inclusionAI/Ring-lite-rl-data huggingface dataset:
There are 37 people lined up in a row, and they are counting off one by one. The first person says 1, and each subsequent person says the number obtained by adding 3 to the previous person’s number. At one point, someone makes a mistake and subtracts 3 from the previous person's number instead. The sum of all the numbers reported by the 37 people is 2011. Which person made the mistake?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Find the initial conditions for which $\lim_{t \rightarrow \infty}y = 4$ for the following differential equation: $$y' = y(y-4)(y+4)(y-6)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the double integral \( \int_0^1 \int_0^1 \frac{x-y}{(x+y)^3} \, dy \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the smallest positive solution of the infinite series equation \(\sum_{n=0}^{\infty}\frac{(-x)^n}{(n!)^2}=0\) accurate to four digits of the fractional part.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.4458}
Calculate the volume of the region $W$ in $\\mathbb{R}^3$ defined by $W=\\{ (x,y,z)\\in\\mathbb{R}^3 ∣ z≥2,\\, x^2+y^2≤6−z,\\, y≤x \\}$ using triple integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Let \( N = {10}^{6} \) . For which integer \( a \) with \( 0 \leq  a \leq  N - 1 \) is the value of maximized?
 Proposed by Lewis Chen 
 $\binom{N}{a+1}-\binom{N}{a}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
a=499499
}
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=n}^{5n}{k-1\choose n-1}\left(\frac{1}{5}\right)^{n}\left(\frac{4}{5}\right)^{k-n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the degree of the field extension \(\mathbb{Q}(\sqrt[4]{3}, \omega_{12})\) over \(\mathbb{Q}\), where \(\omega_{12}\) is a primitive 12th root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the volume of the solid bounded by the surfaces $x + z - 1 = 0$, $z = 1$, and $x^2 + y^2 = 4$ in $\mathbb{R}^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{32}{3}}
Evaluate the expression \( \int_1^a \sqrt[5]{x^5-1}\ dx + \int_0^b \sqrt[5]{x^5+1}\ dx \) given that \( a^5 - b^5 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab}
$\text{Suppose that}20^{21}=2^{n}5^{b}=4^{e}5^{d}=8^{e}5^{f}\text{for positive integers}a,b,c,d,e,\mathrm{and}f.\mathrm{Find}\frac{100bdf}{ace}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
75
}
Find the minimum possible number of distinct terms in the 6-tuple \((a, b, c, \alpha, \beta, \gamma)\) assigned to a scalene triangle, where \(a, b, c\) are the side lengths and \(\alpha, \beta, \gamma\) are the corresponding angles measured in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit: $$ \lim_{n \to \infty} \sum_{k=0}^{3n} \binom{k+n-1}{n-1} \left(\frac{1}{4} \right)^{n} \left(\frac{3}{4}\right)^{k} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
How many polynomials \( P \) with integer coefficients and degree at most 5 satisfy the inequality \( 0 \le P(x) < 120 \) for all \( x \in \{0,1,2,3,4,5\} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86400000}
Find the Fourier transform of the function $f(x) = e^{iax}$, where $a \in \mathbb{R}$, given that $F \delta_a = e^{-iax}$ and $\overline{F} \delta_a = f$, with $F$ denoting the Fourier transform and $\delta_a$ being the Dirac delta function centered at $a$. Assume $f \in S'(\mathbb{R})$. Provide your answer in terms of $a$ and $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(x - a)}
Determine the dimension of the quotient ring $R/J$, where $J$ is the ideal generated by an arbitrary nonconstant monic polynomial $f(x)$ in $K[x]$. Provide your answer in terms of the degree of $f(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\deg f(x)}
Calculate the expected claim size for an insurance policy with the following distribution function: $$F(x)=\begin{cases}0&x\le0\\0.0002x&0<x<1000\\0.4&x=1000\\1-0.6e^{(x-1000)/2000}&x>1000\end{cases}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2100}
Evaluate the sum \( \sum_{n=1}^{\infty} \arccos\left(\sqrt{\frac{n}{n+1}}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the average IQ of Mensa members, given that Mensa is a group of people with an IQ of at least 130. Assume the IQ is normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 15. Use the formula for the mean of a function over an interval to find the average IQ for this group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135.6}
