inclusionAI/Ring-lite-rl-data huggingface dataset:
When the Absent-Minded Scientist comes up with a genius idea, he writes it down on a piece of paper but immediately realizes that the idea is not genius, crumples the paper and throws it under the table where there are two trash bins. The scientist misses the first bin with probability \( p > 0.5 \), and he misses the second bin with the same probability. Over the course of a morning, the scientist throws five crumpled genius ideas under the table. Find the probability that each bin contains at least one of the morning's ideas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2p^5 + p^{10}}
记 $F(x)=\int_{0}^{x}(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{1!}+\cdots+\frac{(x-t)^{n-1}}{(n-1)!}) e^{n t} d t \quad$当 $n=4$, 求 $\left.\frac{d^{4} F(x)}{d x^{4}}\right|_{x=0}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
43. If $a, b, c, d$ are the lengths of the four sides of a quadrilateral, and satisfy $a^{4}+b^{4}+c^{4}+d^{4}=4 a b c d$, then this quadrilateral must be . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Rhombus}
Rectangular tiles each of size 25 cm by 16 cm must be laid horizontally on a rectangular floor of length 180 cm and a certain width, such that the tiles do not overlap and they are placed with edges jutting against each other on all edges. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is 54. What is the width of the floor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
The radian measure of -150° is equal to what fraction of π.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5\pi}{6}}
Determine the order of the pole of the function \( \frac{1}{(e^{2z}-e^{-2z}+z^4-4)^2} \) at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Pete was assigned several physics problems and several math problems at home. All the problems solved by Pete constitute 5% of all the physics problems and 20% of all the math problems. What percentage of the total number of problems did Pete solve?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \%}
Compute the sum of the series:

\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["305175780"]}
What percent of the palindromes between 1000 and 2000 contain at least one 3 or 5, except in the first digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36\%}
In a sequence of 2003 circles, each circle is tangent to the sides of a $60^{\circ}$ angle $\angle M O N$, and each adjacent pair of circles is externally tangent. If the radius of the smallest circle is 1, determine the radius of the 2003rd circle in ascending order of radii.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{2002}}
The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $\mathbb{N}_{>1}$ denote the set of positive integers greater than 1. Let $f: \mathbb{N}_{>1} \rightarrow \mathbb{N}_{>1}$ be a function such that $f(mn)=f(m)f(n)$ for all $m, n \in \mathbb{N}_{>1}$. If $f(101!)=101$!, compute the number of possible values of $f(2020 \cdot 2021)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
When you add 124, each digit is the same number. The sum of the hundreds and ones digits of this number is 5, and the tens digit is 3. If each of these three digit numbers is different, find the three digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{431}
Radii of five concentric circles  $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$  form a geometric progression with common ratio  $q$  in this order. What is the maximal value of  $q$  for which it's possible to draw a broken line  $A_0A_1A_2A_3A_4$  consisting of four equal segments such that  $A_i$  lies on  $\omega_i$  for every  $i=\overline{0,4}$ ?

<details><summary>thanks </summary>Thanks to the user Vlados021 for translating the problem.</details>
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
One of the angles of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and also circumscribed around this trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
In the game Deal or No Deal, participants choose a box at random from a set of $26,$ one containing each of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400,000\\\hline\$400&\$500,000\\\hline\$500&\$750,000\\\hline\$750&\$1,000,000\\\hline\end{tabular} After choosing a box, participants eliminate other boxes by opening them, showing the amount of money in the box to the crowd, and then removing that box (and its money!) from the game. What is the minimum number of boxes a participant needs to eliminate in order to have a half chance of holding at least $\$100,\!000$ as his or her chosen box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
记 $F(x)=\int_{0}^{x}\left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{1!}+\cdots+\frac{(x-t)^{n-1}}{(n-1)!}\right) e^{n t} d t \quad$ 当 $n=4$, 求 $\left.\frac{d^{4} F(x)}{d x^{4}}\right|_{x=0}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Let  $S={1,2,\ldots,100}$ . Consider a partition of  $S$  into  $S_1,S_2,\ldots,S_n$  for some  $n$ , i.e.  $S_i$  are nonempty, pairwise disjoint and  $\displaystyle S=\bigcup_{i=1}^n S_i$ . Let  $a_i$  be the average of elements of the set  $S_i$ . Define the score of this partition by 
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]

Among all  $n$  and partitions of  $S$ , determine the minimum possible score.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the game of Guess the Card, two players each have a  $\frac{1}{2}$  chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered  $1,2,\dots,16$  clockwise. They participate in an  $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players  $1$  and  $9$  face each other in the last round is  $\frac{m}{n}$  where  $m,n$  are positive integers, find  $100m+n$ .

