inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a matrix of size $n\\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.\n\n[i]Proposed by Marcin Kuczma, Poland[\/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{n}{2}$ 

}
Evaluate the double integral \( \iint_D \left|x^2 + y^2 - 2y\right| \, d\sigma \), where \( D = \{(x, y) \mid x^2 + y^2 \le 4\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\pi}
Square $ABCD$ has side length $2$. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$. Compute the area of pentagon $AFECD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given a Markov chain $(X_n)_{n\geq1}$ with state space $S=\{A,B,C,D,E \}$ and the transition matrix
$$ P = \left( \begin{matrix} 0.6 & 0.4 & 0 & 0 & 0 \\ 0.3 & 0.7 & 0 & 0 & 0 \\ 0.2 & 0 & 0.4 & 0 & 0.4 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} \right) $$
compute $\lim_{n \to \infty} P(X_n=A|X_0=C)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
Given the following transition matrix for a Markov chain with state space $S=\{1,2,3,4,5\}$:
\[
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 \\
0 & 0 & \frac{4}{5} & \frac{1}{5} & 0 \\
0 & \frac{1}{6} & \frac{2}{3} & 0 & \frac{1}{6} \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\]
Calculate the probability that the Markov chain is absorbed at state 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Find \(\lim_{x \to \infty} x^2\big(\ln(x\cot^{-1}(x))\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Find a polynomial \( g(t) \) such that the linear map \( g(T) \) has the characteristic polynomial \( p(t) = (11-t)^r(17-t)^{n-r} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17 - 6t}
Find the permutation $x$ such that for $y = (1274)(356)$, the equation $x^{-1}yx = (254)(1736)$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2\ 3\ 7)(4\ 6)}
Find the harmonic conjugate of the function \( v(r,\theta) = -\ln r \) in polar coordinates, where \( r \) and \( \theta \) are the radial and angular coordinates, respectively.
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Answer: \boxed{\theta}
On the sides $AB$ and $BC$ of triangle $ABC$, points $K$ and $M$ are chosen such that the quadrilaterals $AKMC$ and $KBMN$ are cyclic , where $N = AM \cap CK$ . If these quads have the same circumradii, find$\angle ABC$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\angle ABC = 45^{\circ}$}
Calculate the sum \( \sum_{k \in S(m,n)} \varphi(k) \) where \( S(a,b) = \{ k \in \mathbb{Z} \mid a \equiv b \pmod{k} \} \) and \( \varphi \) is Euler's totient function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|m - n|}
Evaluate the line integral \( \int_{C} \frac{x \, dy - y \, dx}{x^{2}+y^{2}} \) along the oriented broken line \( C \) with vertices \((2,-2)\), \((4,4)\), \((-5,5)\) oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Cheburashka has a set of 36 stones with masses of 1 g, 2 g, ..., 36 g, and Shapoklyak has super glue, with which one drop can glue two stones into one (it is possible to glue three stones with two drops and so on). Shapoklyak wants to glue the stones in such a way that Cheburashka cannot choose one or more stones with a total mass of 37 g from the resulting set. What is the minimum number of glue drops she needs to carry out her intention?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given circle \( O: x^{2} + y^{2} = 4 \) and curve \( C: y = 3|x - t| \). Points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^{*})\) are on curve \( C \). It is required that the ratio of the distance from any point on circle \( O \) to point \( A \) to the distance to point \( B \) is a constant \( k \) (\( k > 1 \)). Find the value of \( t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set \( A = \{ f: \mathbb{R} \to \mathbb{R} \mid f \text{ is continuous and } f(\mathbb{Q}) \subset \mathbb{Q} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Find the coefficient of \(x^{30}\) in the expansion of the generating function \(\frac{(1-x^{10})^4}{(1-x)^4}\) using the Negative Binomial Series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Let \( a \) be a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). Determine all natural numbers \( n \) such that \( a^{4} + a^{3} = a^{n} + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the limit: \[ L = \lim_{n \rightarrow \infty} \prod_{r=3}^{n} \frac{r^3 - 8}{r^3 + 8} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The perimeter of the pentagonal square ABCDE is 2000 meters. Person A and person B start from points A and C, respectively, at the same time and walk around the square in the direction A→B→C→D→E→A→… Person A walks at a speed of 50 meters per minute, and person B walks at a speed of 46 meters per minute. After how many minutes will person A and person B be walking on the same side for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
