inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $f: \mathbf{D} \rightarrow \mathbf{D}$ be a holomorphic function that is bijective on $\mathbf{D} \setminus \{0\} \rightarrow \mathbf{D} \setminus \{0\}$. Determine whether $f(0) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Mark has 75% more pencils than John, and Luke has 50% more pencils than John. Find the percentage relationship between the number of pencils that Mark and Luke have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}
Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$. Is the top singular cohomology group $H^n(M, \mathbb{Z})$ zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_C \vec{F} \cdot d\vec{r} \) using Stokes' theorem, where \( \vec{F}=[-z, 5x, -y] \), \( C \) is the ellipse \( x^2 + y^2 = 4 \) with \( z = x + 2 \), and the direction of integration is clockwise as observed from the origin. Determine the surface integral of the curl of \( \vec{F} \), \( \text{curl} \ \vec{F} = (-1, -1, 5) \), over the ellipse's projection in the \( xy \)-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24\pi}
Approximate \( \arctan\frac{8}{9} \) using a first-order Taylor expansion around \( \arctan 1 \). How many digits of accuracy can you guarantee in your approximation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The maximum value is \(\dfrac{1}{27}\).

}
A circle with integer radius \( r \) is centered at(r,r). Distinct line segments of length \( {c}_{i} \) connect points \( \left( {0,{a}_{i}}\right) \) to \( \left( {{b}_{i},0}\right) \) for
 \( 1 \leq  i \leq  {14} \) and are tangent to the circle,where \( {a}_{i},{b}_{i} \) ,and \( {c}_{i} \) are all positive integers and \( {c}_{1} \leq  {c}_{2} \leq  \cdots  \leq  {c}_{14} \) . What is the
 ratio \( \frac{{c}_{14}}{{c}_{1}} \) for the least possible value of \( r \) ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Determine whether the limit \( \lim_{m,n\to\infty} a_{mn} \) exists for the expression \( a_{mn}=\frac{m^2n^2}{m^2+n^2}\left(1-\cos\left(\frac{1}{m}\right)\cos\left(\frac{1}{n}\right)\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine whether the function \( g(x) = \int_x^{x^2} \frac{1}{\ln t} \, dt \) has a finite limit as \( x \to 1 \), given the inequalities \( t - \frac{t^2}{2} \leq \ln(1+t) \leq t \) and \( \frac{1}{t - \frac{t^2}{2}} = \frac{1}{t} + \frac{\frac{1}{2}}{1 - \frac{t}{2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Calculate the sum \( \sum \gcd(x,y) \) where \( x = 3^{a_1}5^{a_2}7^{a_3} \) and \( y = 3^{b_1}5^{b_2}7^{b_3} \), with nonnegative integers \( a_i, b_i \) such that \( a_k + b_k \le 5 \) for all \( k \). The sum is over all possible pairs \( (x, y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1477773}
Given a complete locally convex topological vector space $X$ and a dense subspace $M$ of $X$, determine if $M = X$ when $M$ is quasi-complete (i.e., every bounded closed subset of $M$ is complete).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M = X}
Determine the maximum possible number of discontinuities for a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that has both a left limit and a right limit at each point of \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Evaluate the limit: \[ L = \lim_{x \to 0} \frac{\sin(\ln(x+1)) - \ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Determine the number of incongruent solutions to the congruence equation \(x^2 \equiv 49 \pmod{10^6}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Compute \( \lim_{n\to\infty} \frac{S_1(n)S_5(n)-S_3(n)^2}{S_7(n)} \), where \( S_k(n)=\sum_{r=1}^n r^k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Determine if the function \( f(x) = \begin{cases} \frac{1}{10^n}, & \text{if } x \in (2^{-(n+1)}, 2^{-n}) \\ 0, & \text{if } x = 0 \end{cases} \) is integrable over the interval \([0, 1]\). If it is integrable, find the value of \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{19}}
Determine whether the limit \( \lim_{n \to \infty} S_n = 0 \) holds for the alternating series \( S_n = \sum_{k=0}^{n} (-1)^k {n\choose k} a_k \), where the weights \( a_k \) are non-negative, bounded, and monotonically decreasing (i.e., \( a_{k+1} < a_k \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the equation \(y^{-\frac{1}{4}} + y^{\frac{1}{4}} = 2x\), and the differential equation \((x^2-1)\frac{d^2y}{dx^2} + ax\frac{dy}{dx} + by = 0\), find the value of \(|a-b|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Find the number of positive integers $n$ between $1$ and $1000$, inclusive, satisfying