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
For how many numbers  $n$  does  $2017$  divided by  $n$  have a remainder of either  $1$  or  $2$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
【Question 13】  
A positive integer with $n+3$ digits $144 \cdots 430(n$ 4's$)$, is a multiple of 2015, the smallest positive integer $n$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given a regular hexagon \( A B C D E F \) with side length \( 10 \sqrt[4]{27} \), find the area of the union of triangles \( ACE \) and \( BDF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{900}
A  $4\times4\times4$  cube is composed of  $64$  unit cubes. The faces of  $16$  unit cubes are to be coloured red. A colouring is called interesting if there is exactly  $1$  red unit cube in every  $1\times1\times 4$  rectangular box composed of  $4$  unit cubes. Determine the number of interesting colourings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Evaluate the infinite sum: $$\sum_{r=1}^{\infty}\frac{2^{2r+1}\cdot (r)!\cdot (r+1)!}{(2r+3)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \dots, x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
Compute the number of ways there are to assemble $2$ red unit cubes and $25$ white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly $4$ faces of the larger cube. (Rotations and reflections are considered distinct.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
Given that the positive integer \( n \) does not exceed 2000 and can be expressed as the sum of at least 60 consecutive positive integers, how many such \( n \) exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Three trucks need to transport \(k\) full barrels, \(k\) half-full barrels, and \(k\) empty barrels such that each truck is equally loaded and each truck carries the same number of barrels. In how many ways can this be done for \(k=7\), considering the trucks and barrels with the same fullness indistinguishable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In an isosceles right triangle, the radius of the inscribed circle is 2.

Find the distance from the vertex of the acute angle to the point where the inscribed circle touches the leg opposite to this angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{7 + 4 \sqrt{2}}}
The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Find the smallest positive integer \( m \) such that for any 14-partition \( A = \{1, 2, \cdots, m\} \) into subsets \( A_1, A_2, \cdots, A_{14} \), there always exists a subset \( A_i \) (with \( 1 < i < 14 \)) containing two elements \( a \) and \( b \) where \( b < a \leq \frac{4}{3}b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Let \( S = \{1, 2, \cdots, 1990\} \). If the sum of all the numbers in a 31-element subset of \( S \) is a multiple of 5, it is called a "good subset". Find the number of good subsets of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5} \binom{1990}{31}}
Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner posts. Calculate the area of Carl’s flowerbed, in square yards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
A circle is tangent to two parallel lines and their secant. The segment of the secant enclosed between the parallel lines is divided by the tangent point in the ratio $1:3$. At what angle does the secant intersect each of the parallel lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$, where $B$ is a standard Brownian motion? Specifically, calculate $E[X(t)X(s)]$ for $0<s<t<\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{ts^2}{2} - \frac{s^3}{6}\)}
For how many triples of positive integers $(a, b, c)$ with $1 \leq a, b, c \leq 5$ is the quantity $(a+b)(a+c)(b+c)$ not divisible by $4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
At a certain middle school, there are 110 teachers who know both English and Russian. According to the statistics, there are 75 teachers who know English and 55 teachers who know Russian. How many teachers at this school know English but not Russian?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Find the limit of \( \frac{\sin_{n+1}x}{\sin_nx} \) as \( x \to 0 \), where \( \sin_n \) is defined as in the original question.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( X \) be a countably infinite (or larger) set with the cofinite topology. For every \( x \in X \), does there exist a family \( \xi \subset \tau \) such that \( \{ x \} = \bigcap \xi \)? If so, determine the cardinality of \( \xi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|X|}
Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{651}
Let  $ABCD$  be a parallelogram that is not rhombus. We draw the symmetrical half-line of  $(DC$  with respect to line  $BD$ . Similarly we draw the symmetrical half- line of  $(AB$  with respect to  $AC$ . These half- lines intersect each other in  $P$ . If  $\frac{AP}{DP}= q$  find the value of  $\frac{AC}{BD}$  in function of  $q$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{q}}
Determine the number of natural number triples \((a, b, p)\) with \(a > b > 0\) such that \[ \left(\sum_{k=1}^n k^a\right)^p = \sum_{k=1}^n k^b \] holds for all natural numbers \(n\). Identify these triples.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given angles $\alpha$ and $\beta$ in a triangle, where $\alpha, \beta \in (0, \pi)$, determine if $\alpha = \beta$ given that:

$$\left(\frac{\sin \alpha}{\sin \beta}\right)^{100}= \frac{\sin^2 \frac {\alpha}{2}}{\sin^2 \frac{\beta}{2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha = \beta}
For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2
024}{n} \right]$. Find the value of$$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cdot (1012)^2}
Let  $S$  be a real number. It is known that however we choose several numbers from the interval  $(0, 1]$  with sum equal to  $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5. 

