inclusionAI/Ring-lite-rl-data huggingface dataset:
For which positive integers \( n \) does the determinant \( \mathcal{D}_n \) of an \( n \times n \) matrix, filled row-by-row with the first \( n^2 \) elements of the Thue-Morse sequence, not equal zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand \((m - a)(n - b)\), where \(a, b, m, n\) are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for \(m\) and \(n\), he realizes that the value of ChatSFFT’s result is the same as the real result in terms of \(a\) and \(b\). How many such pairs are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. What is the total number of triangles of any size in the figure? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $E \subset F$ be a field extension with degree $n$. Determine the degree of the extension $E(x) \subset F(x)$, where $E(x)$ and $F(x)$ are the rational function fields over $E$ and $F$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Given proposition $p$: The equation $\frac{x^2}{2} + \frac{y^2}{m} = 1$ represents an ellipse with foci on the $y$-axis; proposition $q$: $\forall x \in \mathbb{R}$, $4x^2 - 4mx + 4m - 3 \geqslant 0$. If $(\neg p) \land q$ is true, find the range of values for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1,2]}
Jane started babysitting when she was a certain age. Whenever she babysat for a child, that child was no more than half her age at the time. Jane is currently 34 years old, and she stopped babysitting 12 years ago. The current age of the oldest person for whom Jane could have babysat is 25. At what age did Jane start babysitting?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Angelina drove at an average rate of $80$ kmh and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ kmh. Altogether she drove $250$ km in a total trip time of $3$ hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80t+100\left(\frac{8}{3}-t\right)=250}
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. What is the minimum number of moves required to reach the destination point from the starting point? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1328}
Calculate the probability of selecting two subsets from a set of 100 integers (50 odd, 50 even) such that both subsets have the same number of odd and even integers, given that the subsets are of different sizes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{761}
Calculate the rank of the symmetric bilinear form \( f(A,B) = \operatorname{tr}A\operatorname{tr}B + n\operatorname{tr}AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1139.71}
There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
In the range of $0^\circ$ to $360^\circ$, the angle(s) with the same terminal side as $-35^\circ$ is/are $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{325^\circ}
Let  $a$  and  $b$  be positive integers satisfying
	\[ \frac a{a-2} = \frac{b+2021}{b+2008} \]
Find the maximum value  $\dfrac ab$  can attain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{312}{7}}
Find the degrees of irreducible divisors of the polynomial $x^4-3$ in $F_{31}[x]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
ii. $n \geqslant 5$ is a positive integer, $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \geqslant 0$. It is known that
$$
\sum_{i=1}^{n} a_{i}^{2}=1, \sum_{i=1}^{n} b_{i}=1
$$