Evaluate the integral \( \int_{-\pi}^\pi \frac{\sin(13x)}{\sin x} \cdot \frac{1}{1+2^x} \, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
What is the relationship between the dual space of the dual space, \(V^{**}\), and the original vector space \(V\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V \cong V^{**}}
What is an upper bound for the difference between the $n$-th moment and the $n$-th power of the expected value of a random variable $X$ constrained to the interval $[0,1]$, for $n \in \mathbb{Z}_+$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine all complex numbers \(a\) for which the function \(f(z) = \frac{z^2 - z +1}{z(z-1)^2}+\frac{a}{\sin z} = \frac{1}{z}+\frac{1}{(z-1)^2}+\frac{a}{\sin z}\) has an antiderivative in the domain \(U = \{z : |z|<2\} \setminus \{0,1\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the contour integral \( \oint_C \sqrt{z^2 - 1} \, dz \) where \( C \) is the circle of radius \( R > 1 \), centered at the origin, in the complex plane. The integrand is defined using a straight branch cut connecting \( z = 1 \) and \( z = -1 \), and \( \sqrt{z^2 - 1} > 0 \) on the line \( y = 0 \) for \( x > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi i}
Find the smallest positive solution of the equation \(\sum_{n=0}^{\infty}\frac{(-x)^n}{(n!)^2}=0\) and provide the answer with four digits of the fractional part.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.4458}
Let \( X \) and \( Y \) be independent random variables with densities \( f_X(x) = 4x^2e^{-2x} \) for \( x > 0 \) and \( f_Y(y) = \frac{8}{3}y^3e^{-2y} \) for \( y > 0 \). Define \( V = \frac{X}{X+Y} \). Find the variance of \( V \), \( \text{Var}(V) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{98}}
Determine the limiting proportion of odd composite positive numbers that are divisible by a given prime number greater than 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{p}}
Let \( n \) be the answer to this problem. In acute triangle \( {ABC} \) , point \( D \) is located on side \( {BC} \) so that \( \angle {BAD} = \angle {DAC} \) and
 point \( E \) is located on \( {AC} \) so that \( {BE} \bot  {AC} \) . Segments \( {BE} \) and \( {AD} \) intersect at \( X \) such that \( \angle {BXD} = {n}^{o} \) : Given that
 \( \angle {XBA} = {16}^{o} \) , find the measure of \( \angle {BCA} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of $$a^abc + b^bca + c^cab.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
729}
There are 2017 distinct points $A_{1}, A_{2}, \ldots, A_{2017}$ marked on a circle. All possible chords connecting these points in pairs are drawn. A line passing through point $A_{1}$, which does not pass through any of the other points $A_{2}, \ldots, A_{2017}$, is drawn. Determine the greatest possible number of chords that can intersect this line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1016064}
A train 110 m long is running with a speed of 84 km/hr. In 4.399648028157747 seconds, it will pass a man who is running at a certain speed in the direction opposite to that in which the train is going. What is the speed of the man in km/hr?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined. What is the maximum number of numbers that can be underlined? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
It is known that there exists a natural number \( N \) such that \( (\sqrt{3}-1)^{N} = 4817152 - 2781184 \cdot \sqrt{3} \). Find \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For a given positive integer \( k \), denote the square of the sum of its digits by \( f_1(k) \) and let \( f_{n+1}(k) = f_1(f_n(k)) \). Determine the value of \( f_{1001}(2^{1990}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Central Park originally had 8 more than half of the number of trash cans as in Veteran's Park, and Liberty Park had twice the number of trash cans as in Central Park. One night, someone took half of the trash cans from Central Park and put 1/3 of them in Veteran's Park and the remaining 2/3 in Liberty Park. Then, they took 1/4 of the original trash cans from Liberty Park and put them in Veteran's Park. If there were originally 24 trash cans in Veteran's Park, how many trash cans are now in Veteran's Park after the exchange?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows: 

