inclusionAI/Ring-lite-rl-data huggingface dataset:
3. In Chernomorsk, there are $n$ residents. All of them formed a party. Then this party split into two non-intersecting factions, each of which declared itself a party. Each subsequent day, each party formed on the previous day, consisting of at least two members, split into two factions. Factions consisting of at least two people immediately declared themselves parties. No other parties were formed. When the process ended, each resident of Chernomorsk paid a membership fee of 1 ruble to each party they belonged to. What is the maximum amount the total fees could have been?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n+1)/2-1}
Calculate the sum of the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{2n+3}\) to within an error of \(\pm 0.02\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-0.12}
The base of an oblique prism is an equilateral triangle with a side length of $a$. One of the lateral faces of the prism is perpendicular to the base plane and is a rhombus with a diagonal of length $b$. Find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ab \sqrt{12a^2 - 3b^2}}{8}}
A shop sells two kinds of products $A$ and $B$ at the price $\$ 2100$. Product $A$ makes a profit of $20\%$, while product $B$ makes a loss of $20\%$. Calculate the net profit or loss resulting from this deal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-175"]}
There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\\%$ solution. If you mix them in a ratio of $5: 4: 3,$ you will get a solution with $50\\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40%}
For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{ \sqrt{5} \right\}}
Rachel and Brian are playing a game on a grid with 1 row of 2011 squares. Initially, there is one white checker in each of the first two squares from the left, and one black checker in the third square from the left. At each stage, Rachel can choose to either run or fight. If Rachel runs, she moves the black checker 1 unit to the right, and Brian moves each of the white checkers one unit to the right. If Rachel chooses to fight, she pushes the checker immediately to the left of the black checker 1 unit to the left; the black checker is moved 1 unit to the right, and Brian places a new white checker in the cell immediately to the left of the black one. The game ends when the black checker reaches the last cell. How many different final configurations are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
Two trains of equal length are running on parallel lines in the same direction at 52 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. A third train of length L meters is running at 60 km/hr on a parallel track in the same direction. Calculate the time it takes for the faster train to overpass both the slower train and the third train one after another.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins or indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/11}
If line $l_1: ax+3y+1=0$ is parallel to line $l_2: 2x+(a+1)y+1=0$, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Find the mass percentage of O in BaO
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.43\%}
Consider a new infinite geometric series: $$\frac{7}{4} + \frac{28}{9} + \frac{112}{27} + \dots$$
Determine the common ratio of this series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{9}}
A natural number \( n \) has its divisors written in ascending order. After erasing some of the first and last numbers, 151 divisors remain. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
A train of length 250 m moving at a speed of 72 kmph takes 54.995600351971845 seconds to cross another train of some length, moving at 36 kmph in the same direction. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{299.96}
Given the function f(x) = ax + sin(x), find the range of the real number a that allows f(x) to have extreme points. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 < a < 1}
A certain unit has set up two buses at a fixed location to pick up employees for work. The departure times and probabilities of the two buses are as follows: The first bus departs at $8:00$, $8:20$, and $8:40$ with probabilities of $\frac{1}{4}$, $\frac{1}{2}$, and $\frac{1}{4}$ respectively. The second bus departs at $9:00$, $9:20$, and $9:40$ with probabilities of $\frac{1}{4}$, $\frac{1}{2}$, and $\frac{1}{4}$ respectively. The departure times of the two buses are independent of each other. If an employee arrives at the boarding point at $8:10$, what is the probability that the employee will wait for more than $50$ minutes? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{16}}
The number \( x \) is chosen randomly from the interval \( (0,1] \). Define \( y = \left\lceil \log_{4} x \right\rceil \). Find the sum of the lengths of all subintervals of \( (0,1] \) for which \( y \) is odd. For any real number \( a, \left\lceil a \right\rceil \) is defined as the smallest integer not less than \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Find the limit of the following expression as n approaches infinity:

