inclusionAI/Ring-lite-rl-data huggingface dataset:
Given a sequence $\{a_{n}\}$ such that $a_{1}+3a_{2}+\ldots +\left(2n-1\right)a_{n}=\frac{1}{3}-\frac{1}{{2n+3}}$, $n\in N^{*}$, then $a_{1}+a_{2}+\ldots +a_{n}=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}-\frac{1}{2(2n+1)(2n+3)}}
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), find the value of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Given that there are \( C \) integers that satisfy the equation \( |x-2| + |x+1| = B \), find the value of \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the distance covered by a man walking uphill for 25 minutes at a speed of 8 km/hr and then downhill for 11 minutes at a speed of 12 km/hr. The uphill slope is at an angle of 15 degrees, and the downhill slope is at an angle of 10 degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.53}
Given that α and β are acute angles, and $\tan \alpha = \frac{2}{t}$, $\tan \beta = \frac{t}{15}$. When $10\tan \alpha + 3\tan \beta$ reaches its minimum value, the value of $\alpha + \beta$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1847}
What is the least positive integer m such that $m\cdot 2! \cdot 3! \cdot 4! \cdot 5! \ldots 16!$ is a perfect square? Listing out all possibilities of m is not allowed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$70$}
Two unit-radius spheres intersect in such a way that the resulting 3 parts have equal volume. How far apart are the centers of the spheres?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\cos\frac{4\pi}{9}\approx0.695}
Louis is making himself a velvet suit for a high-end gala. The velvet fabric he chose costs $24 per yard. He purchases a pattern for $15, two spools of silver thread for $3 each, lining fabric at an unknown price per yard, a set of 12 stylish buttons for $14, and 3 yards of exquisite trim for $19 per yard. Additionally, he has a 10% discount coupon specific to fabrics, which he applies only to the velvet and lining fabric. He spent a total of $310.50 after applying the discount. If Louis bought 5 yards of velvet fabric and 4 yards of lining fabric, what is the cost per yard of the lining fabric?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30.69}
For a sequence  $s = (s_1, s_2, . . . , s_n)$ , define  $$ F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2. $$  Consider the sequence  $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$ . Let  $R$  be the sum of all  $F(m)$  for all non-empty subsequences  $m$  of  $S$ . Find the remainder when  $R$  is divided by  $1000$ .

