inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the minimum value of \( M \) such that the inequality \( \cos \alpha_{1} \cos \alpha_{2} \cdots \cos \alpha_{n} + \sin \alpha_{1} \sin \alpha_{2} \cdots \sin \alpha_{n} \leqslant M \) holds for all angles \( \alpha_1, \alpha_2, \ldots, \alpha_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the eigenvalues of the matrix $A = (p_{ij})$ where $\sum_{i=1}^3 p_{ij} = 1$ for all $j = 1, 2, 3$. Assume $A$ is a $3 \times 3$ matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct ways to 4-color the vertices of a square such that adjacent vertices have different colors, considering colorings equivalent under rotations and reflections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the integral \( \int_{0}^{3} (x^2+1) \, d[x] \), where \([x]\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Tim has a working analog 12-hour clock with two hands that run continuously. He also has a clock that runs at half the correct rate. At noon one day, both clocks show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Determine the number of infinite arithmetic sequences of natural numbers such that the ratio of the sum of the first $2n$ terms to the sum of the first $n$ terms is constant for all $n$, and one of the terms in the sequence is 1971.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e. there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of the players can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
On an island, there are chameleons of five colors. When one chameleon bites another, the color of the bitten chameleon changes according to some rule, where the new color depends only on the color of the biting and the bitten chameleon. It is known that \$ 2023 \$ red chameleons can agree on a sequence of bites such that they all become blue. What is the smallest \$ \mathrm{k} \$ that guarantees that \$ \mathrm{k} \$ red chameleons can agree on a sequence of bites to become blue?

