inclusionAI/Ring-lite-rl-data huggingface dataset:
At the beginning of each hour from 1 o'clock AM to 12 NOON and from 1 o'clock PM to 12 MIDNIGHT a coo-coo clock's coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock's coo-coo bird coo-coos a single time at :30 minutes past each hour. How many times does the coo-coo bird coo-coo from 12:42 PM on Monday until 3:42 AM on Wednesday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
279
}
Calculate the current price of a 2-year derivative contract with a maturity payoff of £90 × I(S2 > £90), where the current share price is £100, the share price can increase or decrease by 20% annually, and the continuously compounded constant annual risk-free interest rate is such that e^r = 1.1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69.73}
How many partitions are there for a set with an uncountable number of elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{\aleph_0}$}
Given that $A=\{x\in \mathbb{N} | -2 \lt x \lt 2\}$ and $B=\{-1,0,1,2\}$, find the intersection $A\cap B$. Express your answer in standard set notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0,1\}}
Given $x < y < 0$, let $M = |x|$, $N = |y|$, and $P = \frac{|x + y|}{2}$. Determine the relationship among $M$, $N$, and $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M > P > N}
Identify a group that is isomorphic to the additive group $\mathbb{R} / \mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1}
Calculate the intersection number of the curves given by \((x^2+y^2)^3-4x^2y^2=0\) and \(x=0\) at the point \((0,0)\). The intersection number is defined as \(\dim_k(O_p/(F,G))\), where \(O_p\) is the local ring at the point \(P\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given an alphabet with \( n > 1 \) letters, a word is defined as any finite sequence of letters where no two neighboring letters are the same. A word is considered good if it is impossible to delete all letters except for four such that the remaining sequence is of the form \( a a b b \), where \( a \) and \( b \) are different letters. Determine the maximum possible number of letters in a good word.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n + 1}
Petya and Vasya came up with ten quadratic trinomials. Then Vasya sequentially named consecutive natural numbers (starting from a certain number), and Petya substituted each named number into one of the trinomials of his choice and wrote the resulting values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in that exact order). What is the maximum number of numbers that Vasya could name?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the minimum value of the integral

$$ \mathcal{I}[y] = \int_{-\infty}^{\infty} \left( \frac{1}{2} y'^2 + 1 - \cos y \right) \, dx $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes the last stick. For $1 \leq k \leq 100$, determine the number of cases in which the first player can guarantee a win.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Suppose $x, y, z$, and $w$ are positive reals such that:
\[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \]
\[ xz + yw = 30. \]
Find the largest possible value of $(xy + wz)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{960}
Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}} = 2$. Find $r^3 - \frac{1}{r^3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2786}
Use the Lagrange Multipliers to find the maximum value of the function $f(x,y,z)= x+2y+3z$ on the curve of intersection of the plane $x-y+z=1$ and the cylinder $x^2+y^2=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + \sqrt{29}}
How many boolean functions $F(w, x, y, z)$ satisfy the property $F(\bar{w}, x, \bar{y}, z) = F(w, \bar{x}, y, \bar{z}) = F(w, x, \bar{y}, \bar{z})$ for all values of the Boolean variables $w, x, y,$ and $z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Calculate the value of $8\cos ^{2}25^{\circ}-\tan 40^{\circ}-4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Find the minimal polynomial of $\zeta = e^{2\pi i / 7}$ over the field $\mathbb{Q}(\cos(2\pi/7))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X^2 - 2\cos(2\pi/7)X + 1}
Three boys played a "Word" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their scores, they found that they each had different scores. Sam had the smallest score (19 points), and James scored the most. How many points did James score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In △ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. If a=1 and the height on side BC is equal to tanA, then the range of the perimeter of △ABC is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{5} + 1, +\infty)}
Let \( p(x) \) be a polynomial of degree \( 3n \) such that \( p(0)=p(3)=\cdots=p(3n)=2 \), \( p(1)=p(4)=\cdots=p(3n-2)=1 \), and \( p(2)=p(5)=\cdots=p(3n-1)=0 \). Given that \( p(3n+1)=730 \), find \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the  $111^{\text{st}}$  smallest positive integer which does not have  $3$  and  $4$  in its base- $5$  representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{755}
The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A group of hikers is planning a trip that will take them up a mountain using one route and back down using another route. The upward route consists of various terrains: the first half is a well-maintained trail with tree cover, while the second half is a rocky, open terrain that exposes them to changing weather conditions. The downward route, on the other hand, is a more consistent trail through the forest, which shields them from the elements, allowing them to travel 1.5 times their rate on the way up.

On the first day, the hikers move at a rate of 4 miles per day on the well-maintained trail, but their speed drops to 2 miles per day when they reach the rocky terrain on the second day due to the rough ground and unpredictable weather. The total time it takes for them to climb the mountain is two days.