Identical regular pentagons are arranged in a ring. Each of the regular pentagons has a perimeter of 65. The regular polygon formed as the inner boundary of the ring has a perimeter of \( P \). What is the value of \( P \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Consider the complex function $f(z) = \frac{1}{(z-\alpha)^2(z-1)}$ where $\alpha \in \mathbb{C}$ with $\operatorname{Im}(\alpha) > 0$. Determine the principal value of the integral
$$P.V. \int_{-\infty}^{+\infty} \frac{\operatorname d x}{(x-\alpha)^2(x-1)}$$
using the definition of Cauchy's principal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{-i\pi}{(\alpha - 1)^2}\)}
Find the upper bound for the index of the core of a subgroup $H$ of finite index in a group $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Given that the line \( y = -2x + 1 \) is tangent to the curve \( f(x, y) = 2 \) at the point \( (1, 3) \), and \( f_x(1, 3) = 4 \), find the maximum directional derivative of \( f(x, y) \) in the direction of the vector \( (1, 3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Given a regular tetrahedron ABCD with edge length a (a > 3), points E, F, G are on edges AB, AC, AD respectively. Calculate the number of triangles EFG that satisfy EF = EG = 3 and FG = 2. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the value of $n$ such that the cyclotomic polynomial $\Phi_n(x)$ is given by $f(x) = (x^4 - x^2 + 1)(x^2 + 1)$. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given that $G/Z(G) \cong \mathbb{Z}_2 \times \mathbb{Z}_2$, determine the structure of the derived subgroup $G'$.
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Answer: \boxed{\mathbb{Z}_2}
Find the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \) where \( f(x) = |1 - 2x| \) and \( x \in [0, 1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the numbers from 1 to 20, players A and B take turns placing either a "+" or "-" sign in front of each number. The order of placing the signs is arbitrary. After all numbers have been assigned a sign, the absolute value of the resulting algebraic sum \( S \) is calculated. Player A aims to minimize \( S \), while Player B aims to maximize \( S \). What is the maximum value of \( S \) that Player B can guarantee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the ratio of the areas of the largest and smallest equilateral triangles that can be inscribed in a unit square, with vertices touching the sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 - 4\sqrt{3}}
In triangle $ABC$, $AB = 42$, $AC = 39$, and $BC = 45$. Points $E$ and $F$ are on sides $AC$ and $AB$ such that $AF = 21$ and $AE = 13$. Lines $CF$ and $BE$ intersect at $P$, and ray $AP$ meets $BC$ at $D$. Let $O$ be the circumcenter of triangle $DEF$, and $R$ its circumradius. Calculate $CO^2 - R^2$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Consider a connected graph $G = (V, E)$ with $n$ vertices, where all edges are part of $3$-cycles. The graph contains exactly $n-2$ $3$-cycles, and every vertex is part of at least one $3$-cycle. The minimum degree of the graph is $2$. Determine the maximum number of vertices in $G$ that have degree $2$. Provide your answer as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 2}
Fill in the three blanks in the denominators of the equation $$\frac {1}{(   )} + \frac {4}{(   )} + \frac {9}{(    )} = 1$$ with three positive integers so that the equation holds. The minimum sum of these three positive integers is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
For any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5040}
Let \( x \) be an irrational number such that \( 0 < x < 1 \). The decimal expansion of \( x \) is denoted as \( 0.c_1c_2c_3\ldots \). Define \( B(x) \) as the set of all subsequences of 6 consecutive digits from \( c_1c_2c_3\ldots \). For example, if \( x = \frac{1}{22} \), then \( B(x) = \{045454, 454545, 545454\} \). Determine the minimum number of elements in \( B(x) \) as \( x \) varies among all irrational numbers with \( 0 < x < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many fractions between \(\frac{1}{6}\) and \(\frac{1}{3}\) inclusive can be written with a denominator of 15?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the expected number of consecutive random number inputs required to unlock a lock with the combination 5556, given that the lock checks for the sequence without needing a reset between inputs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Carol starts with 7,125 blocks. She loses 45% of her blocks and then gives away ⅔ of the remaining blocks. How many blocks does Carol end up with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,306}