For example, the rules could be: if a red chameleon bites a green one, the bitten changes color to blue; if a green bites a red, the bitten remains red; if a red chameleon bites a red one, the bitten changes to yellow, and so on. (The specific color change rules may vary.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the minimum number of squares that must be initially planted with grass in an n x n square garden, divided into a grid by footpaths, such that if at least two neighboring squares have full grass coverage, the grass will eventually spread to the entire garden.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Let \( S = \left\{ p_1 p_2 \cdots p_n \mid p_1, p_2, \ldots, p_n \text{ are distinct primes and } p_1, \ldots, p_n < 30 \right\} \). Assume 1 is in \( S \). Let \( a_1 \) be an element of \( S \). We define, for all positive integers \( n \):
\[ a_{n+1} = \frac{a_n}{n+1} \quad \text{if } a_n \text{ is divisible by } n+1; \]
\[ a_{n+1} = (n+2) a_n \quad \text{if } a_n \text{ is not divisible by } n+1. \]
How many distinct possible values of \( a_1 \) are there such that \( a_j = a_1 \) for infinitely many \( j \)'s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
In triangle \( \mathrm{KLM} \) with angle \( \mathrm{L} = 120^{\circ} \), the angle bisectors \( \mathrm{LA} \) and \( \mathrm{KB} \) of angles \( \mathrm{KLM} \) and \( \mathrm{LKM} \) respectively are drawn. Find the measure of angle \( \mathrm{KBA} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Given that $a_n(X_n-\theta)-a_n(Y_n-\theta)\xrightarrow[d]{}Z$, $a_n\xrightarrow[]{}\infty$, and $X_n, Y_n\xrightarrow[p]{}\theta$, with $o(X_n-\theta)= (X_n-\theta)\cdot o(1)$, determine if $a_n(X_n-\theta)\cdot \phi_n-a_n(Y_n-\theta)\cdot \phi_n\xrightarrow[d]{}0$ when $\phi_n$ is a sequence that converges to zero arbitrarily slowly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A train of a certain length is running with a speed of 60 km/hr. In 14.998800095992323 seconds, it will pass a man who is running at 6 km/hr in the direction opposite to that in which the train is going. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{275}
Determine the rank of the free group that is the fundamental group of the covering space \( \hat{X} \rightarrow X \), where \( X \) is a bouquet of 3 circles and the covering space is of degree 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given the hyperbola \( \frac{(3x-4y-1)^2}{100} - \frac{(4x+3y-1)^2}{225} = 1 \), find the length of the latus rectum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given the circle $O: x^2 + y^2 = 1$ and the functions $f(x) = x^3$, $f(x) = \tan(x)$, and $f(x) = x\sin(x)$, how many of these functions have graphs that can bisect the area of circle $O$ due to their symmetry properties? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the degrees of freedom (DOF) for a natural cubic spline with knots at \{10, 20, 30, 50\}, given that the curve is constrained to be linear before the first knot and after the last knot.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for 8 hours a day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand and the hour hand to coincide once. Therefore, the factory underpays each worker by ____ yuan every day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.6}
Janet, a third grade teacher, is picking up the sack lunch orders from a local deli for the field trip she is taking her class on. There are 35 children in her class, 5 volunteer chaperones, and herself. She she also ordered three additional sack lunches, just in case there was a problem. Among the lunch orders, 10 are vegetarian, 5 are gluten-free, and 2 are both vegetarian and gluten-free. The regular sack lunches cost $7, while the vegetarian and gluten-free ones cost $8 each, and the lunches meeting both dietary restrictions cost $9 each. Calculate the total cost of all the lunches considering the various dietary restrictions and preferences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$323}
20. N6 (POL) ${ }^{1 \mathrm{MO}}$ Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2, \ldots, 2 p\}$ such that the sum of all elements of $A$ is divisible by $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{p}(\binom{2p}{p}-2)+2}
In a school, a certain percentage of students study sociology, 44% study mathematics, and 40% study biology. If 30% of students study both mathematics and sociology, the largest possible number of students who study biology but do not study either mathematics or sociology is 180. How many students are there in the school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
How many 4-permutations can be formed from the letters in the word "aaabbccdef"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{758}
Given the linear regression model with the design matrix \(X = [x_1 \mid x_2 \mid \cdots\mid x_p] \in\mathbb R^{n\times p}\), where the columns of \(X\) are centered (i.e., \(x^T_j 1_n = 0\) for \(j = 1, \ldots , p\)), and the model does not include an intercept, assume the following covariance matrix:
\[
\frac{1}{n}X^TX=\begin{bmatrix}I_{p-1} & z\\z & a^2\end{bmatrix}, z\in \mathbb{R}^{p-1}, a^2\in \mathbb{R}
\]
where \(a^2=10\|z\|^2\). Let \(R^2_j\) be the \(R^2\) obtained when regressing \(x_j\) onto the rest of the covariates \(\{x_1, \ldots , x_{j−1}, x_{j+1}, \ldots , x_p\}\). Find \(R^2_p\), the coefficient of determination resulting from regressing \(x_p\) onto \(\{x_1, \ldots , x_{p−1}\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