Considering the varying terrain and weather conditions, how many miles long is the route down the mountain, given that the hikers will take the same amount of time to complete the descent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A group of adventurers is displaying their loot. It is known that exactly 13 adventurers have rubies; exactly 9 have emeralds; exactly 15 have sapphires; exactly 6 have diamonds. Additionally, the following conditions are known:

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

What is the minimum number of adventurers that can be in such a group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Each month a retailer sells some identical items. On each item, he makes a profit of $30 that constitutes 16% of the item's price to the retailer. If the retailer contemplates giving a 5% discount on the items he sells, he will have to sell at least 156.86274509803923 items each month to justify the policy of the discount. How many items does the retailer sell each month without the discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
How many real solutions are there to the equation $|||| x|-2|-2|-2|=|||| x|-3|-3|-3|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the area of the region \(D\) bounded by the curves

\[ x^{2} + y^{2} = 12, \quad x \sqrt{6} = y^{2} \quad (x \geq 0) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S = 3\pi + 2}
Given a rectangular prism \(ABCD - A'B'C'D'\), how many pairs of skew lines are there among the following twelve lines: \(AB', BA', CD', DC', AD', DA', BC', CB', AC, BD, A'C', B'D'\)? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{97}-19}{20}}
At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the comfort level of the second bus is better than the first, he will board the second bus; otherwise, he will take the third bus. Given that the comfort levels of the buses can be classified as high, medium, and low, the probability of Xiao Jie boarding a bus with a high comfort level is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Task 6. (30 points) A regular triangular prism $A B C A_{1} B_{1} C_{1} \mathrm{c}$ is inscribed in a sphere with base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$. Segment $C D$ is the diameter of this sphere, point $K$ and $L$ are the midpoints of edge $A A_{1}$ and $A B$ respectively. Find the volume of the prism if $D L=\sqrt{2}, D K=\sqrt{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
A dog is tied to a tree by a long nylon cord. There is a triangular-shaped obstacle nearby, with its vertices situated at points A, B, and C, where point A lies on the north side of the tree, point B is 10 feet directly east of point A, and point C is 20 feet diagonally southeast of point A. The dog starts at point A and runs around the triangular obstacle to finally reach point D, the due south side of the tree. During its run, the nylon cord is extended to its full length at all times without interfering with the obstacle. If the dog traveled approximately 60 feet, what was the approximate length of the nylon cord, L, in feet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $x_{n+2}=\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\sum_{n=0}^\infty\dfrac{x_{3n}}{2^n}=\dfrac{p}{q}$ for some positive integers $p$, $q$ with $\text{GCD}(p,q)=1$. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1685}
Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(\frac{n}{2})!}{(\frac{n}{4})!}}
Calculate the probability $P\left(X_{(2)}+X_{(3)}\geq1\right)$ for the order statistics $X_{(1)},X_{(2)},X_{(3)}$ of three independent random variables $X_1,X_2,X_3$ with uniform distribution in $(0,1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
We colored the faces of a regular dodecahedron with red, blue, yellow, and green so that any two adjacent faces have different colors. How many edges of the dodecahedron have one face colored blue and the other green?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Suppose we have a group $G$ of order $pq^3$ with a normal Sylow $q$-subgroup $H$. If $G$ has a subgroup $P$ of order $pq^2$ and a Sylow $p$-subgroup $R$ of $P$ with $|N_P(R)|=p$, can we find the order of $N_G(R)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{pq^3}
Use Horner's method to find the value of V_2 for the polynomial function f(x) = 2x^5 - 3x^3 + 2x^2 - x + 5 when x = 2. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the angle $\alpha$ for which the vectors $(\rho^2 + i)^{16}$ and $(\rho^2 - i)^{16}$ on the complex plane are aligned with the imaginary axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{32}$}
In trapezoid \( ABCD \), point \( X \) is taken on the base \( BC \) such that segments \( XA \) and \( XD \) divide the trapezoid into three similar but pairwise unequal, non-isosceles triangles. The side \( AB \) has a length of 5. Find \( XC \cdot BX \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given circle $O$: $x^2+y^2=1$, and line $l$: $2x-y-4=0$. If point $A$ is on line $l$ such that there exists a point $B$ on circle $O$ and $\angle OAB=30^{\circ}$ (where $O$ is the origin), then the range of the $x$-coordinate of point $A$ is...
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{6}{5}, 2]}
How many such nine-digit positive integers can Ricardo make if he wants to arrange three 1s, three 2s, two 3s, and one 4 with the properties that there is at least one 1 before the first 2, at least one 2 before the first 3, and at least one 3 before the 4, and no digit 2 can be next to another 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{254}
Determine the number of integer solutions to the equation $x^6 - 15x^5 - 10x^3 + 30x + 720 = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Divide the numbers 1 through 10 into three groups such that the difference between any two numbers within the same group does not appear in that group. If one of the groups is {2, 5, 9}, then the sum of all the numbers in the group that contains 10 is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
A rectangle with integer side-lengths is divided into four smaller rectangles. The perimeters of the largest and smallest of these smaller rectangles are 28 cm and 12 cm. What is a possible area of the original rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
给定整数 $n(n \geq 2)$. 对于一个 $2 n$ 元有序数组 $$ T=\left(a_{1}, b_{1}, a_{2}, b_{2}, \cdots, a_{n}, b_{n}\right), $$ 若 $T$ 的每个分量均为 0 或 1 ，且对任意 $p, q(1 \leq p<q \leq n)$ ，均有 $\left(a_{p}, b_{p}, b_{q}\right) \neq(1,0,1)$ 且 $\left(a_{q}, b_{q}, a_{p}\right) \neq(1,0,0)$, 则称 $T$ 为 "有趣数组". 求有趣数组的个数。请提供当 n = 2 时的答案。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Create a fraction set $A_1$ with numerators being the integers in the interval $(0, m)$ ($m > 1$, $m \in \mathbb{N}$) and the denominator being $m$, with the sum of all elements denoted by $a_1$. Construct another fraction set $A_2$ with numerators being the integers in the interval $(0, m^2)$ ($m > 1$, $m \in \mathbb{N}$) and the denominator being $m^2$, such that $A_2$ contains no elements from set $A_1$, with the sum of all elements denoted by $a_2$. Continuing in this manner, create the fraction set $A_n$ using the integers in the interval $(0, m^n)$ as numerators ($m > 1$, $m \in \mathbb{N}$) and $m^n$ as the denominator, ensuring that $A_n$ includes no elements from $A_1$, $A_2$, ..., $A_{n-1}$, with the sum of all elements denoted by $a_n$. Find the sum $a_1 + a_2 + ... + a_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{m^n-1}{2}}
In Wonderland, some cities are connected by roads so that one can reach any city from any other city, possibly by passing through other cities. No two cities are connected by more than one direct road.