Paul made two bank transfers of $90 and $60 respectively. A service charge was added to each transaction. The second transaction was reversed (without the service charge). His account balance is now $307 if it was $400 before he made any of the transfers. What was the percentage of the service charge added to each transaction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\%}
Find the maximum value of the function \( F(x_1, x_2, \ldots, x_N) = - \sum_{i=1}^N x_i \log(x_i) \) subject to the constraint \( \sum_{i=1}^N x_i = 1 \) and \( x_i > 0 \) for all \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log N}
Given the numbers 1 through 5, determine the total number of permutations that respect the order constraints specified by the ordered sets \(s_1 = (1, 2, 3)\) and \(s_2 = (5, 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{m+n}{m}}
Calculate the expected value of the area under the trajectory of a Wiener process, $W_t$, above the line $y=t$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that \( x \) and \( y \) are real numbers greater than 10, the leading digit of \( \lg x \) is \( a \) and the trailing digit is \( b \); the leading digit of \( \lg y \) is \( c \) and the trailing digit is \( d \). Additionally, it is known that \( |1 - a| + \sqrt{c - 4} = 1 \) and \( b + d = 1 \). Find the value of \( x \times y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^7}
Find the largest natural number in which each non-edge digit is less than the arithmetic mean of the adjacent digits.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96433469}
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m + n - 2}
Evaluate the double integral \(\int\int (x-y)^2 \,dx\,dy\) over the region enclosed by the circle \(x^2 + y^2 = 4\) and the line \(y=x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Given an acute-angled triangle \(ABC\). Point \(M\) is the intersection point of its altitudes. Find angle \(A\) if it is known that \(AM = BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Evaluate
$\frac{\log_{10}20^2 \cdot \log_{20}30^2 \cdot \log_{30}40^2 \cdot \cdot \cdot \log_{990}1000^2}{\log_{10}11^2 \cdot \log_{11}12^2 \cdot \log_{12}13^2 \cdot \cdot \cdot \log_{99}100^2}$
.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
\left.\begin{array}{l}\text { The Law of Sines } \\ \text { [Midline of a Triangle] }\end{array}\right]

Two chords, equal to 10 and 12, are drawn from one point on a circle. Find the radius of the circle if the distance from the midpoint of the smaller chord to the larger one is 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{4}}
A mother rabbit bought seven drums of different sizes and seven pairs of drumsticks of different lengths for her seven bunnies. If a bunny sees that both its drum is bigger and its drumsticks are longer compared to those of any of its siblings, it starts drumming loudly. What is the maximum number of bunnies that can start drumming?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit: $$\lim_{n \rightarrow \infty} \int^{n}_{0} \left(1+\frac{x}{n}\right)^{-n} \log\left(2+ \cos\left(\frac{x}{n}\right)\right)\ \mathrm{d}x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
There is one black ball and one white ball in a bag. Each time a ball is taken from the bag, it is not returned, but instead, another black ball is added. What is the probability of drawing a black ball on the $n$-th draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{1}{2^n}}
If 80% of a certain number is greater than 70% of 60.00000000000001 by 30, what is the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90.00000000000001}
Determine a tight upper bound for the expression \(-p \log p\) in terms of the probability \(p\), where \(0 < p < 1\), especially when \(p\) is very close to 0, such as \(p < 1/e\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Consider a $3 \times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $33$. Determine the value of the center square, labeled $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