Note: A subsequence is a sequence that can be obtained from another sequence by deleting some non-negative number of values without changing the order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Compute the surface integral $\iint_S F\cdot dS$ where $F=\langle x,y,z^2\rangle$ and $S$ is the unit sphere centered at the origin. The surface $S$ is parameterized by $r(\phi,\theta)=\langle \sin\theta \cos\phi,\sin\theta \sin\phi, \cos\theta\rangle$ for $0\leq\theta\leq\pi$ and $0\leq\phi\leq2\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{8}{3}\pi\)}
Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$, and the graphs of the functions $y = \sqrt{x^3 + 1}$ and $y = -\sqrt[3]{x^2 + 2x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the integral $\int \operatorname{cth}^{2} x \, dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x - \operatorname{cth} x + C}
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$.  Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1002}
In a group of 100 cars, some cars do not have air conditioning. If at least 53 cars have racing stripes, the greatest number of cars that could have air conditioning but not racing stripes is 47. How many cars do not have air conditioning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
$S$ is the sum of the first 9 terms of an increasing arithmetic sequence $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{5} a_{18} > S - 4$ and $a_{13} a_{10} < S + 60$. Determine all possible values of $a_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10, -9, -8, -7, -5, -4, -3, -2}
Find the exact value of $t$ such that $\frac{d}{dt}\int_{t}^{t+1}\sin e^x \ dx=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln\left(\frac{2\pi}{e-1}\right)}
A train of a certain length passes a man, running at 6 kmph in the direction opposite to that of the train, in 6 seconds. The speed of the train is 77.993280537557 kmph. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{139.99}
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$ , 
$(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots$
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{428}
In convex quadrilateral \(ABCD\) with \(AB = 11\) and \(CD = 13\), there is a point \(P\) for which \(\triangle ADP\) and \(\triangle BCP\) are congruent equilateral triangles. Compute the side length of these triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $R$  be the set of points  $(x, y)$  such that  $x$  and  $y$  are positive,  $x + y$  is at most 2013, and
\[
  \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set  $R$ .  Recall that  $\lfloor a \rfloor$  is the greatest integer that is less than or equal to  $a$ , and  $\lceil a \rceil$  is the least integer that is greater than or equal to  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006.5}
A certain percentage of antifreeze solution and 54 % antifreeze solution were combined to give 39 gallons of 58 % antifreeze solution. The amount of the first solution used was 26 gallons. What was the percentage of the first antifreeze solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6}
The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Given that |a| = 7, |b| = 5, and a < b, calculate all possible values of a - b. Express your answer as a list of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12, -2}
Let  $ABC$  be a triangle with  $\angle BAC=40^\circ $ ,  $O$  be the center of its circumscribed circle and  $G$  is its centroid. Point  $D$  of line  $BC$  is such that  $CD=AC$  and  $C$  is between  $B$  and  $D$ . If  $AD\parallel OG$ , find  $\angle ACB$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^
}
Suppose \(a\), \(b\), and \(c\) are three distinct numbers randomly chosen from \(1, 2, \cdots, 5\), and \(ab + c\) is odd. Find the number of possible values of \(ab + c\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1875}
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{329}
Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
128 students represent x percent of the boys at a school. If the boys at the school make up 50% of the total school population of x students, what is x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
$A, B, G$ are three points on a circle with $\angle A G B=48^\circ$. The chord $AB$ is trisected by points $C$ and $D$ with $C$ closer to $A$. The minor arc $AB$ is trisected by points $E$ and $F$ with $E$ closer to $A$. The lines $EC$ and $FD$ meet at $H$. If $\angle A H B = x^\circ$, find $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Evita wants to write the numbers 1 to 8 in the boxes of the grid shown, so that the sums of the numbers in the boxes in each row are equal and the sums of the numbers in the boxes in each column are equal. She has already written numbers 3, 4, and 8, as shown. What number should she write in the shaded box? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Define a sequence of functions recursively by  $f_1(x) = |x-1|$  and  $f_n(x)=f_{n-1}(|x-n|)$  for integers  $n > 1$ . Find the least value of  $n$  such that the sum of the zeros of  $f_n$  exceeds  $500{,}000$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Two trains move in the same direction at some speeds. The slower train moves at 32 kmph. A man in the slower train observes that 15 seconds elapse before the faster train completely passes by him. The length of the faster train is 75.006 meters. What is the speed of the faster train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50.00144}
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$  Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{492}
A group of 7 fruit baskets contains 9 apples, 15 oranges, 14 bananas, 12 grapes, and 18 strawberries in the first three baskets. The fourth basket has 2 fewer of each fruit than the first basket, the fifth basket has 3 more apples and 5 fewer oranges than the first basket, the sixth basket contains twice as many bananas and half as many grapes as the third basket, and the seventh basket holds 1.5 times the number of strawberries in the second basket and 4 more oranges than the first basket. How many fruits are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{485}
In a survey of parents and caregivers, exactly 5/6 of the mothers, 3/4 of the fathers, and 1/2 of the grandparents held full-time jobs. Part-time jobs were held by 1/6 of the mothers, 1/8 of the fathers, and 1/4 of the grandparents. The remaining parents and grandparents in the survey were either unemployed or not in the labor force. If 55 percent of the respondents surveyed were women, 35 percent were men, and 10 percent were grandparents, what percent of the surveyed individuals did not hold a full-time or part-time job?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.875\%}
Given an ellipse $C$: $\frac{x^2}{8}+\frac{y^2}{4}=1$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. A line $l$ passing through the right focus $F_{2}$ intersects $C$ at points $A$ and $B$. If the center of the circle passing through points $A$, $B$, and $(-\sqrt{7},0)$ lies on the $y$-axis, then the slope of line $l$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\frac{\sqrt{2}}{2}}
Misha drew a triangle with a perimeter of 11 and cut it into parts with three straight cuts parallel to the sides, as shown in the figure. The perimeters of the three shaded figures (trapezoids) turned out to be 5, 7, and 9. Find the perimeter of the small triangle that resulted from the cuts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
设函数

$$f(\:x\:)=\sqrt{2x^2\:+2x\:+41}\:-\sqrt{2x^2\:+4x\:+4}\:(\:x\:\in\:\mathbf{R}\:)\:.$$
则$f(x)$的最大值为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is the minimum total number of cactus species that can be in all collections?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Jane started babysitting when she was 18 years old. Whenever she babysat for a child, that child was no more than half her age at the time. She charged different rates based on the child's age: $5 per hour for children under 2 years old, $7 per hour for children between 2 and 5 years old, and $8 per hour for children older than 5 years. Jane is currently 34 years old. She stopped babysitting for one year when she was 24, then resumed for three years before stopping permanently 6 years ago. During her babysitting career, Jane babysat four children with different ages: Child A was half her age when she started babysitting, Child B was 2 years younger than Child A, Child C was 3 years older than Child B, and Child D was the same age as Child C. Each child was babysat for a different number of hours: 50 hours for Child A, 90 hours for Child B, 130 hours for Child C, and 70 hours for Child D. Calculate the total amount of money Jane earned from babysitting all four children, considering their ages and varying babysitting rates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2720$}
if 5 a = 3125 , then the value of 5 ( a - 3 ) is :
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A new mixed-use development has 60 floors, with varying types of residential floors and commercial spaces. The residential floors are divided into three different categories: regular floors, luxury floors, and penthouse floors. There are 25 regular floors, 20 luxury floors, and 10 penthouse floors. The remaining 5 floors are dedicated to commercial spaces, amenities, and uncounted residential units.

