inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the circle \(\Gamma: (x-2)^{2} + y^{2} = 3\), let the line \(l_{1}: x - m y - 1 = 0\) intersect the circle \(\Gamma\) at points \(A\) and \(C\), and the line \(l_{2}: m x + y - m = 0\) intersect the circle \(\Gamma\) at points \(B\) and \(D\). The segments \(AB\) and \(CD\) are located above and below the x-axis, respectively. When the slope of \(CD\) is \(-1\), find the length of the segment \(AB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{2}}
How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20}$= 0.05 has two digits after the decimal point, so $20$ is a valid N.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3997}
For which $a>0$ does the series $$\sum_{n=1}^\infty \left(2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n} \right) \right)^a $$ converge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 1/4}
In triangle $\triangle ABC$, the sides opposite to angles A, B, and C are of lengths $a$, $b$, and $c$ respectively. Given the vectors $\overrightarrow {m} = (a+b, \sin C)$ and $\overrightarrow {n} = (\sqrt {3} a+c, \sin B-\sin A)$. If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, find the magnitude of angle B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
Father is aged 3 times more than his son Ronit. After some years, he would be two and a half times Ronit's age. After further 8 years, he would be 2 times Ronit's age. How many years later will the father be two and a half times Ronit's age?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Two players, A and B, take turns rolling a standard six-sided die. Player A rolls first. If a player rolls a 1 or 2, they win the game. If a player rolls a 3, 4, or 5, the other player gets a turn. If two consecutive 6's are rolled, the game ends in a tie. What is the probability that player A wins the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{48}{85}}
Evaluate the limit: \[ \lim_{x\to0^+} \frac{(1-x)^{x}-\cos x e^{-\frac{x^2}{2}}}{\arctan(x^2)\sin x +x^3 \ln x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the cardinality of the set \( \{ f \in \mathbb{N}^{\mathbb{N}} \mid f \le h \} \), where \( h: \mathbb{N} \to \mathbb{N} \) is defined by \( h(n) = n + 1 \) and the partial order relation \( f \le g \) is defined as \( \forall n \in \mathbb{N}, f(n) \le g(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the number of rotation matrices that can rotate line A to line B in 3D space, given that both lines are of the same length, centered at the origin, and are not identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the Galois group of the polynomial \( g(x) = x^4 + x^3 + x^2 + x + 1 \) over the finite field \( \mathbb{F}_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/4\mathbb{Z}}
2. A circle with center $O$ is inscribed in triangle $A B C$. Point $L$ lies on the extension of side $A B$ beyond vertex $A$. A tangent line drawn from $L$ to the circle intersects side $A C$ at point $K$. Find $\angle K O L$, if $\angle B A C=50^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\circ$. Determine the area of the shaded region.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{225\pi - 450\sqrt{2}}{4}}
Consider the ellipse given by the equation x^2/12 + y^2/3 = 1. Let F1 denote the left focus of the ellipse, and let P be a point on the ellipse. If the midpoint M of the line segment PF1 lies on the y-axis, then find the y-coordinate of point P. Express your answer as a numerical value, including the ± symbol to indicate both positive and negative values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \frac{\sqrt{3}}{2}}
All the faces of a cube are painted with red color. It is then cut into some equal small cubes. There are 8 small cubes whose three faces are colored. Into how many equal small cubes is the large cube cut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider a series of random variables defined by $a_1=1$, and for $n \geq 2$, $a_n$ is either $a_{n-1}$ or $\frac{1}{2} a_{n-1}$, each with equal probability. What is the probability that the series $\sum_{i=1}^{\infty} a_i$ converges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \( d \) within this segment, there are no more than \( 1 + 1000d^{2} \) points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Calculate the value of \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \) for the functions \( f(x) = \left(\int_{0}^{x} e^{-t^2}dt\right)^2 \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^2(1+t^2)}}{1+t^2} dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of coordinate patches required to cover the boundary of the unit ball in \(\mathbb{R}^n\), which is the sphere \(S^{n-1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $f(x)$ is a differentiable function satisfying the equation $f(x) = \left(1 + \frac{x^3}{3}\right) + \int_0^x e^{-t} f(x-t) \, dt$, find the value of $\int_0^1 f(x) \, dx$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{5}}