1. The first contestant takes $\frac{1}{n}$ of the cake.
2. The second contestant takes $\frac{2}{n}$ of the remaining cake.
3. The third contestant takes $\frac{3}{n}$ of what remains after the first and second contestants have taken their portions.
4. This process continues until the last contestant, who takes all of the remaining cake.

Determine which competitor takes the largest piece of cake.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
19. [10] Define $x \star y=\frac{\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute
$$
((\cdots((2007 \star 2006) \star 2005) \star \cdots) \star 1) .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{15}}{9}}
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Consider a seller who must sell a single private value good to two potential buyers. The valuations of the buyers are uniformly distributed between 0 and 2. The seller will offer the good using a second-price sealed-bid auction with a reserve price of r. What is the optimal reserve price in this case?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the function $f\left( x \right)={x}^{2}+{\left( \ln 3x \right)}^{2}-2a(x+3\ln 3x)+10{{a}^{2}}(a\in \mathbf{R})$, determine the value of the real number $a$ for which there exists ${{x}_{0}}$ such that $f\left( {{x}_{0}} \right)\leqslant \dfrac{1}{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{30}}
In triangle \(ABC\), \(\angle B = 110^\circ\) and \(\angle C = 50^\circ\). On side \(AB\), point \(P\) is chosen such that \(\angle PCB = 30^\circ\), and on side \(AC\), point \(Q\) is chosen such that \(\angle ABQ = 40^\circ\). Find the angle \(\angle QPC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40^\circ}
设对于k=1,2,⋯,n，存在实数x满足如下不等式：$2 ^ { k } < x ^ { k } + x ^ { k + 1 } <$ $2 ^ { k + 1 }$，试求n的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. We know that whatever the arrangement of the edges, from every vertex, we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
Annie has 120 barrettes. She has half as many scrunchies as barrettes. She gifted some bobby pins to her friend and now has 50 fewer bobby pins than three times the number of barrettes. She also has twice as many hairbands as the difference between the number of her bobby pins and scrunchies. What percentage of her total hair decorations are bobby pins, rounded to the nearest percent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31\%}
记 $M_{100}(\mathbb{R})$ 为所有 100 阶实方阵的集合. 设 $S=\left\{A \in M_{100}(\mathbb{R}) \mid A^{4}+A^{2}=\right.$ $O\}$ ，则 S 中最多可以选出 $\qquad$个互不相似的矩阵. (我们称 $A, B \in M_{100}(\mathbb{R})$ 相似,当且仅当 存在可逆的 $P \in M_{100}(\mathbb{R})$ ，使得 $P^{-1} A P=B$ )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1326}
Find the smallest positive integer \( n \) such that for any given \( n \) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \( R_{1}, R_{2}, R_{3} \) such that \( R_{1} \) can be nested inside \( R_{2} \) and \( R_{2} \) can be nested inside \( R_{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Given $a \gt 1$, if for any $x∈[{\frac{1}{3},+∞})$, the inequality $\frac{1}{{3x}}-x+ln3x≤\frac{1}{{a{e^x}}}+lna$ always holds, then the minimum value of $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{e}}
Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF\) of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.5}
In triangle \(ABC\), point \(N\) is taken on side \(AC\) such that \(AN = NC\). Side \(AB\) is twice the length of \(BN\) and the angle between \(AB\) and \(BN\) is \(40^\circ\). Find the angle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110^\circ}
Solve the system of equations in natural numbers:

\(a^{2} + b - c = 100\),

\(a + b^{2} - c = 124\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(12, 13, 57)}
Tolya arranged in a row 101 coins of denominations 1, 2, and 3 kopecks. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kopeck coins could Tolya have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \( P(-2,0) \) to its asymptote is \( \frac{2 \sqrt{6}}{3} \). If a line with slope \( \frac{\sqrt{2}}{2} \) passes through point \( P \) and intersects the hyperbola at points \( A \) and \( B \), intersects the y-axis at point \( M \), and \( P M \) is the geometric mean of \( P A \) and \( P B \), then the half focal length of the hyperbola is what?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Example 4.2.5 How many $1 \times 2$ dominoes must be placed on a $6 \times 6$ chessboard to achieve a saturated cover.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In $\triangle ABC$, given $a=x,b=2,B=60^{\circ}$, if $\triangle ABC$ has two solutions, then the range of $x$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 < x < \frac{4\sqrt{3}}{3}}
1. A mathematician left point A for point B. After some time, a physicist also left point A for point B. Catching up with the mathematician after 20 km, the physicist, without stopping, continued to point B and turned back. They met again 20 km from B. Then each continued in their respective directions. Upon reaching points A and B respectively, they turned and walked towards each other again. How many kilometers from point A will they meet for the third time, if the distance between points A and B is 100 km?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
If $a,b,c,d\geq 0$ and $t\geq \frac{2+\sqrt{2}}{8}(a+b+c+d)$, then
$$(t-a)(t-b)(t-c)(t-d)\leq \left(t-\frac{a+b+c+d}{4}\right)^4.$$
Find the minimum value of $k$ such that the inequality holds for all $a,b,c,d\geq 0$ and $t\geq k(a+b+c+d)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2+\sqrt{2}}{8}\)}
In a ternary tree of height 3, where each node except the root has three children, how many pairs of vertices (u, v) are there such that the distance between u and v is 4? Note that (u, v) and (v, u) are considered the same pair.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Determine the asymptotic behavior of the ratio of the $n$th nice prime to the $n$th prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine all values of the parameter \( a \) for which the system of equations \( \left\{\begin{array}{l} |y+x+8|+|y-x+8|=16 \\ (|x|-15)^{2}+(|y|-8)^{2}=a \end{array}\right. \) has exactly two solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
$\begin{aligned} & \mathrm{For}0\leq t\leq1,\mathrm{define}f(t)=\int_0^{2\pi}|\sin x-t|dx. \\ & \mathrm{Evaluate}\int_0^1f(t)dt. \end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
A hostess baked a pie for her guests. There might be either $p$ people or $q$ people at the table, where $p$ and $q$ are coprime. What is the minimum number of pieces the pie needs to be cut into in advance so that it can be evenly distributed among the guests in either case?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + q - 1}
Calculate the value of the expression

$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4080401}
Given a regular 2007-gon, find the minimal number $k$ such that among every $k$ vertices of the polygon, there always exist 4 vertices forming a convex quadrilateral where 3 sides of the quadrilateral are also sides of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1506}
Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
When w is divided by 13, the remainder is a certain number. If w is 3 more than its value and when divided by 11, its remainder is 0. What is the remainder when w is divided by 13?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the derivative of the trace of the matrix expression \( \operatorname{Trace}\left[\left(AB\right)^{T}Q\left(AB\right)\right] \) with respect to the matrix \( A \), where \( Q = Q^T > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 Q A B B^\top}
A car of the new model travels $4 \frac{1}{6}$ kilometers more per liter of gasoline than the car of the old model. At the same time, the fuel consumption per 100 km for the new car is 2 liters less. How many liters of gasoline does the new car consume per 100 km?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \text{ liters}}
实数 $a 、 b 、 c$ 满足 $a^{2}+b^{2}+c^{2}=\lambda(\lambda>0)$, 试求

$$
f=\min \left\{(a-b)^{2}, \quad(b-c)^{2}, \quad(c-a)^{2}\right\}
$$

的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\lambda}{2}$}
Find the value of \(a_2 + 11a_3 + 70a_4\) for which the function \(f(x) = x^3 + 3(a-7)x^2 + 3(a^2-9)x - 1\) has a positive point of maximum in the interval \((a_1, a_2) \cup (a_3, a_4)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{320}
The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(991,997,1009)}
Calculate the time-averaged power of the continuous signal given by: \[ z(t) = \sqrt{E} \sum_{k=-\infty}^{+\infty} A_k g(t-kT) \] where \( A_k = 1 \) if \(|k|\) is a prime number and \( A_k = -1 \) otherwise, and \( g(t) = 1 \) for \( 0 \leq t \leq T \) and \( g(t) = 0 \) otherwise. \( T \) is the fundamental period. The power of a continuous signal is defined by: \[ \lim_{T\to\infty} \frac{1}{2T} \cdot \int_{-T}^{T} |x(t)|^2 dt \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E}
4. (7 points) Linlin copied a repeating decimal incorrectly as 0.123456. If the digits are correct but the repeating part indicator was mistakenly left out, how many possible original repeating decimals could there be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
一个不均匀的骰子，掷出 1,2,3,4,5,6 点的概率依次成等差数列. 独立地先后掷该骰子两次，所得的点数分别为 a,b. 若事件"a+b=7"发生的概率为 \df{1}{7}，则事件"a=b''发生的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
The numbers \(1,2,3, \ldots, n^{2}\) are arranged in a square table. A random number is selected from this table, and the row and column containing this number are crossed out. Then, from the remaining numbers, another number is selected, and again, the row and column containing this number are crossed out. This process continues until only one number remains in the table, which is automatically included among the selected numbers. What is the sum of all the numbers selected in this way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n^2 + 1)}{2}}