inclusionAI/Ring-lite-rl-data huggingface dataset:
Arthur Gombóc lives at 1 Édes Street, and the chocolate shop is at the other end of the street, at number $n$. Each day, Arthur performs the following fitness routine: he starts from house number 2. If he is standing in front of house number $k$ (where $1 < k < n$), he flips an expired but fair chocolate coin. If it lands heads, he moves to house number $(k-1)$, and if it lands tails, he moves to house number $(k+1)$. If he reaches the chocolate shop, he goes in and eats a chocolate ball, then moves to house number $(n-1)$. If he reaches home, the exercise is over. On average, how many chocolate balls does Arthur eat each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A train 140 m long is running at 60 kmph. It takes 23.998080153587715 seconds to pass a platform of a certain length. How long is the platform?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{259.97}
Compute the number of values of $x$ in the interval $[-11\pi, -2\pi]$ that satisfy \[ \frac{5\cos(x) + 4}{5\sin(x) + 3} = 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the integers \( a, b, c \) that satisfy \( a + b + c = 2 \), and
\[ 
S = (2a + bc)(2b + ca)(2c + ab) > 200,
\]
find the minimum value of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with two foci $F_{1}(-c,0)$, $F_{2}(c,0)$ $(c>0)$, a line passing through $F_{1}$ at an angle of $\frac{π}{4}$ intersects the ellipse at points $A$ and $B$. If the radius of the inscribed circle of $\triangle ABF_{2}$ is $r=\frac{\sqrt{2}}{6}c$, then the eccentricity of the ellipse is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{5}}{5}}
5. (10 points) For a decimal natural number $n, S(n)$ denotes the sum of the digits of $n$. How many three-digit numbers $a$ satisfy $S(a)=S(2a)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n \leq 1000$. Given that 
\[p(1001) = a \cdot 2^b - c,\]
where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
A chunk fell out of a dictionary. The first page of the chunk has the number 213, and the number of the last page is written using the same digits in a different order. How many pages are in the chunk that fell out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Trodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many seconds does it take Trodgor to burn down the entire village?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1920}
对于整数$n≥4$，求出最小的整数$f _ { i } ( n )$,使得对于任何正整数$m$，集合${m,m+1,⋯,m+n-1} $的任一个$f(n)$元子集中，均有至少3个两两互素的元素．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(M\) be the intersection of diagonals of the convex quadrilateral \(ABCD\), where \(m(\widehat{AMB})=60^\circ\). Let the points \(O_1\), \(O_2\), \(O_3\), \(O_4\) be the circumcenters of the triangles \(ABM\), \(BCM\), \(CDM\), \(DAM\), respectively. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the value of 1 / (3 + 1 / (3 + 1 / (3 - 1 / (3 + 1 / (2 * (3 + 2 / 5))))))
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{968/3191}
已知函数$f\left( x \right)=3\sin \left( \omega x+\varphi  \right)$（$\omega >0$，$\left| \varphi  \right|\le \frac{\pi }{2}$），若点$\left( -\frac{\pi }{6},0 \right)$是函数$y=f\left( x \right)$图象的对称中心，直线$x=\frac{\pi }{3}$是函数$y=f\left( x \right)$的对称轴，且$y=f\left( x \right)$在区间$\left( \frac{2\pi }{33},\frac{5\pi }{33} \right)$上单调，则实数$\omega $取最大值时$\varphi $的值为______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\frac{\pi }{6}$}
What is the value of $\frac12\cdot\frac41\cdot\frac18\cdot\frac{16}{1} \dotsm \frac{1}{512}\cdot\frac{1024}{1}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Consider a circle of radius 1 with center $O$ down in the diagram. Points $A$, $B$, $C$, and $D$ lie on the circle such that $\angle AOB = 120^\circ$, $\angle BOC = 60^\circ$, and $\angle COD = 180^\circ$. A point $X$ lies on the minor arc $\overarc{AC}$. If $\angle AXB = 90^\circ$, find the length of $AX$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
A train is made up of 18 carriages. There are 700 passengers traveling on the train. In any block of five adjacent carriages, there are 199 passengers in total. Calculate the total number of passengers in the middle two carriages of the train. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Suppose that a polynomial of the form $p(x)=x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of -1 in $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
求所有的正整数 $n$,使集合 $M=\{1,2, \cdots, 50\}$ 的任意 $35$ 元子集至少存在两个不同的元素 $a 、 b$ 满足 $a+b=n$ 或 $a-b=n$ 。请给出所有满足条件的n值的总和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2415}
In the expression 8k8 + k88 - 16t6, k and t represent non-zero digits within the integers. The value of t is 6. What is the value of k?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
对于正整数n，将其各位数字之和记为s（n），各位数字之积记为p(n),若 (n)+p(n)=n成立，就称n为巧合数，则所有巧合数的和为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{531}
The number $6\,21H\,408\,3G5$ is divisible by $6$. If $H$ and $G$ each represent a single digit, what is the sum of all distinct possible values of the product $HG$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the derivative of the given order.

