Context:
Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is always the same, regardless of whether it is day or night. If Yue Yue counts 56 legs during the day, how many heads will be counted at night?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given are  $n$  points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with  leading coefficient equal to  $1$ . A parabola is called  $good$  if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it).  What is the greatest number of  $good$  parabolas?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a, b, c, d\) be positive integers such that the least common multiple (L.C.M.) of any three of them is \(3^{3} \times 7^{5}\). Determine the number of different sets of \((a, b, c, d)\) possible if the order of the numbers is taken into consideration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
点 M,N 分别在 \triangle ABC 的边 AB,AC 上，且\overrightarrow{AM}=x\overrightarrow{AB},\quad \overrightarrow{AN}=y\overrightarrow{AC}.D 为线段 BC 的中点，G 为线段 MN 与 AD 的交点. 若 \overrightarrow{AG}=\df{3}{5}\overrightarrow{AD}，则 \df{1}{x^2}+\df{4}{y^2} 的最小值为__________.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{}.


Context:
Evaluate$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]-\ln4\right)$$
and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers.
Here $[x]$ is defined to be the integer such that $[x]\leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\left\{a_n\right\}$ such that all terms are integers, the first term $a_1=2019$, and for any positive integer $n$, there always exists a positive integer $m$ such that $a_1 + a_2 + \cdots + a_n = a_n$. Determine the number of such sequences $\left\{a_n\right\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circles with centers \(O\) and \(P\) have radii 2 and 4, respectively, and are externally tangent. Points \(A\) and \(B\) are on the circle centered at \(O\), and points \(C\) and \(D\) are on the circle centered at \(P\), such that \(\overline{AD}\) and \(\overline{BC}\) are common external tangents to the circles. Find the area of hexagon \(AOBCPD\). The original answer is in the form \(k\sqrt{m}\). Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the flux of the vector field \( F = \langle x, y, 1 \rangle \) through the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the plane \( z = 1 \) using the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \geq 0$. Given that $z_{10}=2017$, find the minimum possible value of $|z|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate Kendall's Tau for the random variable vector $(X_1, X_2)$, where $X_1$ follows an Exponential distribution with parameter 1 (Exp$(1)$), $X_2$ follows a standard Normal distribution (N$(0,1)$), and the dependence structure is defined by the copula:

$$ C(u_1, u_2) = \frac{1}{3}W(u_1,u_2) + \frac{2}{3}\Pi(u_1,u_2), \text{ } u \in [0,1]^2 $$

with the contra-montone copula $W(u_1,u_2) = (u_1+u_2+1)_+$ and the independence copula $\Pi(u_1,u_2) = u_1u_2$. Use the formula for Kendall's Tau:

$$ \rho_{\tau}(X_1,X_2) = 4 \int_0^1 \int_0^1 C(u_1,u_2)dC(u_1,u_2)-1 $$

Evaluate this integral to find Kendall's Tau.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of the parameter \( a \) for which the equation \( 5^{x^{2}+2ax+a^{2}} = ax^{2}+2a^{2}x+a^{3}+a^{2}-6a+6 \) has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?

[i]Carl Schildkraut and Milan Haiman[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a plane, 10 equal line segments are drawn, and all their intersection points are marked. It turns out that each intersection point divides any segment passing through it in the ratio $3: 4$. What is the maximum possible number of marked points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A market has 100 baskets, each containing a total of 1000 kilograms of rice and 3000 eggs. A basket is considered a "Stuff Basket" if it contains exactly 10 kilograms of rice and 30 eggs. Initially, the baskets may not meet these criteria. In each step, market workers can select two baskets and transfer any amount of rice or eggs between them. What is the minimum number of steps required to ensure all 100 baskets are Stuff Baskets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 36 (1992 Shanghai High School Competition Question) Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. Find the maximum value of $m$ under the condition that $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=$ 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bruce, Anne, and Carl can clean their house in 6 hours working together at their respective constant rates. If Anne's speed were doubled and Carl's speed were tripled, they could clean their house in 2 hours working at their respective rates. If Bruce and Anne together can clean the house in 4 hours, and Bruce and Carl together can clean the house in 3 hours, how many hours does it currently take Anne to clean the house on her own?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the maximum whole number of minutes Petya's sleep could have lasted?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 10 people sitting around a round table, some of whom are knights and the rest are liars (knights always tell the truth, and liars always lie). It is known that among them there is at least one knight and at least one liar. What is the maximum number of people sitting at the table that can say: "Both of my neighbors are knights"? (A statement is considered false if it is even partially untrue.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the Paradise Park, there is a Ferris wheel with 14 seats. However, 2 of these seats are specially designed to accommodate wheelchairs, so they can only hold 4 passengers each. The remaining seats have varying capacities (in ascending order): 5, 5, 6, 6, 6, 6, 7, 7, 8, and 9 passengers per seat. Additionally, the park staff must ensure that at least 10% of the wheel's total capacity is left available for guests in the queue line who are waiting to ride. Calculate the maximum number of people that can ride the Ferris wheel at the same time, without exceeding the aforementioned constraints.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $ABC$ has $\angle C = 60^{\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\tan{\angle BAD}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the first quarter, the workforce of Company X is 60% female. During the second quarter, the company hired 30 additional male workers, which reduced the female workforce percentage to 57%. In the third quarter, they decided to increase the female representation again by hiring 50 new female employees, resulting in a 62% female workforce. Finally, in the fourth quarter, the company hired 40 more male and 10 female workers, which led to a new gender ratio of 58% female employees. Calculate the total number of employees in Company X at the end of the fourth quarter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mark and William are playing a game. Two walls are placed 1 meter apart, with Mark and William each starting an orb at one of the walls. Simultaneously, they release their orbs directly toward the other. Both orbs are enchanted such that, upon colliding with each other, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb so that when it collides with a wall it instantly reverses direction and goes at double its previous speed (William's reverses direction at the same speed). Initially, Mark's orb is moving at \(\frac{1}{1000}\) meters/s, and William's orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between the two walls. How fast, in meters/s, is his orb going when this first happens?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of $\left(\left((4-1)^{-1} - 1\right)^{-1} - 1\right)^{-1} - 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_M \left|\cos(x+y)\right|\,dx\,dy \), where \( M = [0,\pi] \times [0,\pi] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate: $1+i^2+i^3+\ldots+i^{10}$ (where $i$ represents the imaginary unit).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four distinct integers $a$, $b$, $c$, $d$ have a product of $a \cdot b \cdot c \cdot d = 9$. Find the value of $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
I have a function $f(x)$ that is continuous on the interval $[0, 1]$. I know that $f(0) < 0$ and $f(1) > 0$. I want to find a zero of $f(x)$ numerically using the bisection method. However, the time it takes to evaluate $f(x)$ numerically is different if $f(x) < 0$ or $f(x) > 0$. Let's say that for $f(x) < 0$ the time needed for evaluation is $1$ and for $f(x) > 0$ it's $C$, with $C > 1$. What is the optimal point to choose in a bisection step to minimize the expected time to find a zero of $f(x)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular $4n$-gon $A_{1} A_{2} \ldots A_{4n}$ with area $S$, where $n > 1$. Find the area of the quadrilateral $A_{1} A_{n} A_{n+1} A_{n+2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) \) be an increasing continuous function defined on the interval \([0, 3]\), and let \( g(x) \) be its inverse function, such that \( g(x) > f(x) \) for all positive \( x \) where both functions are defined. Additionally, \( f(0) = 0 \) and \( f(3) = 2 \).

The area under the graph of \( f(x) \) on the interval \([0, 3]\) is 2. Find the area of the figure bounded by the graphs of \( f(x) \) and \( g(x) \), as well as the segment connecting the points \((3, 2)\) and \((2, 3)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is:
Find all positive integers m for which for all  $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jason went to the mall on Saturday to buy clothes for a special event. He decided to purchase a jacket, a pair of shorts, a shirt, and a pair of shoes. The prices of these items before any discounts are as follows:

- Jacket: $45.28
- Shorts: $29.99
- Shirt: $26.50
- Shoes: $84.25

Jason had a 15% off coupon for the jacket and a 20% off coupon for the shoes. The mall was also offering a "buy one, get one 50% off" sale on shirts and shorts. Assume that tax is 8.25%.

