inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1889}
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, and the results are a, b, c, d. What is the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} is equal to 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343/1296}
Define
\[A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and
\[B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dotsb,\]which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3.

Determine $\frac{A}{B}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A and B play a number-changing game on a $5 \times 5$ grid: A starts and both take turns filling empty spaces, with A filling each space with the number 1 and B filling each space with the number 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the maximum sum among these squares is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve? 

(The problem is from the 35th IMO selection test)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2n)f(2n+1)=9f(n)^{2}+3f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2215}
Jack and Jill run \(10\) kilometers. They start at the same point, run \(5\) kilometers up a hill, and return to the starting point by the same route. Jack has a \(10\) minute head start and runs at the rate of \(15\) km/hr uphill and \(20\) km/hr downhill. Jill runs \(16\) km/hr uphill and \(22\) km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Evaluate the limit: $$L=\lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find all solutions for the equation \( \tan(\alpha) = \cos(\alpha + 33.44^\circ) \) within the interval \( 0 < \alpha < 90^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26.56^\circ}
Find the value of \(\theta_0\) such that the distance from the origin to the point \(P(x(\theta_0), y(\theta_0), z(\theta_0))\), where \(x(\theta) = 5\cos \theta\), \(y(\theta) = 5\sin \theta\), and \(z(\theta) = \theta\), is equal to the arclength from the origin to that point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Marge planted 23 seeds in her garden. Five of the seeds never grew into plants. A third of the remaining seeds grew, but the plants were eaten by squirrels and rabbits. A third of the number of uneaten plants were strangled by weeds. Marge pulled some weeds, but liked the flowers on one weed and let the plant grow as part of her garden. Marge ended up with 9 plants. How many weeds did Marge pull?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. It is given that $\angle BAF = 18^\circ$. The lines $EF$ and $GH$ intersect at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
In a modern art museum, visitors who watched a certain Picasso painting were asked to fill in a short questionnaire. According to the results of the survey, all 110 visitors who did not enjoy the painting also did not feel they had understood the painting, and the number of visitors who enjoyed the painting was equal to the number of visitors who felt they had understood the painting. If 440 visitors answered the questionnaire, what fraction of the visitors both enjoyed the painting and felt they had understood the painting?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/8}
At a large round table, 90 people are feasting: 40 barons, 30 counts, and 20 marquises, all facing the center of the table. At a signal, exactly those who have both neighbors (left and right) with the same title should stand up. What is the maximum number of people who can stand up? 

For example, in order for a count to stand up, both of their neighbors must have the same title, though not necessarily the title of count.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
The average of some numbers is 5 more than the average of 10, 60, and 35. If the other numbers are 20 and 60, what is the second number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A burger at Ricky C's weighs $120$ grams, of which $30$ grams are filler. What percent of the burger is not filler?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$75\%$}
In a 5x6 seating arrangement for 30 students, two students (A and B) from the same school are assigned to sit in the same examination room. However, they cannot sit immediately next to each other in front, back, left, or right directions. Calculate the total number of different seating arrangements for students A and B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{772}
Two circles with radius $R$ and centers at points $O_1$ and $O_2$ touch each other. A straight line intersects them at points $A$, $B$, $C$, and $D$ such that $AB = BC = CD$. Find the area of the quadrilateral $O_1AD O_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5R^2 \sqrt{3}}{4}}
Given natural numbers $a$, $b$, and $c$ such that $a, b, c > 100$ and the greatest common divisor $(a, b, c) = 1$, find the minimal value of $b$ under the conditions:
1. $c$ divides $a + b$ (i.e., $c \mid a + b$).
2. $a$ divides $b + c$ (i.e., $a \mid b + c$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10099}
An ant walks an average of 1000 meters in 30 minutes on smooth terrain. However, on forest terrain, which makes up 40% of its walking path, it can only achieve 75% of its smooth terrain speed due to the difficulties of the terrain. A beetle, although smaller, is better equipped for tough terrain and walks 10% less distance on smooth terrain at the same time as the ant but maintains 90% of its speed on forest terrain. 