Find
$$
\sum_{i=1}^{n} a_{i}^{1+b_{i}}
$$

the maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n-1}}
Find the constant $c$ such that the limit \( \lim_{n \to \infty} P\left(\sqrt{\left(\sum_{i=1}^{n}X_i\right)^2+\left(\sum_{i=1}^{n}Y_i\right)^2}>c\sqrt{n}\right) = 0.05 \) holds, where $(X_n, Y_n)$ are independent two-dimensional random variables with each $X_n$ and $Y_n$ having a Laplace distribution with density $f(x) = \frac{1}{2}e^{-|x|}$, and the density function of $\sum_{i=1}^{n}X_i$ and $\sum_{i=1}^{n}Y_i$ is $g(x) = \frac{1}{2^n}e^{-n|x|}$. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{\ln 20}}
The base of the pyramid is a square $ABCD$ with a side length of 1.5, the lateral edge $SC$ is perpendicular to the base plane and equals 1.75. Points $S, B$, and $D$ lie on the lateral surface of a cone with the vertex at point $A$, and point $C$ lies in the base plane of this cone. Find the lateral surface area of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{30\pi}{\sqrt{22}}}
Evaluate the following alternating series using the Leibniz alternative test for convergence:
$$
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}\pi}{6}}
At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 10 minutes, and when she came back, she found Ivan and Foma in front of the door, with Foma having apparently arrived after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the shop to reopen.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
Let $a$ be a real number such that the equation
$$
x^{4}+3 a x^{3}+a\left(1-5 a^{2}\right) x-3 a^{4}+a^{2}+1=0
$$
has a real root, and there are exactly 2 distinct real roots. Determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \frac{2 \sqrt{26}}{13}}
Evaluate the integral \( \int_{-\infty}^{\infty} u(uu')'\,dx \) given that \( u = u(x) \), \( u' = \frac{du}{dx} \), and \( u(x) \rightarrow 0 \) and \( u'(x) \rightarrow 0 \) as \( x \rightarrow \pm \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Each edge of a regular tetrahedron is divided into three equal parts. Through each resulting division point, two planes are drawn, parallel to the two faces of the tetrahedron that do not pass through that point. Into how many parts do the constructed planes divide the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given a floor plan composed of \( n \) unit squares, Albert and Berta want to cover this floor with tiles. Each tile is either a \( 1 \times 2 \) domino or a T-tetromino. Albert has tiles of one color, while Berta has dominoes of two colors and tetrominoes of four colors. Albert can cover the floor plan in \( a \) ways, and Berta in \( b \) ways. Assuming \( a \neq 0 \), find the ratio \( \frac{b}{a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n/2}}
For an agricultural experiment, seeds were planted in five different plots as follows:

Plot 1: 300 seeds with 25% germination rate and 90% transplant success rate
Plot 2: 200 seeds with 35% germination rate and 85% transplant success rate
Plot 3: 400 seeds with 45% germination rate and 80% transplant success rate
Plot 4: 350 seeds with 15% germination rate and 95% transplant success rate
Plot 5: 150 seeds with 50% germination rate and 70% transplant success rate

After germination, the sprouted seeds were transplanted to a separate growth area. What percent of the total number of seeds planted across all five plots successfully germinated and were transplanted to the growth area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26.67\%}
Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Let $C$ be the set of all 100-digit numbers consisting of only the digits $1$ and $2$. Given a number in $C$, we may transform the number by considering any $10$ consecutive digits $x_0x_1x_2 \dots x_9$ and transform it into $x_5x_6\dots x_9x_0x_1\dots x_4$. We say that two numbers in $C$ are similar if one of them can be reached from the other by performing finitely many such transformations. Let $D$ be a subset of $C$ such that any two numbers in $D$ are not similar. Determine the maximum possible size of $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21^5}
Two fleas are jumping on a large sheet of paper. The first flea jumps in a sequence: right, up, left, down, with each jump increasing by 1 cm. The second flea jumps in a sequence: left, up, right, down, also increasing each jump by 1 cm. After 100 jumps, the fleas are 3 meters apart. What was the initial distance between the fleas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
One integer is selected at random from the following list of 15 integers: $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5$. The probability that the selected integer is equal to $n$ is $\frac{1}{3}$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Two particles are positioned at the vertices of a cube. If at a given time $t$ the particles are on the same edge, they remain in the same position up to time $t + 1$. Otherwise, one of them is chosen at random with equal probability, and at time $t + 1$ it is moved to one of the 3 neighboring vertices with equal probability. If the particles start at opposite vertices of the same face, what is the expected value of the least time they are on the same edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Anderson is bored in physics class. His favorite numbers are $1$, $7$, and $33$. He randomly writes $0$., and then writes down a long string consisting of those numbers juxtaposed against one another (e.g., $0.1337173377133$...) on a sheet of paper. Since physics class is infinitely long, he writes an infinitely long decimal number. If the expected value of the number he wrote down is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{392}
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the parabola $y^{2} = 4cx$, where $c = \sqrt{a^{2} + b^{2}}$, find the eccentricity of the hyperbola given that $|AB| = 4c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} + 1}
用代数式表示 $a$ 的倒数的 70% 与 $b$ 的 2 倍的倒数的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{10a} + \frac{1}{2b}}
Determine the multiplicity of the point \( Q = (0:1:0) \) for the curve given by the equation \( G = X^4Z + X^4Y + Y^4Z \) using the method of partial derivatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of $n$, required to eliminate all the soldiers?