Calculate the total amount Jason spent on his shopping trip, taking into consideration the discounts and tax.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a purchase of $118, a store offered a payment plan consisting of a $18 down payment and 12 monthly payments of a certain amount each. The customer paid 15.254237288135593% of the purchase price in interest by using this plan. What was the amount of each monthly payment?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Chords \(AB\) and \(CD\) of a circle with center \(O\) both have a length of 5. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) intersect at point \(P\), where \(DP=13\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL:LC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \( A \) lies on the line \( y = \frac{5}{12} x - 7 \), and point \( B \) lies on the parabola \( y = x^2 \). What is the minimum length of segment \( AB \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{[0,1]} f \, \, d\lambda \), where \( f(x) = e_2(x) \) and \( e_k(x) \) is the \( k \)-th digit after the decimal point of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of infinite planar graphs with the degree sequence \(D=(4^4,6^\infty)\), where the superscripts denote the number of vertices with that degree.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $418 \times 418$ grid, black and white coins are placed on some of the squares. First, remove all black coins that are in the same row as any white coin(s). Then, remove all white coins that are in the same column as any black coin(s). Let $b$ be the number of black coins remaining and $w$ be the number of remaining white coins. Find the remainder when the maximum possible value of $bw$ is divided by $2007$. \( \text{(Express your answer as an integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rhombus is inscribed in $\triangle ABC$ in such a way that one of its vertices is $A$ and two of its sides lie along $AB$ and $AC$. If $\overline{AC} = 6$ inches, $\overline{AB} = 12$ inches, and $\overline{BC} = 8$ inches, the side of the rhombus, in inches, is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that line segments AB and CD are parallel, and angle D is twice angle B, and AD = a and CD = b, find the length of AB. Express your answer as an algebraic expression in terms of a and b.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of convergence for the power series \( \sum_{n=1}^{\infty}\frac{(-1)^{[\sqrt n]}}{n}x^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following integral:

$$\int_{0}^{1}\left(\int_{0}^{\sin^2x}\arcsin(\sqrt{t})\, dt+\int_{0}^{\cos^2x}\arccos(\sqrt{t})\, dt\right)dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the points at which the function \( f(x) = \begin{cases} \cos x & x \in \mathbb{Q} \\ x & x \notin \mathbb{Q} \end{cases} \) is differentiable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 10 birds pecking on the ground, and among any 5 birds, at least 4 must be on the same circle. What is the minimum number of birds that must be on the circle that contains the most birds?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are seven cards in a hat, and on card $k$ there is a number $2^{k-1}$, where $k=1,2,...,7$. Solarin picks the cards at random from the hat, one card at a time, until the sum of the numbers on the cards in his hand exceeds $124$. What is the most probable sum he can get?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $P=\{x|x^2-2x-3<0\}$ and $Q=\{x|x\in\mathbb{N}\}$, then $P\cap Q=$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twenty pairs of integers are formed using each of the integers \( 1, 2, 3, \ldots, 40 \) once. The positive difference between the integers in each pair is 1 or 3. Find the greatest possible sum of the differences.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any real numbers $m$ and $n$, when $0 < n < m < \frac{1}{a}$, the inequality $\frac{\sqrt[m]{n}}{\sqrt[n]{m}} > \frac{n^a}{m^a}$ holds true. Find the minimum value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABC$  be an equilateral triangle.  $A $  point  $P$  is chosen at random within this triangle. What is the probability that the sum of the distances from point  $P$  to the sides of triangle  $ABC$  are measures of the sides of a triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If you set the clock back by 10 minutes, the number of radians the minute hand has turned is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \int_{|z|=R}\frac{1}{P(z)}dz \), where \( P(z) \) is a polynomial with all roots inside a circle of radius \( R \) centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How do you evaluate the limit of the sum $$\lim_{n\to\infty}\sum\limits_{0\leqslant{k}\leqslant{2n}} {\frac{k}{k+n^{2}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\overrightarrow {a}$ and $\overrightarrow {b}$ are unit vectors, and $\overrightarrow {a}$ is perpendicular to $\overrightarrow {b}$, vector $\overrightarrow {c}$ satisfies $| \overrightarrow {c}- \overrightarrow {a}- \overrightarrow {b}|=2$. The range of values for $| \overrightarrow {c}|$ is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a circle such that chords  $\overline{AC}$  and  $\overline{BD}$  intersect at a point  $E$  inside the circle.  Suppose that  $\angle ADE =\angle CBE = 75^\circ$ ,  $BE=4$ , and  $DE=8$ .  The value of  $AB^2$  can be written in the form  $a+b\sqrt{c}$  for positive integers  $a$ ,  $b$ , and  $c$  such that  $c$  is not divisible by the square of any prime. Find  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A guard has detained an outsider and wants to expel him. The outsider then stated that he made a bet with his friends for 100 coins that the guard would not expel him (if the guard expels him, the outsider pays his friends 100 coins; otherwise, his friends pay him 100 coins). The outsider, deciding to buy off the guard, offered him to name a sum. What is the maximum number of coins the guard can demand so that the outsider, guided only by his own benefit, will definitely pay the guard?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $\triangle ABC$  be a triangle with  $AB=85$ ,  $BC=125$ ,  $CA=140$ , and incircle  $\omega$ .  Let  $D$ ,  $E$ ,  $F$  be the points of tangency of  $\omega$  with  $\overline{BC}$ ,  $\overline{CA}$ ,  $\overline{AB}$  respectively, and furthermore denote by  $X$ ,  $Y$ ,  and  $Z$  the incenters of  $\triangle AEF$ ,  $\triangle BFD$ , and  $\triangle CDE$ , also respectively.  Find the circumradius of  $\triangle XYZ$ .

*Proposed by David Altizio*
Please reason step by step, and put your final answer within \boxed{}.