\[ \lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor\]where $\lfloor n \rfloor$ denotes the greatest integer not exceeding $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{535}
Consider the function \(f(x,y,z) = \alpha^2(x-y) + \alpha(z-y) + (x-z)\), where \(\alpha\) is a constant in \(\mathbb{R}\). If \(f(a,b,c) = f(c,a,b) = f(b,c,a) = 0\) for all \(a, b, c \in \mathbb{R}\), does this imply that \(a = b = c\)? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that$$\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.$$What is $k?$


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137
}
Problem A2. Determine all four digit numbers $\overline{a b c d}$ such that

$$
a(a+b+c+d)\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(a^{6}+2 b^{6}+3 c^{6}+4 d^{6}\right)=\overline{a b c d}
$$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning that $B_r(x)\subseteq S$ for some $r>0$. Is it necessarily the case that $\pi_1(S\setminus\{x\})\cong\mathbb{Z}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the values of $a \in \mathbb{N}$ for which the equation $|||x-1|-3|-a|=k$ has exactly 8 distinct real roots for some $k \ge 0$. Provide your answer as a set of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2\}}
Let \( f:(0,1) \rightarrow (0,1) \) be a differentiable function with a continuous derivative such that for every positive integer \( n \) and odd positive integer \( a < 2^n \), there exists an odd positive integer \( b < 2^n \) such that \( f\left(\frac{a}{2^n}\right) = \frac{b}{2^n} \). Determine the set of possible values of \( f'\left(\frac{1}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1,1\}}
Find a possible solution $(B, E, T)$ to the equation $THE + MBMT = 2018$, where $T, H, E, M, B$ represent distinct digits from $0$ to $9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(4,2,6)}
Verify the accuracy of Bretschneider's approximation for \( \pi \) given by \( \frac{13}{50} \sqrt{146} \), assuming the radius of the circle is one. Determine if it is accurate to the fifth decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.14159}
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions  
$(a) 0\le A<B<C\le99$ ,    
$(b)$ there exist integers $a$ , $b$ , and $c$ , and prime $p$ where $0\le b<a<c<p$ ,  
$(c)$ $p$ divides $A-a$ , $B-b$ , and $C-c$ , and  
$(d)$ each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{272}
Determine the number of real solutions to the equation: $$\sin(\sin(\sin(\sin(\sin(x)))))=\frac{x}{3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a 15 by 15 grid of 1 by 1 squares, what is the maximum number of diagonals that can be drawn such that no two diagonals share an endpoint?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Evaluate the limit: \[ \lim_{x\to0}{{\left(\frac{\tan{(1+\tan x)}}{\tan{(1+\sin x)}}\right)}^{1/x^3}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{\sin 2}}}
Three persons $A$, $B$, and $C$ flip three fair coins $n$, $n+1$, and $n+2$ times, respectively. Let $X$, $Y$, and $Z$ be the number of heads for $A$, $B$, and $C$, respectively. Find the probability $P(X<Y<Z)$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Evaluate the integral \( \int_{C} \log(z-z_0) \, dz \) where \( C \) is the unit circle enclosing the origin and \( z_0 > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Lunasa, Merlin, and Lyrica each have an instrument. The following conditions are given about the prices of their instruments:

(a) If the price of Lunasa's violin is increased by 50% and the price of Merlin's trumpet is decreased by 50%, the violin becomes $50 more expensive than the trumpet.
(b) If the price of Merlin's trumpet is increased by 50% and the price of Lyrica's piano is decreased by 50%, the trumpet becomes $50 more expensive than the piano.

Find integers $m$ and $n$ such that if the price of Lunasa's violin is increased by $m\%$ and the price of Lyrica's piano is decreased by $m\%$, the violin becomes exactly $n more expensive than the piano. Calculate $100m+n$. 

Proposed by Yannick Yao.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8080}
Let \( A,B \) and \( C \) be real numbers such that
 (i) \( \sin A\cos B + \left| {\cos A\sin B}\right|  = \sin A\left| {\cos A}\right|  + \left| {\sin B}\right| \cos B \) ,
 (ii) \( \tan C \) and \( \cot C \) are defined.