The "Wuyang" building, if constructed jointly by team A and team B, would need more than 6 months. If constructed by team A and team C, it would need more than 5 months. If constructed by team B and team C, it would need more than 4 months. Calculate the maximum number of months it would take for teams A, B, and C to build it together. Express your answer as a whole number of months.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the least number that must be subtracted from 427398 so that the remaining number is divisible by a certain divisor. The least number is 3. What is the divisor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of zeros in the right half-plane for the polynomial \(z^4 + 2z^3 - 2z + 10\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest  $\lambda \in \mathbb{R}$  such that for all  $n \in \mathbb{N}_+$ , there exists  $x_1, x_2, \ldots, x_n$  satisfying  $n = x_1 x_2 \ldots x_{2023}$ , where  $x_i$  is either a prime or a positive integer not exceeding  $n^\lambda$  for all  $i \in \left\{ 1,2, \ldots, 2023 \right\}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{2024}}
Given a set $X$ with at most $\kappa$ operations, each having arity less than $\kappa$, and a subset $S$ of $X$ with cardinality less than or equal to $\kappa$, determine the maximum cardinality of the substructure of $X$ generated by $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\kappa^{<\kappa}}
Calculate the genus of a three-component unlink using the Seifert surface method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that \( \alpha, \beta, \gamma, \delta \) are the roots of the polynomial \( x^4 + 4x^3 - 6x^2 + 7x - 9 = 0 \), find the value of the product \( \prod(1+\alpha^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given four functions: $①y=-x$, $②y=- \frac{1}{x}$, $③y=x^{3}$, $④y=x^{ \frac{1}{2}}$, determine the probability of the event "the graphs of the two selected functions have exactly one common point" when two functions are selected from these four.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Given the power function $y=(m^2-9m+19)x^{2m^2-7m-9}$, the graph does not pass through the origin. Then the value of $m$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( S = \{1, 2, 3, \ldots, 20\} \) be the set of all positive integers from 1 to 20. Suppose that \( N \) is the smallest positive integer such that exactly eighteen numbers from \( S \) are factors of \( N \), and the only two numbers from \( S \) that are not factors of \( N \) are consecutive integers. Find the sum of the digits of \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
若${{x}_{1}}$满足$3x+{{3}^{x-1}}=7$,${{x2}}$满足$3x+3{{\log }_{3}}\left( x-2 \right)=7$,则${{x1}}+{{x}_{2}}=$__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{13}{3}$}
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 7$.  If $YZ = 100$, then what is $XY$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70\sqrt{2}}
Find the determinant of a matrix that is similar to matrix \( A_n \), but with non-zero elements located on the left below the main diagonal instead of the bottom left corner.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
整数 $\left[\frac{10^{93}}{10^{31}+3}\right]$ 的末两位数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
8. (for 9th grade. 2 points). In triangle $A B C$, angle $A$ is equal to $40^{\circ}$. The triangle is randomly thrown onto a table. Find the probability that vertex $A$ will be to the east of the other two vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{18}}
The linear equation $Ax + By = 0$: if we choose any two different numbers from the set $\{1, 2, 3, 6, 7, 8\}$ for the values of $A$ and $B$, then the number of distinct lines that can be represented is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Evaluate the convolution integral \( \int_{-\infty}^{+\infty} \delta'(a-t) g(t) \, dt \) given that \( \int_{-\infty}^{+\infty} \delta'(t) g(t) \, dt = -g'(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g'(a)}
To investigate the number of students from various classes participating in an extracurricular calligraphy group, 5 classes were randomly selected from the school, and the number of students from each class participating in the group was taken as sample data. It is known that the sample mean is 7, and the sample variance is 4, and the sample data are all different. The maximum value among the sample data is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Suppose that $f(x)$ and $g(x)$ are functions which satisfy the equations $f(g(x)) = 2x^2$ and $g(f(x)) = x^4$ for all $x \ge 1$. If $g(4) = 16$, compute $[g(2)]^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95.44\%}
What is the term for a lattice that is isomorphic to its dual lattice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{self-dual lattice}
Three equal circles are placed in triangle \(ABC\), each touching two sides of the triangle. All three circles have one common point. Find the radii of these circles if the radii of the inscribed and circumscribed circles of triangle \(ABC\) are \(r\) and \(R\), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{rR}{R+r}}
Arrange 6 volunteers $A$, $B$, $C$, $D$, $E$, $F$ to take care of 3 elderly people $X$, $Y$, $Z$. Each pair of volunteers takes care of one elderly person. Considering the distance between the volunteers' and the elderly people's residences, volunteer $A$ cannot be assigned to take care of elderly person $X$, and volunteer $B$ cannot be assigned to take care of elderly person $Y$. How many different arrangements are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given that D is 1.3 meters, C is 0.6 meters, and the angle \( a \) is 30 degrees with a rate of change \( a' = 150 \) rad/s, calculate the rate of change of M. Use the following relationships:

\[ M = \cos(30^\circ) \cdot 0.6 + \cos(13.3424) \cdot 1.2 \]

\[ M' = -\left(\sin(30^\circ) \cdot 0.6 \cdot a' + \sin(13.3424) \cdot 1.3 \cdot B'\right) \]

where \( B' \) is calculated using:

\[ B' = \frac{\cos(30) \cdot 0.6 \cdot 150}{\cos(13.3424) \cdot 1.3} \]

Determine \( M' \) in rad/s or m/s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-63.5}
Evaluate the limit \( \lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1+\cos^2(nx)} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Ned was helping the cafeteria workers pick up lunch trays, but he could only carry 5 trays at a time. He needs to pick up trays from 4 tables with different numbers of trays on each table. The first table has 7 trays, the second table has 10 trays, the third table has 12 trays, and the fourth table has 3 trays. How many trips will he make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \(I\) be the center of the inscribed circle in triangle \(ABC\). We assume that \(AB = AC + CI\). Determine the value of the ratio \(\frac{\widehat{ACB}}{\widehat{ABC}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $ABCDEF$ be a convex hexagon, whose opposite angles are parallel, satisfying $AF = 3$, $BC = 4$, and $DE = 5$. Suppose that $AD$, $BE$, $CF$ intersect in a point. Find $CD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The probability of snow for each of the next four days is $\frac{3}{4}$. However, if it snows any day before the last day, the probability of snow on the last day increases to $\frac{4}{5}$. What is the probability that it will snow at least once during these four days? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1023}{1280}}
Find the singular integral of the differential equation \((xy^{\prime}-y)^{2}=x^{2}(x^{2}-y^{2})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y^2 = x^2}
Let $C$ be the answer to problem $18$. Suppose that $x$ and $y$ are real numbers with $y > 0$ and

$$x + y = C$$
$$x +\frac{1}{y} = -2.$$
Compute $y +\frac{1}{y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{(C+2)^{2}+4}$.}
Is there a closed-form solution for this combinatorial limit of a sum?
$$\lim_{n\to\infty}\sum_{k=0}^n\binom nk\frac{3k}{2^n(n+3k)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/5}
Given real numbers $a$ and $b$ belong to $(0,1)$, and satisfy $(1-a)b > \frac{1}{4}$, then the relationship between $a$ and $b$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < b}
A set $S$ of natural numbers is called good, if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a good set which is a subset of the set $A=\{1,2,3, \ldots, 63\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{1}{4}$}
Determine the region of convergence for the power series \( \sum_{n=1}^{\infty} a_{n}(x-1)^{n} \), given that the sequence \( \{a_{n}\} \) decreases monotonically, \( \lim_{n \rightarrow \infty} a_{n}=0 \), and the series \( S_{n}=\sum_{k=1}^{n} a_{k} \) diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2)}
Alice claims, "The probability that I'm lying is greater than \( p \)." Determine the probability that Alice is lying.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at 0. Every minute, he has a $\frac{1}{3}$ chance of moving 1 unit left, a $\frac{1}{3}$ chance of moving 1 unit right and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to 0 (not including the start) before getting eaten.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{5}-5}{5}}
Evaluate the limit: $$\lim_{n\to\infty}\left(e^n \left(n-ne\sqrt[n]{\ln{(1+e^n)}-n}\right) \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate the sum \(\sum^{7}_{n=2} (n-1)P (X = n)\) for the random experiment of tossing a coin until the same number of heads and tails have been observed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{19}{16}}
On the radius \( A O \) of a circle with center \( O \), point \( M \) is chosen. On the same side of \( A O \) on the circumference, points \( B \) and \( C \) are chosen such that \( \angle A M B = \angle O M C = \alpha \). Find the length \( B C \), given that the radius of the circle is 15 and \( \cos \alpha = \frac{3}{5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In a convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(DB\) are perpendicular to sides \(DC\) and \(AB\) respectively. From point \(B\), a perpendicular is dropped to side \(AD\), intersecting \(AC\) at point \(O\). Find \(AO\) if \(AB = 4\) and \(OC = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of compact Hausdorff spaces of cardinality \( \kappa \), up to homeomorphism, where \( \kappa \) is an uncountable cardinal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^\kappa}
Alice has a deck of 36 cards, consisting of 4 suits with 9 cards each. She selects any 18 cards and gives the remaining 18 cards to Bob. In each turn, Alice picks one of her cards and places it face-up on the table, followed by Bob doing the same with one of his cards. Bob scores a point if the two cards have the same suit or the same value. What is the maximum number of points Bob can guarantee, regardless of Alice's actions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the triple integral of \( f(x,y,z) = (x^8 + y^6 + z^4)\sin(z^3) + 2y - 3 \) over the unit ball.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
Given the joint moment generating function of random variables $X$ and $Y$:
\[ M(t_1,t_2) = \frac{1}{2}e^{t_1+t_2} + \frac{1}{4}e^{2t_1+t_2} + \frac{1}{12}e^{t_2} + \frac{1}{6}e^{4t_1+3t_2} \]
find the probability $P(X < Y)$. Provide your answer as a simplified expression or numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Determine the Galois group of the polynomial \( f = x^5 + x^4 + x^3 + x \) over the field \( \mathbb{Z}_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_3}