Assuming both the ant and the beetle split their walk equally between the two terrains, calculate the beetle's average speed in km/h. Consider that both creatures are not taking a rest and are walking continuously.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.71}
A retail appliance store priced three different video recorders (V1, V2, and V3) at 20 percent, 25 percent, and 30 percent above their respective wholesale costs of $200, $250, and $300. In addition, the store has a fluctuating wholesale discount policy for its employees, where they get 20 percent off on V1, 15 percent off on V2, and 10 percent off on V3 in the first week, and the discount rates change to 25 percent, 20 percent, and 15 percent respectively in the second week. If a store employee intends to purchase all three video recorders, one per week for three weeks, what would be the minimum amount they would have to spend in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$761.50}
Determine the number of solutions the equation ||$|x-a|-1|-1|=|b|$ can have. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A pentagon has vertices labeled $A, B, C, D, E$ in that order counterclockwise, such that $AB$ and $ED$ are parallel, and \( \angle EAB = \angle ABD = \angle ACD = \angle CDA \). Furthermore, suppose that \( AB = 8 \), \( AC = 12 \), \( AE = 10 \). If the area of triangle $CDE$ can be expressed as \( \frac{a \sqrt{b}}{c} \), where $a$, $b$, $c$ are integers such that $b$ is square-free, and $a$, $c$ are relatively prime, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
The equation

$$
(x-1) \times \ldots \times(x-2016) = (x-1) \times \ldots \times(x-2016)
$$

is written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to achieve this objective.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
A car traveled the first quarter of a certain distance at 4 times the speed it traveled the second quarter, and 6 times the speed it traveled the third quarter. During the last quarter of the distance, the car traveled at half the speed of the second quarter. What proportion of the total time traveled, was the time taken to travel the first quarter of the distance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/37}
In triangle $ ABC$ with $ \left|BC\right|>\left|BA\right|$, $ D$ is a point inside the triangle such that $ \angle ABD=\angle DBC$, $ \angle BDC=150{}^\circ$ and $ \angle DAC=60{}^\circ$. What is the measure of $ \angle BAD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
60
}
Determine the value of \( n \in \mathbb{N} \) for which the following limit is a non-zero finite number:

\[ \lim_{x \to 0} \frac{\cos^2x - \cos x - e^x\cos x + e^x - \frac{x^3}{2}}{x^n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
143. Find the general solution of the equation $y^{\prime \prime}-6 y^{\prime}+9 y=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(C_{1}+C_{2}x)e^{3x}}
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Three thin, straight sticks of equal length are bound together in a shape resembling the letter "Π" and tossed into the air. The adjacent sticks are connected by a short string, allowing them to rotate freely relative to each other. When the sticks fall to the ground, they form a random figure. Calculate the probability that the first and third sticks do not intersect when they land.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given that \(a\) is a positive integer and the equation \( \cos^{2} \pi(a-x) - 2 \cos \pi(a-x) + \cos \frac{3 \pi x}{2a} \cos \left(\frac{\pi x}{2a} + \frac{\pi}{3}\right) + 2 = 0 \) in terms of \(x\) has real solutions, find the smallest value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\ldots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t\geq t_0$ by the
following properties:
(a) $f(t)$ is continuous for $t\geq t_0$,and is twice differentiable for all $t>t_0$ other than $t_1,\ldots,t_n;$
(b)$f( t_0) = 1/ 2;$
(c) $\lim _{r\to t_{k}^{+ }}f^{\prime }( t) = 0$ for $0\leq k\leq n;$
(d) For $0\leq k\leq n-1$, we have $f^{\prime\prime}(t)=k+1$ when
$t_k<t<t_{k+1}$, and $f^{\prime\prime}(t)=n+1$ when $t>t_n.$

