inclusionAI/Ring-lite-rl-data huggingface dataset:
A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Suppose $Y$ is a multivariate random variable taking values in $\mathbb{R}^k$, where $k \geq 3$. If the marginal distribution function of each coordinate of $Y$ can only take values 0 and 1, does it imply that $Y$ is constant almost surely? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
There are $\displaystyle{2n}$ students in a school $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$. Each week $\displaystyle{n}$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The minimum number of trips needed is 6.}
On the basis of VSEPR theory, what geometry is predicted for the central sulfur atom in SOCl₂?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
trigonal pyramidal
}
What is the least integer whose square is 75 more than its double?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
Find all real numbers $x$ such that $x - \lfloor x \rfloor = \frac{2}{\frac{1}{x} + \frac{1}{\lfloor x \rfloor}}$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Determine if your solution set includes integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Find the smallest 10-digit number such that the sum of its digits is greater than that of any smaller number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1999999999}
In a bowl, there are 19 marbles of varying sizes and weights. Out of these marbles, 5 are yellow, and the remainder are blue and red marbles in the ratio 3:4 respectively. The yellow marbles have a combined weight of 8 grams. The red marbles weigh twice as much as the yellow marbles per gram. The blue marbles are not counted for this problem. Calculate the difference in total weight between the red marbles and the yellow marbles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \(U\) be a finite set, \(D\) a set, and \(F = \{(X_\lambda, Y_\lambda)\}_{\lambda \in \Lambda}\) a finite family of pairs of subsets of \(U\). Suppose \(a \in U \setminus \cup_{\lambda \in \Lambda} (X_\lambda \cup Y_\lambda)\) and \(K \subseteq U\) satisfies that for any finite subset \(I\) of \(D^U\), \(I \vDash F\) implies \(P(I, K, U)\). Determine if \(a \in K\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \in K}
Given that for any non-zero real number $m$, the inequality $|2m-1|+|1-m| \geqslant |m|(|x-1|-|2x+3|)$ always holds, find the range of the real number $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,-3] \cup [-1,+\infty)}
Let $x$ and $y$ be non-negative real numbers that sum to 1. Compute the number of ordered pairs $(a, b)$ with $a, b \in\{0,1,2,3,4\}$ such that the expression $x^{a} y^{b}+y^{a} x^{b}$ has maximum value $2^{1-a-b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos t \sin( \sqrt{1+t^2})}{\sqrt{1+t^2}}dt \) using contour integration or other suitable methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Given a triangle with sides 2, 3, and 4, find the radius of the smallest circle from which this triangle can be cut out.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.00}
Find the least number \( c \) such that \( \sum_{i=1}^n x_i^2 \leq cn \) for all real numbers \( x_1, x_2, \ldots, x_n \) where each \( x_i \geq -1 \) and \( \sum_{i=1}^n x_i^3 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. Find the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the asymptotic equivalence of the function \( f(x) = \sum_{n=1}^\infty n^\varepsilon \frac{x^n}{n!} \) as \( x \to \infty \), where \( \varepsilon \in (0, 1) \). Find a function \( g(x) \) such that \( f(x) \sim g(x) e^x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^\varepsilon}
Given that the complex number $z$ and ${{(z-2)}^{2}}-8i$ are both purely imaginary, then $z=\boxed{\_\_\_\_\_}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2i}
Determine whether the series \( \sum_{n=0}^\infty \frac{\sin(2n+1)}{2n+1} \) converges, and if so, find its sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Find the area of a wall where the base is part of the circle of radius 1 centered at the origin, lying in the first quadrant. The height of the wall is given by the function \( f(x, y) = y + 4x \). Use line integrals to determine the area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine which inverse trigonometric function $g$ has the derivative $g'(x) = -\frac{1}{x^2+1}.$


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cot^{-1}(x)}
已知向量 \(\vec{a}=(0,1), \vec{b}=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right), \vec{c}=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\)，且 \(x \vec{a}+y \vec{b}+z \vec{c}=(1,1)\)，求 \(x^{2}+y^{2}+z^{2}\) 的最小值。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \leq x \leq 4$. Find the smallest \( x \) for which \( f(x) = f(2022) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
NASA’s Perseverance Rover was launched on July $30,$ $2020.$ After traveling $292{,}526{,}838$ miles, it landed on Mars in Jezero Crater about $6.5$ months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
60,000
}
18.2. A segment 20 cm long is divided into two segments, and a square is constructed on each of them as a side. Find the lengths of the sides of these squares, if the difference in the areas of the squares is $120 \mathrm{~cm}^{2}$.

