inclusionAI/Ring-lite-rl-data huggingface dataset:
Three investors have bought shares in a company that offers varying percentages of dividends. Investor A receives a 12.5% dividend, Investor B receives a 14% dividend, and Investor C receives a 15% dividend. The investors have purchased shares with face values of Rs. 50, Rs. 100, and Rs. 150, respectively. The returns on investment for Investor A, B, and C are 25%, 22% and 18%, respectively. Calculate the average buying price of the shares for all three investors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71.21}
12. For a regular tetrahedron \(ABCD\), construct its circumscribed sphere, where \(OC_1\) is the diameter of the sphere. Then the angle between the line \(AC_1\) and the plane \(BCD\) is \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan\frac{\sqrt{2}}{2}}
设 $a_{n}$ 是与 $\sqrt{\frac{n}{2}}$ 的差最小的整数， $b_{n}$ 是与 $\sqrt{2 n}$ 的差最小的整数，记 $\left\{\frac{1}{a_{n}}\right\}$ 的前 $n$ 项和为 $S_{n},\left\{\frac{1}{b_{n}}\right\}$的前 $n$ 项和为 $T_{n}$ ，则 $2 T_{100}-S_{100}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given sets $A=\{x\in \mathbb{N}\ |\ x-4 < 0\}$ and $B=\{0,1,3,4\}$, find the intersection $A\cap B$. Express your answer as a set of elements enclosed in curly brackets and separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0, 1, 3\}}
Jenny has 6 Pokemon cards, where 50% of them are rare. Orlando has 2 more cards than Jenny, and 40% of his cards are rare. Richard has three times as many cards as Orlando, with 25% of them being rare. Jenny receives 4 additional cards from her friend, with 2 of them being holographic and worth 15 points each, while the other 2 are first edition cards worth 8 points each. If a rare card has a value of 10, and a non-rare card has a value of 3, and taking into account the new cards Jenny received, what is the total value of all of their Pokemon cards combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{244}
## 3. Before the Competition

A group of cyclists decided to train on a small road before the competition. All of them are riding at a speed of 35 km/h. Then one of the cyclists suddenly breaks away from the group and, moving at a speed of 45 km/h, covers 10 km, after which he turns back and, without slowing down, rejoins the others (who, ignoring him, continue to move at the same speed).

How much time has passed from the moment the cyclist broke away from the group until he rejoined them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Let $a \star b=ab-2$. Compute the remainder when $(((579 \star 569) \star 559) \star \cdots \star 19) \star 9$ is divided by 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given the function f(x) = x^3, find the value(s) of x_0 such that f'(x_0) = 3. Express your answer using the ± symbol if there are multiple solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{35}}
Two individuals, A and B, participate in a mathematics test simultaneously. The test consists of $10$ multiple-choice questions, each with $4$ options. Correct answers are awarded $3$ points, while incorrect or unanswered questions receive $0$ points. Both A and B answered all the questions, and upon comparison, they only differed in their choice for one question. If A's final score is $27$ points, what is the set of all possible scores B could have received?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{24, 27, 30\}}
There are six empty slots corresponding to the digits of a six-digit number. Claire and William take turns rolling a standard six-sided die, with Claire going first. They alternate with each roll until they have each rolled three times. After a player rolls, they place the number from their die roll into a remaining empty slot of their choice. Claire wins if the resulting six-digit number is divisible by 6, and William wins otherwise. If both players play optimally, compute the probability that Claire wins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{43}{192}}
The domain of the function $y=\tan\left(\frac{\pi}{6}x+\frac{\pi}{3}\right)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{x\mid x\neq 1+6k, k\in \mathbb{Z}\}}
Martha has 56 butterflies in her collection. She has twice as many blue butterflies as yellow butterflies, three times as many red butterflies as yellow butterflies, and the rest of her butterflies are black. If Martha has 12 blue butterflies and 10 red butterflies, how many black butterflies does she have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
The diagram shows a large circular dartboard with four smaller shaded circles, each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle and are tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown such that it sticks to a point randomly chosen on the dartboard, then the probability that the dart sticks to a point in the shaded area is $\frac{m}{n}$, 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{31}