$$\lim_{n\rightarrow \infty} \sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}-\sqrt n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many six-letter words formed from the letters of "AMC" do not contain the substring "AMC"? For example, "AMAMMC" has this property, but "AAMCCC" does not.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{622}
Let $f$ be a continuous real function such that $f(11) = 10$ and for all $x$, $f(x)f(f(x)) = 1$. What is the value of $f(9)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The remainder of $1991^{2000}$ modulo $10^6$ is \underline{\hspace{2cm}}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+3=0$ are parallel, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87341}
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{52}}
In preparation for the 2010 Shanghai World Expo, the organizing committee needs to select four volunteers from among Zhang, Liu, Li, Song, and Wang to perform four different tasks: translation, tour guiding, etiquette, and driving. If Zhang and Zhao can only work in the first two roles, while the other three can work in any of the four roles, how many different assignment plans are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to the volume of Bonnie's prism. Find the ratio of the total length of Bonnie's wire to Roark's wire. Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
Compute the value of the expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1022}
What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3051}
Find the 2020th term of the following sequence:
$$
1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, \ldots
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Compute the sum of all positive integers $n$ such that $50 \leq n \leq 100$ and $2 n+3$ does not divide $2^{n!}-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
The area of a cross-section of a regular tetrahedron, which is in the shape of a square, is \(m^2\). Find the surface area of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4m^2 \sqrt{3}}
Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Simplify the expression \(\sin 10\alpha \sin 8\alpha + \sin 8\alpha \sin 6\alpha - \sin 4\alpha \sin 2\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cos 2\alpha \sin 6\alpha \sin 10\alpha}
The value of $\log_{10}{28471}$ is between the consecutive integers $a$ and $b$.  Find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The time on a cell phone is $3:52$. How many minutes will pass before the phone next shows a time using each of the digits 2, 3, and 5 exactly once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Let \(A B C D E\) be a square pyramid of height \(\frac{1}{2}\) with a square base \(A B C D\) of side length \(A B = 12\) (so \(E\) is the vertex of the pyramid, and the foot of the altitude from \(E\) to \(A B C D\) is the center of square \(A B C D\)). The faces \(A D E\) and \(C D E\) meet at an acute angle of measure \(\alpha\) (so that \(0^{\circ}<\alpha<90^{\circ}\)). Find \(\tan \alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{144}}
Let \( A B C D E F \) be a regular hexagon. A frog starts at vertex \( A \). Each time, it can jump to one of the two adjacent vertices. If the frog reaches vertex \( D \) within 5 jumps, it stops jumping; if it cannot reach vertex \( D \) within 5 jumps, it stops after making 5 jumps. How many different sequences of jumps can the frog make from the start until it stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Given $y=a\sin x+b\cos x+c$ has a lowest point $(\frac{11π}{6},1)$ on its graph. If the ordinate of each point on the graph remains unchanged, the abscissa is shortened to $\frac{3}{π}$ of the original, and then shifted to the left by 1 unit length to obtain the graph of $y=f\left(x\right)$, and all roots of $f\left(x\right)=3$ differ by 3 units in increasing order, then $a+b+c=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4-\sqrt{3}}
A circle intersects an equilateral triangle $\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $X$, $A$, $B$, $Y$, $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. Given:

- $AC^2 + CE^2 + EA^2 = 1900$
- $BD^2 + DF^2 + FB^2 = 2092$

Compute the positive difference between the areas of triangles $\triangle ACE$ and $\triangle BDF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find the value of $m + n + p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{752}
The adult human body has 206 bones. Each foot has 26 bones. Approximately what fraction of the number of bones in the human body is found in one foot?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{8}$}
求最小的正整数 n，使得\sqrt{\df{n-2011}{2012}}-\sqrt{\df{n-2012}{2011}}<\sqrt[3]{\df{n-2013}{2011}}-\sqrt[3]{\df{n-2011}{2013}}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4024}
There is a chain of 5 people - A, B, C, D, and E. A buys a bicycle and includes a 10% tax in the cost price, then, A sells the bicycle to B at a profit of 60%. B sells the bicycle to C at a profit of 25%. C sells the bicycle to D at a loss of 15%. Finally, D sells the bicycle to E at a profit of 30%. If the final price for E is Rs. 500, find the original cost price (excluding tax) of the bicycle for A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{205.68}
Find the smallest multiple of $2019$ that is of the form $abcabc\dots abc$, where $a$, $b$, and $c$ are digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673673673}
A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The piece of length \(a\) is bent to form an equilateral triangle, and the piece of length \(b\) is bent to form a regular hexagon. The triangle and the hexagon have equal area. The original answer is in the form \(\frac{\sqrt{p}}{n}\) where the fraction is in its simplest form. Please give the value of \(p + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
During a particular month in Orlando, the temperatures recorded were:

Week 1: 55, 62, 58, 65, 54, 60, 56
Week 2: 70, 74, 71, 77, 64, 68, 72
Week 3: 82, 85, 89, 73, 65, 63, 67 
Week 4: 75, 72, 60, 57, 50, 55, 58

Additionally, the temperatures were 69, 67, and 70 in the first three days of the following week.