\boxed{The circumradius of $\triangle XYZ$ is 30.}


Context:
A merchant must mix salt at a certain price per pound with 40 lbs of salt that costs 25 cents/lb so that they will get a 20% profit by selling the mixture at 48 cents/lb. The merchant needs to mix 60 lbs of salt at this price. What is the price per pound of the salt that needs to be mixed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $(X-2)^8 = a + a_1(x-1) + \ldots + a_8(x-1)^8$, then the value of $\left(a_2 + a_4 + \ldots + a_8\right)^2 - \left(a_1 + a_3 + \ldots + a_7\right)^2$ is (Answer in digits).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marcella has some pairs of shoes. If she loses 9 individual shoes and has 15 matching pairs left, how many pairs of shoes did she have initially?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( \lim_{n \rightarrow \infty} \int_{0}^{\infty} \left(1+\frac{x}{n}\right)^{-n}\sin\left(\frac{x}{n}\right) dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. A circle is inscribed with 2019 numbers. For any two adjacent numbers $x$ and $y$, the inequalities $|x-y| \geqslant 2, x+y \geqslant 6$ are satisfied. Find the smallest possible sum of the recorded numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
James gets bored with his game so decides to play a different one. That game promises 100 hours of gameplay but a certain percentage of that is boring grinding. However, the expansion does add another 30 hours of enjoyable gameplay. James gets 50 hours of enjoyable gameplay. What is the percentage of the gameplay that is boring grinding?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a bus station, a bus leaves at a certain interval for 12 hours a day. For 5 days, 120 buses leave the station. What is the time interval between each bus leaving the station?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) be a nonnegative, continuous, and strictly increasing function on \([0,1]\). For \( p > 0 \), let \( x_p \) be the number in \((0,1)\) that satisfies \( f^p(x_p) = \int_0^1 f^p(x) \, dx \). Determine \( \lim_{p \to \infty} x_p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a diagram, several circles are connected by segments. Choose a natural number \( n \) and place different natural numbers in the circles such that the following conditions are met: If two numbers \( a \) and \( b \) are not connected by a segment, then \( a - b \) is coprime with \( n \). If they are connected, then \( a - b \) and \( n \) share a common divisor greater than 1. Determine the smallest \( n \) for which such an arrangement is possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
I bought a TV for $1700 excluding tax. The store offers two discounts: a 10% discount on the original price, and a store membership discount of 5%. There are three different taxes and fees to consider: a 15% value-added tax (VAT), an 8% sales tax, and a $25 recycling fee. I am eligible for the store membership discount. What is the final price of the television after applying both discounts, accounting for all taxes, and adding the recycling fee?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given points $A(-2,3)$ and $B(3,2)$, if line $l$: $y=kx-2$ has no intersection with line segment $AB$, then the range of the slope $k$ of line $l$ is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数$f\left( x \right)=\sin x$．若存在${{x}_{1}}$，${{x2}}$，$\cdot \cdot \cdot $，${{x}_{m}}$满足$0\le {{x1}}<{{x}_{2}}<\cdot \cdot \cdot <{{xm}}\le 6\pi $，且$\left| f\left( {{x}_{1}} \right)-f\left( {{x2}} \right) \right|+\left| f\left( {{x}_{2}} \right)-f\left( {{x3}} \right) \right|+\cdot \cdot \cdot +\left| f\left( {{x}_{n-1}} \right)-f\left( {{xn}} \right) \right|=12$（$m\ge 2$，$m\in {{\Nu }^{*}}$），则$m$的最小值为_________．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if there exists a constant \(C_h\) such that \(\|\partial_h x\| \leq C_h \|x\|\) for all \(x \in \ell^2(\mathbb{N})\), where \(\partial_h\) is the finite difference operator defined as \((\partial_h(x))_j = (x_{j+1}-x_j)/h\) for \(h>0\). If such a constant exists, find the optimal value of \(C_h\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the probability of reaching a state of 100 or more, starting from 20, assuming an optimal strategy is used. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the constant $c_1$ in the steady state solution $u(x) = c_1 - 2x$ for the following PDE problem:
\begin{align}
u_{t} &= u_{xx} \\
u_{x}(t,0) &= u_{x}(t,4) = -2 \\
u(0,x) &= \begin{cases} 0 & 0 \leq x \leq 2 \\ 2x-4 & 2 \leq x \leq 4 \end{cases}
\end{align}
Choose $c_1$ such that the total heat energy as $t \rightarrow \infty$ is the same as when $t=0$.}
Please reason step by step, and put your final answer within \boxed{}.


Context:
14. Teacher $D$ placed cards with the numbers $1, 2, \cdots, 13$ in a clockwise order on a circle. Initially, all cards are face up. Each time, one card can be flipped to face down (once a card is face down, it cannot be flipped again). Teacher $D$'s rule for flipping cards is: if a face-up card has the number $A$, and the card that is 2 cards away is also face up, then Teacher $D$ can flip the card with the number $A$. For example: as long as the cards with the numbers 9 or 2 are face up, Teacher $D$ can flip the card with the number 12 (of course, provided that the card with the number 12 is still face up). In the end, as long as Teacher $D$ flips 12 cards to face down, it is considered a success. How many different sequences of flipping can Teacher $D$ use to achieve success?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $A$ is an abelian group of order $p^4$, where $p$ is a prime number. If there exists a subgroup $N$ such that $A/N \simeq \mathbb{Z}/p^3\mathbb{Z}$, how many distinct groups $A$ are there satisfying this condition? Given the classification of abelian groups of order $p^4$, which are: $\mathbb{Z}/p^4\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^3\mathbb{Z}$, and $\mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$, how many of these admit a subgroup $N$ with the given property?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mr. Wong harvested 560 mangoes from the tree outside his house. He sold x mangoes at the market and gave 50 mangoes to his family. He divided the rest of the mangoes evenly among 12 of his neighbors. If x is half the number of mangoes harvested, how many mangoes does each neighbor receive?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设集合 $A=\{1,2, \cdots, 10\}, A$ 到 $A$ 的映射 $f$ 满足下列两个条件:
(1)对任意 $x \in A, f_{30}(x)=x$.
(2) 对每个 $k \in \mathbf{N}_{+}, 1 \leqslant k \leqslant 29$, 至少存在一个 $a \in A$, 使 $f_k(a) \neq a$.
求这样映射 $f$ 的个数(约定 $\left.f_1(x)=f(x), f_{k+1}(x)=f\left(f_k(x)\right), k=1,2, \cdots\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 8.4. Points $D$ and $E$ are marked on sides $A C$ and $B C$ of triangle $A B C$ respectively, such that $A D=E C$. It turns out that $B D=E D, \angle B D C=\angle D E B$. Find the length of segment $A C$, given that $A B=7$ and $B E=2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the spring round of the 2000 Cities Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, but no two students together solved all six problems. What is the minimum possible number of high school students in country $N$ who participated in the spring round?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a, b, c, d \) be any positive real numbers. Determine the range of the sum
$$
S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}.
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola y^2 = 4x and points A(1,0), B(-1,0), if point M is on the parabola, calculate the maximum value of angle MBA in radians, using pi notation for precision.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\pi$ be a permutation of the numbers from 1 through 2012. What is the maximum possible number of integers $n$ with $1 \leq n \leq 2011$ such that $\pi(n)$ divides $\pi(n+1)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}$: $(m+1)x+2y+2m-2=0$ and $l_{2}$: $2x+(m-2)y+2=0$, if $l_{1} \parallel l_{2}$, then $m=$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = |\log_{3}x|$, if $0 < m < n$ and $f(m) = f(n)$, then the range of values for $2m+n$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在矩形 $A B C D$ 中, $A B=2, A D=1$, 在 $D C$ 边上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$延长线上（包含 $B$ ）的动点 $Q$ 满足 $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$ ，则向量 $\overrightarrow{P A}$ 与向量 $\overrightarrow{P Q}$ 的数量积 $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ 的最小值为 $\qquad$.
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{}.


Context:
Calculate the volume of the region in the first octant bounded by the cylinder \(x^2 + z^2 = 4\) and the planes \(x = 0\), \(y = z\), and \(z = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{\sqrt{(x^2+2x+2)^3}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ten-digit positive integers with the following properties:

- Each of the digits $0, 1, 2, \ldots, 8, 9$ is contained exactly once.
- Each digit, except $9$, has a neighboring digit that is larger than it.

(Note: For example, in the number $1230$, the digits $1$ and $3$ are the neighboring digits of $2$, while $2$ and $0$ are the neighboring digits of $3$. The digits $1$ and $0$ have only one neighboring digit.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( a \) for which the improper integral \( \int_0^{\frac{1}{2}} \frac{1}{x|\log(x)|^a} \, dx \) is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest known constant \(c\) such that for any integer \(n \geq 2\), there are at least \(\frac{n}{\log_2{n}}\) primes in the interval \([n, cn]\]?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the volumes of the solids formed by the rotation of the figures bounded by the graphs of the functions. The axis of rotation is $O y$.