The polynomial $a x^{2} + b x + c$ has integer coefficients and two distinct roots that are greater than 0 but less than 1. How small can $|a|$ be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
12.313. The height of the cone is $H$, and the angle between the generatrix and the base plane is $\alpha$. The total surface area of this cone is divided in half by a plane perpendicular to its height. Find the distance from this plane to the base of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2H\sin^{2}\frac{\alpha}{4}}
For lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$ to be parallel, $a=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
The bookseller receives a discount from the publisher which, expressed as a percentage, is 1.5 times smaller than the profit percentage. What percentage of the purchase price is the discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33 \frac{1}{3} \%}
A pedestrian walks along a highway at a speed of 5 km/h. Buses travel along this highway in both directions at the same speed, meeting every 5 minutes. At 12 o'clock, the pedestrian noticed that the buses met near him and, continuing to walk, began counting the passing and overtaking buses. At 2 o'clock, the buses met again near him. It turned out that during this time, the pedestrian encountered four more buses than those overtaking him. Find the speed of the bus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30 \text{ km/h}}
For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$) with four colors. A coloring is called [i]multichromatic[/i] if whenever $a$, $b$ and $\gcd(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Positive integers a and b satisfy a * b = 2010. If a > b, find the smallest possible value of a - b. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
In a rectangle trapezoid, there are two circles. One of them, with a radius of 4, is inscribed in the trapezoid, and the other, with a radius of 1, touches two sides of the trapezoid and the first circle. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{196}{3}}
Find the gradient of the function $$g(X) = u^T X^T B X v$$ with respect to $X$, where $u$ and $v$ are vectors and $B$ and $X$ are matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{BXvu^T + B^TXuv^T}
Using each of the eight digits  $1,3,4,5,6,7,8$  and  $9$  exactly once, a three-digit number  $A$ , two two-digit numbers  $B$  and  $C$ ,  $B<C$ , and a one digit number  $D$  are formed. The numbers are such that  $A+D=B+C=143$ . In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
How many ways are there to arrange the array of positive integers such that:
- The number $2012$ is in the first position.
- The number $1$ is in the last position.

Note: Each number following $n$ must be a positive integer less than $\sqrt{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
5. In a $2008 \times 2008 \times 2008$ cube, if a line is drawn such that it passes through the centers of 2008 unit cubes, then the number of such lines that can be drawn is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12108244}
Two dice (each die has sides numbered from 1 to 6) are rolled sequentially and land on a horizontal surface, recording the numbers facing up as x and y. Let event A be “x+y is an even number”, and event B be “one of x or y is an even number, and x!= y”. Calculate the conditional probability P(B|A). Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
The cells of a $9 \times 9$ board are colored in a checkerboard pattern with black and white colors (the corner cells are white). What is the minimum number of rooks that must be placed on this board so that all the white cells are under attack by these rooks? (A cell is under attack by a rook if it lies in the same row or column as the rook.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A cyclist rides a bicycle 8 km at an average speed of 11 km/hr and again travels 11 km at an average speed of 8 km/hr. What is the average speed for the entire trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.04}
Let \( x \geqslant y > 0 \). If there exist real numbers \( a \) and \( b \) satisfying \( 0 \leqslant a \leqslant x \), \( 0 \leqslant b \leqslant y \), and \((x-a)^2 + (y-b)^2 = x^2 + b^2 = y^2 + a^2\), then what is the maximum value of \( \frac{x}{y} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2 \sqrt{3}}{3}\)}
