inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that $f(x)$ is an odd function that is increasing and defined on $\mathbb{R}$, and $\{x_n\}$ is an arithmetic sequence with a common difference of $2$, satisfying

$f(x_7) + f(x_8) = 0$, find the value of $x_{2017}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4019}
A line that always passes through a fixed point is given by the equation $mx - ny - m = 0$, and it intersects with the parabola $y^2 = 4x$ at points $A$ and $B$. Find the number of different selections of distinct elements $m$ and $n$ from the set ${-3, -2, -1, 0, 1, 2, 3}$ such that $|AB| < 8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
On an $8 \times 8$ grid, 64 points are marked at the center of each square. What is the minimum number of lines needed to separate all of these points from each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find a function, other than $e^x$ and $\cos x$, whose derivative is a function of itself, similar to how $f'(x) = f(x)$ for $f(x) = e^x$ and $f'(x) = f(x+\pi/2)$ for $f(x) = \cos x$. Provide one example of such a function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin x}
Find the area of the surface S, which is the portion of the cylinder defined by \(x^2+y^2=a^2\) that is bounded below by the \(xy\)-plane and above by the cone \(z=\sqrt{(x-a)^2+y^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8a^2}
正整数 $9+95+995+\cdots+\underbrace{99\cdots95}_{2016\text{个}9}$ 的十进制表示中数字 1 的个数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Determine the smallest positive integer \( N \) such that for any initial arrangement of \( 2n \) balls, each colored with one of \( n \) colors (two balls per color), Phoe Wa Lone can always sort the balls into \( n \) boxes, each containing two balls of the same color, using at most \( N \) moves. A move consists of either reversing the order of two balls in a box or moving a top ball to an empty box or to a box with a ball of the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Solve the equation

$$
\sin ^{4} x+\cos ^{4} x=\sin ^{4} 2 x+\cos ^{4} 2 x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n\pi}{6}}
Given that $\triangle ABC$ is an inscribed triangle in the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and the side $AB$ passes through the point $P(1,0)$. Find the maximum area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16 \sqrt{2}}{3}}
Let \( a, b, c \) be non-negative real numbers satisfying
\[ a + b + c = 8 \]
and
\[ ab + bc + ca = 16.\]
Let \( m = \min \{ ab, bc, ca \} \). Find the maximum possible value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16/9}
A club with 75 members ordered fruit juices. Three-sevenths of them ordered lemon juice. One-fourth of the remaining members ordered mango juice, and the rest ordered orange juice. How many members ordered orange juice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
The CMU Kiltie Band is attempting to crash a helicopter via grappling hook.  The helicopter starts parallel (angle $0$ degrees) to the ground.  Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x$ degrees with the ground.  Causing the helicopter to tilt to $90$ degrees or beyond will crash the helicopter.  Find the expected number of times the band must pull the hook in order to crash the helicopter.

[i]Proposed by Justin Hsieh[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{269}{32}}
Find another element of order 2 in the dihedral group \(D_{2p}\) besides the element \(b\), where \(p\) is a prime greater than 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab}
For any integer k > 1, the term "length of an integer" refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. If x and y are positive integers such that x > 1, y > 1, and the maximum possible sum of the length of x and the length of y is 16, what is the maximum value of x + 3y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98306}
14. Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14 th jump lands at the origin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{613470}
Using $$\int_0^{\pi/2} \log\cos x\,\mathrm dx= -\dfrac{\pi}{2} \log 2$$, find $$I=\int_0^{\infty} \log{(x^2+1/x^2)}\,\frac{dx}{1+x^2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\pi\log(2)\)}
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \geq 1$ and $a+b+c \leq 7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given that \( x, y, z \) are the side lengths of a triangle and \( x + y + z = 1 \). Find the minimum value of the real number \( \lambda \) such that the inequality
$$
\lambda(x y + y z + z x) \geqslant 3(\lambda + 1)x y z + 1
$$
always holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such that there are infinitely many positive integers $n$ for which $S(n) - S(n + a) = 2018$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let \( S = \{1, 2, 3, 4, \ldots, 100000\} \). Find the least possible value of \( k \) such that any subset \( A \) of \( S \) with \( |A| = 2010 \) contains two distinct numbers \( a \) and \( b \) with \( |a - b| \leq k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Each row of a  $24 \times 8$  table contains some permutation of the numbers  $1, 2, \cdots , 8.$  In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 * (8!)^3}
In how many ways can the natural numbers from 1 to 10 (each used exactly once) be arranged in a $2 \times 5$ table so that the sum of the numbers in each of the five columns is odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["460800"]}
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\left\lfloor\frac{N}{3}\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
There are three piles of stones: the first has 10 stones, the second has 15 stones, and the third has 20 stones. Each turn, one is allowed to split any pile into two smaller piles. The player who cannot make a move loses. Who will win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Second}
Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.