Calculate the average temperature of Orlando for the entire month, including the additional three days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66.55}
A rectangular piece of paper measures 17 cm by 8 cm. It is folded so that a right angle is formed between the two segments of the original bottom edge. Calculate the area of the new figure formed after folding the paper. Express your answer in square centimeters (cm²) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
数列 \{a_n\} 为：1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, \cdots. 即先取 a_1=1，接着复制该项粘贴在后面作为 a_2，并添加后继数 2 作为 a_3，再复制所有项 1, 1, 2 并粘贴在后面作为 a_4, a_5, a_6，并添加后继数 3 作为 a_7, \cdots. 依次继续下去，则 a_{2021}=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
已知函数$f(x)=-\sin 2\omega x(\omega >0)$的图象关于点$M(\frac{5\pi }{4},0)$对称，且在区间$[0,\frac{\pi }{2}]$上是单调函数，则$\omega $的值为__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2}{5}$}
Given a triangle $ABC$ where the height on side $AB$ is equal to the length of $AB$, find the maximum value of $$\frac{AC}{BC} + \frac{BC}{AC} + \frac{AB^2}{BC \cdot AC}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
A rectangle with a perimeter of 96 cm is divided by eight vertical cuts and eleven horizontal cuts into 108 identical smaller rectangles. What is the perimeter of each smaller rectangle if the sum of the lengths of all the cuts is 438 cm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
What is the maximum number of solutions that the equation \(\max \left\{a_{1} x+b_{1}, \ldots, a_{10} x+b_{10}\right\} = 0\) can have, if \(a_{1}, \ldots, a_{10}, b_{1}, \ldots, b_{10}\) are real numbers, with all \(a_{i} \neq 0\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{311}
The Fibonacci sequence is defined recursively by $F_{n+2}=F_{n+1}+F_{n}$ for $n \in \mathbb{Z}$ and $F_{1}=F_{2}=1$. Determine the value of:

$$
\left(1-\frac{F_{2}^{2}}{F_{3}^{2}}\right)\left(1-\frac{F_{3}^{2}}{F_{4}^{2}}\right) \cdot \ldots \cdot\left(1-\frac{F_{2019}^{2}}{F_{2020}^{2}}\right)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{F_{2021}}{2F_{2019}F_{2020}}}
The sum of the interior angles of a polygon is twice the sum of its exterior angles. Starting from a vertex of this polygon, connect it to the other vertices. How many triangles can this polygon be divided into at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
During the Tree Planting Festival, students from a class are planting trees. There are three different types of saplings, labeled as A, B, and C, available in the nursery, with an ample supply of each type. They need to plant 5 saplings in a row in 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Calculate the total number of planting methods available.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given $S_n$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, if the ratio of $a_2$ to $a_4$ is $7:6$, then the ratio of $S_7$ to $S_3$ equals \_\_\_\_\_\_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
How many convex 33-sided polygons can be formed by selecting their vertices from the vertices of a given 100-sided polygon, such that the two polygons do not share any common sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{67}{33} + \dbinom{66}{32}}
Let $S(n)$ be the sum of the squares of the positive integers less than and coprime to $n$. For example, $S(5) = 1^2 + 2^2 + 3^2 + 4^2$, but $S(4) = 1^2 + 3^2$. Let $p = 2^7 - 1 = 127$ and $q = 2^5 - 1 = 31$ be primes. The quantity $S(pq)$ can be written in the form \( \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) \) where $a$, $b$, and $c$ are positive integers, with $b$ and $c$ coprime and $b < c$. Find $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7561}
Find the sum of the first 100 even numbers divisible by 6 between 100 and 300 inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39,900}
Find the number of integers $a$ with $1 \le a \le 2012$ for which there exist nonnegative integers $x, y, z$ satisfying the equation:
\[ x^2(x^2 + 2z) - y^2(y^2 + 2z) = a. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1256}
Given that bus types $A$, $B$, and $C$ all depart at 6:00 AM from a certain station, with departure intervals of 10 minutes, 12 minutes, and 15 minutes, respectively, if Xiaoming arrives at the station at some time between 8:00 AM and 12:00 PM to catch one of these buses, what is Xiaoming's average waiting time in minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{6}}
Cai and Lai are eating cookies. Their cookies are in the shape of $2$ regular hexagons glued to each other, and the cookies have an area of $18$ units. They each make a cut along the $2$ long diagonals of a cookie; this now makes four pieces for them to eat and enjoy. What is the minimum area among the four pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $X_i$ be i.i.d. exponential random variables with mean $1$. Compute the variance of the ratio: $$ Y = \dfrac{X_i}{\sum\limits_{j=1,j\neq i}^{n}X_j}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n-1}{(n-2)^2(n-3)}}
What integer values can the expression 

$$
(\sqrt{5}-\sqrt{2})(\sqrt{a}+\sqrt{b})
$$ 

take if \(a\) and \(b\) are any non-negative integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3k}
Temerant is a spherical planet with a radius of $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as $a\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3008}
The consignment shop received for sale cameras, clocks, pens, and receivers totaling 240 rubles. The sum of the prices of the receiver and one clock is 4 rubles more than the sum of the prices of the camera and the pen, and the sum of the prices of one clock and the pen is 24 rubles less than the sum of the prices of the camera and the receiver. The price of the pen is an integer not exceeding 6 rubles. The number of accepted cameras is equal to the price of one camera in rubles divided by 10; the number of accepted clocks is equal to the number of receivers, as well as the number of cameras. The number of pens is three times the number of cameras. How many items of the specified types were accepted by the store in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapsed. What is the expected value of Kelvin's score? For example, if Kelvin flips two tails in the first minute, the game ends and his score is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{64}{9}}