In the idioms "虚有其表" (empty show), "表里如一" (consistent inside and out), "一见如故" (feel like old friends at first sight), and "故弄玄虚" (deliberately mystifying), each Chinese character represents one of eleven consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. It is also given that "表" > "一" > "故" > "如" > "虚", and the sum of the numbers represented by the four characters in each idiom is 21. What is the maximum number that "弄" can represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Positive integers $a$ and $b$ satisfy $a b=2010$. If $a>b$, what is the smallest possible value of $a-b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given the function $f\left(x\right)=\frac{1}{1+{x}^{2}}-\ln |x|$, if the inequality $f\left(-ax+\ln x+1\right)+f\left(ax-\ln x-1\right)\geqslant 2f\left(1\right)$ holds for $x\in \left[1,3\right]$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{1}{e}, \frac{2+\ln 3}{3}\right]}
A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members such that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be happy if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
The measure of angle $ACB$ is 45 degrees. If ray $CA$ is rotated 510 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. Determine the number of women in the luxury compartment if the train carries 300 passengers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Suppose that $a_1 = 1$, and for all $n \ge 2$, \[ a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1. \] Furthermore, suppose that \[ b_n = a_1 + a_2 + \ldots + a_n \] for all $n$. If \[ b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k \] for some $k$, find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
Given proposition $P$: The equation $\frac{x^2}{3+a} - \frac{y^2}{a-1} = 1$ represents a hyperbola, and proposition $Q$: The point $(2,a)$ is inside the circle $x^2 + (y-1)^2 = 8$. If $P \land Q$ is a false proposition, and $\neg Q$ is also a false proposition, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 < a \leqslant 1}
6. Variant 1 Sasha wrote the number 765476547654 on the board. He wants to erase several digits so that the remaining digits form the largest possible number divisible by 9. What is this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7654765464}
Given proposition $p$: The equation $\frac{x^2}{2} + \frac{y^2}{m-1} = 1$ represents an ellipse with foci on the $y$-axis, and proposition $q$: The equation in $x$, $x^2 - 2x + 4m = 0$, has no real roots. If $\neg p$ is true, and $p \lor q$ is true, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{1}{4},3\right]}
(Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \ldots$ in that order. You start at the square labeled $0$. You want to end up at the square labeled $3$. In how many ways can this be done in exactly $15$ moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Calculate the minimum value of $a$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Given that \( a \) and \( b \) are integers, and the equation \( x^{2} + a x + 2 b^{2} = 0 \) has two roots \( \sin \alpha \) and \( \cos \alpha \) where \( 0 \leqslant \alpha < 2 \pi \), then what are all possible values of \( a + b \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
规定抽球试验规则如下：盒子中初始装有白球和红球各一个，每次有放回地任取一个，连续取两次，将以上过程记为一轮.如果每一轮取到的两个球都是白球，则记该轮为成功，否则记为失败.在抽取过程中，如果某一轮成功，则停止；否则，在盒子中再放入一个红球，然后接着进行下一轮抽球，如此不断继续下去，直至成功.

$附：线性回归方程\hat{y}=\hat{b}x+\hat{a}中，\hat{b}=\frac{\sum \limits^{n}_{{i=1}}x_iy_i-n\overline{x} \cdot \overline{y}}{\sum \limits^{n}_{{i=1}}x^2_i-n{\overline{x}}^2}，\hat{a}=\overline{y}-\hat{b}\overline{x}；$

$参考数据：\sum \limits^{5}_{{i=1}}x^2_i=1.46，\overline{x}=0.46，\overline{x}^2=0.212 \left(\begin{matrix}\text{其中}x_i=\frac{1}{t_i},\overline{x}=\frac{1}{5}\sum \limits^{5}_{{i=1}}x_i\end{matrix}\right).$
$某人进行该抽球试验时，最多进行三轮，即使第三轮不成功，也停止抽球，记其进行抽球试验的轮次数为随机变量X，求X的数学期望；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{29}{12}$}
in business , a and c invested amounts in the ratio 2 : 1 , whereas the ratio between amounts invested by a and b was 2 : 3 , if rs 15,0000 was their profit , how much amount did b receive .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75,000}
A man rows at a speed of $2$ mph in still water. He sets out on a trip towards a spot $2$ miles downstream. He rows with the current until he is halfway there, then turns back and rows against the current for $15$ minutes. Then, he turns around again and rows with the current until he reaches his destination. The entire trip takes him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph, where $p$ and $q$ are relatively prime positive integers. Find $10p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
a typist uses a sheet measuring 20 cm by 30 cm lengthwise . if a margin of 2 cm is left on each side and a 3 cm margin on top and bottom , then percent of the page used for typing is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\%}