$$
\text { (7-8 grades) }
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Let $a$ and $b$ be the roots of the polynomial $x^2 - 10cx - 11d$, and let $c$ and $d$ be the roots of the polynomial $x^2 - 10ax - 11b$. Given that $a, b, c, d$ are distinct real numbers, find the value of $a + b + c + d$. \[ a + b + c + d = ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1210}
Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^o$. Find the degree measure of $\angle BAC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
At $12:34$ and $12:43$, the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$13:02$.}
In triangle \(ABC\), the angle bisectors at vertices \(A\) and \(C\) intersect at point \(D\). Find the radius of the circumcircle of triangle \(ABC\) if the radius of the circumcircle of triangle \(ADC\) with center at point \(O\) is \(R = 6\), and \(\angle ACO = 30^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the largest squarefree positive integer \( n \) such that the ring \( \mathbb{Q}(\sqrt{-n}) \cap \overline{\mathbb{Z}} \) is a unique factorization domain (UFD).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
Determine the parameters \((p, d, q)\) for the \(ARIMA(p, d, q)\) process of the time series \(Y_t\), given the stationary process \(X_t\) defined by the equation \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\), where \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\), and the process \(Y_t\) defined as \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1, 3)}
Find the orthogonal trajectories of the family of curves given by the equation \(3x^2 - y^2 = c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x y^3 = C}
Mia buys 10 pencils and 5 erasers for a total of $2.00. Both a pencil and an eraser cost at least 3 cents each, and a pencil costs more than an eraser. Determine the total cost, in cents, of one pencil and one eraser.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Evaluate the double integral \( \int_0^1 \int_0^x \left( 1 - \frac{\sqrt{3}}{2 + \cos(2\pi t)}\right) dt \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( A B C D \) be a quadrilateral and \( P \) the intersection of \( (A C) \) and \( (B D) \). Assume that \( \widehat{C A D} = 50^\circ \), \( \widehat{B A C} = 70^\circ \), \( \widehat{D C A} = 40^\circ \), and \( \widehat{A C B} = 20^\circ \). Calculate the angle \( \widehat{C P D} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
Real numbers $x, y, z$ satisfy $$x+x y+x y z=1, \quad y+y z+x y z=2, \quad z+x z+x y z=4$$ The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5272}
Determine the cardinality of the set of functions from the set of natural numbers \( \mathbb{N} \) to the set of all subsets of the natural numbers \( \mathcal{P}(\mathbb{N}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the number of edges in the graph $G_n$, where the vertex set $V(G_n)$ consists of all binary strings of length $n$ with at most one continuous block of $1$'s, and two vertices are adjacent if they differ in exactly one position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Determine the value of \( n = m \) such that the width of the 95% confidence interval for \( p_1 - p_2 \) is at most 0.1, regardless of the sampling results. The confidence interval is given by:
\[ \hat{p}_1 - \hat{p}_2 \pm z_{0.025} \sqrt{\frac{\hat{p}_1\hat{q}_1}{m} + \frac{\hat{p}_2\hat{q}_2}{n} } \]
where \( \hat{p}_1 = \frac{X}{m} \) and \( \hat{p}_2 = \frac{Y}{n} \), with \( X \sim \text{Binomial}(m, p_1) \) and \( Y \sim \text{Binomial}(n, p_2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{769}
Compute \( \mathbb{E}\left[\left(X_1 + 2X_2 + 3X_3\right)^2\right] \), where \( X_t \) is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Let $G$ be a simple connected graph with 2016 vertices and $k$ edges. Determine the maximal value of $k$ such that there exists a set of vertices whose removal disconnects the graph, regardless of the arrangement of the $k$ edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Sam and Ben are each flipping fair coins. If Sam flips a single coin until he gets a tails, and Ben flips $10$ coins in total, what is the probability Sam and Ben get the same number of heads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ {\dfrac{3^{10}}{2^{21}}} \]}
Evaluate the expression: \[ \frac{2012!}{2^{2010}} - \sum_{k=1}^{2010} \frac{k^2 k!}{2^k} - \sum_{k=1}^{2010} \frac{k \cdot k!}{2^k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of $n$ such that the limit \( \lim\limits_{x \to 0} \frac{x^n}{\sin(\cos x) -\sin x} \) is a non-zero finite real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Identify the elements of the unit group of the polynomial ring $F[x]$, where $F$ is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F^\times}
Determine the number of points \((x, y)\) over the finite field \(\mathbb{F}_p\), where \(p\) is a prime number such that \(p \equiv 2 \pmod{3}\), that satisfy the equation \(y^2 = x^3 + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
Given $n$ balls, where $\frac{n}{3}$ are blue and $\frac{2n}{3}$ are red, each ball is thrown randomly into one of $k$ bins. Define $M_k$ as 1 if a bin has more red balls than blue balls, and 0 otherwise. Let $A = \sum_{i=0}^k M_i n_i$, where $n_i$ is the number of balls in bin $i$. What is the upper bound of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Simplify the expression: \[ \frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt[4]{5}}
Let $ABC$ be a triangle with $AB = 20$ and $AC = 18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE = AF = 8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC = m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{305}
