inclusionAI/Ring-lite-rl-data huggingface dataset:
A triangle has two sides of lengths 4 cm and 9 cm. The radius of the inscribed circle that touches all three sides of the triangle is 1 cm. Determine the length of the third side of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Jerry made 14 dollars mowing lawns and 31 dollars weed eating. He also earned an additional 20 dollars from trimming bushes. If he spends 5 dollars on snacks, 10 dollars on transportation, and saves 8 dollars every week, how long would his total earnings last him for?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose we roll a die with $n$ faces repeatedly and let $T_k$ be the number of rolls until some number appears $k$ times in a row. Find the expected value of $T_3$ for a die with $10$ faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks an integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing. The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65819}
Example 3 (China Girls Mathematical Olympiad) Find all positive real numbers $a$ such that there exists a positive integer $n$ and $n$ pairwise disjoint infinite sets $A_{1}, A_{2}, \cdots, A_{n}$ satisfying $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=\mathbf{Z}$, and for any two numbers $b>c$ in each $A_{i}$, we have $b-c \geqslant a^{i}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the error order of the second Richardson extrapolation of the trapezoidal rule in Romberg integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Gavril traveled in Africa. On a sunny windy day, at noon, when the rays from the Sun were falling vertically, the boy threw a ball from behind his head with a speed of 5 m/s against the wind at an angle to the horizon. After 1 second, the ball hit his abdomen 1 meter below the point of release. Determine the maximum distance the ball's shadow moved away from Gavril's feet. The force exerted on the ball by the air is directed horizontally and does not depend on the position and speed. The acceleration due to gravity $g$ is $10 \mathrm{~m} / \mathrm{s}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75 \, \text{m}}
关于x的二次函数$f ( x ) = x ^ { 2 } - ( m + 1 ) x + 4$的图象与x轴上从0到3的长为3的一段线段只有一个交点，求m的取值范围．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{｛3｝U$( \frac { 1 0 } { 3 } , + \infty ) .$}
There are  $N{}$  points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum  $N{}$  for which this is possible?

*Proposed by E. Bakaev*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\sqrt{2}}
Dylan filled his glass with ice cubes before he poured his soda into it. He counted and there were 8 ice cubes in his glass. Later when he prepares a pitcher of lemonade, he puts twice as many ice cubes per glass into the pitcher. He expects to serve 5 guests, and each guest will receive a glass of lemonade as well as his own glass. Then it is time to refill the ice cube trays, which have 14 spaces each for ice cubes. Dylan realizes that he has used 80% of the total ice cubes they initially had. How many trays does Dylan need to fill to return to the initial number of ice cubes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Pete chooses $1004$ monic quadratic polynomials $f_1, \cdots, f_{1004}$ such that each integer from $0$ to $2007$ is a root of at least one of them. Vasya considers all equations of the form $f_i = f_j$ for $i \neq j$ and computes their roots. For each such root, Pete has to pay Vasya $1$ ruble. Find the least possible value of Vasya's income.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$

Find all possible values of $a_{2021}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ <i>quasi-increasing</i> if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$486$}
On a roulette table with 52 numbers (26 red and 26 green), you observe the selected numbers over N spins. The null hypothesis states that the table is fair, with equal probabilities for red and green (p = 0.5). The alternative hypothesis suggests the table is biased, favoring red with a probability of p = 0.5005. Given a significance level of 1% and a target power of 90%, determine the minimum number of spins (N) required for the test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13017664}
Given the function $f(x)=ax-x^{2}-\ln x$, if the function $f(x)$ has an extreme value, and the sum of all extreme values is less than $5+\ln 2$, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2\sqrt{2},4)}
In $\triangle ABC$, $\angle ABC = 50^\circ$, $\angle ACB = 20^\circ$, and $N$ is a point inside $\triangle ABC$. If $\angle NAB = 40^\circ$ and $\angle NBC = 30^\circ$, find the measure of $\angle NCB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let \( S = \{1, 2, \ldots, 50\} \). Determine the smallest natural number \( K \) such that in any subset of \( S \) with \( K \) elements, there exist two distinct numbers \( a \) and \( b \) such that \( (a+b) \mid a b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Determine the maximum number of non-attacking $2 \times 2$ squares that can be placed on a $1000 \times 1000$ board. Two $2 \times 2$ squares are said to attack each other if they lie in the same pair of adjacent rows (or columns), and the cells between them are all empty. What is the largest number of such squares that can be placed on the board without any two attacking each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250000}
Find an equivalent of the function $f(x)$ as $x \to 0^+$, where $f(x)$ is defined as:

$$f(x) = \int_{0}^{\infty} \frac{e^{-xt}}{(1 + t^3)^{1/3}} \, dt$$

This means finding a simple function $g(x)$ such that when $x \to 0^+$, we have:

$$\lim_{x \to 0^+} \frac{f(x)}{g(x)} = 1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{x}}
For every non-negative integer \( n \), a polynomial \( p_n(x) \) exists such that \( \int x^n e^x \,dx = p_n(x) e^x + C \). Determine \( \lfloor 1000 \cdot L \rfloor \), where \( L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Calculate the expected length of a coin-tossing game where Player A has n coins and Player B has m coins. In each round, if two heads or two tails appear, Player A takes both coins. Otherwise, Player B takes them. What is the expected number of rounds until all coins are taken?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{nm}
For a nilpotent group $G$ of class $c$ with the upper central series $1=Z_{0}(G)<Z_1(G)<Z_2(G)<\ldots<Z_c(G)=G$, determine the nilpotency class of the quotient group $G/Z_i(G)$ for a given integer $i$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c - i}
Given that the eccentricity of an ellipse is \( e \), with foci \( F_1 \) and \( F_2 \). A parabola \( C \) has \( F_1 \) as its vertex and \( F_2 \) as its focus. Point \( P \) is an intersection point of the parabola and the ellipse. If \( e|PF_2| = |PF_1| \), find the value of \( e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Define a permutation of the set $\{1,2,3,...,n\}$ to be \textit{sortable} if, upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when \(\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)\) is divided by $1000$. Note that the empty set is considered sortable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
Determine the computational complexity of verifying whether a matrix is orthogonal using the matrix product method and fast matrix multiplication algorithms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^\omega)}
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9702}
Find a $3\times 3$-matrix whose null space is the $x$-axis and whose column space is the $yz$-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix}0&0&0\\0&1&0\\0&0&1\end{pmatrix}}
A train moves at a speed of 60 kilometers per hour, making stops every 48 kilometers. The duration of each stop, except for the fifth stop, is 10 minutes. The fifth stop lasts half an hour. What distance did the train travel if it departed at noon on September 29 and arrived at its destination on October 1 at 22:00?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2870}
Evaluate the expression \(\sum_{k=1}^4 k \sum_{1 \le j_1 < j_2 < \ldots < j_k \le 4} (n_{j_1} + n_{j_2} + \ldots + n_{j_k})\) given that \(n_1 = 2\), \(n_2 = 3\), \(n_3 = 4\), and \(n_4 = 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Determine the asymptotic behavior of the function \( h(n) = \sum_{p_k < n} \frac{\log (1 + p_k)}{p_k} \), where \( p_k \) is the \( k \)th prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log n}
Two ships are moving as follows: Ship A is 1 mile due north of a pier and is traveling N $30^o$ E at 3 mph. Ship B is $\frac{3}{4}$ mile east of the pier and is traveling east at 7 mph. Determine the rate at which the angle between the two ships' paths is changing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{2}}
A city has \( n \) schools, and the \( i \)-th school sends \( C_{i} \) students \( \left(1 \leqslant C_{i} \leqslant 39, 1 \leqslant i \leqslant n\right) \) to watch a game in an auditorium. The total number of students from all schools is \( \sum_{i=1}^{n} C_{i} = 1990 \). Each row of seats in the stand has 199 seats. It is required that all students from the same school must sit in the same row. What is the minimum number of rows needed to ensure that all students can be seated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the structure of all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x+y) = f(x) + f(y) + 2xy \) for all real numbers \( x \) and \( y \), given that \( f(0) = 0 \) and \( f(x+1) = f(x) + f(1) + 2x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = x^2 + kx}
What is the remainder when the sum
\[\binom{2013}{1} + 2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\]
is divided by $41$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Al told Bob that he was thinking of  $2011$  distinct positive integers. He also told Bob the sum of those  $2011$  distinct positive integers. From this information, Bob was able to determine all  $2011$  integers. How many possible sums could Al have told Bob?

