inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a rectangular grid of  $ 10 \times 10$  unit squares. We call a *ship* a figure made up of unit squares connected by common edges. We call a *fleet* a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Samuel is driving to San Francisco's Comic-Con in his car and he needs to travel 1,200 miles to the hotel where he made a reservation. At first, he drives at 60 miles/hour for 2 hours straight. Heavy rain starts, forcing Samuel to slow down to 40 miles/hour for the next hour. After the rain stops, he takes a 30-minute break. 

He then speeds up to 70 miles/hour for 3 hours, but due to traffic, he only maintains this speed for 2 hours and 30 minutes.  He takes a 1 hour break before switching to a different car. In the new vehicle, he drives at 50 miles/hour for 4 hours before stopping at a friend's place for a 2-hour break. 

Upon leaving his friend's place, Samuel switches to a motorcycle and maintains a speed of 80 miles/hour for the first hour. However, the motorcycle begins to malfunction, and he is forced to reduce his speed to 60 miles/hour for the next 3 hours. How far is he from the hotel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
In the figure below, there are 4 identical circles placed inside an equilateral triangle. If the radii of the circles are 1 unit, what is the value of the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \sqrt{3}}
Determine the center, $Z(G)$, of the group $G$ generated by two elements, $\sigma$ and $\tau$, with orders 5 and 4, respectively, given that $\tau\sigma\tau^{-1}=\sigma^2$. The group $G$ has order 20, and every element $g \in G$ can be uniquely expressed as $g=\sigma^k\tau^m$ for $0 \le k < 5$ and $0 \le m < 4$. The commutator subgroup of $G$, $[G:G]$, is equal to $\langle\sigma\rangle$. What is $Z(G)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{e\}}
Determine the number of roots of the equation $\displaystyle \frac{\mathrm d^{40}}{\mathrm dx^{40}}\left(x^3-8\right)^{20}=0$ within the interval $[-5,5]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Kiera is making trail mix out of 16 bags of nuts and some bags of dried fruit. She wants each new portion of trail mix to be identical, containing the same combination of bags of nuts and bags of dried fruit, with no bags left over. The greatest number of portions of trail mix Kiera can make is 2. How many bags of dried fruit does Kiera have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
On a board, the product \(a_{1} a_{2} \ldots a_{100}\) is written, where \(a_{1}, \ldots, a_{100}\) are natural numbers. Consider 99 expressions, each obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given 3 distinct natural numbers whose sum is 55, and the sum of any two of these numbers is a perfect square, what are the three natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19, 6, 30}
Calculate the area of the region bounded by the curves $xy = 4$, $xy = 8$, $xy^3 = 5$, and $xy^3 = 15$. Express your answer in terms of natural logarithms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 3}
As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respectively, find the area of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
A standard deck of 52 cards is shuffled and dealt. Let \(X_{1}\) be the number of cards appearing before the first ace and \(X_{2}\) be the number of cards between the first and second ace (not counting either ace). Calculate the probability \(P(X_{2} = 10 | X_{1} = 15)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{119}}
In an acute-angled triangle \(ABC\), heights \(AP\) and \(CQ\) are drawn from vertices \(A\) and \(C\) to sides \(BC\) and \(AB\), respectively. Given that the area of triangle \(ABC\) is 18, the area of triangle \(BPQ\) is 2, and \(PQ = 2\sqrt{2}\), find the radius of the circumcircle of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
A 600 meter long train crosses a signal post in 40 seconds. How long will it take to cross a bridge of a certain length at the same speed if it takes 6 minutes to cross the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4800}
Let $X_k$, $k \geq 0$, be iid random variables with pmf $p_j = P(X_1 = j)$ for $1 \leq j \leq m$, where $\sum_{j = 1}^{m} p_j = 1$. Let $T = \min\{n > 0 : X_n = X_0\}$ denote the first time we observe the initial value again. Compute $E[T]$. \(\text{Express your answer in terms of } p_j.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Evaluate the integral \( \int_{0}^{\infty} x \exp\left(-\frac{x^2+a^2}{2}\right) I_0(ax) \, dx \), where \( I_0(x) \) is the modified Bessel function of the first kind of order zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1890}
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find an integer solution to the quadratic congruence equation \(x^2 \equiv 7 \pmod{787}\), where \(1 \leq x \leq 786\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. What is the probability that Nyusha wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.5"]}
Two concentric circles have radii 2006 and 2007. $ABC$ is an equilateral triangle inscribed in the smaller circle and $P$ is a point on the circumference of the larger circle. Given that a triangle with side lengths equal to $PA, PB$, and $PC$ has area $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4020}
Let $m$ be the largest real solution to the equation
$\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$
 There are positive integers $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{263}
Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28812}
