inclusionAI/Ring-lite-rl-data huggingface dataset:
How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$ where $i,j,$ and $k$ are positive integers not exceeding four?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$76$
}
Determine the number of times a biased coin must be flipped to estimate the probability of it landing heads with an accuracy of 0.01 and a confidence level of 99%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16590}
Determine the number of real numbers required to describe all possible arrangements of a quantum register consisting of 3 qubits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
48%}
Let $X_1, X_2, X_3$ be independent random variables each uniformly distributed on the interval $(0, 1)$. Find the expected value of the expression $E\left[\frac{X_1 + X_2}{X_1 + X_2 + X_3}\right]$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
In a programming contest with 2008 participants, the participants are divided into two groups in each round. Determine the minimum number of rounds required so that every pair of participants has been in the same group at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the norm of the operator \( A: L^{1}[1,2] \to L^{1}[0,1] \) defined by \( (Af)(x) = \int_{1}^{2} \frac{f(t)}{x+t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
In triangle $\triangle ABC$, the in-centers are $I(0,0)$, $I_1(2,3)$, and $I_2(5,7)$, and the ex-center is $I_3$. Calculate the distance between the orthocenters of triangles $\triangle II_1I_3$ and $\triangle I_1I_2I_3$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{74}}
Find the second derivative $F''(\pi)$ for the function $$F(x)=\int_{x}^{\sin(x)} \left(\int_{0}^{\sin(t)}\sqrt{1+u^4}\text{d}u\right)\text{d}t.$$ Use the Fundamental Theorem of Calculus and the chain rule to evaluate this double integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of endomorphisms on the direct sum of the groups \( \mathbb{Z}_2 \) and \( \mathbb{Z}_4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Determine the minimum width required for an infinite strip of paper to be able to cut out any triangle with an area of 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{3}}
Given that the equation \( ax^3 + bx^2 + cx + d = 0 \) has three distinct solutions, determine the number of distinct solutions for the equation:
\[
4(ax^3 + bx^2 + cx + d)(3ax + b) = (3ax^2 + 2bx + c)^2.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the service time follows an exponential distribution with a rate of 12 customers per hour, calculate the probability that a customer waits for less than 3 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.4512}
Consider a square matrix \( M \) with dimensions \( n \times n \), containing only 0s and 1s. Each column of \( M \) sums to a constant \( k \). Additionally, the following equation holds: \( M^T M = kI + J \), where \( I \) is the identity matrix and \( J \) is a matrix of all 1s. Determine the relationship between \( n \) and \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n = k^2 - k + 1}
Solve the initial value problem: \( y' + y = \delta(t) \) with \( y(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-t} u(t)}
Let $L^2[0,10]=\{f:[0,10]\rightarrow \mathbb{R} \mid f \text{ is Lebesgue measurable and } \int_0^{10} f^2(x)\,dx < \infty \}$ be equipped with the norm $\| f \|=\left(\int_0^{10} f^2(x)\,dx \right)^{1/2}$. Let $T$ be the linear functional on $L^2[0,10]$ given by $T(f)=\int_0^2 f(x)\,dx - \int_3^{10} f(x)\,dx$. Find the norm $\| T \|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
An explorer is trapped in a cave with three tunnels. The first tunnel leads to the surface after 1 day, the second leads back to the cave in 4 days, and the third leads back to the cave in 2 days. The explorer chooses the first, second, and third tunnels with probabilities 0.2, 0.3, and 0.5, respectively. Given that the expectation of the number of days until the explorer is free has been calculated, compute the variance of the number of days until the explorer is free.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Determine the values of $p$ for which the series \( \sum_{n=N}^\infty \frac{1}{n(\log(n))(\log\log(n))\dotsm(\log_{j}(n))(\log_{j+1}(n))^p} \) converges, where $N$ is a positive integer and $\log_j(n)$ denotes the $j$th iteration of the logarithm function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
Given that each particle maintains a direction towards the particle at the next corner, the time when particles will meet each other is given by \( \frac{2a}{\sqrt{n} v} \). Find the value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that the solution set of $f(\cos x) \geq 0$ is $\left[0, \frac{\pi}{2}\right]$, find the solution set of the inequality $f(\sin x) \geq 0$. Provide your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \pi]}
For each integer $  n \geq 2  $, let $  S_n  $ be the sum of all products $  j k  $, where $  j  $ and $  k  $ are integers and $  1 \leq j < k \leq n  $. What is the sum of the 10 least values of $  n  $ such that $  S_n  $ is divisible by 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197