Solve the equation \((\cos x + 2 \cos 6x)^{2} = 9 + \sin^{2} 3x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi n}
Given 4 distinct positive integers, how many ways are there to sum them, considering the order of addition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
给定 $n$ 个不同实数，其所有全排列组成的集合为 $A_{n}$ ，对于 $\left(a_{1}, a_{2}, \cdots, a_{n}\right) \in A_{n}$ ，若恰有两个不同的整数 $i, j \in\{1,2, \cdots, n-1\}$ 使得 $a_{i}>a_{i+1}, a_{j}>a_{j+1}$ 成立，则称该排列为 "好排列"。求 $A_{n}$ 中 "好排列"的个数。当 $n = 5$ 时，请给出 "好排列" 的个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Solve the equation: $x^x = 10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.506}
30. On the Magic Island, the divine monkeys feed on immortal peaches. Every noon, each divine monkey eats one immortal peach; by evening, both the monkeys and the peaches on the island can clone themselves. If there are 2000 divine monkeys and 50000 immortal peaches on the first morning, then by the afternoon of the $\qquad$th day, all the immortal peaches will be eaten up.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Three cones are placed on a table, standing on their bases and touching each other. The radii of their bases are \(2r\), \(3r\), and \(10r\). A truncated cone (frustum) is placed on the table with its smaller base downward, sharing a slant height with each of the other cones. Find \(r\), if the radius of the smaller base of the truncated cone is 15.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
A company sells a brand of cars in two locations, A and B, with profits (in terms of ten thousand yuan) given by $L_1 = -x^2 + 21x$ and $L_2 = 2x$, respectively, where $x$ is the number of cars sold. If the company sells a total of 15 cars between the two locations, what is the maximum profit they can achieve?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
When a certain number is divided by 179, the quotient is 89 and the remainder is 37. What is the number being divided?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15968}
If x is less than y by some percentage, then y exceeds x by 25%. What is the percentage by which x is less than y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
A cylinder with a volume of 9 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 63. What is the volume of the original cone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Evaluate the limit: \[ L = \lim_{x \rightarrow \infty} \left(\sqrt[11]{\prod_{n=1}^{11}\left(2x+\frac{n(2n+1)}{2}\right)}-2x\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Given sets $A=\{x|x^2+x-6=0\}$ and $B=\{x|x^2+x+a=0\}$, and $B \subseteq A$, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-6\} \cup \left(\frac{1}{4}, +\infty\right)}
In triangle ABC, points D, E, and F are located on sides BC, AC, and AB, respectively. Given that \( \frac{AE}{EB} = \frac{3}{2} \) and \( \frac{AF}{FC} = \frac{3}{1} \), and the intersection of lines AD and EF is point O, determine the ratio \( \frac{EO}{OF} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{5}}
【Question 15】  
Zhao, Qian, Sun, and Li, four children, all like playing chess. Among them, someone likes playing Chinese chess, someone likes playing Go, and someone likes both types of chess. Therefore, the different ways these four people can like these two types of chess are $\qquad$ kinds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
There are 15 students, with 3 students each from 5 different high schools. Five admission officers, each from a different college, will each select 3 students to attend their college. In how many ways can the officers select the students such that no officer selects all 3 students from the same high school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159252480}
$已知O为坐标原点，点E(\frac{3}{4},0)，过动点W作直线x=-\frac{1}{4}的垂线，垂足为F，\overrightarrow{OW}\cdot \overrightarrow{EF}=0，记W的轨迹为曲线C。$
$若A_1，B_1，A_2，B_2均在C上，直线A_1B_1，A_2B_2的交点为P(\frac{1}{4},0)，A_1B_1\perp A_2B_2，求四边形A_1A_2B_1B_2面积的最小值.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2$}
A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules:
(a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$;
(b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$.