For a permutation \( p = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{9}}\right) \) of the digits \( 1,2,\ldots ,9 \) ,let \( s\left( p\right) \) denote the sum of the three3-digit numbers \( {a}_{1}{a}_{2}{a}_{3} \) ,
 \( {a}_{4}{a}_{5}{a}_{6} \) ,and \( {a}_{7}{a}_{8}{a}_{9} \) . Let \( m \) be the minimum value of \( s\left( p\right) \) subject to the condition that the units digit of \( s\left( p\right) \) is 0 . Let \( n \) denote the number of permutations \( p \) with \( s\left( p\right)  = m \) . Find \( \left| {m - n}\right| \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
162
}
Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{729}
Evaluate the integral \( \frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz \) where \(\mathcal{C}\) is the unit circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Compute the limit of the integral \( I = \lim_{n \to +\infty} \int_0^n \frac{x^n \text{e}^{-x}}{n!} \text{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given $a$, $b$, $c \in \mathbb{R}$, if $|a\cos^2x + b\sin x + c| \leq 1$ holds for $x \in \mathbb{R}$, then the maximum value of $|a\sin x + b|$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Simplify the expression \(\sum_{k=0}^{10} k \binom{10}{k} \binom{20}{10-k}\) using the binomial identity \(\sum_{k=i}^{n-j}\binom{k}{i}\binom{n-k}{j}=\binom{n+1}{j+1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 \dbinom{29}{9}}
Example 4.23 Make all permutations of 5 distinct elements $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$, where $a_{1}$ is not in the 1st or 2nd position, $a_{2}$ is not in the 2nd or 3rd position, $a_{3}$ is not in the 5th position, $a_{4}$ is not in the 4th or 5th position, and $a_{5}$ is not in the 3rd or 4th position. How many different permutations can be made?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given that \(\alpha\) and \(\beta\) are continuous and differentiable functions satisfying the following conditions:
\[\alpha'(x) = \beta(x),\]
\[\beta(x)^3 = (\alpha(x) - 1)^2,\]
\[\alpha(a) = 0,\]
\[\alpha(b) = 2,\]
compute the definite integral:
\[\int_a^b{\alpha(t)^2 \beta(t)^4 \,dt}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{15}}
Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\triangle AFD$ to the area of square $ABCE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{8}}
Inside a truncated cone with a height of 8, there is a sphere O1 with a radius of 2. The center of sphere O1 lies on the axis of the truncated cone, and sphere O1 is tangent to both the upper base and the lateral surface of the truncated cone. Another sphere O2 with a radius of 3 can be placed inside the truncated cone such that sphere O2 is tangent to sphere O1, the lower base, and the lateral surface of the truncated cone, each at a single point. Excluding sphere O2, what is the maximum number of additional spheres with a radius of 3 that can be placed inside the truncated cone? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Using the digits 1, 2, 5, 6, and 9 exactly once, form five-digit even numbers in which no digit is repeated. Calculate the difference between the largest and smallest of these numbers. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83916}
Define the sequence of decimal integers \( a_n \) as follows: \( a_1 = 0 \); \( a_2 = 1 \); \( a_{n+2} \) is obtained by writing the digits of \( a_{n+1} \) immediately followed by those of \( a_n \). When is \( a_n \) a multiple of 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 1 \pmod{6}}
You have $20,000 to invest in 5 different stocks, with the condition that you can only invest in multiples of $1,000 and not all the money can be put into one stock. How many different ways can you invest the $20,000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10621}
For what values of $m$ is the function \[ f\left(\begin{array}{c} x\\ y \end{array}\right)=\begin{cases} \frac{y^{m}\left(1-\cos x\right)}{x^{4}+2y^{2}} & \left(\begin{array}{c} x\\ y \end{array}\right)\neq\left(\begin{array}{c} 0\\ 0 \end{array}\right)\\ 0 & \left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} 0\\ 0 \end{array}\right) \end{cases} \] continuous at the point \(\left(\begin{array}{c} 0\\ 0 \end{array}\right)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > 1}
Determine the values of \( p > 0 \) for which the integral \[ T = \sqrt{\frac{1}{2g}}\int_{0}^{b}\frac{\sqrt{1+y'(x)^2}}{\sqrt{-y(x)}} \, dx \] is convergent, given that \( y(x) = -2x^p \) and \( g \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}