inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, given $a \geqslant \frac{b+c}{3}$, it holds that
$$
a c + b c - c^{2} \leqslant \lambda\left(a^{2} + b^{2} + 3 c^{2} + 2 a b - 4 b c\right).
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{2} + 1}{7}}
Find the value of the integral $$\int_\gamma (z+1)e^{z+1}dz$$ where $\gamma$ is the semicircle of radius 1 centered at the origin, traversed from $-1$ to $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^2+1}
247. Two opposite edges of a tetrahedron have lengths $b$ and $c$, while the others are equal to $a$. What is the minimum value of the sum of the distances from an arbitrary point in space to the vertices of this tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4a^2+2bc}}
Mr. Langsley commutes to work every day by bus. On his way to work, he needs to transfer between three different buses. The first bus picks him up at 6:00 a.m. and takes forty minutes to arrive at the first transfer station. However, there is a traffic jam that causes a delay of ten minutes during this ride. He then waits ten minutes before taking the second bus, which takes fifty minutes to arrive at the second transfer station, with a five-minute delay due to weather conditions. After waiting another fifteen minutes for the third bus, he rides for an hour and thirty-five minutes until he finally arrives at work. This last bus encounters a delay of fifteen additional minutes. If Mr. Langsley arrives at work at 9:00 a.m., what is the total time taken in minutes from the first station to Mr. Langsley's workplace, including the waiting time at transfer stations and the unexpected delays in his journey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
In the relay race from Moscow to Petushki, two teams of 20 people each participated. Each team divided the distance into 20 not necessarily equal segments and assigned them to the participants so that each ran exactly one segment (each participant's speed is constant, but the speeds of different participants may vary). The first participants of both teams started simultaneously, and the baton pass occurs instantaneously. What is the maximum number of overtakes that could occur in such a race? Overtaking at segment boundaries is not considered an overtake.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Compute the number of sequences of integers $(a_1, \ldots, a_{200})$ such that the following conditions hold:

1. $0 \leq a_1 < a_2 < \cdots < a_{200} \leq 202$.
2. There exists a positive integer $N$ with the following property: for every index $i \in \{1, \ldots, 200\}$, there exists an index $j \in \{1, \ldots, 200\}$ such that $a_i + a_j - N$ is divisible by $203$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20503}
In triangle $XYZ$, $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$. What is the measure of $\angle XYW$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^{\circ}}
The seats in a theater hall are divided into three categories according to their distance from the stage. "Category I" seats are the closest to the stage, make up two-fifths of the hall's capacity, and sell for 220 CZK. "Category II" seats make up the next two-fifths of the hall's capacity and sell for 200 CZK. The remaining "Category III" seats sell for 180 CZK. Before the pre-sale for the grand premiere, 150 tickets were given for free to invited guests. The tickets were distributed sequentially from the front seats of the hall to the back. All other tickets were then sold. If the free tickets had been distributed sequentially from the back seats to the front instead, the revenue would have been 4320 CZK higher. How many seats are in the hall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
9. Two cars, A and B, depart from locations $A$ and $B$ respectively at the same time, and travel back and forth between $A$ and $B$ at a constant speed. If after the first meeting, car A continues to drive for 4 hours to reach $B$, while car B only drives for 1 hour to reach $A$, then when the two cars meet for the 15th time (meetings at $A$ and $B$ are not counted), they have driven $\qquad$ hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Calculate the digit in the fifth decimal place of (1.0025)^10, accurate to five decimal places. Express your answer as a single digit (0-9).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
10. (10 points) The teacher used the ten digits from 0 to 9 to form five two-digit numbers, with each digit used exactly once; then gave these five two-digit numbers to five clever and honest students $A, B, C, D, E$, each student could only see their own two-digit number, and the following conversation took place:
$A$ said: “My number is the smallest, and it is a prime number.”
$B$ said: “My number is a perfect square.”
$C$ said: “My number is the second smallest, and it has exactly 6 factors.”
$D$ said: “My number is not the largest, and I already know two of the numbers that $A, B, C$ have.”
$E$ said: “My number is three times someone else’s number.”
What is the sum of these five two-digit numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
In triangle $ABC$, the angle bisectors of $\angle BAC$ and $\angle ABC$ intersect at point $O$. Find $\angle ACB$ if $\angle AOB$ is $125^\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^\circ}
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$48$}
Let $U(x)=\sum_{n=0}^{\infty} u_nx^n$, where $u_n$ is the number of partitions of $n$ into at most two parts. For example, $u_4=3$ because $4$ can be partitioned into at most two parts as $4$, $3+1$, or $2+2$. Use the convention that $u_0=1$. Then $\frac 1{U(x)}$ is a polynomial. What polynomial is it? (Enter your answer in expanded form.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3-x^2-x+1}
Beth has 4 packs of crayons and 10 friends including herself. She divides the packs among everyone. She then finds some packs in a drawer. She keeps these for herself and has a total of 6.4 packs. How many packs did she find in the drawer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A set of natural numbers is called "bad" if from it one can select several numbers such that their sum equals 2012. Find the smallest number \( n \) such that the numbers \( 503, 504, \ldots, 2011 \) can be partitioned into \( n \) sets in such a way that none of these sets are "bad."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the Hausdorff distance between $C_{n+1}$ and $C_{n+2}$, where $C_n$ represents the nth stage in the construction of the Cantor ternary set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2 \times 3^{n+1}}\)}
Let $a > 0$, $a \neq 1$, and the function $f(x) = ax^2 + x + 1$ has a maximum value. Then, the solution set of the inequality $\log_a(x - 1) > 0$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 < x < 2}
Among the rye seeds, there are $0.4\%$ weed seeds. What is the probability of detecting 5 weed seeds when randomly selecting 5000 seeds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.000055}
Consider the sum $$S = \sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3031}
In triangle \(PQR\) with side \(PQ = 3\), a median \(PM = \sqrt{14}\) and an altitude \(PH = \sqrt{5}\) are drawn from vertex \(P\) to side \(QR\).