Let \( P(z) \) be a monic polynomial with complex coefficients of degree \( n \) such that \( |P(z)| \le 1 \) for all \( |z| \le 1 \). Is it true that \( P(z) = z^n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P(z) = z^n}
Given that the function $f(x) = -x^3 + 1 + a$ $(\frac{1}{e} \leq x \leq e, e$ is the base of natural logarithms$)$ and the graph of $g(x) = 3\ln(x)$ have points that are symmetric about the $x$-axis, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, e^3 - 4]}
Apply the operator \( \hat{u} \) to the derivative \( \dfrac{\partial f(x)}{\partial x} \), given that \( \hat{u}(f(x)) = f(u(x)) \) for any function \( f(x) \) over the real numbers \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\partial f}{\partial x}(u(x))}
The distance (as the crow flies, as are all distances in this problem) from Chalons to Vitry is 30 km, from Vitry to Chaumont is 80 km, from Chaumont to Saint-Quentin is 236 km, from Saint-Quentin to Reims is 86 km, from Reims to Chalons is 40 km. Calculate the distance from Reims to Chaumont in this closed polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150 \, \text{km}}
A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 AM, also with equal probability. Let $\frac{p}{q}$ for $p$, $q$ coprime be the probability that they fall in love. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the value of $\tan{t}$ if the series $\sum_{i=1}^{\infty} \tan^{-1}\left(\frac{1}{2i^{2}}\right)$ is equal to $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$, if it is an increasing function on the interval $(0, +\infty)$, then $m = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
试求正整数k，使$f ( x ) = \sin k x \cdot \sin ^ { k } x + \cos k x \cdot \cos ^ { k } x - \cos ^ { k } 2 x$的值不依赖于x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $V$ be an $n$-dimensional real vector space with two different bases $B=\{v_1,\ldots,v_n\}$ and $B'=\{w_1,\ldots,w_n\}$. Consider an endomorphism $f:V\to V$ such that $f(v_i)=w_i$ for every $i=1,\ldots,n$. Is the matrix $M_{B',B}(f)$, associated with $f$ from the basis $B$ to the basis $B'$, equal to the identity matrix $I_n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_n}
Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27.55}
Let \( S_{n} \) denote certain subsets of the set of positive integers \( \{1, 2, 3, \cdots, 100\} \) that satisfy the condition: no number in the subset is twice any other number. What is the maximum number of elements that such a subset can contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
A noodle chef first rolls a piece of dough into a 1.6-meter-long cylindrical noodle stick, folds it in half, and then stretches it back to 1.6 meters; folds it in half again, and stretches it back to 1.6 meters;... doing this 10 times, and stretching it back to 1.6 meters each time, he makes noodles. At this point, if the noodles in his hands are stretched out and considered as a cylinder of uniform thickness, find the ratio of the final thickness (diameter) to the original thickness (diameter) of the noodle stick. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{32}}
Let \( \{x_n\} \) be a non-decreasing sequence of integers such that \( \lim_{n\to\infty} \left( \frac{\text{number of elements in } \{x_i\}_{i\leq n} \text{ that are } \leq n }{n} \right) = \frac{1}{2} \). Determine if \( \lim_{n\to\infty} \left( \frac{\sum_{i=1}^n x_i}{n^2} \right) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of non-constant polynomial factors with leading coefficient 1 of the polynomial \(x^{2015} + 18\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The full time score of a football match was $3$-$2$. How many possible half time results can we have for this match?

 


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12
}
A bug moves along a circle of radius 200 cm centered at the origin in the $xy$-plane with a constant speed of 3 cm/s. The temperature at any point $(x, y)$ is given by the function $T(x, y)$. Calculate the rate of change of temperature experienced by the bug at time $t = \frac{\pi}{3}$, given that the circle is parameterized by $\gamma(t) = \langle 200 \cos t, 200 \sin t \rangle$. Provide your answer in cm/s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
−6[−W]
}
Combine the following radicals: $\sqrt{a^{4} b^{3} c^{4}} \sqrt[3]{a b^{2} c}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$a^{2} b^{2} c^{2} \sqrt[6]{a^{2} b c^{2}}$}
Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Workers were laying down a floor of size \( n \times n \) (\(10 < n < 20\)) with two types of tiles: \( 2 \times 2 \) and \( 5 \times 1 \). It turned out that they managed to completely cover the floor using the same number of tiles of each type. For which \( n \) could this happen? (Cutting the tiles and overlapping them is not allowed.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12, 15, 18}
In order to pass $B$ going $40$ mph on a two-lane highway, $A$, going $50$ mph, must gain $30$ feet. Meanwhile, $C$, who is $210$ feet from $A$, is headed toward him at $50$ mph. If $B$ and $C$ maintain their speeds, then, in order to pass safely, $A$ must increase his speed by how many mph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $X$ be a $p \times q$ matrix with characteristic polynomial $(x-c)^r(x-d)^s$, where $c$ and $d$ are distinct complex numbers. Determine the dimension of the complex vector space $W$ of all $q \times p$ matrices $Y$ such that $XY=YX$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^2 + s^2}