*Author: Ray Li*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate $\int_{-1}^{3} \left[ x+\frac{1}{2} \right] dx$, where $\left[ . \right]$ denotes the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit: $$\lim_{x \to \infty} \left( x - \ln\left(\frac{1+x}{x}\right)^{x^2} \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a game, two players take turns removing cells from a $5 \times 7$ grid. To remove a cell, a player selects an uneaten cell, places a piece on that cell, and then all cells in the "first quadrant" (the right-angled region created with the lower-left corner of that cell extending upwards and to the right) are removed. For example, placing a piece on a shaded cell in the right image will remove the 4 cells marked with $\times$ along with the shaded cell itself, where the cells represented by dotted lines were removed earlier. The goal of the game is to force the opponent to remove the last cell. How many different possible situations can appear during the entire course of the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{792}
How many ways can you color the squares of a \( 2 \times 2008 \) grid in 3 colors such that no two squares of the same color share an edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \times 3^{2007}}
What is the coefficient of $a^2b^2$ in $(a+b)^4\left(c+\dfrac{1}{c}\right)^6$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Daniel wrote all the positive integers from 1 to $n$ inclusive on a piece of paper. After careful observation, he realized that the sum of all the digits that he wrote was exactly 10,000. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{799}
Let  $ k$  be a positive constant number. Denote  $ \alpha ,\ \beta \ (0<\beta <\alpha)$  the  $ x$  coordinates of the curve  $ C: y\equal{}kx^2\ (x\geq 0)$  and two lines  $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$ .　Find the minimum area of the part bounded by the curve  $ C$  and two lines  $ l,\ m$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{2}{3}}}
What is the minimum number of operations required to transform 2014 into 1 using only addition, subtraction, multiplication, or division by single-digit numbers (1-9)? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute the number of nonempty subsets $S$ of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max S + \min S}{2}$ is an element of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{234}
Evaluate the double integral \( \int\int 2xy \, dA \) over the region bounded by \( y = 1 - x^2 \), \( y = 1 + x \), and \( y = 0 \) in the \( xy \) plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Given positive integers \(n\) and \(x_1, x_2, \ldots, x_n \in (0, 1)\) such that for all \(i = 1, 2, \ldots, n\), the equation \(\sum_{i \ne j}\frac{1}{x_i - x_j} = \frac{1}{1 - x_i} - \frac{1}{x_i}\) holds, calculate the value of \(\sum_{i = 1}^n x_i\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{2}}
Consider a random sample $X_1, X_2, \ldots, X_n$ from a uniform distribution $U(\theta, \theta + 1)$. We want to test the hypotheses $H_0: \theta = 0$ versus $H_1: \theta = \theta_1$ for some $0 < \theta_1 < 1$. Using Neyman-Pearson's lemma, find the critical value $k$ for the likelihood ratio test $\Lambda \leq k$ such that the test has size $\alpha$, i.e., $P_{\theta = 0}(\Lambda \leq k) = \alpha$. Assume $\Lambda = \frac{1}{I_{(\theta_1, \infty)}(y_1)}$, where $y_1 = \min(x_1, \ldots, x_n)$. Determine the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine a tight asymptotic upper bound for the expected maximum load \( \mathbb{E}(X_n) \) when \( n \) balls are thrown into \( n \) bins. Each ball is placed in the least full of two bins chosen uniformly and independently at random, with the bins selected with pairwise independence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log \log n)}
Find the positive integer $q \leq 50$ such that the following equation holds: $$\binom{98}{30} + 2\binom{97}{30} + 3\binom{96}{30} + \dots + 68\binom{31}{30} + 69\binom{30}{30} = \binom{100}{q}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A shipping company charges $0.30l + 0.40w + 0.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l, w, h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l, w, h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $8.10, and two different ways to label $B$ to get a fee of $8.70. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{276}
设 $A, B, C$ 是 $\triangle A B C$ 的三个内角, 则 $3 \cos A+2 \cos 2 B+\cos 3 C$ 的取值范围为. 请给出该范围的端点之和的整数部分。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $a$, $b$, $c$ be positive real numbers for which:
\[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b.
\]
If $a+b+c = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Determine the number of zeros of the polynomial \( z^{17} + z - 1 \) that lie within the region \( |z| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Through a point located inside a triangle, lines are drawn parallel to the sides of the triangle.

These lines divide the triangle into three smaller triangles and three quadrilaterals. Let \( a \), \( b \), and \( c \) be the heights of the three smaller triangles parallel to the sides of the original triangle. Find the height of the original triangle parallel to these.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + b + c}
Determine the Picard group of the projective space over the integers, \( \operatorname{Pic}(\mathbb{P}^n_{\mathbb{Z}}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Two five-digit numbers were written on the wall. Pat added a one in front of one of these numbers, and Mat added a one after the other. This resulted in two six-digit numbers, one of which was three times larger than the other.

Which five-digit numbers were originally written on the wall?

(L. Hozová)

Hint. Which of the newly created numbers was larger?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42857}
Calculate the area of the region enclosed by the polar curve \( r = \arctan(r) \) from \( \phi = 0 \) to \( \phi = \frac{\pi}{\sqrt{3}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Ivan and Foma in front of the store, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait for the store to open for no more than 4 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
Each square of a chessboard is occupied by a knight. What is the minimum number of knights that need to be removed so that there is no knight left on the board that attacks exactly three other knights? (A knight attacks squares that are one square away horizontally and two squares vertically, or vice versa.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
It is given that  $2^{333}$  is a 101-digit number whose first digit is 1.  How many of the numbers  $2^k$ ,  $1\le k\le 332$  have first digit 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
1. On the first day, $2^{n}$ schoolchildren played ping-pong in a "knockout" format: first, two played, then the winner played with the third, the winner of this pair played with the fourth, and so on until the last schoolchild (there are no draws in ping-pong). On the second day, the same schoolchildren played for the cup: first, they arbitrarily split into pairs and played, the losers were eliminated, and the winners again arbitrarily split into pairs and played, and so on. It turned out that the sets of pairs that played on the first and second days were the same (possibly with different winners). Find the largest possible value of $n$.

B. Frenkin
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
a dealer offers a cash discount of 15 % and still makes a profit of 36 % when he further allows 25 articles to be sold at the cost price of 20 articles to a particular sticky bargainer . how much percent above the cost price were his articles listed ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60\%}