$$
y = e^{x} + e, \ln \sqrt{3} \leq x \leq \ln \sqrt{15}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \frac{1}{2} \ln \left( \frac{9}{5} \right)}
An isosceles triangle can be divided into two isosceles triangles (not necessarily equal). Find the possible values for the smallest angle of such a triangle. Provide the smallest of these values in degrees, multiplied by 6006.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{154440}
Two people, A and B, start from the same point on a 300-meter circular track and run in opposite directions. A runs at 2 meters per second, and B runs at 4 meters per second. When they first meet, A turns around and runs back. When A and B meet again, B turns around and runs back. Following this pattern, after how many seconds will the two people meet at the starting point for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Determine the remainder of the 81-digit integer \(\overline{111 \cdots 1}\) divided by 81.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length \(\frac{1}{2}\) and edges parallel to those of the unit cube that contains all three points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given real numbers $x_1$, $x_2$, $y_1$, $y_2$ satisfying: $x_1^2+y_1^2=1$, $x_2^2+y_2^2=1$, $x_1x_2+y_1y_2=\frac{1}{2}$, find the maximum value of $\frac{|x_1+y_1-1|}{\sqrt{2}} + \frac{|x_2+y_2-1|}{\sqrt{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$,
$$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n-1 + \sqrt{n-1}}{\sqrt{n}}}
Find a three-digit number that is divisible by 45, given that the difference between this number and its reversed version is 297.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
7. (10 points) There are 37 people standing in a line to count off. The first person reports 1, and each subsequent person reports a number that is 3 more than the previous person's number. During the counting, one person made a mistake and reported a number that was 3 less than the previous person's number. The sum of the numbers reported by these 37 people is exactly 2011. Then, the person who made the mistake is the $\qquad$th person to report a number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Find the smallest positive integer $n$ for which $315^2 - n^2$ evenly divides $315^3 - n^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Given a mall with four categories of food: grains, vegetable oils, animal products, and fruits and vegetables, with 40, 10, 20, and 20 varieties, respectively, calculate the total sample size if 6 types of animal products are sampled.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
If the function $f(x)=(a-2)x^{2}+(a-1)x+3$ is an even function, then the interval in which $f(x)$ is increasing is $\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,0]}
Select four students from five students named A, B, C, D, and E to participate in a volunteer service activity. Each person is assigned one of four jobs: translator, tour guide, etiquette, or driver. Each job is taken by one person. A cannot drive, B cannot translate, but both can do the other three jobs, while C, D, and E are capable of doing all four jobs. The number of different arrangement schemes is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
According to legend, the poet Li Bai from the Tang Dynasty went to buy wine, singing as he walked down the street. Each time he passed a store, the amount of his wine doubled, and each time he saw a flower, he drank 2 cups. After encountering four stores and flowers in total, he still had 2 cups of wine left at the end. How many cups of wine did he originally have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$P$是$ΔABC$内一点，引线段$APD$，$BPE$和$CPF$，使$D$在$BC$上，$E$在$AC$上，F在AB上.已知$AP = 6 , BP = 9 ,PD = 6 , PE = 3 , CF = 20$，求$ΔABC$的面积.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
The length of a red bus is four times the length of an orange car. The length of the orange car is also 3.5 times shorter than the length of a yellow bus. A green truck is parked between the red bus and the yellow bus, and the green truck is as long as two orange cars. If the red bus is 48 feet long, and the yellow bus and green truck are parked end to end next to the red bus, what length of the yellow bus may the red bus driver see?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Andrew installed hardwood flooring in his house. His bedroom took eight wooden planks, his living room took twenty planks, his kitchen took eleven planks, and his dining room took thirteen planks. The guest bedroom took two fewer planks than Andrew’s bedroom, and each of his two hallways took four planks. The study took the same number of planks as the guest bedroom, plus three. He accidentally ruined three planks in each bedroom, two planks in the living room, and one plank in the study by dragging furniture over them and had to replace them. He ended up with seven leftover planks at the end. How many wooden planks did Andrew buy to start with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Source: 2017 Canadian Open Math Challenge, Problem B4
-----
Numbers  $a$ ,  $b$  and  $c$  form an arithmetic sequence if  $b - a = c - b$ . Let  $a$ ,  $b$ ,  $c$  be positive integers forming an arithmetic sequence with  $a < b < c$ . Let  $f(x) = ax2 + bx + c$ . Two distinct real numbers  $r$  and  $s$  satisfy  $f(r) = s$  and  $f(s) = r$ . If  $rs = 2017$ , determine the smallest possible value of  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Taylor and his friends decide to bring their pets together for a walk. Six of Taylor's friends join him. Three of his friends have twice as many pets as Taylor, who has 4 cats. One friend has 3 dogs and a bird, while another brings 5 dogs and 2 cats. The last friend brings a unique pet collection of 2 reptiles, 3 birds, and a cat. Calculate the total number of cats, dogs, birds, and reptiles the friends have together.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
For $f \in L^2([0, 1]) $ and $x \in [0, 1]$ we have the following $$ A(f)(x) = \int_0^1 (x - y) f(y)dy $$ Calculate the operator norm $\| A \|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{3}}{6}\)}
Find the splitting field of the polynomial $x^4 + x^3 + 1$ over the finite field $\mathbb{F}_{32}$. Determine the degree of the extension field over $\mathbb{F}_{32}$ that contains all the roots of the polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The deputies in a parliament were split into  $10$  fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. 