 Find the minimum value of \( {\left( \tan C - \sin A\right) }^{2} + {\left( \cot C - \cos B\right) }^{2} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 \( 3 - 2\sqrt{2} \)}
Evaluate the Riemann-Stieltjes integral \( \int_{0}^2 x \, d\alpha \), where \( \alpha(x) = x \) for \( 0 \le x \le 1 \) and \( \alpha(x) = 2 + x \) for \( 1 < x \le 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the largest positive integer value of $n<1000$ such that $\phi(36n)=\phi(25n)+\phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{996}
Evaluate the integral \( \int_0^1 \frac{\ln(x)}{\sqrt{1-x}} \, dx \) using a method that does not involve rearranging terms of a series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 (\ln 2 - 1)}
Determine whether the integral \( \int_{0}^{1} |\ln (x)|^n \, dx \) converges for an arbitrary natural number \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Converges}
Let \( \Omega \subset \mathbb{R}^N \) be a bounded domain and suppose that \( \mu_n \) is a sequence of non-negative Radon measures that converges to a Radon measure \( \mu \) in the weak star sense, i.e., \( \int_\Omega \phi \, d\mu_n \to \int_\Omega \phi \, d\mu \) for all \( \phi \in C_0(\Omega) \), where \( C_0(\Omega) \) denotes the set of continuous functions with compact support. Can you conclude that \( \mu \geq 0 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu \geq 0}
Determine all integer values of n for which the number \( \frac{{14n} + {25}}{{2n} + 1} \) is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n = -2, 0, 4
}
3. For a rational number $r$, its period is the length of the smallest repeating block in its decimal expansion. For example, the number $r=0.123123123 \cdots$ has period 3 . If $S$ denotes the set of all rational numbers $r$ of the form $r=0 . \overline{a b c d e f g h}$ having period 8 , find the sum of all the elements of $S$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49995000}
Let $m$ and $n$ be positive integers. List all the integers in the set $\left\{ 20 ,21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31\right\}$ that $\underline{cannot}$ be written in the form $m+n+m \cdot n$. As an example, $20$ $\underline{can}$ be so expressed since $20 = 2 + 6 + 2\cdot 6$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{22,28,30}}
Determine the element $a \in k$ such that for all $x \in k$, the additive character $\psi_k(x)$ satisfies $\psi_k(x) = \mathrm{Tr}_k(ax)$. Is it possible for $a$ to be equal to 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
7.4. The steamship "Raritet" after leaving the city moves at a constant speed for three hours, then drifts for an hour, moving with the current, then moves for three hours at the same speed, and so on. If the steamship starts its journey from city A and heads to city B, it takes 10 hours. If it starts from city B and heads to city A, it takes 15 hours. How long would it take to travel from city A to city B on a raft?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$.  Compute the probability that $X = Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{20}}
Stephanie enjoys swimming. She goes for a swim on a particular date if, and only if, the day, month (where January is replaced by '01' through to December by '12') and year are all of the same parity (that is they are all odd, or all are even). On how many days will she go for a swim in the two-year period between January 1st of one year and December 31st of the following year inclusive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
Determine a configuration of Anya's triangle on a circle with 29 points such that any volley of K different shots always hits the triangle, and find the minimal value of K.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The same eight people sit in a certain church pew every week, but not always in the same order. Every week, each person hugs the people immediately to his or her left and right. How many weeks does it take (at a minimum) for every pair of people to hug at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15,20,25}
Let $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$. If the maximum element of $S$ is $p_0/q_0$ in reduced form, find $p_0+q_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{595}
Calculate the number of labeled binary trees with \( n \) nodes where the labels decrease along any path from the root, assuming the root is labeled with \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Determine the number of different Fano planes that can be constructed with points labeled from 1 to 7, such that no two planes share any common blocks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the limit: \[ \lim_{n \to \infty} n \sin(2\pi en!) \] where \( e \) is the base of the natural logarithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Find the smallest real number \( p \) such that any line segment with endpoints and midpoint between the parabolas \( y = x^2 \) and \( y = x^2 + 1 \) lies entirely between the parabolas \( y = x^2 \) and \( y = x^2 + p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{8}}
Calculate the expectation \( \mathbf{E}[B_s^2B_t^4 - 2B_s^3B_t^3 + B^4_sB_t^2] \) for a continuous Brownian motion \( \mathbf{B} = \{B_t\}_{t \ge 0} \) given that \( t \ge s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 s t (t - s)}