Find the greatest possible value of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.5}
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$471$}
若点M到x轴的距离是它到y轴距离的2倍，则点M的轨迹方程是？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = \pm 2x}
Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2^{-3/2}"]}
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{8}}
Consider an inverse system of modules over an integral domain $R$ defined by the maps $A_{i+1} \to A_i, a \mapsto r_{i+1} a$, where $r_i \neq 0,1$. Determine if the inverse limit of this system is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( A M : A A_{1} = 1 : 2, B N : B B_{1} = 1 : 3, \) and \( C K : C C_{1} = 1 : 4 \). Point \( P \) belongs to the prism. If the volume of the prism is 16, find the maximum possible volume of the pyramid \( M N K P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine if the function \( g(x) = \int_{0}^{\infty} t^{x^2+1} e^{-3t} \, dt \) can be expressed as a power series in a neighborhood of zero. If it can, find the radius of convergence of this series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine all values of $a$ for which the function \[ h(x) = \begin{cases} x^3 - x^2, & \text{if } x \in \mathbb{Q} \\ 2x - 2x^2, & \text{otherwise} \end{cases} \] is differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the largest integer  $n\geq 3$  for which the edges of the complete graph on  $n$  vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n = 4}
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers of consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Determine whether the Kullback-Leibler divergence between the standard Gaussian distribution \( f(x) = \frac{1}{\sqrt{2\pi}}\exp{(-\frac{x^2}{2})} \) and the standard Cauchy distribution \( g(x) = \frac{1}{\pi(1+x^2)} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Finite}
A burger at Ricky C's weighs $120$ grams, of which $30$ grams are filler. What percent of the burger is not filler?



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
75%}
In a group $G$, given elements $x, a, y$ such that $xay = a^{-1}$, determine if $yax = a^{-1}$ must also hold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yax = a^{-1}}
Find the largest integer $r$ and the smallest integer $q$ such that $\exp(r) < \int_{1}^{2} \exp(x^3 + x) \, dx < \exp(q)$. Provide your answer as a pair $(r, q)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(7, 8)}
Carlos Montado was born on Saturday, November 9, 2002.  On what day of the week will Carlos be 706 days old? (Spell out the entire day of the week.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Friday}}
When $x$ is added to both the numerator and denominatorof the fraction $\frac{a}{b},a\neb,b\ne0$ (Error compiling LaTeX. Unknown error_msg),the value of the fraction is changed to $\frac{c}{d}$. Then $x$ equals:

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{ad-bc}{c-d}$}
Determine whether the ring \(\mathbb{Z}[x]/(23,x-2)\) is an integral domain or isomorphic to another ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_{23}}
Given two sequences of events $\{A_n\}$ and $\{B_n\}$ with $\mathbb{P}(\liminf_n A_n) = \mathbb{P}(\limsup_n B_n) = 1$, determine if $\mathbb{P}(\limsup_n A_n \cap B_n) = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine whether the subspace of $[0,1]^\mathbb{N}$, consisting of monotone (not necessarily strictly) real sequences with values in $[0,1]$ and equipped with the product topology, is connected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Connected}
Determine whether the paths $p \ast f \ast \bar{p}$ and $h \ast f \ast \bar{h}$ are path homotopic, where $p(s) = (1, s)$, $f(s) = (e^{2\pi is}, 1)$, and $h(s) = (e^{2\pi is}, s)$ for $s \in [0, 1]$. Here, $g \ast k$ denotes the path $\sigma(s) = \begin{cases} g(2s), & s \in [0, 1/2] \\ k(2s-1), & s \in [1/2, 1] \end{cases}$, and $\bar{g}(s)$ denotes the path $g(1-s)$. Answer "yes" if they are path homotopic and "no" otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $G:[0,\infty)\to \mathbb{R}^n$ be a continuous and differentiable function such that $G(0) = 0\in\mathbb{R}^n$. Suppose there exists a constant $c>0$ such that for all $t>0$, the inequality $$||DG(t)||\leq c||G(t)||$$ holds, where $DG(t)$ is the derivative of $G$ and $||\cdot||$ denotes the Euclidean norm. Does this imply that $G(t) \equiv 0$ for all $t \geq 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G(t) \equiv 0}
Find all real numbers  $x,y,z$  such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = y = z = 1}
Determine whether the following sum approaches zero as \( n \to \infty \): \[ \sum_{k=1}^{n}\frac{k^{1-n}}{n(n-1)}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}