inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $a$ be the number of numbers that satisfy the condition "there is no other number whose absolute value is equal to the absolute value of $a$," $b$ be the number of numbers that satisfy the condition "there is no other number whose square is equal to the square of $b$," and $c$ be the number of numbers that satisfy the condition "there is no other number that, when multiplied by $c$, results in a product greater than 1." Then, $a+b+c=$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \(\mathcal{V}\) be the volume enclosed by the graph
\[ 
x^{2016} + y^{2016} + z^{2} = 2016 
\]
Find \(\mathcal{V}\) rounded to the nearest multiple of ten.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Given an odd function $y=f\left(x\right)$ defined on $R$, which is strictly decreasing on the interval $\left[0,+\infty \right)$. If for any $x\in R$, we always have $f(kx^{2}+2)+f\left(kx+k\right)\leqslant 0$ holds, then the range of real number $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,+\infty)}
Natural numbers are arranged in a spiral, turning the first bend at 2, the second bend at 3, the third bend at 5, and so on. What is the number at the twentieth bend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
In \(\triangle ABC\), \(AB = 9\), \(BC = 8\), and \(AC = 7\). The bisector of \(\angle A\) meets \(BC\) at \(D\). The circle passing through \(A\) and touching \(BC\) at \(D\) cuts \(AB\) and \(AC\) at \(M\) and \(N\) respectively. Find \(MN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180^{\circ}}
6. (10 points) Doudou's family has 4 people. This year, Doudou's brother is 3 years older than Doudou, and Doudou's mother is 2 years younger than Doudou's father. 5 years ago, the total age of the family was 59 years, and 5 years later, the total age of the family will be 97 years. Doudou's mother is $\qquad$ years old this year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given that $a_n$ is the coefficient of the term containing $x^n$ in the expansion of $(1+x)^{n+2}$, find the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n}\}$. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{n+2}}
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. Find the value of $c_1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$431$}
All three vertices of an equilateral triangle are on the parabola $y = x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A circle is circumscribed around a regular $2n$-gon. A set of three distinct vertices of the polygon is called one-sided if there exists a semicircle on which these vertices lie (the endpoints of the semicircle are included). What is the probability that a randomly chosen set of three vertices is one-sided?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3n}{2(2n-1)}}
Because of the migration of salmon, the total number of salmon in river Trout has increased by ten times the number initially in the river. Now, there are 5500 salmons in the river. How many salmons were initially in the river?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
4. Let $n \in \mathbf{Z}_{+}$. A volleyball team has $n$ male players and $n$ female players. Initially, each player is assigned to one of the positions numbered $1,2, \cdots, 2 n$, with only positions 1 and $n+1$ being outside the court. During the game, position swaps occur, and each swap involves moving the player in position $i(i=1,2, \cdots, 2 n)$ to position $i+1$, where position $2 n+1$ is the same as position 1. How many different initial allocation methods are there such that, regardless of how many rounds of swapping occur, at least $n-1$ female players are on the court?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n}(n!)^{2}}
The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A cube has an edge length of 3. A square hole with a side length of 1, which goes through to the opposite side, is drilled in the exact center of each face, and the edges of the hole are parallel to the edges of the cube. Find the volume of the resulting solid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
There are 10 children, 5 boys and 5 girls, imagining playing the following game: First, let 4 children (not all boys) stand at 4 equidistant positions on a circle. If two adjacent children are of the same gender, then a boy is placed between them; otherwise, a girl is placed between them. Then, the original 4 children temporarily exit, which counts as one activity. This activity continues according to the above rules until the 4 children standing on the circle are all boys. The maximum number of times this activity can be performed is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given real numbers $a$ and $b$ satisfy $\ln (b+1)+a-3b=0$, and real numbers $c$ and $d$ satisfy $2d-c+ \sqrt {5}=0$, then the minimum value of $(a-c)^{2}+(b-d)^{2}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In how many ways can a lottery ticket be filled out such that the sum of the 5 marked numbers is at least 228?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21974634}
A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{998}.}
求最小的正整数 k，使得对集合 S=\{1,2,\cdots,2012\} 的任意一个 k 元子集 A，都存在 S 的三个不同的元素 a,b,c，使得 a+b, b+c, c+a 均在 A 中.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
From point \(A\), lying on a circle, chords \(AB\), \(AC\), and tangent \(AD\) are drawn. The angle between the chords is \(\frac{\pi}{6}\), and the angle between the chord \(AC\) and the tangent \(AD\), which does not contain the chord \(AB\), is \(\frac{5\pi}{12}\). Compute the integer part of the radius of the circle if the area of the triangle \(ABC\) is 32.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle ABC, the side lengths opposite to angles A, B, and C are a, b, and c, respectively. Given that the height h from vertex A to side BC equals c - a, calculate the value of sin((C - A)/2) + cos((C + A)/2).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On each non-boundary unit segment of an $8 \times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A set of five-digit numbers $\{N_1, ..., N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1, ..., N_k$. Find the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the greatest number which is such that when 794, 858 and a certain number are divided by it, the remainders are all the same. The greatest number is 1. What is the third number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{922}
In bridge, a standard 52-card deck is dealt to 4 players. Each hand is assigned a number of "points" based on the formula
\[ 4 \times(\# \text{A's}) + 3 \times(\# \text{K's}) + 2 \times(\# \text{Q's}) + 1 \times(\# \text{J's}). \]

Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{197}{1820}}
Of the land owned by a farmer, 85 percent was cleared for planting. Of the cleared land, 10 percent was planted with grapes, 70 percent was planted with potatoes, and 5 percent was planted with corn. If the remaining 500 acres of cleared land was planted with tomatoes, how many acres did the farmer own?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3921.57}
Find the maximum value of the function $f(x_1, x_2, ..., x_T) = \sum_{t=1}^T 2^{-t}\sqrt{x_t}$ subject to the constraints $\sum_{t=1}^T x_t \leq 1$ and $x_t \geq 0$ for all $t = 1, 2, ..., T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{1-4^{-T}}{3}}}
3. In the Cartesian coordinate system, the set of points $(x, y)$ that satisfy $x<y,|x|<3,|y|<3$, and make the equation $\left(x^{3}-y^{3}\right) t^{4}+$ $(3 x+y) t^{2}+\frac{1}{x-y}=0$ have no real roots is denoted as $N$. Then the area of the region formed by the point set $N$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{81}{5}}
In a certain kingdom, there were 32 knights. Some of them were vassals of others (a vassal can have only one liege, and a liege is always wealthier than their vassal). A knight with at least four vassals held the title of baron. What is the maximum number of barons that could exist under these conditions?

(In the kingdom, the law stated: "a vassal of my vassal is not my vassal.")
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{33}{105}}
There are 111 StarCraft progamers. The StarCraft team SKT starts with a given set of eleven progamers on it, and at the end of each season, it drops a progamer and adds a progamer (possibly the same one). At the start of the second season, SKT has to field a team of five progamers to play the opening match. How many different lineups of five players could be fielded if the order of players on the lineup matters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4015440}
Lizzy's mother gave her 80 cents, and her father gave her 40 cents for keeping to her promise. She went to her uncle's and spent 50 cents buying candy for her little cousin. Her uncle gave her another 70 cents. Then, she went to a toy store, saw a toy she wanted to buy. The price tag on the toy was 90 cents but there was a discount of 20% on toys that day. With the discount applied, she paid for the toy and the cashier gave her the change in dollars and cents: 1 dollar and 10 cents. How many cents does she have now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178}
Given the hyperbola $\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$, and a chord AB with a length of 6 connected to the left focus F₁, calculate the perimeter of △ABF₂ (F₂ being the right focus).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Find the argument of $Z_4$, given that $O, Z_1, Z_2, Z_4$ are vertices of a parallelogram with $O$ as the origin, $|Z_1|=|Z_2|=5$, $|Z_4|=7$, and $\arg(Z_2-Z_1)=\dfrac{\pi}{4}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
Simplify: $\overrightarrow{OP} + \overrightarrow{PQ} - \overrightarrow{MQ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\overrightarrow{OM}}
The charge for a single room at hotel P is 20 percent less than the charge for a single room at hotel R and some percent less than the charge for a single room at hotel G. The charge for a single room at hotel R is 12.500000000000004 percent greater than the charge for a single room at hotel G. What is the percentage difference between the charges for a single room at hotel P and hotel G?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The Razorback t-shirt Shop sells their t-shirts for $16 each. Last week, they sold 45 t-shirts. They had a "buy 3 get 1 free" promotion in addition to a 10% discount on every t-shirt sold. After applying these offers, a 6% sales tax was added to the final price. How much money did they make in total after applying the promotion, discount, and sales tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$518.98}
2. For a natural number ending not in zero, one of its digits (not the most significant) was erased. As a result, the number decreased by 9 times. How many numbers exist for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
The value of the expression $ \frac{(304)^5}{(29.7)(399)^4}$ is closest to

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
若曲线 $y=x\ln x$ 上一点 $P$ 到直线 $y=\frac{1}{2}x-1$ 的距离最小，则点 $P$ 的横坐标为 $\frac{e}{e}$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{e}{e}}
Determine if \(\exp(-\bar{X})\) is the maximum likelihood estimator (MLE) of \(\pi(\theta) = \exp(-\theta)\) for a Poisson distribution, where \(\bar{X}\) is the sample average.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exp(-\bar{X})}
If $P$ is a positive definite matrix, and $A^TPA-P$ is negative definite, does it follow that $APA^T-P$ is also negative definite? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
There are five courses at my school. Students take the classes as follows: 243 take algebra. 323 take language arts. 143 take social studies. 241 take biology. 300 take history. 213 take algebra and language arts. 264 take algebra and social studies. 144 take algebra and biology. 121 take algebra and history. 111 take language arts and social studies. 90 take language arts and biology. 80 take language arts and history. 60 take social studies and biology. 70 take social studies and history. 60 take biology and history. 50 take algebra, language arts, and social studies. 50 take algebra, language arts, and biology. 50 take algebra, language arts, and history. 50 take algebra, social studies, and biology. 50 take algebra, social studies, and history. 50 take algebra, biology, and history. 50 take language arts, social studies, and biology. 50 take language arts, social studies, and history. 50 take language arts, biology, and history. 50 take social studies, biology, and history. 20 take algebra, language arts, social studies, and biology. 15 take algebra, language arts, social studies, and history. 15 take algebra, language arts, biology, and history. 10 take algebra, social studies, biology, and history. 10 take language arts, social studies, biology, and history. 5 take all five. None take none.