The red segments in the diagram have equal lengths. They overlap by identical small segments of length \( x \) cm. What is \( x \) in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles  $ABH ,CBH$  respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Calculate the expected number of rolls required when rolling a fair 6-sided die repeatedly until the product of the last two rolls equals 6, using an absorbing Markov chain. Note that the process restarts if the previous roll was a 4 or 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{2}}
Find the minimum value of the expression \( \frac{x^4 + y^4 + z^2}{xyz} \) for real numbers \(x\), \(y\), and \(z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Determine the number of ring homomorphisms from \( \mathbb{Z}[X]/(X^2-X) \) to \( \mathbb{Z}/6\mathbb{Z} \), excluding the zero homomorphism. Assume that any non-zero homomorphism satisfies \( f(1) = 1 \mod 6 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
On the map shown, there are a number of routes from Mathville to Algebratown. Each route must travel along the roads in the direction marked by the arrows. What is the total number of routes from Mathville to Algebratown? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the cardinality of the set \( S = \{ f: \mathbb{R} \to \mathbb{R} \mid f(x) \neq 0, \text{ only for finitely many } x \in \mathbb{R} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Identify the set of non-invertible elements in the formal power series ring $K[[x]]$ where $K$ is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x)}
A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages.  Any pair of members always talk to each other in only one language.  Suppose that there were no three members such that they use only one language among them.  Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages.  Find the maximum possible value of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(2k+1)k(k-1)}{6}}
The equation \( ax^{2} - bx + c = 0 \) has two distinct roots \( x_{1}, x_{2} \). What is the maximum number of elements in the set \( \{a, b, c, x_{1}, x_{2}\} \) that can be prime (and hence natural) numbers? If a prime number appears twice in the set, it must be counted twice. Note that the number 1 is not considered a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property:
设 $A$ 为集合 $\{1,2,\ldots,n\}$ ， $n\geq 2$ 。 查找具有以下属性的集合 $A$ 中存在排列 $\alpha$ ， $\beta$ ， $\gamma$ ， $\delta$ 的最小数 $n$ ：\[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n = 28}
Suppose that  $ABCD$  is a rectangle with sides of length  $12$  and  $18$ . Let  $S$  be the region of points contained in  $ABCD$  which are closer to the center of the rectangle than to any of its vertices. Find the area of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The area of the region S is 54.}
In the triangle \(ABC\), \(AD = 2BD\), \(AD = EC\), and \(BC = 18\). The area of triangle \(AFC\) is equal to the area of quadrilateral \(DBEF\). What is the length of \(AB\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the increments of the argument and the function for \( y = 2x^2 + 1 \) when the argument \( x \) changes from 1 to 1.02.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.0808"]}
Determine the Galois group of the splitting field of the polynomial \( f(x) = x^8 - 1 \) over the finite field \( \mathbb{F}_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
Consider all functions  $f: \mathbb{Z} \rightarrow \mathbb{Z}$  satisfying \[f(f(x) + 2x + 20) = 15. \]

Call an integer  $n$   $\textit{good}$  if  $f(n)$  can take any integer value. In other words, if we fix  $n$ , for any integer  $m$ , there exists a function  $f$  such that  $f(n) = m.$  Find the sum of all good integers  $x.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-35}
In triangle $\triangle ABC$, $AB=4cm$, $AC=6cm$. If the length of $BC$ is an integer, apply the triangle inequality theorem to find the possible length(s) of $BC$. Express your answer as a list or range of integers in centimeters (cm).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 5, 6, 7, 8, 9}
Alice can bake a pie in 5 minutes. Bob can bake a pie in 6 minutes. Charlie can bake a pie in 7 minutes. Compute the difference between how many pies Alice can bake compared to Bob and Charlie in 90 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A digital display shows the current date as an 8-digit integer, consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date within the month. For how many dates in 2023 will each digit appear an even number of times in the digital display for that date?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$9$}
What is the volume of the pyramid whose net is shown, if the base is a square with a side length of $1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{6}}
In the geometric sequence {a_n}, the product of the first three terms is 27, and the sum of the second and fourth terms is 30. Find the common ratio q. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±3}
In the pyramid \(S A_{1} A_{2} \ldots A_{n}\), point \(O\) is located at the base. It is given that \(S A_{1}=S A_{2}=\ldots=S A_{n}\) and \(\angle S A_{1} O = \angle S A_{2} O = \ldots = \angle S A_{n} O\). Determine the smallest value of \(n\) such that \(SO\) is the height of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Hermione wants to maximize her earnings from selling badges to support the Society for the Promotion of Elfish Welfare (S.P.E.W.). The production cost for the badges is given by the function \( C(Q) = 5Q^2 \), where \( Q \) is the number of badges produced. The demand for badges from all houses except Slytherin is \( q_{ws} = 26 - 2P \), and the demand from Slytherin is \( q_s = 10 - P \). If the price of the badges must be the same for all houses, what is the maximum amount Hermione can earn from selling the badges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{169}{22}}