A road is called "strange" if closing it would prevent travel between some pair of cities. It is known that there are 45 roads in total in the country, out of which 42 roads are strange. How many cities can there be in the country? If there are multiple correct answers, list them separated by commas or semicolons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
You are playing a game called "Hovse." Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ is written on the board, you can perform one of the following operations:

- Replace $x$ with $3x + 1$
- Replace $x$ with $9x + 1$
- Replace $x$ with $27x + 3$
- Replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor$

You are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
设 $\ln \left(1+x+x^{2}+x^{3}\right)$ 的考克劳林展开式为 $\sum_{n=0}^{\infty} a_{n} x^{n}$, 则 $\left[\left|a_{2024}\right|^{-1}\right]=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{674}
Siva has the following expression, which is missing operations:  $$ \frac12 \,\, \_  \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_  \,\,\frac{1}{16} \,\, \_  \,\,\frac{1}{32}. $$   For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than  $\frac12$  can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a$ ,  $b$ , then find  $a + b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
A wheel has a certain radius. The distance covered by the wheel in making 100 revolutions is 91.77 meters. What is the radius of the wheel in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.61}
A bus departed from city $M$ to city $N$ at a speed of 40 km/h. After a quarter of an hour, it encountered a car traveling from city $N$. This car reached city $M$, waited for 15 minutes, and then returned to city $N$, overtaking the bus 20 km from city $N$. Find the distance between cities $M$ and $N$ if the car's speed is 50 km/h.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160 \, \text{km}}
Let  $I, T, E, S$  be distinct positive integers such that the product  $ITEST  =  2006$ .  What is the largest possible value of the sum  $I  +  T  +  E  +  S  +  T  +  2006$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2086}
What is the minimum percentage increase in the mean of set G { -7, -5, -4, -1, 0, 6, 9, 11 } if its four smallest elements are replaced with four different prime numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{377.78\%}
Find the number of positive integers $k \leq 2018$ for which there exist integers $m$ and $n$ such that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+d e f$ 是奇数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Positive integers  $a_1, a_2, \ldots, a_{101}$  are such that  $a_i+1$  is divisible by  $a_{i+1}$  for all  $1 \le i \le 101$ , where  $a_{102} = a_1$ . What is the largest possible value of  $\max(a_1, a_2, \ldots, a_{101})$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \geq 3$, $$A_{n}=\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\frac{1}{n^{4}-n^{2}}$$ Compute $\lim _{N \rightarrow \infty} A_{N}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{6}-\frac{\pi^{2}}{12}}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $ac = \frac{1}{4}b^2$, $\sin A + \sin C = t \sin B$, and angle $B$ is acute, the range of the real number $t$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{\sqrt{6}}{2}, \sqrt{2} \right)}
A cat broke into a wine store at night. It jumped onto a shelf where wine bottles were arranged in a long row - the first third of bottles were priced at 160 CZK each, the next third were priced at 130 CZK each, and the last third at 100 CZK each. First, the cat knocked over a bottle priced at 160 CZK, which was at the very beginning of the row, and then continued knocking over one bottle after another without skipping any. It knocked down a total of 25 bottles before it stopped. In the morning, the owner regretted that the cat did not start its rampage from the other end of the shelf. If it had broken the same number of bottles from the other end, the damage would have been 660 CZK less. How many bottles were originally on the shelf?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
John decides to buy some birds. He received $50 each from his 4 grandparents. In the pet store, parrots cost $30, pigeons cost $20, and canaries come at $15 each. Assuming all birds have 2 wings, and he spent all his money, how many wings did all the birds he bought have, if he bought at least one of each type of bird?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}