[i]David Yang.[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil \log_2(n) \rceil}
Find the point of maximum for the function \( f_n(x) = x^n (\ln x)^2 \) on the interval \((0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-2/n}}
A hexagon has each angle measuring $120^{\circ}$, and its sides alternately measure 1 cm and $\sqrt{3} \mathrm{~cm}$. What is the area of the hexagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + \sqrt{3} \text{ cm}^2}
Santa Claus has 36 identical gifts divided into 8 bags. The number of gifts in each of the 8 bags is at least 1 and all are different. You need to select some of these bags to evenly distribute all their gifts to 8 children, such that all gifts are distributed completely (each child receives at least one gift). How many different selections are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Determine the minimum number of terms required in the Taylor series expansion of \( \ln(1+x) \) about 1 to approximate \( \ln(4) \) with an error less than \( 2^{-10} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
In a financial market with a bond $A$, a stock $S$, and call options $f$ on the stock $S$, consider the following values: $A(0) = 100$, $A(T) = 120$, $S(0) = 50$, $S^u(T) = 70$, and $S^d(T) = 30$. The call options have a strike price $K = 50$ and expire at $T$. Construct a portfolio $\\phi$ consisting of shares of $S$ and $f$ such that its value $V(T)$ is $120$, regardless of whether $S$ goes up or down. Calculate the price of the call option at time $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}
Stella is filling up a truck with a capacity of 6000 stone blocks. In the first 2 hours, she can put 250 blocks per hour, while Twinkle can put 200 blocks per hour. After 2 hours, their rates change: Stella can put 220 blocks per hour, and Twinkle can put 230 blocks per hour. They work for four hours and are then joined by 6 other people. For the first hour, three of these people have a work rate of 300 blocks per hour, and the other three have a work rate of 180 blocks per hour. After one hour, their work rates change: the first three now have a rate of 280 blocks per hour, and the other three have a rate of 190 blocks per hour. How many hours did filling the truck take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$: $$x_{n+1} = \sum_{k=1}^{n} \sqrt[n]{x_k}.$$ Find the value of $$\lim_{n\to\infty} \frac{x_n - n}{\ln n},$$ assuming it exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2}{3}\\pi"]}
Inside a convex pentagon, a point is marked, and it is connected to all the vertices. What is the maximum number of the ten segments (five sides and five segments connecting the marked point with the vertices of the pentagon) that can have a length of 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A rabbit runs in a perfect circle of radius $r$ with a constant speed $v$. A fox chases the rabbit, starting from the center of the circle and also moves with a constant speed $v$, always positioning itself between the center of the circle and the rabbit. Determine the time it will take for the fox to catch the rabbit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi r}{2v}}
Write the four numbers \(1, 9, 8, 8\) in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, then write the difference between these two numbers in between them. This completes one operation. Then, perform the same operation on the new series of 7 numbers. Continue doing this a total of 100 times. Find the sum of the numbers in the final row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{726}
Hunter was observing the wildlife in the pond and noticed an intriguing mix of amphibians. He saw 4 different species of frogs, including:

- 3 x 2 = 6 green frogs sitting on lily pads,
- 5 tree frogs, which made up 1/3 of the total tree frogs in the area,
- 2 bullfrogs,
- 8 exotic tree frogs, where half of them were blue and half were red.

Apart from these, he saw:

- 3 salamanders swimming near the surface,
- 2/5 of the water surface filled with 50 tadpoles moving in a group,
- 10 baby frogs hopping onto a big rock jutting out from the pond,
- An additional group of 20 tadpoles that were 20% smaller in number than the first group,
- A newt, two toads, and a caecilian hiding under the vegetation at the water's edge.