Consider the sequences \((a_{n})\) and \((b_{n})\) defined by the initial conditions \(a_{1} = \alpha\) and \(b_{1} = \beta\), and the recurrence relations \(a_{n+1} = \alpha a_{n} - \beta b_{n}\) and \(b_{n+1} = \beta a_{n} + \alpha b_{n}\) for all \(n > 0\). Determine the number of pairs \((\alpha, \beta)\) of real numbers such that \(a_{1997} = b_{1}\) and \(b_{1997} = a_{1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1999}
Ludwig works 7 days a week with varying daily salaries depending on the day of the week. He earns $12 on weekdays (Monday to Thursday) and $15 on weekends (Friday to Sunday). He usually works half of the day on weekends but if he works more than 48 hours a week, he is also entitled to an overtime pay of 1.5 times the normal daily rate for any additional hours worked beyond that. In a given week, Ludwig worked 52 hours. Calculate his total earnings for that week considering his daily rate, weekend rate, and overtime pay.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$115.50}
Find the set of all points where the Taylor series of the function \( f(x)=\sum_{n=0}^{\infty} \frac {x^2}{(1+x^2)^n} \) around the point \( x=e \) converges to \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R} \setminus \{0\}}
Three identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\frac{1}{r})^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A pebble is shaped as the intersection of a cube of side length 1 with the solid sphere tangent to all of the cube's edges. What is the surface area of this pebble?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6 \sqrt{2}-5}{2} \pi}
Calculate the volume of the set \( \{ \begin{pmatrix} 2 & 33 & 444 \\ 0 & 3 & 4 \\ 0 & 0 & 4 \end{pmatrix}\begin{pmatrix}x \\ y \\ z \end{pmatrix} : x^2 + y^2 + z^2 \leq 1 \} \) in \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi}
Evaluate the definite integral \( \int_{-\pi/2}^{\pi/2} \sqrt{\sin^3x - \sin^5x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{5}}
Evaluate the limit of the integral: \[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx. \] Determine whether the integral converges to a finite value or diverges to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find all real numbers $x$ between 0 and 360 such that $\sqrt{3} \cos 10^\circ = \cos 40^\circ + \sin x^\circ.$  Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70,110}
Company P had 15 percent more employees in December than it had in January, Company Q had 10 percent more employees in December than in January, and Company R had 20 percent more employees in December than in January. If Company P had 500 employees in December, Company Q had 550 employees in December, and Company R had 600 employees in December, how many employees did these three companies have combined in January?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1435}
Evaluate the infinite-nested radical expression \(\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\cdots}}}}}}\) and determine if it converges to a specific value or oscillates between two values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the dimension of an irreducible finite-dimensional representation of the algebra \( A=\frac{k[x,y]}{\langle yx-xy-1\rangle} \), where \( k \) is an algebraically closed field of characteristic \( p \), and the action of \( x \) and \( y \) is given by the Jordan normal form of \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
In a 4-dimensional space divided into 16 equal regions about the origin, determine the maximum number of regions that a plane can intersect.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine a formula for the total number of bit changes when counting from 1 to a power of 2, n, in binary. The time to add 1 to the current number i is proportional to the number of bits that change from i to i + 1 in their binary representation. For example, transitioning from 0000 to 0001 involves changing one bit, and from 0001 to 0010 involves changing two bits. What is the formula for the total number of bit changes as we increment from 1 to n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 2}
Determine the cardinality of the set of all different partitions of the set of natural numbers \( \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3}{26}$}
Evaluate the contour integral \(\oint_{C}\frac{e^z+\sin z}{z^5}dz,\) where \(C\) is the path defined by \(r(t) = 3 + e^{2\pi i t}\) for \(t \in [0,1]\), using Cauchy's integration formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A policeman is trying to catch a thief who is initially $a$ meters away. The thief moves with a constant velocity $v$ in a straight line to the right, starting at $(0, a)$. The policeman, starting at $(0, 0)$, moves with a constant velocity $k \cdot v$, where $k > 1$, and always runs directly towards the thief. The thief is caught at the point $(a, a)$. Determine the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a $(0,1)$-matrix $A$ with $m$ rows and $m^2$ columns, where each column contains exactly three $1$'s. What is the minimum number of rows that need to be removed to ensure that no column has more than two $1$'s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m - 2}
In the diagram, \( AB \parallel EF \parallel DC \). Given that \( AC + BD = 250 \), \( BC = 100 \), and \( EC + ED = 150 \), find \( CF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
We call a number beautiful if it reads the same from left to right as from right to left. For example, the number 12321 is beautiful. Find all beautiful five-digit numbers that use only the digits 1 and 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10001, 10101, 11011, 11111}
Let point P be a common focus $F_1(-2, 0)$, $F_2(2, 0)$ of an ellipse and a hyperbola, and $\cos\angle F_1PF_2 = \frac{3}{5}$. It is known that the length of the major axis of the ellipse is four times the length of the real axis of the hyperbola. Calculate the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}