Find side \(PR\), given that \(\angle QPR + \angle PRQ < 90^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{21}}
Colston's teacher has a thermos that she fills with coffee and 1/2 cup of milk. She does this twice a day for a five-day school week. After learning how much coffee she drinks, she decides to drink 1/4 of what she normally drinks and now drinks 40 ounces of coffee a week. What is the capacity of her thermos in ounces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $ABCDA'B'C'D'$ be a rectangular parallelepiped, where $ABCD$ is the lower face and $A, B, C$, and $D$ are below $A', B', C'$, and $D'$, respectively. The parallelepiped is divided into eight parts by three planes parallel to its faces. For each vertex $P$, let $V_P$ denote the volume of the part containing $P$. Given that $V_A = 40$, $V_C = 300$, $V_{B'} = 360$, and $V_{C'} = 90$, find the volume of $ABCDA'B'C'D'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Tom buys 20 shares of stock A for $3 each, and 30 shares of stock B for $5 each. He also buys 15 shares of stock C for $10 each, paying a 2% commission on the total purchase. Later, he sells 10 shares of stock A for $4 each, 20 shares of stock B for $7 each, and 5 shares of stock C for $12 each, with a 2% commission on the total sales. The remaining shares of stock A double in value, while the remaining shares of stock B increase in value by 20% and the remaining shares of stock C decrease in value by 10%. Calculate Tom's overall profit or loss.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$78}
Two opposite edges of a tetrahedron are perpendicular and their lengths are \(a\) and \(b\); the distance between them is \(c\). A cube is inscribed in the tetrahedron such that four edges of the cube are perpendicular to these two edges of the tetrahedron, and exactly two vertices of the cube lie on each face of the tetrahedron. Find the side length of the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{abc}{ab + bc + ac}}
Given $f(x) = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1$, calculate $v_2$ using the Horner's method when $x = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
9.1. At a round table, 10 people are sitting, some of them are knights, and the rest are liars (knights always tell the truth, while liars always lie). It is known that among them, there is at least one knight and at least one liar. What is the maximum number of people sitting at the table who can say: "Both of my neighbors are knights"? (A false statement is one that is at least partially not true.)