$$
y=\sqrt{x-1}, y=0, y=1, x=0.5
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\int_{-\infty}^0 \frac{dx}{(x-2)\sqrt{1-x}}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a finite group, and let $H$ be a normal subgroup of $G$ with order $p$, where $p$ is a prime number. Let $K$ be a subgroup of $G$ such that $\gcd(|K|, p-1) = 1$. Determine the structure of the subgroup $HK$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a sequence: some number, 15, 17, 51, 53, 159161. What is the first number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume bounded by the surfaces $x^2+z=4$ and $y^2+z=4$ in the first octant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal polynomial of the linear transformation \( \phi: V \to V \) on the vector space \( V = \mathbb{R}^n \), where \( \phi(v_1,v_2,\dots,v_n)^t = (v_2,v_3,\dots,v_1)^t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a casino game involving fair coin flips, the game ends when the sequence $HTH$ appears. Using the concept of martingales and the optional stopping theorem, calculate the expected number of flips required for the sequence $HTH$ to occur.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the game "Clap 7", natural numbers starting from 1 that contain the digit "7" are called "visible 7s", and multiples of 7 are called "invisible 7s". Among the natural numbers from 1 to 100, what is the total number of "visible 7s" and "invisible 7s"? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $i$ is the imaginary unit, and the complex number $z = \frac{a+i}{1-i}$, where $a$ is a real number, and $|z| = 1$, calculate the value of $a$. Express your answer as a real number, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最小的质数 $p$, 满足 $(p, N)=1$,其中, $N$ 为所有满足以下条件的 $\left(a_{0}, a_{1}, \cdots\right.$, $a_{2012}$ ）的个数：
(1) $\left(a_{0}, a_{1}, \cdots, a_{2012}\right)$ 是 $0,1, \cdots, 2012$的一个排列;
(2) 对 2013 的任意一个正约数 $m$ 及所有的 $n(0 \leqslant n<n+m \leqslant 2012)$ 有
$m \mid\left(a_{n+m}-a_{n}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $\displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by 14% while buying and by 20% while selling. What is his percentage profit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a right tetrahedron \(ABCD\), \(DA\), \(DB\), and \(DC\) are mutually perpendicular. Let \(S\) and \(R\) represent its surface area and the radius of the circumscribed sphere, respectively. What is the maximum value of \(\frac{S}{R^2}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
6.1. Find the sum of all integers \( x \in [-3 ; 13] \) that satisfy the inequality

$$
\left(1-\operatorname{ctg}^{2} \frac{\pi x}{12}\right)\left(1-3 \operatorname{ctg}^{2} \frac{\pi x}{12}\right)\left(1-\operatorname{tg} \frac{\pi x}{6} \cdot \operatorname{ctg} \frac{\pi x}{4}\right) \leqslant 16
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知双曲线 $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a 、 b>0)$的左、右焦点分别为 $F_{1} 、 F_{2}$, 双曲线 $C$ 与圆 $x^{2}+y^{2}=r^{2}(r>0)$ 的一个交点为 $P$ 。若 $\frac{\left|P F_{1}\right|+\left|P F_{2}\right|}{r}$ 的最大值为 $4 \sqrt{2}$, 则双曲线 $C$的离心率为 $\qquad$。原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
I have  $8$  unit cubes of different colors, which I want to glue together into a  $2\times 2\times 2$  cube.  How many distinct  $2\times 2\times 2$  cubes can I make?  Rotations of the same cube are not considered distinct, but reflections are.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The power function $$y=(m^{2}-m-1)x^{m^{2}-2m-3}$$ is monotonically decreasing on the interval \[0, +∞), then the value of the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define \( a_3 = \frac{2 + 3}{1 + 6} \), and for \( n \geq 4 \), \( a_n = \frac{a_{n-1} + n}{1 + n a_{n-1}} \). Find \( a_{1995} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given points C and D on segment AB, where M and N are the midpoints of AC and DB respectively, and given that AB = 10 and CD = 2, calculate the length of MN. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{7^{\sqrt{n+1}-\sqrt{n}}\cdot(\frac{n+1}{2})!\cdot(\frac{n+1}{2})!}{(n+1)\cdot(\frac{n}{2})!\cdot(\frac{n}{2})!}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a complex polynomial \(P_n(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0\), where \(\alpha_i\) are its complex roots, calculate the product \((\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of $\mathbb{Q}$-morphisms from the algebraic closure of $\mathbb{Q}$ to $\mathbb{C}$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
After the "I Love Beijing" knowledge competition, A, B, and C made predictions about their scores. A said: "My score is higher than B's." B said: "C's score is higher than mine and A's." C said: "My score is higher than B's." After the scores were announced, the scores of the three people are all different, and only one person predicted correctly. In this case, the order of the three people from highest to lowest score is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the same time, a car departs from point \( A \) to point \( B \) at a speed of \( 90 \) km/h, and a train departs from point \( B \) to point \( C \) at a speed of \( 60 \) km/h. After 2 hours from the start, they are at the shortest distance from each other. Find the distance between the points if all three points are equidistant from each other and connected by straight roads.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate: $\displaystyle\int_{0}^{1}\frac{\ln(x+1)}{x^2+1}\,\mathrm{d}x$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the output of the given program when the input is 6? Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A container can hold a certain number of orange slices. If a company had three hundred twenty-nine orange slices to put into containers, they would need 3 more slices to fill up the last container. How many orange slices can one container hold?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\Omega,\Omega^*$ be bounded domains in $\mathbb{R}^n$, and $B_r(y_0)$ be a small ball of radius $r$ centered at $y_0\in\Omega^*$. In $\Omega^*$, define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on $B_r(y_0)$. We want to calculate the function obtained by applying the Legendre transform on $\psi$, namely $$u_0(x)=\sup\{x\cdot y-\psi(y):y\in B_r(y_0)\}$$ What is the explicit expression of $u_0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find $\int \frac{\sin ^{3} x}{\cos ^{5} x} \, dx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$.  What is $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Logan makes $65,000 a year. He spends a certain amount on rent every year, $5000 on groceries every year, and $8000 on gas every year. If he wants to have at least $42000 left and needs to make $10,000 more each year, how much does he spend on rent every year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In tetrahedron \(ABCD\), the dihedral angle between face \(ABC\) and face \(BCD\) is \(60^\circ\). The orthogonal projection of vertex \(A\) onto face \(BCD\) is \(H\), which is the orthocenter of \(\triangle BCD\). Point \(G\) is the centroid of \(\triangle ABC\). Given that \(AH = 4\) and \(AB = AC\), find the length of \(GH\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}$. Find the value of $CE^2+DE^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum of all positive irreducible fractions less than $1$ with the denominator $1991$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Between them, Mark, Sarah, and John have 36 traffic tickets. Mark has twice as many parking tickets as Sarah, and they each have an equal number of speeding tickets. John has one third the number of parking tickets as Mark, but twice as many speeding tickets as Sarah. If Sarah has 6 speeding tickets, how many parking tickets does Mark have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cylinder with a base radius of $r$ and a height of $h$, its volume is 2, and its surface area is 12. Find the value of $\frac {1}{r} + \frac {1}{h}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $PQR,$ $PQ = 25$ and $QR = 10.$ Find the largest possible value of $\tan B.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a unitary matrix $U$, determine the condition on the $i$-th column vector of $U$ that ensures the $i$-th entry of the product $U\mathbf{x}$ is non-zero, where $\mathbf{x}$ is the $i$-th unit vector.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series \( \displaystyle \sum_{n=0}^{\infty} \left(\int_{|z|=1} \frac{\cos{\left(\frac{1}{z-10}\right)}}{z^{n+1}} \,dz\right) z^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real-valued solutions to the equation \( f(f(x)) = 25 \) for \( 0 \le x \le 2023 \), where \( f(x) = x\{x\} \) and \( \{x\} \) is the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the subset \( \left\{ \begin{pmatrix} 2 & 33 & 444\\ 0 & 3 & 4 \\ 0 & 0 & 4 \end{pmatrix}\begin{pmatrix}x \\y \\ z \end{pmatrix}:x^2+y^2+z^2 \leq 1\right\} \) of \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles, each with a radius of 6 cm, have their centers on the surface of a sphere with a radius of 10 cm. These centers are also the points of tangency between the circles and the sphere. Determine if the circles can intersect, and if so, find the number of possible intersection points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Ramsey number \( R(T, K_{1,n}) \) where \( T \) is a tree with \( m \) vertices and \( m-1 \) is a divisor of \( n-1 \), with \( m, n \ge 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the integers from 1 to 25, Ajibola wants to remove the smallest possible number of integers so that the remaining integers can be split into two groups with equal products. What is the sum of the numbers which Ajibola removes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$\int_{0}^{\infty}\text{sech}^2(x+\tan(x))dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \) be a given positive integer. Find the smallest positive integer \( u_n \) such that, for every positive integer \( d \), the number of numbers divisible by \( d \) in any \( u_n \) consecutive positive odd numbers is no less than the number of numbers divisible by \( d \) in the set of odd numbers \( 1, 3, 5, \ldots, 2n-1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. Let $\Sigma(n)$ denote the sum of the digits of the number $n$. Find the smallest three-digit $n$ such that $\Sigma(n)=\Sigma(2 n)=\Sigma(3 n)=\ldots=\Sigma\left(n^{2}\right)$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he "only" solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively, with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be the sequence defined by
$$
a_{0}=1, a_{n+1}=\frac{1}{n+1} \sum_{k=0}^{n} \frac{a_{k}}{n-k+2}
$$
Find the limit
$$
\lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{a_{k}}{2^{k}}
$$
if it exists. The original answer is in the form $f\left(\frac{a}{b}\right)=\frac{e}{c}$, please give the value of a+b+c.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are  $800$  marbles in a bag. Each marble is colored with one of  $100$  colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.