Find the smallest possible number of independent deputies after the vacation.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3000}
Given the vector space $V = M_{2\times 3}(\mathbb{R})$ and the bilinear form $f: V \times V \to \mathbb{R}$ defined by $f(X, Y) = \text{Tr}(X^T A Y)$, where $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, find the matrix representation of $f$ with respect to the standard basis of $M_{2\times 3}(\mathbb{R})$. What is the order of this matrix?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine if the dimension of the symplectic group Sp(2n, \(\tilde{F}\)) over the algebraic closure of a finite field \(F\) is equal to \(2n^2 + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^2 + n}
Determine whether the cardinality of the set of all real-valued functions is $2^{\mathfrak c}$, where $\mathfrak c$ is the cardinality of the real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\mathfrak{c}}}
Let $n > 1$ be an odd number. Consider an $n \times n$ matrix $A$ defined by the following rule for its entries:
\[
a_{i, j} =
\begin{cases}
1, & \text{if } i - j \equiv \pm 2 \pmod n,\\
2, & \text{if } i = j,\\
0, & \text{otherwise}.
\end{cases}
\]
Calculate the determinant of the matrix $A$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
You arrange $12$ biologists of $12$ different heights in three rows of $4$, with the same conditions on height as in part 10(a) for all three rows. How many different arrangements are possible?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462
}
If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{line
}
Balls of $3$ colours — red, blue, and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The product of six distinct integers is 36. Find the sum of these six integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many integer triples(x,y,z)are there such that
 \( x - y{z}^{2} \equiv  1\;\left( {\;\operatorname{mod}\;{13}}\right) \)
 \( {xz} + y \equiv  4{\;(\operatorname{mod}\;{13})} \)
 where \( 0 \leq  x < {13} \) , \( 0 \leq  y < {13} \) ,and \( 0 \leq  z < {13} \) ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
23
}
Determine the packing density of the heptomino P on a square grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
Evaluate the sum \( \sum_{k=1}^{2023}(-1)^{^{\binom {k+3}4}}k \) and determine if the result is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
Determine whether the dimension of any open subset of \(\mathbb{R}^n\) is equal to \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Find the limit of the following expression as $n$ approaches infinity, given that $b > 0$:
\[ \lim_{n \to \infty} \frac{\sum_{i=0}^{n/3-1}{\frac{(bn^3)^{i}}{i!}}}{e^{bn^3}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
设函数 $f(x)=e^{x}-x^{2} e^{x}-a x-1$.
若对于任意的 $x_{1}, x_{2} \in(1,+\infty), x_{1} \neq x_{2}$, 都有 $\frac{f\left(x_{1}\right)-f\left(x_{2}\right)}{x_{1}-x_{2}}<x_{1}+x_{2}$ 恒成立, 求实数 $a$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[-2 e-2, +\infty)$}
G20 (7-3, Czechoslovakia) A tetrahedron $ABCD$ is divided into two parts by a plane parallel to $AB$ and $CD$. It is known that the ratio of the distance from $AB$ to this plane and the distance from $CD$ to the plane is equal to $k$. Find the ratio of the volumes of the two parts into which the tetrahedron is divided.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^2(k+3)}{3k+1}}
There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let $(B_{t})_{t \geq 0}$ be a one-dimensional Brownian motion and $(\phi(t,0))_{t \geq 0}$ be its local time at $0$. Determine whether $\lim_{t \uparrow \infty}\phi(t,0)=\infty$ almost surely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/11}
In triangle \( ABC \), angle \( A \) is \( 60^\circ \). A bisector \( AD \) is drawn, and the radius of the circumcircle of triangle \( ADC \) with center at point \( D \) is \( \frac{2\sqrt{3}}{3} \). Given that \( AB = 1 \), find the length of segment \( BM \), where \( M \) is the intersection point of segments \( AD \) and \( BO \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the Hamiltonian \( H = \frac{1}{2}(S_1^2+S_2^2+\beta S_3^2)+R_1 \) and the first integrals \( f_1 = R_1^2+R_2^2+R_3^2 \) and \( f_2 = R_1S_1+R_2S_2+R_3S_3 \), compute the Poisson bracket \( \{f_1, f_2\} \) using the relations \( \{S_i, S_j\} = \epsilon_{ijk}S_k \), \( \{R_i, S_j\} = \epsilon_{ijk}R_k \), and \( \{R_i, R_j\} = 0 \). Verify if \( \{f_1, f_2\} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}