inclusionAI/Ring-lite-rl-data huggingface dataset:
Three different cars (A, B, and C) have different gasoline consumptions while traveling various distances. 

Car A uses 12 gallons of gasoline to travel 180 miles.
Car B uses 15 gallons of gasoline to travel 225 miles.
Car C uses 18 gallons of gasoline to travel 270 miles.

In order for all three cars to travel their respective distances using 10 gallons of gasoline each, by how many miles per gallon must each car's gas mileage be increased on average?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Emma is a straight-A student who loves spending time in the library. She has 5 favorite books that she reads repeatedly. One day, she observes her friend Alex, a social butterfly, confidently talking to a group of 12 students about their favorite books. Inspired by Alex’s confidence, Emma decides to step out of her comfort zone and join the conversation. To prepare, she decides to read 3 new books and share her thoughts on them. Each new book takes her 2 days to read, and she spends 1 day summarizing her thoughts on each book. If Emma starts reading on a Monday, on which day of the week will she be ready to join the conversation with Alex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Tuesday}
A 5x5 square is divided into 25 cells, all initially white. Neighbouring cells share a common edge. On each move, the colors of two neighbouring cells are changed to the opposite color (white cells become black and black ones become white). What is the minimum number of moves required to obtain a chess-like colouring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that \( b\left[\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{1999 \times 2001}\right]=2 \times\left[\frac{1^{2}}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{1000^{2}}{1999 \times 2001}\right] \), find the value of \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
We say a $2023$-tuple of nonnegative integers $\left(a_{1}, a_{2}, \ldots a_{2023}\right)$ is sweet if the following conditions hold:  
- $a_{1}+a_{2}+\ldots+a_{2023}=2023$,  
- $\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leqslant 1$.  

Determine the greatest positive integer $L$ such that  
$$  
a_{1}+2 a_{2}+\ldots+2023 a_{2023} \geqslant L  
$$  
holds for every sweet 2023 -tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22228}
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{382}
Find the number of real solutions to the equation \( \frac{x}{100} = \sin x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Given integers $m, n \geqslant 2$. Color each cell of an $m$-row by $n$-column grid $S$ with either red or blue such that the following condition is satisfied: For any two cells in the same row, if both are colored red, then one column must have all cells colored red, and the other column must have at least one cell colored blue. Find the number of different coloring methods. Please provide the value when $m = 2$ and $n = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Evaluate the integral

$$\dfrac{1}{\pi} \int e^{itn}\cos^2\left(\dfrac{1}{2}t \right)dt$$

over the interval $(-\pi, \pi)$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{n+1}}
Calculate the definite integral:

$$
\int_{2 \operatorname{arctg} 2}^{2 \operatorname{arctg} 3} \frac{d x}{\cos x(1-\cos x)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6} + \ln 2 - \ln 3}
Let \( A = \{1, 2, 3, \cdots, 4n+2\} \) and \( M = \{2n+1, 4n+3, 6n+5\} \). A subset \( B \) of \( A \) is called an \( M \)-free set if the sum of any two numbers in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \( (A_1, A_2) \) is called an \( M \)-partition of \( A \). Determine the number of all \( M \)-partitions of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n+1}}
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条, 则由这 3030 条线段能构成不全等的三角形的个数为 $\qquad$ (用数字作答).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510555}
A circle passes through the vertices \( A \) and \( B \) of triangle \( ABC \) and intersects its sides \( AC \) and \( BC \) at points \( Q \) and \( N \) respectively. Given \( AQ : QC = 5 : 2 \) and \( CN : NB = 5 : 2 \), find \( AB \) if \( QN = 5\sqrt{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\sqrt{5}}
Determine the number of duplication formulas that exist for the Mordell curve family $Y^2-X^3=c$. Given that Bachet and Réalis have provided two such formulas, how many additional duplication formulas are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On Lineland, there are 2018 bus stations numbered from 1 to 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers). 