How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340}
James has some barrels that he uses to store water. They store 3 gallons more than twice as much as a large cask. He has a certain number of barrels and a cask stores 20 gallons. He can store 172 gallons of water. How many barrels does he have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the number of ways to divide 13 people into 5 indistinguishable groups with sizes 2, 2, 3, 3, and 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600600}
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$. Find the least $k$ so that for any $b \in M$, there exist $a_{i}, a_{j}\in A$, satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ($a_{i}$ and $a_{j}$ do not have to be different) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Solve the inequality in integers:

$$
\frac{1}{\sqrt{x-2y+z+1}} + \frac{2}{\sqrt{2x-y+3z-1}} + \frac{3}{\sqrt{3y-3x-4z+3}} > x^2 - 4x + 3
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 1, -1)}
Given the equation \( \int_0^1 (1+\cos^8 x)(ax^2+bx+c) \, dx = \int_0^2 (1+\cos^8 x)(ax^2+bx+c) \, dx \), determine the interval in which the root of the quadratic equation \( ax^2 + bx + c = 0 \) lies, where \( a, b, \) and \( c \) are non-zero numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Evaluate the limit: \[ \lim\limits_{n\to \infty}\lim\limits_{x\searrow 0}f^{(n)}(x) \] where the function \( f:[0,\infty)\to \mathbb{R} \) is given by \( f(x)=\exp(\sqrt{x})+\exp(-\sqrt{x}) \). Assume that \( f \) and its derivatives satisfy the differential equation \( 2f'(x)+4xf''(x)=f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find whole numbers $\heartsuit$ and $\clubsuit$ such that $\heartsuit \cdot \clubsuit = 48$ and $\heartsuit$ is even, then determine the largest possible value of $\heartsuit + \clubsuit$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
For a standard Brownian motion $(B_t)_{t\geq0}$, let $T$ be the first time the process hits $1$ or $-1$. Compute the expected value \( \mathbb{E}\left(\int_{0}^{T} |B_{t}|^{2} \, dt\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Using Qin Jiushao's algorithm, calculate the polynomial expression for the given input values: n = 5, v = 1, and x = 2. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset".
    The probability that 10 minutes pass without either baby crying can be expressed as $\frac{p}{q}$. Compute $p+q$.

[i]2022 CCA Math Bonanza Team Round #7[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{337}
Evaluate the limit: $$\lim\limits_{x \to -\infty} \left(\frac{\left((x+3)^2+x\cos(x)\right)(x)!}{(x-1)!+2^x}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Define the sequence  $ (a_p)_{p\ge0}$  as follows:  $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$ .