}
Determine the center of gravity of a hemispherical shell with radius $a$, assuming a uniform density of 1. The shell is positioned with its base on the $xy$-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a}{2}}
Solve the equation for $x$ without using inverse functions or a calculator: $$2 - 3 \cos 2x - 5 \sin 2x + \cos 4x - \sin 4x = 0$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = \pi n}
Determine all possible values of \(\frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}\) for positive reals \(a, b, c, d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Reese joined a Pinterest group where members contributed an average of 10 pins per day. The group owner deleted older pins at the rate of 5 pins per week per person. The group has 20 people. After Reese had been a member for a month, there were 6,600 pins. What was the initial total number of pins in the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Find \( g(2022) \) if for any real numbers \( x, y \) the following equality holds:

\[ g(x-y) = g(x) + g(y) - 2021(x+y) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4086462}
$M$ is the midpoint of the height $B D$ in an isosceles triangle $A B C$. Point $M$ serves as the center of a circle with radius $M D$. Find the angular measure of the arc of the circle enclosed between the sides $B A$ and $B C$, given that $\angle B A C = 65^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100^\circ}
设 $a, b$ 为实数, 若一元二次不等式 $x^{2}+b<a x$ 的解集为 $(1,5)$, 请计算一元二次不等式 $x^{2}-a>b x$ 的解集的区间端点和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A tournament in Anchurian checkers is conducted over multiple rounds. If the number of participants in a round is even, they are split into random pairs. If the number is odd, random pairs are created, and one player is randomly left out and does not participate in the round. No ties are possible, and the loser of each pair is eliminated from the tournament, while the winners and the player without a pair, if any, move to the next round under the same rules. This continues until only two players remain, who then play a final match to determine the tournament winner.

The tournament started with 26 participants, all of whom play equally well, meaning each player has an equal chance of winning against any opponent. Among these participants are Denis and Oleg. Find the probability that Denis and Oleg will play against each other at some point during the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{13}}
What is the minimum number of sides that an odd-sided polygon (not necessarily convex) can have, which can be divided into parallelograms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the phrases "虚有其表", "表里如一", "一见如故", and "故弄玄虚", each Chinese character represents one of 11 consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. Additionally, "表" > "一" > "故" > "如" > "虚", and the sum of the numbers represented by the four characters in each phrase is 21. What is the maximum value that "弄" can represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
How many distinct 3x3 matrices with entries in \(\{0,1\}\) are there such that each row is non-zero, distinct, and each matrix is of real rank 2 or 3, considering that if a matrix \(M\) is included in the count, \(PM\) where \(P\) is a permutation matrix, is not included?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$  $7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$ When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$ $N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
The integers \(a, b,\) and \(c\) satisfy the equations \(a + 5 = b\), \(5 + b = c\), and \(b + c = a\). What is the value of \(b\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10}
Given the numbers \(\log _{\left(\frac{x}{2}-1\right)^{2}}\left(\frac{x}{2}-\frac{1}{4}\right)\), \(\log _{\sqrt{x-\frac{11}{4}}}\left(\frac{x}{2}-1\right)\), \(\log _{\frac{x}{2}-\frac{1}{4}}\left(x-\frac{11}{4}\right)^{2}\). For which \(x\) are two of these numbers equal, and the third one greater than them by 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute: $\left(\frac{1}{2} \right)^{3} \cdot \left(\frac{1}{7} \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{56}}
Given that $\tan \alpha = \frac{1}{3}$ and $\tan (\alpha + \beta) = \frac{1}{2}$, find the value of $\tan \beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{7}}
Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A collection of coins includes only pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). What is the smallest possible number of coins in the collection that can be used to create any amount of money less than one dollar (100 cents)? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1, 1.5, and 3 years. What is the average length of time the boat can remain at sea?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.35}
Find the binormal vector for a curve given by the parametric equations: $x = 5\sin t$, $y = 2$, $z = 5\cos t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1,0)}
A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents.