How many total amphibians (including different species of frogs, salamanders, tadpoles, toads, newts, and caecilians) did Hunter observe in the pond?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{138}
正整数$a _ { 1 } ,a _ { 2 } ,\cdots,a_{2006}$（可以有相同的）使得$\frac { a _ { 1 } } { a _ { 2 } },frac { a _ { 2 } } { a _ { 3 } },\cdots, \frac { a _ { 2 0 0 5 } } { a _ { 2 0 0 6 } }$两两不相等.问：$a _ { 1 } , a _ { 2 },\cdots,a_{2006}$中最少有多少个不同的数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Jimmy and Irene go shopping for clothes on a Tuesday, where senior citizens get a 10% discount on their purchases. There is also a "buy 2, get 1 free" promotion on all summer clothes, and there is a 5% sales tax applied on the final sale amount. Jimmy picks out 3 shorts from the $15 rack, and Irene grabs 5 shirts from the $17 rack. Taking into account the senior citizens' discount, sales tax, and the "buy 2, get 1 free" promotion, how much money do they give to the cashier?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$76.55}
Omi is twice as old as Kimiko. Arlette is 3/4 times as old as Kimiko. Xander is 5 years younger than the square of Kimiko's age, and Yolanda's age is the cube root of Xander's age. If Kimiko is 28 years old, calculate the average age of the five individuals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178.64}
Natural numbers \(a\), \(b\), and \(c\) greater than 2022 have the following properties:
- \(a + b\) is divisible by \(c - 2022\)
- \(a + c\) is divisible by \(b - 2022\)
- \(b + c\) is divisible by \(a - 2022\)

What is the greatest possible value of \(a + b + c\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022 \cdot 85}
In the Cartesian coordinate system, the coordinates of the point $\left(2,-1\right)$ with respect to the $x$-axis are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(2,1\right)}
利用24小时制计时，可得到某一时刻的四个数字之和比时刻$20:21$的四个数字之和小的概率为_______。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{301}
已知圆锥面 $x^{2}+y^{2}=\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\sqrt{3}, 3,6)$ 到 $Q(\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 $I$, 则 $[10 I]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Calculate the value of $6(7+1)(7^2+1)(7^4+1)(7^8+1)+1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7^{16}}
Given the recurrence relation $$T(n) = nT(\sqrt{n}) + n^2$$ with the initial condition $$T(2) = 1$$ and considering $n = 2^{2^k}$, find a closed-form solution for $T(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac 14n^2 + n^2\log_2(\log_2(n))$$}
A teacher teaches one class for each of three grades in a day, with a total of 8 periods in a day, 5 in the morning and 3 in the afternoon. The teacher cannot teach three consecutive classes (periods 5 and 6 do not count as consecutive). Calculate the total number of possible schedules for the teacher in a day. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{312}
Find any four distinct natural numbers that have the following property: if you add the product of any two of them to the product of the remaining two numbers, you will get a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2, 3, 5}
On weekdays, a Scatterbrained Scientist commutes to work on the circular line of the Moscow metro from the "Taganskaya" station to the "Kievskaya" station, and in the evening, he returns back. Upon entering the station, the Scientist boards the first train that arrives. It is known that trains run at approximately equal intervals in both directions. The northern route (through "Belorusskaya") takes 17 minutes from "Kievskaya" to "Taganskaya" or vice versa, while the southern route (through "Paveletskaya") takes 11 minutes. The Scientist, by habit, calculates everything. One day, he calculated based on many years of observations:
- A train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after a train traveling clockwise arrives there. The same is true for "Taganskaya".
- The commute from home to work takes, on average, 1 minute less than the commute from work to home.

Find the expected interval between trains traveling in one direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider a standard Brownian motion $\{B(t); t\ge 0\}$ (zero mean and $\sigma^{2} = 1 $) at times $0 < u < u+v < u+v+w < u+v+w+x $, where $u, v, w, x > 0$. Evaluate the product moment: $$E[B(u)B(u+v)B(u+v+w)B(u+v+w+x)] $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3u^2+3uv+uw}