Suppose Anna has not stopped after drawing  $699$  marbles. Compute the probability that she stops immediately after drawing the  $700$ th marble.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The negation of the proposition "$\forall x > 2$, $x^{2} > 4$" is ______.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\exists x_{0} > 2, x_{0}^{2} \leqslant 4}


Context:
The given expression $\boldsymbol{F}(n)$, where $n$ denotes a variable positive integer, should be assigned a plus sign when $n$ is even and a minus sign when $n$ is odd. How can this be expressed concisely in an algebraic form without providing any verbal explanation of the formula?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Divide the sequence $\{2n+1\}$ cyclically into one-term, two-term, three-term, four-term groups as follows: $(3), (5,7), (9,11,13), (15,17,19,21), (23), (25,27), (29,31,33), (35,37,39,41), (43), \cdots$. What is the sum of the numbers in the 100th group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$  Find $a + b + c + d.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of \( \sum_{n=2}^{\infty} \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)=x^{3}+a x^{2}+b x+2015$ be a polynomial all of whose roots are integers. Given that $P(x) \geq 0$ for all $x \geq 0$, find the sum of all possible values of $P(-1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A country is shaped like a square and is divided into 25 identical square counties. In each county rules either a knight-count, who always tells the truth, or a liar-count, who always lies. One day, each count said: "Among my neighbors, there are an equal number of knights and liars." What is the maximum number of knights that there could be? (Counts are considered neighbors if their counties share a common side.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express

$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$

as a rational number.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2  \cdot a_3 \cdot  ...  \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times does $111$ appear in this sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The line connecting the center of the circumscribed circle and the intersection point of the altitudes of a scalene triangle is parallel to the bisector of one of its angles. What is the measure of this angle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects $AC$ at $X$. Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An aquarium holds clownfish, blowfish, and angelfish in a ratio of 1:1:2, respectively. 26 of the blowfish stay in their own tank, and the remaining blowfish swim into a display tank. An equal number of clownfish join the blowfish in the display tank, but then a third of these clownfish swim back into their own tank. The number of angelfish in the display tank is 1.5 times the number of clownfish that remain in the display tank, and the angelfish that stay in their own tank is half the initial number of blowfish. If the aquarium holds a combined total of 180 fish, how many clownfish are now in the display tank?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Niles is collecting his book club's annual fees. Each of the six members pays $150/year towards snacks, plus $30 each for six hardcover books and $12 each for six paperback books. There is a 10% discount on snacks for long-standing members and three of them qualify for this discount. Additionally, members need to pay a 5% sales tax on all books. Moreover, the book club chooses three special event activities during the year, costing a total of $60 for each member. Lastly, in case the chosen books are of the mystery genre, the hardcover books cost $35 each, while if the books are pre-2010 publications, the paperback books cost only $10 each. During this year, four hardcover books are mysteries and two paperback books are pre-2010 publications. Calculate the total amount of money Niles collects from all the members.
Please reason step by step, and put your final answer within \boxed{}.


Context:
(9) (20 points) The angles of triangle $ABC$ satisfy: $\frac{A}{B}=\frac{B}{C}=\frac{1}{3}$; Find the value of $T=\cos A+\cos B+\cos C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can we find the closed form of the following series? $$\sum_{k=0}^n\frac{1}{k+1}{\binom n k}^2$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x) \cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Golu started from his house towards north. After covering a distance of 8 km, he turned towards left (west) and covered a distance of 6 km. Then, he took another left (south) and walked for 10 km. Following that, he turned right (west) and walked for 3 km. Finally, he took a 45-degree turn towards southwest and covered an additional 5 km. What is the shortest distance now from his house?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct sets are there such that each set contains only non-negative powers of  $2$  or  $3$  and sum of its elements is  $2014$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \leq k \leq N$, with each possible $k$ having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let $p$ be the probability that the number of rocks left in the pile after each round is a multiple of 5. If $p$ is of the form $5^{a} \cdot 31^{b} \cdot \frac{c}{d}$, where $a, b$ are integers and $c, d$ are positive integers relatively prime to $5 \cdot 31$, find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\pi_1$, $\pi_2$, $\pi_3$ be three planes in a projective space such that $\pi_1 \cap \pi_2 \cap \pi_3$ = $\emptyset$, $\dim\,(\pi_1 \cap \pi_2) = 1$, $\dim\,(\pi_1 \cap \pi_3) = \dim (\pi_2 \cap \pi_3) =0.$ Find the dimension of $\pi_1 \lor \pi_2\lor \pi_3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $C$: $mx^{2}+ny^{2}=1(mn < 0)$, one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Determine the eccentricity of $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive integers \(a_{1}, a_{2}, \ldots, a_{2023}\) such that \(a_{k}=\sum_{i=1}^{2023}\left|a_{k}-a_{i}\right|\) for all \(1 \leq k \leq 2023\), find the minimum possible value of \(a_{1}+a_{2}+\cdots+a_{2023}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The greatest common divisor of 217 and 155 is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Mad Hatter's clock gains 15 minutes per hour, and the March Hare's clock loses 10 minutes per hour. Once, they set their clocks by the Dormouse's clock (which stopped and always shows 12:00) and agreed to meet at 5 o'clock in the evening for a traditional five-o'clock tea. How long will the Mad Hatter wait for the March Hare if each arrives exactly at 17:00 by their own clocks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
将与 105 互素的所有正整数从小到大排成数列, 试求出这个数列的第 1000 项.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Add 78.621 to 34.0568 and round to the nearest thousandth.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence $\{b_n\}$ defined recursively as $b_n = \sqrt{b_{n-1}b_{n-2}}$ for $n≥3$, where $b_1$ and $b_2$ are positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each real number \(a\) with \(0 \leq a \leq 1\), let numbers \(x\) and \(y\) be chosen independently at random from the intervals \([0, a]\) and \([0, 1]\), respectively, and let \(P(a)\) be the probability that
\(\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\).
Find the maximum value of \(P(a)\). The original answer is in the form \(k - \sqrt{m}\), please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知集合 $A=\{1,5\}, B=\left\{x^{2}-3 x+1, x^{2}-4 x+5\right\}$ ，若集合 $A \cup B$ 的元素个数是 3 ，请给出可能的实数 $x$ 中的最大值与最小值之差。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(ABCD\) be a cyclic quadrilateral, and suppose that \(BC = CD = 2\). Let \(I\) be the incenter of triangle \(ABD\). If \(AI = 2\) as well, find the minimum value of the length of diagonal \(BD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Machine A takes some hours to complete a certain job and starts that job at 7 am. After two hours of working alone, Machine A is joined by Machine B and together they complete the job at 12 pm. It would have taken Machine B 18.07228915662651 hours to complete the job if it had worked alone for the entire job. How long does it take for Machine A to complete the job alone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral $$\iint_D \frac{\partial}{\partial y}\frac{y}{(x^2+y^2)^2} \, dx \, dy$$ where the domain $D$ is defined as $D=\left\{ (x,y) \in \mathbb{R}^2 \mid 1\leq x^2+y^2 \leq 4 \right\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $\{a_n\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\cos \frac {2n\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Calculate $S_{24}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the circle $x^2+y^2=r^2$ ($r>0$) intersects with the circle $(x+3)^2+(y-4)^2=36$, then the range of $r$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Milo's parents tell him that he can win cash rewards for good grades. He will get $5 times the average grade he gets. He gets three 2s, four 3s, a 4, and a certain number of 5s. He gets $15 cash. How many 5s did he get?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The solution to the inequality

$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$

is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.