Find the ratio of the area of the largest circle to the combined area of all the other circles. The original answer is in the format \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The walls of a room are in the shape of a triangle  $ABC$  with  $\angle ABC = 90^\circ$ ,  $\angle BAC = 60^\circ$ , and  $AB=6$ .  Chong stands at the midpoint of  $BC$  and rolls a ball toward  $AB$ .  Suppose that the ball bounces off  $AB$ , then  $AC$ , then returns exactly to Chong.  Find the length of the path of the ball.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{21}}
Let the set \( I = \{1, 2, \cdots, n\} (n \geqslant 3) \). If two non-empty proper subsets \( A \) and \( B \) of \( I \) satisfy \( A \cap B = \varnothing \) and \( A \cup B = I \), then \( A \) and \( B \) are called a partition of \( I \). If for any partition \( A \) and \( B \) of the set \( I \), there exist two numbers in \( A \) or \( B \) such that their sum is a perfect square, then \( n \) must be at least \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A flea jumps on the number line. It starts at 0, each jump has a length of 1, and the next jump has a probability $p$ of being in the same direction as the previous one, and a probability $1-p$ of being in the opposite direction. What is the probability that it returns to 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Convex quadrilateral $ABCD$ satisfies $\angle{CAB} = \angle{ADB} = 30^{\circ}, \angle{ABD} = 77^{\circ}, BC = CD$ and $\angle{BCD} =n^{\circ}$ for some positive integer $n$. Compute $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68^\circ}
A modified octahedron consists of two pyramids, each with a pentagonal base, glued together along their pentagonal bases, forming a polyhedron with twelve faces. An ant starts at the top vertex and walks randomly to one of the five adjacent vertices in the middle ring. From this vertex, the ant walks again to another randomly chosen adjacent vertex among five possibilities. What is the probability that the second vertex the ant reaches is the bottom vertex of the polyhedron? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
In every acyclic graph with 2022 vertices we can choose  $k$  of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1517}
The base of a pyramid is an isosceles triangle with a side length of \(a\) and a vertex angle of \(\alpha\). All the lateral edges are inclined to the plane of the base at an angle \(\beta\). Find the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{a^3 \sin \frac{\alpha}{2} \tan \beta}{6} }
数列 $\left\{a_{n}\right\}$ 满足: $a_{1}=\frac{5}{2}, a_{n+1}=a_{n}^{2}-2$, 求 $\left[a_{2023}\right]$ 除以 7 的余数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A train 120 m long is running at 60 kmph. It takes 21.598272138228943 seconds to pass a platform of a certain length. How long is the platform?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Bret sets off on his journey to Boston, which includes a total of five train transfers. His entire trip takes 20 hours. On the first train, he reads a book for 2 hours and spends another 1 hour eating dinner. On the second train, he watches movies on his computer for 3 hours and spends 1 hour chatting with a fellow passenger. During the transfer between the second and third trains, he spends 45 minutes browsing a souvenir shop and 30 minutes waiting for the next train.

On the third train, he spends 2 hours working on his laptop and takes a 1-hour break to read a magazine. When transferring to the fourth train, he encounters a 1-hour delay and spends the time listening to music. On the fourth train, he spends an additional 1 hour browsing the internet on his phone. During the final transfer, he has a 30-minute phone call with a friend while waiting for the last train.