Define any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u < j_v$ for all $u,v \in \{1,2\}$ as a good group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
Construct a curve of constant width \( h \), composed of five, seven, or any odd number of arcs of circles with radius \( h \). What is the length of each of these curves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi h}
Five numbers, $a_1$, $a_2$, $a_3$, $a_4$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 50\}$. Four other numbers, $b_1$, $b_2$, $b_3$, $b_4$, are then drawn randomly and without replacement from the remaining set of 46 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3 \times a_4$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3 \times b_4$, with the sides of the brick parallel to the sides of the box. Compute $p$ in lowest terms and determine the sum of the numerator and denominator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
A deck consists of 25 cards: 10 black cards, and 5 sets of 3 identical cards each in green, red, yellow, blue, and pink. How many different hands of 10 cards can be drawn from this deck?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99404}
Given the vectors $\overrightarrow{a} \cdot (\overrightarrow{a}+2\overrightarrow{b})=0$ and the magnitudes $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
There is a store that sells each of the 26 letters. All letters of the same type cost the same amount (i.e. any 'a' costs the same as any other 'a'), but different letters may or may not cost different amounts. For example, the cost of spelling "trade" is the same as the cost of spelling "tread," even though the cost of using a 't' may be different from the cost of an 'r.' If the letters to spell out 1 cost \$ $1001$, the letters to spell out 2 cost \$ $1010$, and the letters to spell out 11 cost \$ $2015$, how much do the letters to spell out 12 cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
如果 $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right), \theta \in[0,2 \pi)$, 那么 $\theta$ 的取值范围是 \qquad. 请给出 m + n + k + j 的和，其中原始答案格式为$\left(\frac{m\pi}{n}, \frac{k \pi}{j}\right)$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given two random variables $X$ and $Y$ with joint distribution function $f_{X,Y}(x,y) = \frac{8xy}{3}$ for $0 < x < 1$ and $x < y < 2x$, determine the covariance of $X$ and $Y$. Consider how the inequality constraint $x < y < 2x$ should be applied when integrating over $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{28}{675}}
Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and:
\[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]
What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15230}
Let $f$ be continuous for all $x$. Find the value of the double integral $$\int_0^1\int_x^{1-x}f(t)dt\ dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A [i]pseudo-ultrafilter[/i] $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then $\operatorname{lcm} (a,b) \notin U$.
\end{itemize}
How many such pseudo-ultrafilters are there?

[i]Proposed by Evan Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let  $S=\{1,2,\ldots ,98\}$ . Find the least natural number  $n$  such that we can pick out  $10$  numbers in any  $n$ -element subset of  $S$  satisfying the following condition: no matter how we equally divide the  $10$  numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A wire of length $4$ meters is bent into a square. A bug starts crawling from one corner of the square to the adjacent corner at time $t = 0$, maintaining a constant speed. It continues along the remaining sides until it returns to its starting point at $t = 1$ second. Let $A$ be the starting point of the bug, $O$ the center of the square, and $B$ its position at time $t$. Calculate the rate at which the angle $\angle AOB$ is changing at $t = 0.3$ seconds, with angle measures in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{100}{17}}
Given that \(f(x) = x^3 + x + 3\) is irreducible over \(\mathbb{Q}[x]\) and has exactly one real root \(\theta\), determine the degree of the extension \(\mathbb{Q}(\theta, \phi')\) over \(\mathbb{Q}(\phi)\), where \(\phi\) and \(\phi'\) are the other two roots of \(f(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
At the parade, the drummers stand in an even square formation of 50 rows by 50 drummers. The drummers are dressed in either blue or red costumes. What is the largest number of drummers that can be dressed in blue costumes such that each drummer dressed in blue can only see drummers in red costumes? Consider the drummers to be looking in all directions (360 degrees) and as point-like.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}
Determine the subspace of polynomials of degree at most 2, denoted as \( P^2 \), that is orthogonal to the polynomial \( y(x) = x \) with respect to the inner product \( \langle f, g \rangle = \int_{-1}^{1} f(x)g(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{ a x^2 + c \mid a, c \in \mathbb{R} \}}
For how many ordered triples $(x, y, z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1 + i)^y, z\}$, where $i = \sqrt{-1}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$, \[a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.\] What is $|a_{2009}|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$0$}
Let \( n > 1 \) be an odd positive integer and \( A = {\left( {a}_{ij}\right) }_{i,j = 1..n} \) be the \( n \times  n \)
 matrix with
 \( {a}_{ij} = \left\{  \begin{array}{ll} 2 & \text{ if }i = j \\  1 & \text{ if }i - j \equiv   \pm  2\;\left( {\;\operatorname{mod}\;n}\right) . \\  0 & \text{ otherwise } \end{array}\right. \)
 Find det \( A \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4

}
Determine the maximum size of a set \( S \) of irrational real numbers such that among any subset of five numbers in \( S \), there exist two numbers whose sum is irrational.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A person's age at the time of their death was one 31st of their birth year. How old was this person in 1930?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
As shown in the figure, in the triangular prism \(P-ABC\), \(PA \perp \) plane \(ABC\), and \(\angle ABC = 120^\circ\), with \(PA = 4\). If the radius of the circumscribed sphere of the triangular prism \(P-ABC\) is \(2\sqrt{2}\), then the tangent of the angle between line \(PC\) and plane \(ABC\) is \(\quad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{3}}{3}}
If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Avery puts 25 shirts in the donation box. He adds eight times as many pants as shirts in the box. He throws in one-third as many shorts as pants in the box. In total, how many pieces of clothes is Avery donating?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{292}
I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
If a real number $a$ satisfies $a^3 < a < a^2$, then the solution set of the inequality $x + a > 1 - ax$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x < \frac{1 - a}{1 + a}}
A waiter at 'The Gourmet Kitchen' restaurant had 70 customers to wait on during breakfast. Among these, 20 customers came in early at 8 am while the remaining arrived later. Then during the lunch rush he served another 100 customers, 60 of whom were priority customers. In the evening, he served only 45 customers, half of whom were regulars.