On the regular residential floors, there are 14 units on odd-numbered floors and 12 units on even-numbered floors. Luxury floors have a range of 6 to 10 units per floor, with the number of units decreasing as the floor number increases. Penthouse floors have only 2 units each.

There are 3 floors of commercial units with 5 shops each and 2 separate floors for amenities and uncounted residential units. The floor with amenities has 4 uncounted residential units, and the remaining floor has 6 uncounted residential units. 

Calculate the total number of residential and commercial units in the entire mixed-use development, considering the varying number of units per floor type and the uncounted residential units on the specified floors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{531}
If we increase one leg of a right triangle by 5 and decrease the other leg by 5, the area of the triangle increases by 5. How does the area of the square drawn on the hypotenuse change in the meantime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Each cell of an infinite table (infinite in all directions) is colored with one of  $n$  given colors. All six cells of any  $2\times 3$  (or  $3 \times 2$ ) rectangle have different colors. Find the smallest possible value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $m$ be a real number where $m > 0$. If for any $x \in (1, +\infty)$, the inequality $2e^{2mx} - \frac{ln x}{m} ≥ 0$ always holds, then find the minimum value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2e}}
If you increase a number  $X$  by  $20\%,$  you get  $Y.$  By what percent must you decrease  $Y$  to get  $X?$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\frac{2}{3}\%}
A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3003}
Find the greatest common divisor in $\mathbb{Z}[i]$ of $11+7i$ and $18-i$ using the Euclidean algorithm or ideal membership tests.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
将等差数列 $1,5,9,13, \ldots, 2017$ 排成一个大数 $15913 \ldots 2017$, 则该数被 9 除的余数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( A = \{1,2,3,4,5\} \). How many functions \( f: \mathscr{P}(A) \backslash \{\varnothing\} \rightarrow A \) are there that satisfy \( f(B) \in B \) and \( f(B \cup C) \in \{f(B), f(C)\} \) for all non-empty subsets \( B \) and \( C \) of \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
The natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2023\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
An  $m \times n$  chessboard where  $m \le n$  has several black squares such that no two rows have the same pattern. Determine the largest integer  $k$  such that we can always color  $k$  columns red while still no two rows have the same pattern.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n - m + 1 }
Suppose \(S\) cans of soda can be purchased from a vending machine for \(Q\) quarters. Find the expression that describes the number of cans of soda that can be purchased for \(D\) dollars, where \(1\) dollar is worth \(4\) quarters. The original answer is in \(\frac{kDS}{mQ}\) format, please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order). Express your answer as a ratio of three numbers, separated by colons (e.g., a:b:c).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3:4}
Let  $ABC$  be a triangle (right in  $B$ ) inscribed in a semi-circumference of diameter  $AC=10$ . Determine the distance of the vertice  $B$  to the side  $AC$  if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/2}
Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
On the lateral side $CD$ of trapezoid $ABCD (AD \parallel BC)$, a point $M$ is marked. From vertex $A$, a perpendicular $AH$ is dropped to the segment $BM$. It is found that $AD = HD$. Find the length of segment $AD$ if it is known that $BC = 16$, $CM = 8$, $MD = 9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
If  $x$  is a real number, let  $\lfloor x \rfloor$  be the greatest integer that is less than or equal to  $x$ .  If  $n$  is a positive integer, let  $S(n)$  be defined by
\[
  S(n) 
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor} 
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.)  How many integers  $n$  from 1 to 2011 (inclusive) satisfy  $S(S(n)) = n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
As shown in the figure, \( C \) and \( D \) are two points on segment \( AB \), and \( M \) and \( N \) are the midpoints of \( AC \) and \( DB \) respectively. If \( AB = 10 \) and \( CD = 2 \), then find the length of \( MN \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
## Problem Statement

Calculate the definite integral:

$$
\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\cos x \, dx}{1+\sin x-\cos x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}\cdot\ln2-\frac{\pi}{12}}