\[ y = \frac{\log_{2} x}{x^3}, \quad y^{\prime \prime \prime} = ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{47 - 60 \ln x}{\ln 2 \cdot x^6}}
Given a function f(x) defined on R, where f''(x) < f''(-x) for any x in (0, +inf), and non-zero real numbers a and b satisfying f(a) - f(b) > f(-b) - f(-a), determine the relationship between a^2 and b^2. Express your answer as a relationship between a^2 and b^2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 < b^2}
Given that $O$ is the circumcenter of an acute triangle $\triangle ABC$ with $\tan A = \frac{1}{2}$. If $\frac{\cos B}{\sin C} \cdot \overrightarrow{AB} + \frac{\cos C}{\sin B} \cdot \overrightarrow{AC} = 2m \cdot \overrightarrow{AO}$, then the real number $m$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{5}}
Let $E$ be a three-dimensional ellipsoid. For a plane $p$, let $E(p)$ be the projection of $E$ onto the plane $p$. The minimum and maximum areas of $E(p)$ are $9 \pi$ and $25 \pi$, and there exists a $p$ where $E(p)$ is a circle of area $16 \pi$. If $V$ is the volume of $E$, compute $V / \pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games did Innokentiy play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given that the generatrix of a cone makes an angle of $\frac{\pi}{3}$ with the axis, and the length of the generatrix is 3, the maximum value of the sectional area passing through the vertex is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
$1 \cdot 19$ For a given $n \in N$, find the number of all different natural number triples whose sum is $6 n$.



The above text has been translated into English, preserving the original text's line breaks and format.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n^2}
$f(x), g(x)$ are two quadratic trinomials and $a, b, c, d$ are some real numbers. Given the following values:
- $f(a) = 2$
- $f(b) = 3$
- $f(c) = 7$
- $f(d) = 10$
- $g(a) = 16$
- $g(b) = 15$
- $g(c) = 11$

Find the value of $g(d)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $N = 1890 \times 1930 \times 1970$. Find the number of divisors of $N$ which are not divisors of $45$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
The numbers from 1 to 1000 are written in a circle. Starting from the first one, every 15th number (i.e., the numbers 1, 16, 31, etc.) is crossed out, and during subsequent rounds, already crossed-out numbers are also taken into account. The crossing out continues until it turns out that all the numbers to be crossed out have already been crossed out before. How many numbers remain uncrossed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{800}
\([a]\) represents the largest integer not greater than \(a\). For example, \(\left[2 \frac{1}{3}\right]=2\). Given that the sum of the roots of the equation \([3x + R] = 2x + \frac{3}{2}\) is \(S\), find the value of \(S\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are  $10$  seats in each of  $10$  rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{55}}
Given a sequence of ten distinct positive integers a_1, a_2,..., a_{10} satisfying |a_{i+1} - a_i| = 2 or 3 for i = 1, 2,..., 10, with a_{11} = a_1, find the maximum possible difference between the maximum value M and the minimum value m in the sequence. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A man rows a boat along a river that has multiple streams converging into it. He takes twice as long to row upstream through one of the streams as he does to row downstream through the main river. In addition, he must navigate the confluence of two other streams, one of which affects his rowing speed by a factor of 1/4, while the other influences it by a factor of 3/4. Considering these factors, find the ratio of the speed of the boat in still water to the speed of the combined streams' currents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
The annual interest rate earned by an investment increased by 10 percent from last year to this year. If the annual interest rate earned by the investment last year was 8.256880733944953 percent, what was the annual interest rate this year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.082568807339448\%}