Recall that $[x]$ denotes the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Natasha climbs up a hill with varying elevation levels and changing weather conditions. The hill has three distinct elevation levels: the first section has an inclination of 15 degrees, the second section has an inclination of 30 degrees, and the third section has an inclination of 45 degrees. Additionally, during her journey, Natasha experiences four weather conditions: clear weather, light rain, heavy rain, and strong winds.

The weather conditions affect her average speed as follows:
- Clear weather: no change in speed
- Light rain: reduces her speed by 10%
- Heavy rain: reduces her speed by 35%
- Strong winds: reduces her speed by 25%

It takes Natasha 4 hours to reach the top, during which she experiences clear weather in the first section, light rain in the second section, and strong winds in the third section. On her descent, it takes her 2 hours, with heavy rain affecting her speed throughout the entire descent. If her average speed along the whole journey is 3.5 kilometers per hour, what was her average speed (in kilometers per hour) while climbing to the top under these varying conditions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
94. The base of the pyramid $ABCD$ is an equilateral triangle $ABC$ with a side length of 12. The edge $BD$ is perpendicular to the plane of the base and equals 10 $\sqrt{3}$. All vertices of this pyramid lie on the lateral surface of a right circular cylinder, whose axis intersects the edge $BD$ and the plane $ABC$. Determine the radius of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知三个月球探测器$\alpha $，$\beta $，$\gamma $共发回三张月球照片$A$，$B$，$C$，每个探测器仅发回一张照片．甲说：照片$A$是$\alpha $发回的；乙说：$\beta $发回的照片不是$A$就是$B$；丙说：照片$C$不是$\gamma $发回的．若甲、乙、丙三人中有且仅有一人说法正确，则照片$B$是探测器_____发回的．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $ABC$ has $\overline{AB} = 5$, $\overline{BC} = 4$, $\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$, and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $f_{2023}(2^{2023})$ given that $f_n(k)=f_1(f_{n-1}(k))$ and that $f_x(2^x) = 1$ whenever $x$ is a multiple of 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular table of size \( x \) cm by 80 cm is covered with identical sheets of paper of size 5 cm by 8 cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that points \( B \) and \( C \) are in the fourth and first quadrants respectively, and both lie on the parabola \( y^2 = 2px \) where \( p > 0 \). Let \( O \) be the origin, and \(\angle OBC = 30^\circ\) and \(\angle BOC = 60^\circ\). If \( k \) is the slope of line \( OC \), find the value of \( k^3 + 2k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The monotonic decreasing interval of the function $f(x) = x^2 - 2|x|$ is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two quadratic functions $y=x^{2}-2x+2$ and $y=-x^{2}+ax+b$ $(a > 0,b > 0)$, if their tangent lines at one of their intersection points are perpendicular to each other, find the maximum value of $ab$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kira has  $3$  blocks with the letter  $A$ ,  $3$  blocks with the letter  $B$ , and  $3$  blocks with the letter  $C$ . She puts these  $9$  blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence  $ABCAABCBC$  the blocks with the letter A have distances  $1, 3$ , and  $4$  between one another, the blocks with the letter  $B$  have distances  $2, 4$ , and  $6$  between one another, and the blocks with the letter  $C$  have distances  $2, 4$ , and  $6$  between one another. Altogether, we got distances of  $1, 2, 3, 4$ , and  $6$ ; these are  $5$  distinct distances. What is the maximum number of distinct distances that can occur?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ancient Greek mathematicians from the Pythagorean school studied various polygonal numbers, such as triangular numbers 1, 3, 6, 10, ..., with the $n$-th triangular number being $\frac{n(n+1)}{2} = \frac{1}{2}n^{2} + \frac{1}{2}n$. Let the $n$-th $k$-sided polygon number be denoted as $N(n, k)$ ($k \geq 3$). Below are the expressions for the $n$-th number of some $k$-sided polygon numbers:
Triangular numbers $N(n, 3) = \frac{1}{2}n^{2} + \frac{1}{2}n$
Square numbers $N(n, 4) = n^{2}$
Pentagonal numbers $N(n, 5) = \frac{3}{2}n^{2} - \frac{1}{2}n$
Hexagonal numbers $N(n, 6) = 2n^{2} - n$
...
From this, we can deduce the expression for $N(n, k)$ and calculate $N(8, 12) = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point $P$ is a moving point on circle $C_{1}$: $\left(x-1\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-1\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate plane $(xOy)$, point $A(-2,0)$ is given. Point $B$ is on the circle $C$: $(x-2)^{2}+y^{2}=4$. Point $M$ is the midpoint of $AB$. If there exists a point $P$ on the line $l$: $y=kx-\sqrt{5}k$ such that $\angle OPM = 30^{\circ}$, find the range of values for the real number $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The diagonal \( BD \) of quadrilateral \( ABCD \) is the diameter of the circle circumscribed around this quadrilateral. Find the diagonal \( AC \) if \( BD = 2 \), \( AB = 1 \), and \( \angle ABD : \angle DBC = 4 : 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the chessboard, $ 32$ black pawns and $ 32$ white pawns are arranged. In every move, a pawn can capture another pawn of the opposite color, moving diagonally to an adjacent square where the captured one stands. White pawns move only in upper-left or upper-right directions, while black ones can move in down-left or in down-right directions only; the captured pawn is removed from the board. A pawn cannot move without capturing an opposite pawn. Find the least possible number of pawns which can stay on the chessboard.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the second-order partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text {if }(x,y) \neq (0,0) \\ 0 & \text{otherwise}\end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 275 meter long train crosses a man standing on the platform in 7 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Equilateral \(\triangle ABC\) has side length \(1\), and squares \(ABDE\), \(BCHI\), \(CAFG\) lie outside the triangle. Find the area of hexagon \(DEFGHI\). The answer is in the form \(k+\sqrt{m}\), please give the value of \(k+m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence of length 15 consisting of zeros and ones, find the number of sequences that have either all zeros consecutive or all ones consecutive, but not both.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Machine A can finish a job in some hours, machine B can finish the job in 2 hours, and machine C can finish the job in 6 hours. It takes 1.090909090909091 hours for A, B, and C together to finish the job. How many hours does it take for machine A to finish the job alone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company occupies three floors in a tower: the 13th, 14th, and 25th floors. A cleaning crew is hired to clean these spaces. They work for four hours on the 13th and 14th floors. Then, the crew splits in half: one half cleans the 25th floor while the other half continues cleaning the 13th and 14th floors. After four more hours, the cleaning of the lower two floors is complete, but the 25th floor requires an additional cleaner who will finish it in 8 more hours the next day. Assuming that the cleaning time is directly proportional to the area being cleaned, how many cleaners were in the crew initially?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在三角形 $CAT$ 中，$\angle ACT =\angle ATC$ 且 $\angle CAT = 36^\circ$。如果 $\overline{TR}$ 平分 $\angle ATC$，求 $\angle CRT$ 的度数。如果答案是 $x^\circ$，求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify $\tan \frac{\pi}{24} + \tan \frac{7 \pi}{24}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfies the equation \( f\left(x^{2}\right) f^{\prime \prime}(x) = f^{\prime}(x) f^{\prime}\left(x^{2}\right) \) for all real \( x \). Given that \( f(1) = 1 \) and \( f^{\prime \prime \prime}(1) = 8 \), find the value of \( f^{\prime}(1) + f^{\prime \prime}(1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
定义在实数上的函数 $f(x)=\frac{\sin \pi x}{\sqrt{1+x+x^{2}}}$ 的最小值是以 \frac{m \sqrt{n}}{k} 的形式表示，请给出 m + n + k 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ray B is the perpendicular bisector of AC. Triangle ABC has a perimeter of $41.82$. What is the measure of AB?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence {a_n} with common difference d, if the variance of a_1, a_2, a_3, a_4, a_5, a_6, a_7 is 1, find the value of d. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a random variable $\xi$ that follows the normal distribution $N(0, \sigma^2)$, if $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangles are drawn between the rays of a 30-degree angle as shown in the figure. Points \(A_{i}\) lie on one ray, and points \(B_{i}\) lie on the other ray. The angles \(\angle O A_{i} B_{i}\) are 120 degrees, and the angles \(\angle O B_{i} A_{i+1}\) are right angles. Determine the ratio of the areas of triangles \(A_{1} B_{1} A_{2}\) and \(A_{48} B_{48} A_{49}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which \( n \) can we find a cyclic shift \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) (i.e., \( i, i+1, i+2, \ldots, n, 1, 2, \ldots, i-1 \) for some \( i \)) and a permutation \( b_1, b_2, \ldots, b_n \) of \( 1, 2, 3, \ldots, n \) such that \( 1 + a_1 + b_1 = 2 + a_2 + b_2 = \ldots = n + a_n + b_n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a class of 40 students, the first group of 15 students has an average score of 70% on an exam. The scores of the second group of 10 students have a weighted average of 85%, where each of their score counts twice as much as the scores of the first group. The third group of 10 students has an average of 92%, where each of their score counts thrice as much as the scores of the first group. The last group of 5 students have an average score of 60%, with each of their score counting half as much as the scores of the first group. What is the overall weighted average score in percentage for the entire class?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $\int_0^{\frac{\pi}{2}} \sqrt{\tan\theta} + \sqrt{\cot\theta} d\theta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the expression $$\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$$ given that $a$, $b$, and $c$ are positive real numbers such that $3a + 4b + 5c = 12$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $(a_n)$ is defined by the following initial conditions and recurrence relations:

- Initial conditions:
  \[ a_1 = 1, \quad a_2 = 0 \]

- Recurrence relations for $k \in \mathbb{N}$:
  \[ a_{2k+1} = a_k + a_{k+1}, \quad a_{2k+2} = 2a_{k+1} \]

Find the value of $a_m$ for $m = 2^{19} + 91$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the New Year's table, there are 4 glasses arranged in a row: the first and third contain orange juice, and the second and fourth are empty. In anticipation of guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings, after which the configuration becomes the reverse: the first and third are empty, and the second and fourth are full.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the tetrahedron \(ABCD\), \(\angle ACB = \angle CAD = 90^{\circ}\) and \(CA = CB = AD / 2\) and \(CD \perp AB\). What is the angle between the faces \(ACB\) and \(ACD\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 250 m long train running at a certain speed crosses another train running in opposite direction at the speed of 80 km/hr in 9 sec. The length of the other train is 250.04 m. What is the speed of the first train in km/hr?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point $O$ is a point in the plane of $\triangle ABC$, $4\overrightarrow{OA}•\overrightarrow{AC}={\overrightarrow{OC}^2}-1$, $4\overrightarrow{OC}•({\overrightarrow{OB}+\overrightarrow{OC}})=1-{\overrightarrow{OB}^2}$, then the maximum value of the angle between the vectors $4\overrightarrow{OA}+\overrightarrow{OB}$ and $2\overrightarrow{OA}-\overrightarrow{OC}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many positive integers $x$ is $(x-2)(x-4)(x-6) \cdots(x-2016)(x-2018) \leq 0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Peter wants to create a unique die where each face displays a different positive integer. For any two adjacent faces, the numbers must differ by at least two. Determine the minimum possible sum of the six numbers on the die.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?
Please reason step by step, and put your final answer within \boxed{}.


Context:
To shift the graph of the function $y=\sin \left(2x- \frac{\pi}{4}\right)$ to the graph of the function $y = \sin(2x)$, determine the horizontal shift required.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Zhenya took a $3 \times 3$ board and placed a column of blue and red cubes on each cell. Then he drew a diagram of the resulting arrangement: he wrote down the number of cubes of both colors in each column (the order of the cubes is unknown).

What is the maximum number of blue cubes Zhenya can see if he looks at the construction from the front? (For example, if a column of 8 cubes stands behind a column of 5, he will see all 5 cubes of the front column and only 3 upper cubes of the rear column.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The time on a digital clock reads 3:33. What is the shortest length of time, in minutes, until all of the digits are again equal to each other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S \) be a subset of \(\{1,2,3, \ldots, 199,200\}\). We say that \( S \) is pretty if, for every pair of elements \( a \) and \( b \) in \( S \), the number \( a - b \) is not a prime number. What is the maximum number of elements in a pretty subset of \(\{1,2,3, \ldots, 199,200\}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a dice game, you start by rolling a single fair six-sided die. If you roll an even number, you then roll two dice. If the sum of the two dice is even, you roll three dice, and this process continues, increasing the number of dice by one each time, until you roll an odd sum. The first odd sum you roll is your score. Calculate the average score per turn in this game.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is 35 7/17. What is the value of the erased number? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the 2-form $dxdy+3dxdz$ on the oriented triangle with vertices $(0,0,0)$, $(1,2,3)$, and $(1,4,0)$ in that order.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Xiaohong walks up a mountain, resting for 10 minutes every 30 minutes. Walking down the mountain, she rests for 5 minutes every 30 minutes. The speed of walking down is $1 \frac{1}{2}$ times the speed of walking up. If Xiaohong took 3 hours and 50 minutes to walk up, how much time did it take to walk down?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \text{ hours } 15 \text{ minutes}}


Context:
Given points M and N on the parabola y = 4x^2, F is the focus of the parabola, and ∠MFN = 2π/3. The midpoint of chord MN is P, and the distance from P to line l (directrix): y = -1/16 is denoted as d. If |MN|^2 = λ * d^2, calculate the minimum value of λ.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 1000 m race, A beats B by 70 m and B beats C by some meters. In the same race, A beats C by 163 m. By how many meters does B beat C?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of the $\frac{1}{z^2}$ term in the Laurent series expansion of the function $f(z) = \frac{e^{2z}}{(z-1)^3}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group $G$ of order $pq^3$, where $H$ is a normal Sylow $q$-subgroup, there exists a subgroup $P$ of order $pq^2$ and a Sylow $p$-subgroup $R$ of $P$ such that $|N_P(R)|=p$. Determine the order of $N_G(R)$. \( \text{(Express your answer as a single number.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the sides $AC$ and $BC$ of triangle $ABC$, points $P$ and $Q$ are marked respectively. It turns out that $AB = AP = BQ = 1$, and the intersection point of segments $AQ$ and $BP$ lies on the incircle of triangle $ABC$. Find the perimeter of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, let $\overrightarrow {CA} = a$ and $\overrightarrow {CB} = b$. Point $M$ is the midpoint of $CB$, and point $N$ is the midpoint of $AB$. Lines $CN$ and $AM$ intersect at point $P$. Express $\overrightarrow {AP}$ in terms of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an integer  $n\geq 3$ . For each  $3\times3$  squares on the grid, call this  $3\times3$  square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white.

Now suppose one can paint an infinite grid by white or black, so that one can select an  $a\times b$  rectangle which contains at least  $n^2-n$  isolated  $3\times 3$  square. Find the minimum of  $a+b$  that such thing can happen.

(Note that  $a,b$  are positive reals, and selected  $a\times b$  rectangle may have sides not parallel to grid line of the infinite grid.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each integer in  $\{1, 2, 3, . . . , 2020\}$  is coloured in such a way that, for all positive integers  $a$  and  $b$ such that  $a + b \leq 2020$ , the numbers  $a$ ,  $b$  and  $a + b$  are not coloured with three different colours.
Determine the maximum number of colours that can be used.

*Massimiliano Foschi, Italy*
Please reason step by step, and put your final answer within \boxed{}.