If 30% of the early morning customers, 60% of the priority customers, 50% of the regular customers and 25% of all remaining customers did not leave him a tip, how many customers in total left him a tip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134}
Find the integers \( n \) such that 5 divides \( 3n - 2 \) and 7 divides \( 2n + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35k + 24}
If the vector $\lambda \overrightarrow{e_{1}}- \overrightarrow{e_{2}}$ is collinear with $\overrightarrow{e_{1}}-\lambda \overrightarrow{e_{2}}$, where $\overrightarrow{e_{1}}, \overrightarrow{e_{2}}$ are non-collinear unit vectors, then the value of the real number $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm1}
Farmer James invents a new currency, such that for every positive integer $n \leq 6$, there exists an $n$-coin worth $n$ ! cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be nice if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Given the positive real numbers \(x\) and \(y\) such that

\[ x^5 + 5x^3y + 5x^2y^2 + 5xy^3 + y^5 = 1, \]

what positive values can \(x + y\) take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the set $P=\{x|x^2-8x-20\leqslant 0\}$ and non-empty set $S=\{x|1-m\leqslant x\leqslant 1+m\}$, if $x\notin P$ is a necessary but not sufficient condition for $x\notin S$, find the range of values for the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[9,+\infty)}
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_1 = 1$, and $S_6 = 4S_3$, find the value of $a_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the smallest positive integer that ends in 3 and is divisible by 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \, \mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both meeting the cyclist again. 

How fast were the buses traveling?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46 \mathrm{~km} / \text{hr}}
Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{90\sqrt{261}}{261}}
Find the third derivative of the inverse function $g(x)$ of $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$ at $x=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A group of fifty people went to a theater performance consisting of twenty-five adults, fifteen seniors, and ten children. Adult tickets cost $15 each, senior tickets have a 25% discount off the adult price, and the total bill for the entire group was $600. If x dollars is the price of one child's ticket, how much does one child's ticket cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$5.63}
The graph of the functions $y=-3x$ and $y=kx+1$ intersect at point $A(m,4)$. If $m=4$, then the solution set for the inequality $kx+b \lt -3x$ in terms of $x$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \lt -\frac{4}{3}}
Compute the number of positive integers that divide at least two of the integers in the set $\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
For the final act, the circus brought out dancing animals wearing crowns. There are two groups of animals performing. The first group of animals is wearing crowns made with 7 different colored feathers, while the second group of animals is wearing crowns with 5 additional feathers, which are gold. If there are 934 animals in the first group and 425 animals in the second group, how many feathers are needed in total to create crowns for all the animals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11638}
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 95 }
Using the digits 1, 2, 3, 4, and 5 without repetition, find the number of unique five-digit "wave numbers" that can be formed, where the tens and thousands digits are greater than the digits immediately adjacent to them. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Kaleen filled a bucket with 4 quarts of water. A few minutes later, she realized only 0.33 gallons of water remained. _____ water had leaked out of the bucket.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.68}
Call a positive integer \( N \geq 2 \) "special" if for every \( k \) such that \( 2 \leq k \leq N \), \( N \) can be expressed as a sum of \( k \) positive integers that are relatively prime to \( N \) (although not necessarily relatively prime to each other). How many special integers are there less than 100?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
If there exist a real number $a$ and a positive integer $n$, such that $f\left(x\right)=\cos 2x-a\sin x$ has exactly $2022$ zeros in the interval $\left(0,n\pi \right)$, then the total number of positive integers $n$ that satisfy this condition is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given 1987 sets, each containing exactly 45 elements, and the union of any two sets contains exactly 89 elements, calculate the number of elements in the union of these 1987 sets. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87429}
Given that \( I \) is the incenter of \( \triangle ABC \), and the following equation holds:
\[
3 \overrightarrow{IA} + 4 \overrightarrow{IB} + 5 \overrightarrow{IC} = \mathbf{0}
\]

Determine the measure of \( \angle C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
(10 points) A swimming pool sells student swim cards for 240 yuan each. The usage rules are as follows: no name registration, each card allows only one person per use, and each card can be used only once a day. A class consists of 48 students, and the teacher plans to organize a swimming trip. Besides purchasing a number of swim cards, they need to rent a bus for each trip. The cost of renting a bus is 40 yuan each time, regardless of the number of passengers. If each student is to swim 8 times, what is the minimum amount of money each student should contribute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80 \text{ yuan}}
Define an odd function $f(x)$ on $\mathbb{R}$. When $x \in (-\infty, 0)$, $f(x) + xf'(x) < 0$ always holds. Let $a = 3f(3)$, $b = (\log_{\pi}e)f(\log_{\pi}e)$, and $c = -2f(-2)$. Determine the relationship among $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b < c < a}