Find  $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
\[
\left(\frac{4-2x+x^{2}}{4-2x}+\frac{6x^{2}+8+12x}{4-x^{2}}-\frac{x^{2}+2x+4}{2x+4}\right)^{-1/3} \cdot (x+2)
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{4-x^2}}
In a cube labeled with vertices \(A, B, C, D, E, F, G, H\), where \(ABCD\) is the top face and \(H, G, F, E\) are directly below \(A, B, C, D\) respectively, a real number is assigned to each vertex. At each vertex, the average of the numbers at the three adjacent vertices is calculated. The averages at vertices \(A, B, C, D, E, F, G, H\) are 1, 2, 3, 4, 5, 6, 7, and 8 respectively. Determine the number assigned to vertex \(F\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Inside a right angle with vertex \(O\), there is a triangle \(OAB\) with a right angle at \(A\). The height of the triangle \(OAB\), dropped to the hypotenuse, is extended past point \(A\) to intersect with the side of the angle at point \(M\). The distances from points \(M\) and \(B\) to the other side of the angle are \(2\) and \(1\) respectively. Find \(OA\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Lotta is a consulting mathematician with 100 clients ranked 1 through 100 in order of importance. Each day, she visits the most important client who feels mistreated. A client feels mistreated if Lotta has never visited them, or if she has visited someone less important since the last time she visited them. On the first day, she visits client 1; on the second day, she visits client 2; on the third day, she goes back to client 1, and so on. Determine the number of days it will take for Lotta to retire when none of her clients feel mistreated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{100} - 1}
Determine if there exists an integer $n > 3$ such that the average digit of $2^n$ is greater than or equal to twice the average digit of $5^n$. If such an $n$ exists, provide an example.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
设 $n(n \geqslant 2)$ 为给定的整数。求最小的实数 $\lambda$ ，使得对于任意实数 $x_{1}, x_{2}, \cdots, x_{n} \in$ $[0,1]$ ，存在 $\varepsilon_{1}, \varepsilon_{2}, \cdots, \varepsilon_{n} \in\{0,1\}$ ，满足：对
于任意的 $1 \leqslant i \leqslant j \leqslant n$ ，均有
$$
\left|\sum_{k=i}^{j}\left(\varepsilon_{k}-x_{k}\right)\right| \leqslant \lambda. 
$$
若$n=3$，答案写为$\frac{m}{n}$，其中$m,n$互质，求$m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The monotonic increasing interval of the function $y=2\cos \left(2x- \frac{\pi}{4}\right)\left(x\in\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]\right)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[- \frac{3\pi}{8}, \frac{\pi}{8}\right]}
The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much should each resident contribute to collect exactly the required rent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.54}
A car traveled the first quarter of a 300-mile journey at 75 mph, the second quarter at 45 km/h, the third quarter at 50 km/h, and the last quarter at 90 mph. What was the average speed of the car in miles per hour for the entire trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.29}
Cyclic quadrilateral $ABCD$ has $AC \perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Compute the second mixed partial derivative \( \frac{\partial^2 f}{\partial y \partial x}(0,0) \) for the function \( f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2} \) when \((x,y) \neq (0,0)\), and \( f=0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1985}
Calculate the probability of observing $x$ particles given that $y$ particles were created, where $x > y$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a 6 by 5 grid, how many 10-step paths are there from $W$ to $X$ that must pass through a point $H$? Assume $W$ is located at the top-left corner, $X$ at the bottom-right corner, and $H$ is three squares to the right and two squares down from $W$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\( 12 + 10\sqrt{3} \)

}
Two balls, A and B, are dropped simultaneously from the same height. Ball A bounces back to 90% of its previous height after each bounce, while Ball B bounces back to 80% of its previous height. How many times will the two balls cross each other during their bounces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A diagnostic test for a disease has a 95% chance of returning a true positive and a 10% chance of a false positive. The true proportion of diseased individuals in the population is \(d\). Everyone is screened, and those who test positive are tested again. Of these, 25% test positive on the second test, assuming the tests are independent. What is the value of \(d\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{136}}
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day’s job.At what time will the doughnut machine complete the job?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4:30 PM
}
A certain debt will be paid in 104 installments from January 1 to December 31 of a certain year. Each of the first 24 payments is to be $520; each of the remaining payments is to be $95 more than each of the first 24 payments. What is the average (arithmetic mean) payment that will be made on the debt for the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$593.08}
Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a, b, c)$ such that $2a + 3b + 5c = 200$ and $a + b + c = N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
In a strip of size \(1 \times 2021\), some cells contain one chip each. In each empty cell, a number is recorded equal to the absolute value of the difference in the number of chips to the left and right of that cell. All recorded numbers are distinct and non-zero. Determine the minimum number of chips that can be placed in the cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1347}
Using Clarke's Theorem, determine the number of labeled trees with 12 vertices where one specified vertex has a degree of 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{372027810}
Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{707}