inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the smallest natural number \( n \) such that in any set of \( n \) different natural numbers, each not exceeding 1000, there exist two numbers where the larger number is not divisible by the smaller one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A quadratic polynomial with positive integer coefficients and rational roots
 can be written as \( {196}{x}^{2} + {Bx} + {135} \) for some integer \( B \) . What is the sun
 of all possible values of \( B \) such that \( \gcd \left( {B,{196} \cdot  {135}}\right)  = 1 \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
42000
}
A cube with edge length 1 is denoted as AC<sub>1</sub>. A moving point P is on its surface, and the distance from point A is $\frac {2 \sqrt {3}}{3}$. The set of points P forms a curve. The length of this curve is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {5 \sqrt {3}}{6}\pi}
Given that the positive real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) satisfy the conditions \(a_{1} \geqslant a_{2} a_{3}^{2}, a_{2} \geqslant a_{3} a_{4}^{2}, a_{3} \geqslant a_{4} a_{1}^{2}, a_{4} \geqslant a_{1} a_{2}^{2}\), find the maximum value of \(a_{1} a_{2} a_{3} a_{4}\left(a_{1}-a_{2} a_{3}^{2}\right)\left(a_{2}-a_{3} a_{4}^{2}\right)\left(a_{3}-a_{4} a_{1}^{2}\right)\left(a_{4}-a_{1} a_{2}^{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/256}
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{7}{36}$}
Find the number of ordered 17-tuples $(a_1, a_2, a_3, \dots, a_{17})$ of integers, such that the square of any number in the 17-tuple is equal to the sum of the other 16 numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12378}
Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the basketball shooting competition. How many students participate in all three events?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given the sets $A=\{y\in \mathbb{N}|y=-x^{2}+4x,x\in \mathbb{Z}\}$ and $B=\{x|\ln x > 1\}$, find the intersection of $A$ and the complement of $B$ in $\mathbb{R}$, denoted as $A\cap (\complement_{\mathbb{R}}B)$. Express your answer as a set of numbers using standard set notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Using the Sturm-Picone comparison theorem, determine the maximum number of zeroes of the solution to the differential equation $(1+x^2)y''+(1+4x^2)y=0$ in the interval $[0, 2\pi]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let  $ABCDEF$  be a regular hexagon with sidelength  $6$ , and construct squares  $ABGH$ ,  $BCIJ$ ,  $CDKL$ ,  $DEMN$ ,  $EFOP$ , and  $FAQR$  outside the hexagon. Find the perimeter of dodecagon  $HGJILKNMPORQ$ .

*Proposed by Andrew Wu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
A set $D$ of positive integers is called indifferent if there are at least two integers in the set, and for any two distinct elements $x, y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1257}
Use Stoke's Theorem to evaluate the line integral \( \int_{C} \mathbf{F} \cdot \mathrm{d}\mathbf{r} \), where \( \mathbf{F}(x,y,z) = -x^2\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) and \( C \) is the boundary of the surface \( S \), which is the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{(x,y) \mid -2 < x < 3, 0 < y < 2\} \). The curve \( C \) is oriented counterclockwise when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
A horse stands at the corner of a chessboard, on a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally, like a knight moves. The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to the rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A bicycle costs 389 yuan, and an electric fan costs 189 yuan. Dad wants to buy a bicycle and an electric fan. He will need approximately \_\_\_\_\_\_ yuan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Martha gets prize points every time she shops at her local grocery store. She gets 50 points per $10 spent, plus a 250 point bonus if she spends more than $100. Additionally, Martha is a member of the store's loyalty program which allows her to earn 100 bonus points for every 5th visit to the store, and today is her 5th visit.

During her shopping trip, Martha buys the following items:
- 3 pounds of beef for $11 each, with an item-specific 10% discount applicable on each;
- 8 pounds of fruits and vegetables for $4/pound, with a distinct category-wise 8% discount on the total amount;
- 3 jars of spices for $6 each, with a "buy 2, get the 3rd one free" offer and a separate 5% discount on the total price;
- Other groceries totaling $37 before a $3 off coupon is applied.

Calculate the number of points Martha earns during this shopping trip, taking into account all the individual discounts, the "buy 2, get 1 free" offer, and the loyalty program bonus points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{850}
Find the partial derivative of the function \( f = \text{tr}((ABC)(ABC)^T) \) with respect to the matrix \( B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 A^\top A B C C^\top}
Percival's castle had a square shape. One day Percival decided to expand his estate and added a square annex to the castle. As a result, the perimeter of the castle increased by $10\%$. By what percentage did the area of the castle increase?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\%}
Find the maximum possible area of a triangle with side lengths at most 2, 3, and 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( S = \{1, 2, 3, 4\} \), and a sequence of \( n \) terms \( a_1, a_2, \ldots, a_n \) has the following property: For any non-empty subset \( B \) of \( S \), there exist \( |B| \) consecutive elements in the sequence that exactly form the set \( B \). Find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the limit \( \lim_{n \to \infty} I_n^n \), where \( I_n = \int_1^{1+\frac{1}{n}}\left\{[(x+1)\ln x +1] e^{x (e^x \ln x +1)}+n \right\}\, dx \) for \( n=1,2,\ldots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^e}
Find the value of: $$\lim_{x\to 0}\left\{\dfrac 2{x^3}(\tan x- \sin x )\right\}^{2/x^2 }$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Find the equation of the tangent line to the function \( f(x) \) at \( x = 1 \), where:

\[ f(x) = \int_0^{g(x)} \frac{1}{1+\sin(t) + e^t} \, dt \]

and \( g(x) \) is given by:

\[ g(x) = \int_{-x}^{x^2} h(t) \, dt \]

with \( h(t) \) being an odd function and \( h(1) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = x - 1}
Pete's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars. What is the maximum amount Pete can withdraw from the account if he has no other money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{498}
How many times will a clock strike over the course of 12 hours if it chimes on the half-hours as well?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Determine the Galois group of the polynomial \( g(x) = x^6 + 2x^3 - 3 \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
How many different permutations $(\alpha_1 \alpha_2\alpha_3\alpha_4\alpha_5)$ of the set $\{1,2,3,4,5\}$ are there such that $(\alpha_1\dots \alpha_k)$ is not a permutation of the set $\{1,\dots ,k\}$, for every $1\leq k \leq 4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71
}
Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 chests, with the knowledge that only one contains the treasure, and the others trigger a stone slab to fall on the head of anyone who tries to open them. The chests are numbered from left to right.

- The first, fourth, and fifth chests are made of cedar.
- The second and third chests are made of sandalwood.
- The inscriptions on the chests are as follows:
  - On the first chest: "The treasure is in me or in the 4th chest."
  - On the second chest: "The treasure is in the chest to the left of me."
  - On the third chest: "The treasure is in me or in the chest at the far right."
  - On the fourth chest: "There is no treasure in the chests to the left of me."
  - On the fifth chest: "All the inscriptions on other chests are false."

The last guardian of the temple, before dying, revealed to Indiana a secret: an equal number of false statements are written on both cedar and sandalwood chests. 

In which chest is the treasure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Choose a permutation of $ \{1,2, ..., 20\}$  at random. Let  $m$  be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of  $2^m$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine the number of digits in the largest natural number $N$ that does not contain the digit $0$, has a digit sum of $2021$, and where the digit sum of $2N$ does not exceed $1202$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{757}
A professional pet groomer who specializes in the care of long-haired cats, like the Siberian, has 3 grooming stations in her shop. On a busy Saturday, she has appointments to groom 12 Siberian cats. Each cat takes 45 minutes to groom. If the groomer works 9 hours that day, how many grooming stations does she need to use at the same time to finish all the appointments within her working hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate \( \sum_{d\mid N}\Lambda(d) \) for a positive integer \( N \), where \( \Lambda(n) \) is defined as follows:

\[\Lambda(n) = \begin{cases} \log p & \text{if } n = p^r, \ p \text{ a prime and } r \in \mathbb{N}, \\ 0 & \text{otherwise.} \end{cases}\]

Here, the sum ranges over all divisors \( d \) of \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log N}
Let \( f \) be the function defined for all \((x,y)\) as follows: \[ f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2}, &\text{if }(x,y)\ne(0,0)\\ 0, &\text{if }(x,y)=(0,0) \end{cases} \] Determine the value of \( \frac{\partial^2 f}{\partial x \partial y} \) at \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of hyperreal numbers that have a standard part equal to 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the boundary value of \(\alpha\) such that \(0 < \alpha < 1\) and \(n^{\alpha}\) never surpasses \(\log(n)\) for sufficiently large \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the expansion of (mx-1)^n, where n is a positive integer, and the sum of the coefficients of all terms is equal to the sum of the binomial coefficients, and the binomial coefficient of the 6th term is the largest, find the number of different ordered real number pairs (m,n) that satisfy these conditions. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that day, told Xiao Xiao: The total age of our family this year is 7 times your age. If we add a younger brother or sister next year, the total age of our family in 2020 will be 6 times your age at that time. How old is Xiao Xiao this year? $\qquad$ years old.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then,$$S = \sum\limits_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$Find $\lfloor 80S \rfloor.$

(Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
98}
8,9

Points $K, L, M$ divide the sides of the convex quadrilateral $A B C D$ in the ratio $A K: B K=C L: B L=C M: D M=1$ : 2. The radius of the circle circumscribed around triangle $K L M$ is $\frac{5}{2}, K L=4, L M=3$. What is the area of quadrilateral $A B C D$, given that $K M<K L$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{189}{25}}
In a circle with center \(O\) and radius \(\sqrt{5} + 1\), a secant \(PBA\) and a tangent \(PE\) are drawn from an external point \(P\). Given that \(PE = AB = 2\), calculate the measure of \(\angle POB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18^\circ}
The number 2019 is expressed as a sum of different odd natural numbers. What is the maximum possible number of terms in this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Evaluate the limit \( \lim\limits_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \), where \( \delta_{a} \) denotes the Dirac distribution at point \( a \). Express the result of the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta''}
Arrange the numbers 1, 2, 3, 4, 5, 6 in the form of rows where the first row contains 1 number, the second row contains 2 numbers, and the third row contains 3 numbers, randomly. Let $N_i$ ($i=1,2,3$) denote the largest number in the $i$-th row, then the number of all permutations satisfying $N_1<N_2<N_3$ is _______ (answer in numerals).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
In a cinema hall with $N$ numbered seats, each seat is assigned to a specific spectator. The first $N-1$ spectators arrive early and occupy seats randomly. The last spectator, who has a designated seat, attempts to occupy it. If her seat is occupied, she asks the person sitting there to move to their assigned seat. This process continues until all spectators are in their correct seats. What is the expected number of seat changes (displacements) that occur during this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(N-1)/2}
Three stations are distributed uniformly over a distance of 10 units. Calculate the probability that the maximum distance between any two stations is at most 1 unit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{250}}
Find the least possible degree of the polynomial \( f(x) \) in the expression \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \), where \( p(x) \) is a polynomial that does not have the factor \( x^3-x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given two circles $\odot M$ and $\odot N$ intersecting at points $A$ and $B$, and a line $CD$ passing through point $B$ and intersecting circles $\odot M$ and $\odot N$ at points $C$ and $D$ respectively, with $\angle ABD = 60^{\circ}$, find the ratio of the area of quadrilateral $AMBN$ to the area of triangle $ACD$. Express your answer as a ratio in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Determine the minimum number of real variables required to describe all 300 by 300 matrices that are their own inverse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45000}
Find the integer \( n \) such that \( \mathbb{Z}[i]/(-2+2i) \cong \mathbb{Z}/n\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Minh enters the numbers from 1 to 81 in a $9\times9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
In a restaurant, men and women arrive at the bathroom at an average rate of one person every 20 minutes. On average, men take 60 seconds to use the facilities, while women take 10 minutes. Determine the long-term ratio of the average queuing time at the women's bathroom to the average queuing time at the men's bathroom.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{190}
Find the function $f(z)$ given that $f = u + iv$ and $if = -v + iu$. Additionally, it is given that $\Re[(1+i)f(z)] = e^{x} (\cos y - \sin y)$. Determine $f(z)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^z}
Determine the maximum number of consecutive positive integers that are singular, where a singular number is defined as a number whose prime factorization contains no even powers other than 0. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9 \sqrt{3}}{4}}
Given that the derivative of a function f(x) is f'(x), if for any x in R, f'(x) - f(x) < 1, and f(0) = 2022, find the solution set of the inequality f(x) + 1 > 2023e^x. Express your answer as an interval or a set of real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
In the rectangular coordinate system $(xOy)$, the equations of the ellipses $(C_{1})$ and $(C_{2})$ are given by $( \frac{x^{2}}{4}+y^{2}=1)$ and $( \frac{y^{2}}{16}+ \frac{x^{2}}{4}=1)$ respectively. The ray $(OA)$ intersects $(C_{1})$ and $(C_{2})$ at points $A$ and $B$ respectively, with $( \overrightarrow{OB}=2 \overrightarrow{OA})$. Find the slope of the ray $(OA)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Out of the 21 numbers $8, 9, \cdots, 28$, randomly select three different numbers. What is the probability that their sum is exactly divisible by 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{133}}
Find an axis of symmetry for the function $f(x) = \cos(2x + \frac{\pi}{6})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{12}}
Find the last two digits of $\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g., if the last two digits are $05$, just write $5$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, $c$, respectively, and $C=\frac{π}{3}$, $c=2$. Then find the maximum value of $\overrightarrow{AC}•\overrightarrow{AB}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{3}}{3} + 2}
What is the value of the integer $n$ in the expression for the area of the intersection of the surface $x^2 + y^2 + z^2 = 4$ (a sphere with radius 2) and the surface $x^2 + y^2 = 2x$ (a cylinder with radius 1), given that the area can be represented as $n(\pi - 2)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the greatest real number $a$ such that the inequality

$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} \geq a\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}\right)
$$

holds for every five real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\sqrt{3}}}
How do you evaluate the line integral $\int_C xdy - ydx$, where C is the curve composed of a straight line segment from $(−2, 0)$ to $(0, 0)$, a straight line segment from $(0, 0)$ to $(0, −2)$, and the part of the circle of radius 2, centered at the origin, traversed counterclockwise starting from $(0,−2)$ and ending at $(−2, 0)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6π}
Given that the random variable ξ follows a normal distribution N(4, 6^2), and P(ξ ≤ 5) = 0.89, find the probability P(ξ ≤ 3). Express your answer as a decimal value between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.11}
In an infinite sequence of natural numbers, the product of any fifteen consecutive terms is equal to one million, and the sum of any ten consecutive terms is equal to \(S\). Find the maximum possible value of \(S\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208}
Two players play a game on a pile of \( n \) beans. On each player's turn, they may take exactly \( 1 \), \( 4 \), or \( 7 \) beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many \( n \) between \( 2014 \) and \( 2050 \) (inclusive) does the second player win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}