Considering all choices of $n$ and $t_0,t_1,\ldots,t_n$ such that $t_k\geq t_{k-1}+1$ for l$\leq k\leq n$,what is the least possible value of $T$ for which $f(t_0+T)=2023?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
The sequence $\left\{a_{n}\right\}$ is defined by the following conditions: $a_{1}=1, a_{n+1}=a_{n}+\frac{1}{a_{n}}$ for $n \geq 1$. Find the integer part of $a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the number of integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Determine the number of linear functions from \( \{0, 1\}^n \) to \( \{0, 1\} \). A Boolean function \( F: \{0, 1\}^n \rightarrow \{0, 1\} \) is said to be linear if \( F(x \oplus y) = F(x) \oplus F(y) \) for all \( x, y \in \{0, 1\}^n \), where \( x \oplus y \) denotes the component-wise exclusive-or of \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
556
}
A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth selected student is ▲.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Real numbers between \(0\) and \(1\), inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is \(0\) if the second flip is heads, and \(1\) if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval \([0,1]\). Two random numbers \(x\) and \(y\) are chosen independently in this manner. Find the probability that \(|x-y| > \tfrac{1}{2}\). The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics  $B$  test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of the questions correctly, what is the average possible value of your friend’s score? Recall that each question is worth the point value shown at the beginning of each question.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Simplify the expression \(\frac{(a-b)^{2}+ab}{(a+b)^{2}-ab} \div \frac{a^{5}+b^{5}+a^{2}b^{3}+a^{3}b^{2}}{\left(a^{3}+b^{3}+a^{2}b+ab^{2}\right) \left(a^{3}-b^{3}\right)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - b}
Ten years ago, the claim size under homeowner insurance policies at a particular insurance company had an exponential distribution, with 25% of claims being less than \$1000. Due to inflation, today's claims are twice the size of equivalent claims from a decade ago. What is the probability that a claim made today is less than \$1000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.134}
9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.11}
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Given the sets \( M = \left\{(x, y) \mid \tan \pi y + \sin^2 \pi x = 0\right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leqslant 2\right\} \), find the number of elements in \( M \cap N \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a triangle with internal angles A, B, C and their respective opposite sides a, b, c, if a = $\sqrt {2}$, $b^2 - c^2 = 6$, find the maximum area of triangle ABC when angle A is at its maximum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {2}}
Calculate the covariance between the first and third order statistics of three independent random variables with uniform distribution in $[0,1]$, given that the variance of the $k$th order statistic is $\frac{k(4-k)}{80}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/80}
A book is bought for $600 and has an additional government tax of 5%, a shipping fee of $20, and a 3% seller's discount. If the book is sold for $624, what is the profit in percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1.27\%}
The area enclosed by the curve $y=\sin x (0\leqslant x\leqslant \pi)$ and the line $y=\frac{1}{2}$ is $\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} - \frac{\pi}{3}}
Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at 4¢ per glob and $J$ blobs of jam at 5¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is \$ 2.53. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1.65}
Four students participate in a knowledge contest, each student must choose one of the two questions, A or B, to answer. Correctly answering question A earns 60 points, while an incorrect answer results in -60 points. Correctly answering question B earns 180 points, while an incorrect answer results in -180 points. The total score of these four students is 0 points. How many different scoring situations are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
If the function $f(x) = |\frac{e^{x}}{2} - \frac{a}{e^{x}}| (a \in \mathbb{R})$ is monotonically increasing in the interval $[1, 2]$, the range of the real number $a$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\frac{e^{2}}{2}, \frac{e^{2}}{2}]}
从正整数列 $1,2,3,4, \cdots$ 中划去 3 及 4 的倍数, 但其中凡是 5 的倍数均保留 (例如 $15,20,60,120$ 均不划去), 划完后, 剩下的数依原有次序组成一个数列: $a_1=1, a_2=$ $2, a_3=5, a_4=7, \cdots$. 试求 $a_{2003}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3338}
Determine the number of subsets $S$ of $\{1,2, \ldots, 1000\}$ that satisfy the following conditions: - $S$ has 19 elements, and - the sum of the elements in any non-empty subset of $S$ is not divisible by 20 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \cdot\binom{50}{19}}
The coordinates of the point symmetric to the point (-1, 1) about the line x - y - 1 = 0 are _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, -2)}
A standard six-sided die is rolled twice, and the uppermost numbers are recorded as m and n. When m is either 2 or 4, find the probability that m + n > 5. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
If angle $α$ is in the second quadrant, then point $P(\tan α, \sec α)$ is located in the \_\_\_\_\_\_ quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{third}
Sisyphus has three boxes of stones. Each hour, he moves a stone from one box to another. For each transfer, he earns coins equal to the number of stones in the box from which the stone is taken minus the number of stones in the recipient box, excluding the stone just moved. If this number is negative, he pays that amount. After 1000 years, all stones are back in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8760000}
In the cells of a $3 \times 3$ square, the numbers $0,1,2, \ldots, 8$ are placed. It is known that any two consecutive numbers are located in neighboring (side-adjacent) cells. What number can be in the central cell if the sum of the numbers in the corner cells is 18?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every $i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No. $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series in one row on his shelf in such a way that there do not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
给定正整数 $6$ 。求最大的实数 $M$ ，使得对于任意的实数 $x_{i}(1 \leqslant i \leqslant 2 n+1)$ ，若 $x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{2 n+1}$, 则 $\sum_{i=1}^{2 n+1} x_{i}^{2} \geqslant M x_{n+1}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $ABCD$  be a cyclic quadrilateral with circumradius  $100\sqrt{3}$  and  $AC=300$ . If  $\angle DBC = 15^{\circ}$ , then find  $AD^2$ .

*Proposed by Anand Iyer*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60000}
In a complex division problem, you are given that the quotient is 256, the divisor is 3892, and the remainder is 354. Furthermore, you must multiply the quotient by 3, add the product to the remainder, and then subtract 1500 from this result. What is the dividend of this operation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{996706}
Let's call a number greater than 25 semi-prime if it is the sum of two distinct prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of group homomorphisms from \( S_3 \) to \( S_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}