How many people are in my school?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
472
}
Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

$[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label("R_1",(3/2,1/2)); label("S_3",(4,1)); label("S_2",(5/2,3/2)); label("S_1",(1,2)); label("R_2",(7/2,5/2)); [/asy]$



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{651
}
小明家有20个鸡蛋，并且养了一只鸡，这只鸡每天下一个蛋，如果小明每天要吃2个蛋，那么小明家的鸡蛋可以够小明连续吃几天？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\sum\limits_{i=1}^{216} x_i=1$ and $\sum\limits_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum\limits_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$. The maximum possible value of $x_2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{863
}
In triangle \( KIA \), a point \( V \) is marked on side \( KI \) such that \( KI = VA \). A point \( X \) is marked inside the triangle such that the angle \( XKI \) is half of the angle \( AVI \), and the angle \( XIK \) is half of the angle \( KVA \). Let \( O \) be the point of intersection of the line \( AX \) and the side \( KI \). Determine if \( KO = VI \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{KO = VI}
Determine whether there exists a natural number \( M \) such that for all \( c \geq M \), there is no prime \( p \) satisfying \( p^c \mid (p-1)! + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
$$
\frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16} + \frac{33}{32} + \frac{65}{64} - 7 =
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$-\frac{1}{64}$}
Given the joint distribution of $X_1$ and $X_2$ with the density function $h(x_1, x_2) = 8x_1x_2$ for $0 < X_1 < X_2 < 1$, find the joint distribution of $Y_1 = \frac{X_1}{X_2}$ and $Y_2 = X_2$. Then, determine the marginal distribution of $Y_2$. Provide your answer in terms of the probability density function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4y_2^3}
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4$}
If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, $\cdots$, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
15
}
In the binary expansion of $\dfrac{2^{2007}-1}{2^{225}-1}$, how many of the first $10,000$ digits to the right of the radix point are $0$'s?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810}
Suppose that $\sin a + \sin b = \sqrt{\frac{5}{3}}$ and $\cos a + \cos b = 1$. What is $\cos (a - b)$?




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{1}{3}$

}
Given that the Fourier transform of a function $f(x)$, defined over $(-\infty, \infty)$, is approximately $g(w) \approx 1-w^2$ for small $w$ ($w \ll 1$), but $g(w)$ over the full range of $w$ is not provided, can we calculate the value of \(\int_{-\infty}^\infty x^2f(x) \,dx\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
10. (20 points) Given real numbers $a_{0}, a_{1}, \cdots, a_{2015}$, $b_{0}, b_{1}, \cdots, b_{2015}$ satisfy
$$
\begin{array}{l}
a_{n}=\frac{1}{65} \sqrt{2 n+2}+a_{n-1}, \\
b_{n}=\frac{1}{1009} \sqrt{2 n+2}-b_{n-1},
\end{array}
$$

where $n=1,2, \cdots, 2015$.
If $a_{0}=b_{2015}$, and $b_{0}=a_{2015}$, find the value of the following expression
$$
\sum_{k=1}^{2015}\left(a_{k} b_{k-1}-a_{k-1} b_{k}\right) .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Given a square matrix $A \in \mathbb{K}^{n \times n}$, where $a$ columns and $b$ rows are chosen such that $a + b > n$, and the entry at the intersection of any chosen row and column is zero, is the determinant of such a matrix always zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the number field $L = \mathbb{Q}(\sqrt{-11})$, with the ring of integers $\mathcal{O}_L = \mathbb{Z}\left[\frac{1+\sqrt{-11}}{2}\right]$, determine if the norm of an element $x + y\sqrt{-11}$ is given by $x^2 + 11y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + 11y^2}
From the natural numbers 1 to 1239, 384 distinct numbers are selected such that the difference between any two of them is neither 4, nor 5, nor 9. Determine whether the number 625 is among the selected numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}
A regular triangular pyramid is intersected by a plane passing through a vertex of the base and the midpoints of two lateral edges. The intersecting plane is perpendicular to one of the lateral faces. Determine the ratio of the lateral surface area of the pyramid to the area of the base, specifying which lateral face the plane is perpendicular to.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
Does the Fourier series of any continuous periodic function with period 1 converge pointwise to the function? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Consider the function defined as follows: \[ h(x) = \begin{cases} 0, & x \in \mathbb{R} \setminus \mathbb{Q} \\ x^2, & x \in \mathbb{Q} \end{cases} \] Determine if this function is differentiable at the point where it is continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Maisy Airlines sees $n$ takeoffs per day. Find the minimum value of $n$ such that theremust exist two planes that take off within aminute of each other.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1441}