maximum possible numbers of $winners$ $of$ $tournament$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left\lfloor \frac{n+1}{2} \right\rfloor }
Calculate the discriminant of the number field \( \mathbb{Q}(\sqrt{-13}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-52}
A board has six coins arranged such that every alternate coin shows heads and tails initially. A die is thrown, and the coin corresponding to the number rolled is flipped. Compute the expected number of die throws required for all coins to show the same side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given two independent random variables, $X$ and $Y$, with distribution function $F_X(x) = \frac{1}{2}x$ for $0 < x < 2$ and density function $f_Y(y) = \frac{2}{9}y$ for $0 < y < 3$, respectively, find the probability that their product $XY$ is greater than or equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{36}}
Evaluate the limit of the Fejer kernel on the real line, defined as \( \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2 \), as \( R \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(t)}
Evaluate the integral: \[ \int_0^1 \frac{dx}{\sqrt{x + \sqrt{x^2 + \sqrt{x^3}}}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral of the function \( f(z) = \frac{2z + j}{z^3(z^2 + 1)} \) along the positively oriented circle \( \Gamma \) defined by \( |z - 1 - i| = 2 \) using residue theory.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Evaluate the integral expression: $$\int_0^4\sqrt{x^2+4}\,dx+\int_2^{2\sqrt{5}}\sqrt{x^2-4}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{5}}
Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^p(x-b)^q$, where $a$ and $b$ are distinct real numbers. Let $V$ be the real vector space of all $n \times n$ matrices $B$ such that $AB=BA$. Determine the dimension of $V$. \( \text{Express your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + q^2}
In an infinite binary sequence, each bit is independently '0' or '1' with a probability of \( \frac{1}{2} \). However, if three consecutive '0's appear, the next bit must be '1'. Given this rule, what is the probability that a randomly chosen bit in the sequence is '1'? Express your answer as a limit as the sequence length approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{15}}
Find the sum of all positive integers that cannot be expressed as the sum of two abundant numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4179871}
Let \( f: D(0,1) \to \mathbb{C} \) be a holomorphic function, where \( D(0,1) = \{ z \in \mathbb{C} \mid |z| < 1 \} \), and assume that \( |f(z)| \leq 2 \) for all \( z \) in \( D(0,1) \). Estimate \( |f^{(3)}(i/3)| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{2}}
Let $σ(n)$ be the number of positive divisors of $n$, and let $rad n$ be the product of the distinct prime divisors of $n$. By convention, $rad 1 = 1$. Find the greatest integer not exceeding
$$100 \left( \sum\limits_{n=1}^{\infty} \frac{σ(n)σ(n rad n)}{n^2 σ(rad n)} \right)^{\frac{1}{3}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
164
}
Determine the maximum possible order of a group $G$ in which every element $g$ satisfies $g^3 = e$, where $e$ is the identity element.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Evaluate the integral \( \int_{-\infty}^{+\infty} \left[ f(t+1) - f(t) \right] dt \) for a continuous function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( \lim_{t \rightarrow -\infty} f(t) = l_1 \) and \( \lim_{t \rightarrow +\infty} f(t) = l_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{l_2 - l_1}
Provide an example of a set $G$ with a binary operation $\cdot$ such that $G$ has a two-sided identity element $e$, every element in $G$ has a left inverse, and the operation $\cdot$ is associative but not commutative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Evaluate the improper integral: \[ \int_{0}^{\infty} \frac{\log(1+x^2)}{1+x^2}~dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \ln 2}
Determine the number of free indices in the result when taking the derivative of a matrix-valued function with respect to a matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a study, 100 people are randomly selected each day for 1095 days. Calculate the probability that at least 50% of these people have been selected at least twice during this period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $\triangle ABC$, points $X$, $Y$, and $Z$ are located on sides $BC$, $CA$, and $AB$, respectively, such that lines $AX$, $BY$, and $CZ$ intersect at point $P$. Given that $AX = 10$, $BY = 8$, $CZ = 6$, $XP = 5$, $YP = 4$, and $ZP = 3$, determine the ratio of the area of triangle $\triangle XYZ$ to the area of triangle $\triangle ABC$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
Let $p(x)$ and $q(x)$ be two polynomials with positive integer coefficients, each of degree 4, such that $p(1) + q(1) = 26$. Determine the number of unordered tuples $(p(x), q(x))$ for which the polynomial $W(x) = (p(x)q(x))^7$ has exactly one odd coefficient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8975}
Given a set with 25 elements, determine the minimum number of subsets of size 5 required such that every possible subset of size 3 is included in at least one of these subsets of size 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}