On the fifth train, he plays a game on his phone for an hour and 30 minutes before deciding to take a nap. How many hours does he have left to take a nap?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.75}
Let \( n \) be a positive integer. There are \( 2018n + 1 \) cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city \( C \) and for each integer \( 1 \leq i \leq 2018 \), there are exactly \( n \) cities that are a distance \( i \) away from \( C \). (The distance between two cities is the least number of roads on any path between the two cities.) For which \( n \) is it possible for Mark to achieve this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ even}}
A circle given by the equation \(x^{2}+(y+1)^{2}=3\) is rotated about the line \(kx-y-1=0\) for one full revolution. Determine the surface area of the resulting geometric shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12π}
Tina is a professional boxer. She wins her first 10 fights of her career. She then goes on to win some more before losing her first fight, and then doubles her number of wins before losing again. She then retires. At the end of her career, she has 28 more wins than losses. How many fights did she win before losing her first fight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
3. An electronic clock displays time from 00.00.00 to 23.59.59. How much time during a day does the clock display exactly four digit 3s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
One hundred and one of the squares of an $n \times n$ table are colored blue. It is known that there exists a unique way to cut the table into rectangles along the boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:
[list]
[*] no two consecutive digits are equal, and
[*] the last digit is a prime.
[/list]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\cdot9^{n-1}}
已知数列 $\left\{a_{n}\right\}$ 满足 $a_{n}=\sqrt{\frac{2 n-1}{4 n^{2}+1}}$, 前 n 项和为 $S_{n}$, 求与 $S_{128}-S_{32}$ 最接近的整数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What fractional part of the entire square is shaded?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Let \( \{a_n\} \) be a sequence of positive numbers with the sum of its first \( n \) terms being \( b_n \). Let \( \{b_n\} \) be such that the product of its first \( n \) terms is \( c_n \), and \( b_n + c_n = 1 \). Find the number in the sequence \( \{1/a_n\} \) that is closest to 2002.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1980}
Determine the number of inequivalent spin structures on the manifold \( M = \#_{k}(S^2 \times S^1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
Your math teacher asks you to rationalize the denominator of the expression $\frac{a}{b + \sqrt{c}}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. You find that $\frac{a}{b + \sqrt{c}}$ is equal to $\frac{30 - 5\sqrt{14}}{11}$. Compute the triple $(a,b,c)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
(5,5,14)}
Determine the signature of the quadratic form given by \( f(x, y, z) = xy + yz + xz \). The quadratic form is represented as a matrix product: \( f(x,y,z) = \begin{bmatrix}x&y&z\end{bmatrix} \cdot \begin{bmatrix}0&1/2&1/2\\1/2&0&1/2\\1/2&1/2&0\end{bmatrix} \cdot \begin{bmatrix}x\\y\\z\end{bmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Find the largest positive integer \( n \) such that for any real number \( c \), there exists at most one base-\( n \) polynomial \( P(x) \) for which \( P(\sqrt{2} + \sqrt{3}) = c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the group that \( \langle (135)(246), (12)(34)(56) \rangle \subset S_6 \) is isomorphic to, using only the groups \( D_n, S_n, \mathbb{Z}_n \), and their direct sums, where \( S_n \) is the permutation group of \( n \) elements, and \( D_n \) is the dihedral group of order \( 2n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_6}
A fisherman continues fishing until the weight of the last fish caught is less than the one before. Calculate the probability that he catches exactly $k$ fish.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{k - 1}{k!}}
Let $X=(X,d)$ be a metric space and $r \ge 0$. Determine the smallest possible upper bound for $\inf_T \text{Lip}(T)$, where $\text{Lip}(T) := \underset{x,x' \in X,\;x \ne x'}{\sup}\; \frac{d(T(x),T(x'))}{d(x,x')}$, and the infimum is taken over all functions $T:X \rightarrow X$ such that $\underset{x \in X}{\sup}\;d(T(x),x) \le r$. Provide your answer as a single expression or number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}