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:

- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adventurer has rubies, they either have emeralds or diamonds (but not both simultaneously).

What is the minimum number of adventurers that could be in this group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The polynomial \((x+y)^9\) is expanded in decreasing powers of \(x\). The second and third terms have equal values when evaluated at \(x=p\) and \(y=q\), where \(p\) and \(q\) are positive numbers whose sum is one. 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{9}
在平面直角坐标系 $x O y$ 中， $F_{1}, F_{2}$ 分别是椭圆 $\frac{x^{2}}{2}+y^{2}=1$ 的左、右焦点，设不经过焦点 $F_{1}$ 的直线 $l$ 与椭圆交于两个不同的点 $A, B$ ，焦点 $F_{2}$ 到直线 $l$ 的距离为 $d$ 。如果直线 $A F_{1}, l, B F_{1}$ 的斜率依次成等差数列，求 $d$ 的取值范围。答案标准形式为$(\sqrt{m}, n)$，请给出m+n的和的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Two of the lightest kittens weigh a total of 80 grams. Four of the heaviest kittens weigh 200 grams. The total weight of all the kittens is \(X\) grams. What is the smallest value of \(X\) for which it is not possible to uniquely determine the number of kittens born?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
如果一个十位数 $F$ 的各数位数字之和等于 81 ，则称 $F$ 为一个"好数"，则所有好数的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48619}
Let \( S=\{1,2, \ldots, n\} \). Define \( A \) as an arithmetic sequence with a positive common difference, containing at least two elements, such that all its elements are within \( S \). Additionally, rearranging the other elements of \( S \) before or after \( A \) should not form a new arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \). (An arithmetic sequence with just two elements is also considered an arithmetic sequence).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{n^2}{4} \right\rfloor}
如果一个正整数集合中没有 3 个数是两两互质的, 则称之为 "异质"的. 问从 1到 16 的整数的集合中 "异质"的子集合的元素的最大数目是多少?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The leader of an IMO team chooses positive integers $n$ and $k$ with $n>k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader's in exactly $k$ positions. (For example, if $n=3$ and $k=1$, and if the leader chooses $101$, the deputy leader would write down $001,111$and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader's string. What is the minimum number of guesses (in terms of $n$ and $k)$ needed to guarantee the correct answer? Please give the sum of guesses when $n = 6$ and $k = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given two points on the sphere and the circle $k$, which passes through exactly one of the two points. How many circles are there on the sphere that pass through both points and are tangent to $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( A = \{a_1, a_2, \cdots, a_7\} \). Here \( a_i \in \mathbf{Z}' \). Let \( n_A \) represent the number of triples \((x, y, z)\) such that \( x < y \), \( x + y = z \), and \( x, y, z \in A \). What is the maximum possible value of \( n_A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series expansion about $x=0$ of $(1+x)^{\alpha}$. Evaluate  
$$  
\int_{0}^{1} C(-y-1)\left(\frac{1}{y+1}+\frac{1}{y+2}+\frac{1}{y+3}+\cdots+\frac{1}{y+1992}\right) d y.  
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1992}
In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 : \sqrt{2}}
By what factor should the expression 

$$
\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}}
$$

be multiplied so that the product is equal to 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{\frac{2}{3} (5 \sqrt{3} + 3 \sqrt{7})}}
In the 8th grade class "Я," there are quite a few failing students, but Vovochka is the worst. The pedagogical council decided that either Vovochka must improve his grades by the end of the term, or he will be expelled. If Vovochka improves his grades, then the class will have $24\%$ failing students, but if he is expelled, the percentage of failing students will become $25\%$. What is the current percentage of failing students in the 8th grade class "Я"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28\%}
The side of the base of a regular quadrilateral pyramid \( \operatorname{ABCDP} \) (with \( P \) as the apex) is \( 4 \sqrt{2} \), and the angle between adjacent lateral faces is \( 120^{\circ} \). Find the area of the cross-section of the pyramid by a plane passing through the diagonal \( BD \) of the base and parallel to the lateral edge \( CP \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{6}}
In an inscribed quadrilateral \(ABCD\) whose diagonals intersect at point \(K\), it is known that \(AB = a\), \(BK = b\), \(AK = c\), and \(CD = d\). Find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ac + bd}{a}}
Find the limit of the following expression as k approaches infinity:

$$(k^2-1)\ln\left(\frac{k-1}{k+1}\right) + 2k$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The largest animal to have ever lived on earth is the blue whale. The tongue of an adult blue whale can weigh 2700 kilograms. If one ton is 2000 pounds and 1 kilogram is equal to 2.20462 pounds, how many tons can the tongue of an adult blue whale weigh? First, convert the weight from kilograms to pounds and then calculate the weight in tons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.976237}
A quadratic polynomial with real coefficients and leading coefficient \(1\) is called \(\emph{disrespectful}\) if the equation \(p(p(x))=0\) is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \(\tilde{p}(x)\) for which the sum of the roots is maximized. The original answer for \(\tilde{p}(1)\) 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{21}
The general term formula of the sequence $\{a_n\}$ is given by ${a_n = \frac{8}{3} \cdot \left(\frac{1}{8}\right)^n - 3 \cdot \left(\frac{1}{4}\right)^n + \left(\frac{1}{2}\right)^n}$. If $a_n \leqslant a_m$, then $m=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the integer $n$, $-180 < n < 180$, such that $\tan n^\circ = \tan 1500^\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
\(\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\) 等于 \(x \pi\)，其中答案以\(\frac{k}{m}\)的最简分数形式给出，请求出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A typist uses a sheet measuring 20 cm by 30 cm lengthwise. If a margin of 2 cm is left on each side and a margin of some cm on top and bottom, then 64% of the page is used for typing. What is the width of the margin on top and bottom?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
A passenger train operates between Harbin and Station A, stopping at 5 stations along the way. Calculate the number of different tickets that need to be arranged between Harbin and Station A. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Let $a_1$, $a_2$, \dots, $a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence "good" if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \] Find the minimum possible number of good subsequences across all possible $(a_i)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19320}
Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given $a{e}^{ax}-ln(x+\frac{2}{a})-2≥0$ always holds for $(-\frac{2}{a},+∞)$, the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[e,+\infty \right)}
In a box, there are 10 red cards and 10 blue cards, each color containing cards labeled with numbers $1, 3, 3^{2}, \cdots, 3^{9}$, one for each number. The total sum of the labels on the cards of both colors is denoted as $S$. For a given positive integer $n$, if it is possible to select some cards from the box such that the sum of their labels is exactly $n$, this is called a "scheme for $n$", and the number of different schemes for $n$ is denoted as $f(n)$. Find $P=$ $\sum_{n=1}^{1000} f(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6423}
10. [8] A real number $x$ is chosen uniformly at random from the interval $[0,1000]$. Find the probability that
$$
\left\lfloor\frac{\left\lfloor\frac{x}{2.5}\right\rfloor}{2.5}\right\rfloor=\left\lfloor\frac{x}{6.25}\right\rfloor .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{10}}
Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941$. What is the largest possible number whose digits Terry could have swapped in the summation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find the number of distinct integers that can be written as the sum of two, not necessarily different, special fractions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Given an isosceles triangle with two sides of lengths $x$ and $y$, and $x$ and $y$ satisfy $|x-3|+\left(y-1\right)^{2}=0$, then the perimeter of this isosceles triangle is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the angle that is equivalent to $2023^{\circ}$ within a full rotation. Express your answer in degrees ($^{\circ}$) and include the negative sign if the angle is negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-137^{\circ}}
Find the maximum real number \( \theta \) (\(\theta < \pi\)) that satisfies
\[ 
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0 
\]
and 
\[ 
\prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2046 \pi}{2047}}