Context:
A population consists of 20 individuals numbered 01, 02,..., 19, 20. 6 individuals are selected using the following random number table. The selection method is to start from the numbers in the 5th and 6th columns of the 1st row of the random number table and select two numbers from left to right as the individual numbers.

| 7816 | 6572 | 0802 | 6314 | 0702 | 4311 |
|------|------|------|------|------|------|
| 3204 | 9234 | 4935 | 8200 | 3623 | 4869 |

Determine the number of the 5th selected individual. Express your answer as a two-digit number (e.g., 07, 11, etc.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $(a-x)^8 = a_0 + a_1x + a_2x^2 + \ldots + a_8x^8$ ($a \in \mathbb{R}$), and $a_5 = 56$, then $a_0 + a_1 + a_2 + \ldots + a_8 = \boxed{\text{\_\_\_\_\_\_}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If 60% of (x - y) = 30% of (x + y), 40% of (x + z) = 20% of (y + z), and 50% of (x - z) = 25% of (x + y + z), then what percent of x is the sum of y and z?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume that the Earth is a perfect sphere.  A plane flies between $30^\circ$ N $45^\circ$ W and $30^\circ$ N $45^\circ$ E along the shortest possible route along the surface of the Earth.  Let $\theta$ be the northernmost latitude that the plane flies over.  Compute $\sin \theta.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is UMD certified if each of these 16 squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated 90, 180, or 270 degrees about its center. How many distinct UMD certified quilts are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sides opposite to angles $A$, $B$, and $C$ in triangle $ABC$ are $a$, $b$, and $c$ respectively. It is given that $10a\cos B=3b\cos A$, and $\cos A= \frac {5 \sqrt {26}}{26}$. Find $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit as \( n \to \infty \) of the sum:
\[ S_n = \frac{1}{n}\left(\frac{1}{2} + \sum_{k=1}^{n}\cos(kx)\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_i$ be positive integers for $i=1,2,...,11$. If $x_i + x_{i+1} \geq 100$ and $|x_i - x_{i+1}| \geq 20$ for $i=1,2,...,10$, and $x_{11} + x_1 \geq 100$ and $|x_{11} - x_1| \geq 20$, what is the minimum possible value of $\sum_{i=1}^{11} x_i$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $4 \times 4$ table, numbers are written such that the sum of the neighbors of each number is equal to 1 (neighboring cells share a common side).

Find the sum of all the numbers in the table.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 10.7. Olya drew $N$ different lines on the plane, any two of which intersect. It turned out that among any 15 lines, there are always two lines such that the angle between them is $60^{\circ}$. What is the largest $N$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A travel agency held a promotion: "Buy a trip to Egypt, bring four friends who also buy a trip, and get the cost of your trip back." During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of them brought exactly 4 new clients, while the remaining 100 did not bring anyone. How many tourists traveled to the Land of the Pyramids for free?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A(x, y)$ denote the number of points ($m, n$) with integer coordinates $m$ and $n$ that satisfy $m^{2}+n^{2} \leqslant x^{2}+y^{2}$ in the plane. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. If
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y=\pi(ag-b)^{c}$$
Find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$A B C D$ is a parallelogram satisfying $A B=7, B C=2$, and $\angle D A B=120^{\circ}$. Parallelogram $E C F A$ is contained in $A B C D$ and is similar to it. Find the ratio of the area of $E C F A$ to the area of $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when \( \left \lfloor \frac{149^{151} + 151^{149}}{22499} \right \rfloor \) is divided by \(10^4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The distance between location A and location B originally required a utility pole to be installed every 45m, including the two poles at both ends, making a total of 53 poles. Now, the plan has been changed to install a pole every 60m. Excluding the two poles at both ends, how many poles in between do not need to be moved?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The hundreds digit of the 3 digit integer x is a, b is the tens digit of x, and c is the units digit of x. There is a relationship between a, b, and c such that a > 0 and the difference between the two greatest possible values of x is 241. What is the ratio of b to a?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tetrahedron $A B C D$ with volume 1 is inscribed in circumsphere $\omega$ such that $A B=A C=A D=2$ and $B C \cdot C D \cdot D B=16$. Find the radius of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$, and let $P$ be a point on the line $x= \frac{3a}{2}$. If $\triangle F_1PF_2$ is an isosceles triangle with a base angle of $30^\circ$, find the eccentricity of the ellipse $E$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly \( k \) chocolate chip cookies. Find the sum of the \( k \) for which Pearl is guaranteed to succeed regardless of how the cookies are arranged.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), and point \(B\) lies on the parabola \(y=x^{2}\). What is the minimum length of segment \(AB\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A straight line intersects parallelogram \(ABCD\) at a single point \(B\). The vertices \(A\) and \(C\) are at distances \(a\) and \(b\) from this line, respectively.

What is the distance of vertex \(D\) from this line?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Nini's report card shows the scores of 6 exams. The average score of the 6 exams is 74 points; the most frequent score among the 6 exams is 76 points; the median score of the 6 exams is 76 points; the lowest score is 50 points; the highest score is 94 points. There is only one score that appears twice, and no score appears more than twice. Assuming her scores are all integers, how many possible values are there for the second lowest score among the 6 exams?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of $m$ so that the domain of
\[ c(x) = \frac{3mx^2 - 4x + 1}{7x^2 - 6x + m} \]
is the set of all real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Janet has 'n' number of blankets, and each blanket has a distinct color. The thickness of each blanket is initially 3 inches. For each colored blanket, she folds it 'c' times (where 'c' represents the color's numeric code, ranging from 1 to 5). Calculate the combined thickness of all the folded blankets.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to\infty}\left((x(x+1)(x+2))^{\frac{1}{3}}-\left(2(x+1)-\left(x^x(x+1)^{x+1}(x+2)^{x+2}\right)^{\frac{1}{3x+3}}\right)^{\frac{3}{2}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for $n$ such that a graph $G$ with $n$ vertices can have more components and edges than another graph $G'$ with the same number of vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of moves required to solve any solvable configuration of the 8-puzzle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a \(7 \times 7\) field that is divided into 49 unit squares. Into this field, we want to place tiles in the shape of a Swiss cross, consisting of 5 unit squares. The edges of the crosses should lie on the lines of the field. Determine the smallest possible number of squares that must be marked on the field so that each cross, no matter where it is placed on the field, covers at least one marked square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the sequence $\{a\_n\}$ has positive terms, with the first term $a\_1=1$, $S\_n$ is the sum of the first $n$ terms of the sequence, and it satisfies: $a\_nS_{n+1}-a_{n+1}S\_n+a\_n-a_{n+1}= \frac {1}{2}a\_na_{n+1}$, then $\frac {3}{34}S_{12}=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of \( \alpha \) such that the inequality \( \ln(1+x) \leq x - \frac{x^2}{2} + \alpha x^3 \) holds for all \( x > -1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Distribute 5 student volunteers to participate in volunteer activities in 4 projects: Idiom Competition, Poetry Contest, Youth Song Concert, and Charity Sale. Each volunteer is assigned to only 1 project, and each project must have at least 1 volunteer. Calculate the total number of different distribution schemes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations.  One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink.  What percent of the flowers are carnations?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of detectors required to uniquely identify the location of a radiation point in an n-dimensional space using a given metric $d$. Provide your answer as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the relationship between $a$ and $b$ such that in any partition of the set $X = \{1, 2, 3, \ldots, a\}$ into two subsets, at least one subset contains an arithmetic progression of length $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the weight of 6 moles of Aluminum bromide when it is utilized in the synthesis of a complex metal-organic framework compound during a specific chemical reaction?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{a_{n}\}$ and $\{b_{n}\}$ be arithmetic sequences with the sums of the first $n$ terms denoted as $S_{n}$ and $T_{n}$, respectively. If for any $n\in N^{*}$, $\frac{{S}_{n}}{{T}_{n}}=\frac{2n-3}{4n-3}$, then calculate $\frac{{a}_{2}}{{b}_{3}+{b}_{13}}+\frac{{a}_{14}}{{b}_{5}+{b}_{11}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Team A has a probability of $$\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular tetrahedron \(ABCD\) with edge length 2, there is a point \(P\) on edge \(AB\) such that \(AP < 1\). A cut is made through point \(P\) perpendicular to the edge \(AB\) and continues through the faces but stops at a certain point. When the cut stops, the length of the cut on face \(ABD\) is \(PM = 1\) and the length of the cut on face \(ABC\) is \(PN = \frac{2}{3}\). Find the length \(MN\).
Please reason step by step, and put your final answer within \boxed{}.