Let $n$ be the 200th smallest positive real solution to the equation $x-\frac{\pi}{2}=\tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{314}
If a non-negative integer \( m \) and the sum of its digits are both multiples of 6, then \( m \) is called a "Liuhe number." Find the number of Liuhe numbers less than 2012.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Find the maximum value of the positive real number \( A \) such that for any real numbers \( x, y, z \), the inequality
$$
x^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \geq 0
$$
holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
How many ordered pairs of integers \((x, y)\) satisfy the equation:
\[
8(x^3 + x^2y + xy^2 + y^3) = 15(x^2 + y^2 + xy + 1)
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given an angle equal to $\alpha$. A point $K$ is taken on its bisector; $P$ and $M$ are the projections of $K$ onto the sides of the angle. On the segment $PM$, a point $A$ is taken such that $KA = a$. A line passing through $A$ perpendicularly to $KA$ intersects the sides of the angle at points $B$ and $C$. Find the area of the triangle $BKC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{2} \cot \left( \frac{\alpha}{2} \right)}
Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then    
$$a_{k+1} = \frac{m + 18}{n+19}.$$
Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Evaluate the limit using L'Hopital's rule: $$\lim_{w \to 0} \frac{(1+w^2)\ln\left(\frac{1+w}{1-w}\right)- 2w}{w^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the range of values for the real number $k$ such that the line $y=k(x-2)+4$ intersects the curve $y=\sqrt{4-x^2}$ at some point(s).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \in \left[\frac{3}{4}, +\infty\right)}
In the arithmetic sequence $\{a_n\}$, $a_3=0$. If $a_k$ is the geometric mean of $a_6$ and $a_{k+6}$, then $k=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A circular flower bed has a circumference of 314 meters. A 2-meter wide path is built around this flower bed. What is the area of this path in square meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{640.56}
For the expression \(\left[\left(\frac{\sqrt{3}+i}{2}\right)^{8}+1\right]^{n}\), where \(n\) takes on the values from 1 to 100, calculate the number of distinct values the expression can take. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a cyclic quadrilateral \(ABCD\) where \(AB = BC = AD + CD\) and \(\angle BAD = \alpha\), with \(AC = d\). Find the area of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} d^2 \sin(\alpha)}
Given the hyperbola $x^2-y^2+1=0$ and the parabola $y^2=(k-1)x$, find the range of $k$ such that the two curves have at most two points in common. Express your answer as an interval, including the endpoints if they are included in the range.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1,3]}
On the sides $BC$ and $CD$ of the square $ABCD$ with side length $1$, points $E$ and $F$ are chosen such that $\angle EAB = 20^\circ$ and $\angle EAF = 45^\circ$. Calculate the distance from point $A$ to the line $EF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
已知数列 $\left\{a_{n}\right\}:$
$a_{n}=2^{n}+3^{n}+6^{n}+1\left(n \in \mathbf{Z}_{+}\right)$.
是否存在整数 $k \geqslant 2$ ，使得 $k$ 与数列 $\left\{a_{n}\right\}$中的所有数均互素？若存在，找出最小的整数 $k$ ；若不存在，说明理由.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
How many consecutive tickets must be purchased at a bus cashier to be sure to get a lucky one? A ticket is called lucky if the sum of its first three digits equals the sum of its last three digits. The number of tickets at the cashier is unlimited.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
Zhang Hua, Li Liang, and Wang Min each sent out x, y, z New Year's cards, respectively. If it is known that the least common multiple of x, y, z is 60, the greatest common divisor of x and y is 4, and the greatest common divisor of y and z is 3, then how many New Year's cards did Zhang Hua send out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let  $p(x)$  be a polynomial of degree strictly less than  $100$  and such that it does not have  $(x^3-x)$  as a factor. If   $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$   for some polynomials  $f(x)$  and  $g(x)$  then find the smallest possible degree of  $f(x)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Abbot writes the letter $A$ on the board. Every minute, he replaces every occurrence of $A$ with $A B$ and every occurrence of $B$ with $B A$, hence creating a string that is twice as long. After 10 minutes, there are $2^{10}=1024$ letters on the board. How many adjacent pairs are the same letter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}