Evaluate the integral \( \int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)\,dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Evaluate the limit \( S_0 = \lim_{x \to \infty} S(x) \) where \( S(x) = \sum_{k=1}^\infty \frac{\sin(kx)}{kx} \) and is given by the closed form \( S(x) = \frac{1}{x}\left(\frac{1}{2}i\left(\ln(1-\exp(ix)\right)-\ln\left(1-\exp(-ix)\right)\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the support of the random variable \( Y = M\left( 1 - \frac{1 + N}{e^{a \times P \times X} + N} \right) \), where \( X \) is an exponentially distributed random variable with PDF \( f_X(x) = \frac{1}{\lambda} \exp\left( -\frac{x}{\lambda} \right) \). Given that \( N > 1, a > 0, P > 0, M > 0 \), find the interval over which the PDF \( f_Y(y) \) integrates to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, M)}
Let $ABCD$ be a square with side length $6$. Points $E$ and $F$ are on rays $AB$ and $AD$ such that segments $EF$ and $BC$ intersect at point $L$, with $D$ between $A$ and $F$. The area of $\triangle AEF$ is 36. Triangle $PQR$ is constructed with $PQ=BL$, $QR=CL$, and $RP=DF$, and its area is $\sqrt{6}$. If the sum of all possible values of $DF$ is $\sqrt{m} + \sqrt{n}$ for positive integers $m \ge n$, compute $100m+n$. \(\text{Based on a proposal by Calvin Lee}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1806}
A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2 - \sqrt{2}$}
Calculate the surface area of the cylinder defined by the equation \(x^2+y^2=2x\), centered at \((1,0)\), and bounded by the cone \(z=\sqrt{x^2+y^2}\) and the plane \(z=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Compute \( \int_{\gamma} y \, dx \) where \( \gamma \) is a curve in \( \mathbb{R}^2 \) with self-intersecting points at \((0,0)\), \((2,1)\), and \((4,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the integral of the function \( f(x,y,z) = -4 \) over the region \( D = \{(r,x,\theta): r^2 \le \min \{3|x|, 4x-x^2\}\} \), where \( y = r\cos\theta \), \( z = r\sin\theta \), \( r \ge 0 \), and \( 0 \le \theta < 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-42\pi}
Consider the graph $G = (V, E)$ where $V$ is the set of all functions $f: \mathbb{N} \to \mathbb{N}$. Two distinct functions $f, g: \mathbb{N} \to \mathbb{N}$ form an edge if and only if they differ at exactly one input $n \in \mathbb{N}$. Determine if the chromatic number $\chi(G)$ is $\aleph_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Determine whether the improper integral \( \int_1^\infty \frac{(\ln (x))^b}{x^2} \, \mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > -1}
Consider a sequence of complex Hilbert spaces \( H_n \) such that \( H_{n+1} \subsetneq H_n \) and \( \bigcap_{n=1}^\infty H_n = \{v_0\} \). Let \( T_1: H_1 \to \mathbb{C} \) be a continuous linear functional with \( T_1(v_0) = 0 \). For each \( n \in \mathbb{N} \), define \( T_n: H_n \to \mathbb{C} \) as the restriction of \( T_1 \) to \( H_n \). Determine whether \( \lim_{n \to \infty} \lVert T_n \rVert = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a sequence of independent random variables \(X_n \sim U (1, 1+ 1/n)\). Determine whether this sequence converges almost surely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Patty has $ 20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $ 70$ cents more. How much are her coins worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1.15
}
A die is rolled randomly from one face to one of four adjacent faces. Starting with the number 6 on top, find the probability $P_n$ that after $n$ rolls, the die will show the number 6 on top again. Determine the limit $\lim_{n\rightarrow \infty} P_n$. \( \text{Express your answer as a single value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Determine the conditions that a function $g(x)$ must satisfy for there to exist a real-valued function $f(x)$ such that the functional equation \[ f(x+y) = f(x) + f(y) + g(xy) \] holds for all real numbers $x$ and $y$. Provide the necessary and sufficient conditions on $g(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = ax + b}
Find the minimum value of  $c$  such that for any positive integer  $n\ge 4$  and any set  $A\subseteq \{1,2,\cdots,n\}$ , if  $|A| >cn$ , there exists a function  $f:A\to\{1,-1\}$  satisfying  $$ \left| \sum_{a\in A}a\cdot f(a)\right| \le 1. $$  
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{2}{3} }
Let $M$ be a compact, connected $n$-manifold with a non-empty boundary $\partial M$. Determine whether the $n$-th cohomology group, $H^n(M)$, is always trivial, regardless of whether the manifold is orientable or not.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the line $l: ax+2by-1=0$ is intersected by the circle $x^2+y^2=4$ with a chord length of $2\sqrt{3}$, the maximum value of $3a+2b$ is $\boxed{\sqrt{10}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Find the extrema of the function $f(x, y) = e^{xy}$ under the restriction $(x, y) \in \mathcal{D}$ where $\mathcal{D}=\{(x,y)\mid x^2+y^2\leq 2\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e, e^{-1}}
find all positive integer  $a\geq 2 $  and  $b\geq2$  such that  $a$  is even and all the digits of  $a^b+1$  are equals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 2), (2, 3), (2, 5), (6, 5)}