Given an arithmetic sequence $\{a_n\}$, where $a_2 + a_3 = 6$, find the value of $3a_4 + a_6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the derivative of $f(x)= \int_x^1 \frac{\cos(xt)}{t} dt$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{x} ( \cos(x) - 2\cos(x^2) )\)}
Determine the maximal length $L$ of a sequence $a_{1}, \ldots, a_{L}$ of positive integers satisfying both the following properties:
- every term in the sequence is less than or equal to $2^{2023}$, and
- there does not exist a consecutive subsequence $a_{i}, a_{i+1}, \ldots, a_{j}$ (where $1 \leqslant i \leqslant j \leqslant L$ ) with a choice of signs $s_{i}, s_{i+1}, \ldots, s_{j} \in\{1,-1\}$ for which
$$
s_{i} a_{i}+s_{i+1} a_{i+1}+\cdots+s_{j} a_{j}=0
$$
Please provide the value of $n+1$ if the answer can be expressed in the form $2^{n} - 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2025}
Consider the odd numbers that are greater than 200 and less than 405. How many such numbers have a sum greater than 20000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
If $x>y>0$ , then $\frac{x^y y^x}{y^y x^x}=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{\left(\frac{x}{y}\right)}^{y-x}}
Two distinct numbers \(a\) and \(b\) are chosen randomly from the set \(\{2, 2^2, 2^3, \ldots, 2^{25}\}\). What is the probability that \(\log_{a} b\) is an integer? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{331}
What is the temperature coefficient of the reaction if the rate grows 15.6 times when the temperature is increased by 30 K?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller is $r$, then the difference between the radii is best approximated by:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.73r}
At Deluxe Paint Store, fuchsia paint is made by mixing 6 parts of red paint with 3 parts of blue paint. Mauve paint is made by mixing 4 parts of red paint with 5 parts blue paint. How many liters of blue paint must be added to a certain amount of fuchsia to change it to mauve paint?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}F}
## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(\frac{1+\sin x \cdot \cos \alpha x}{1+\sin x \cdot \cos \beta x}\right)^{\operatorname{ctg}^{3} x}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{\beta^{2}-\alpha^{2}}{2}}}
Given a wire of length \(150 \mathrm{~cm}\) that needs to be cut into \(n (n>2)\) smaller pieces, with each piece being an integer length of at least \(1 \mathrm{~cm}\). If any 3 pieces cannot form a triangle, what is the maximum value of \(n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
How many homomorphisms are there from the symmetric group $S_{3}$ to the direct product of cyclic groups $\Bbb Z_{2} \times \Bbb Z_{4}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the expected length of the longest common prefix when generating a set of $n$ random strings of length $k$ over an alphabet of $m$ letters, assuming that each letter is picked with equal probability?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1-m^{(1-n)k}}{m^{n-1}-1}}
Calculate the integral $$\int_{0}^{\infty}x\exp\left(-\frac{x^2+a^2}{2}\right)I_0(ax)dx,$$ where $I_0(x)$ is the modified Bessel function of the first kind of order zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
确定最小自然数k，使得对任意的$a \in [ 0 , 1 ]$及任意的$n \in N$，有$a ^ { k } ( 1 - a ) ^ { n } < \frac { 1 } { ( n + 1 ) ^ { 3 } } .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$K_{min}=4$}
Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{880}
Given plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $\langle \overrightarrow{a}, \overrightarrow{b} \rangle = 60^\circ$, and $\{|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{c}|\} = \{1, 2, 3\}$, calculate the maximum value of $|\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}+3}
What is the probability that all 4 blue marbles are drawn before all 3 yellow marbles are drawn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{7}}
A six-digit number (base 10) is considered "squarish" if it satisfies the following conditions:

1. None of its digits are zero.
2. It is a perfect square.
3. The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit numbers.

How many squarish numbers are there? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the integral: $$\int_{0}^{2\pi} e^{R \cos t} \cos(R \sin t + 3t) \, dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
4. 148 Given
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{\omega^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{\omega^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{\omega^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{\omega^{2}}{8^{2}-7^{2}}=1 .
\end{array}\right.
$$

Find the value of \( x^{2}+y^{2}+z^{2}+\omega^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing and the rent of the pasture is Rs. 225, then how much amount should C pay as his share of rent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.32}
Two people are playing "Easter egg battle." In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome of each round depends only on which egg has the stronger shell; the winning egg retains its strength). It is known that the first player won the first ten rounds. What is the probability that they will also win the next round?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{12}}
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1$. Determine how many complex numbers $z$ exist such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute values at most $15$ and $\text{Re}(f(z)) = \text{Im}(f(z))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
A thief has found a secret safe. The safe's password is the sum of the largest and smallest of the three digits created by using all 6,2 and 5 once. What is the password for the safe?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{908}
Using exactly 18 matches, Petya constructed a ladder with three steps. How many matches will he need to construct a ladder with 25 steps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Given a rhombus \(ABCD\) with side length \(a\) and \(\angle A = \frac{\pi}{3}\), the rhombus is folded along its diagonals to form a dihedral angle \(\theta\), where \(\theta \in \left[\frac{\pi}{3}, \frac{2\pi}{3}\right]\). What is the maximum distance between the diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{4}a\)}
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Find the Maclaurin expansion of $$\displaystyle y=\frac{1+x+x^2}{1-x+x^2}\,\,\text{to } x^4$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2x + 2x^2 - 2x^4}
Consider the eighth-sphere $\left\{(x, y, z) \mid x, y, z \geq 0, x^{2}+y^{2}+z^{2}=1\right\}$. What is the area of its projection onto the plane $x+y+z=1$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3}}{4} \pi}
A large high school has 1500 students. Of these students, 300 are taking music, 200 are taking art, 100 are taking dance, and 50 are taking theater. Additionally, 80 students are taking both music and art, 40 are taking both music and dance, 30 are taking both music and theater, 25 are taking both art and dance, 20 are taking both art and theater, 10 are taking both dance and theater. Furthermore, 50 students are taking music, art, and dance, 30 students are taking music, art, and theater, 20 students are taking art, dance, and theater, and 10 students are taking music, dance, and theater. Lastly, 5 students are taking all four classes. How many students are taking neither music, art, dance nor theater?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{950}
A classroom has $30$ students, each of whom is either male or female. For every student $S$, we define their ratio to be the number of students of the opposite gender as $S$ divided by the number of students of the same gender as $S$ (including $S$). Let $\Sigma$ denote the sum of the ratios of all $30$ students. Find the number of possible values of $\Sigma$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\frac{m}{n}$, 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{59}
Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $\displaystyle (1+x)^2+(1+x)^3+.....+(1+x)^{49}+(1+mx)^{50}$ is $\displaystyle (3n+1)\binom{51}{3}$ for some positive integer $n$. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a mathematics competition, there are four problems carrying 1, 2, 3, and 4 marks respectively. For each question, full score is awarded if the answer is correct; otherwise, 0 mark will be given. The total score obtained by a contestant is multiplied by a time bonus of 4, 3, 2, or 1 according to the time taken to solve the problems. An additional bonus score of 20 will be added after multiplying by the time bonus if one gets all four problems correct. How many different final scores are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}