Context:
From a large grid paper, cut out 5 connected squares (two squares with only one common vertex are not considered connected), such that the resulting shape can be folded into an open-top cube. How many different shapes can be cut out, considering shapes that are the same after rotation or flipping as the same? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{x^k\}$ that converges superlinearly to a point $x^*$, meaning that \( \lim_{k\to\infty}\frac{||x^{k+1} - x^*||}{||x^k - x^*||^p}=r, \) where $0<p<2$ and $r$ is a constant, determine the value of \( \lim_{k\to\infty}\frac{||x^{k+1} - x^k||}{||x^k - x^*||} \) according to the definition of superlinear convergence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $a$ for which the equation $a^x = x + 2$ has exactly two real solutions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the digits of $n$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 1. The angle $\alpha$ is greater than its complementary angle by as many times as it is smaller than its supplementary angle. Calculate the angle $\alpha$.

翻译完成，但请注意，原文本中的数学问题已经用英文表述。如果您需要解题帮助，请告知。
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the pyramid \( T A B C D \) is the trapezoid \( A B C D \) where \( B C \parallel A D \). The distances from points \( A \) and \( B \) to the plane \( T C D \) are \( r_1 \) and \( r_2 \) respectively. The area of triangle \( T C D \) is \( S \). Find the volume of the pyramid \( T A B C D \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{S \left( r_1 + r_2 \right)}{3}}


Context:
Li Jiang, a student, buys a piece of sportswear at a sports counter in a mall and receives a 100 yuan voucher. This voucher can be used to purchase one of three different types of sports socks, priced at 18 yuan, 30 yuan, and 39 yuan respectively. It is required that the entire amount of the voucher must be used in one go, and any remaining balance cannot be exchanged for cash. Li Jiang does not want to add any additional cash and wishes to utilize the voucher with an efficiency greater than 95%. Calculate the number of different ways to make a selection that fulfills these conditions. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( s \) for which the integral \[ \int\limits_{e_k}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln(x) \cdot \ln^2(x) \cdot \ln^3(x) \cdots \ln^k(x)^s} \] converges, where \( \ln^k(x) \) denotes the logarithm applied \( k \) times and \( e_k = e^{e^{e^{\ldots e}}} \) is the \( k \)-th iterated exponential.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such that $2^{k_2} \mid N$ and $3^{k_3} \mid N$ respectively. Find $k_2 + k_3.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the letters $ A$, $ M$, $ O$, $ S$, and $ U$, we can form $ 120$ five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $ USAMO$ occupies position
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知$n$个正整数$x_{1}, x_{2}, \cdots, x_{n}$的和为2016。若这$n$个数既可分为和相等的32个组，又可分为和相等的63个组，求$n$的最小值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements of order 4 in the symmetric group $S_6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer solution to the indeterminate equation \(195y = 221x + 65\) using the Indian method of kuttaka.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U \subset W \subset V$ be three linear spaces with dimensions 3, 6, and 10, respectively. Define $E \subset L(V,V)$ as the set of linear transformations $f: V \rightarrow V$ such that $f(U) \subset U$ and $f(W) \subset W$. Determine the dimension of $E$. \( \dim E \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many cyclic subgroups of order $p$ does ${\bf Z}_{p^n}\oplus{\bf Z}_p$ have, where $p$ is a prime number and $n \geq 1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix}dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} \int_{[0,n]} \left(\frac{n+x}{n+2x}\right)^n \, d\lambda \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the degree of the field extension $[\mathbb{Q}(\beta + 1):\mathbb{Q}((\beta + 1)^2 - 4)]$, where $\beta$ is a root of the polynomial $q(x) = x^2 + 3x - 3$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \( f(x) = \tan^2 x - 4 \tan x - 8 \cot x + 4 \cot^2 x + 5 \) on the interval \( \left( \frac{\pi}{2}, \pi \right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the values of $q$ for which the curve $y = x^p \sin \frac{\pi}{x}$, where $0 < x \leq 1$, has finite length.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many seven-digit numbers are divisible by 11 and have a sum of their digits equal to 59?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the automorphism group of the symmetric group of degree three, $Sym(3)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$\int_0^\infty \frac{\cos{kt}}{\pi}\,\mathrm{d}k.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the difference between the number of acyclic digraphs and the number of acyclic transitive digraphs on 5 labeled vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the slope of the curve formed by the intersection of the surface \( z = x^3 + xy^2 \) and the plane \( 2x - 2y = 1 \) at the point \( (1, \frac{1}{2}) \). Use the parametrization \( x = 1 + t \) and \( y = \frac{1}{2} + t \), and calculate the derivative with respect to \( t \) at \( t = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the generating function \( P(x) = \sum_{n=0}^{\infty} p_n x^n = 1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + 11x^6 + \cdots \), which represents the partition generating function, and \( Q(x) = \sum_{n=0}^{\infty} q_n x^n \), where \( q_n \) is the number of partitions of \( n \) containing no 1s, find the polynomial \( \frac{Q(x)}{P(x)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circle $\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA = 13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB + AC$ given that $BC = 7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest number of digits $N$, consisting of ones and twos, arranged in a circle such that every possible four-digit number composed only of the digits 1 and 2 can be represented as a sequence of consecutive digits. What is this smallest value of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0^-} \frac{5x - \sqrt{x}}{\sqrt{e^{2x} - 1}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum: $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k+1)^2 \Gamma (n+1) }{\Gamma (2 k+n+2)}((\psi ^{(0)}(n+1)-\psi ^{(0)}(2 k+n+2)) (\psi ^{(0)}(k+1)-\psi ^{(0)}(2 k+n+2))-\psi ^{(1)}(2 k+n+2)).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sarah is driving to the airport. After driving at 15 miles per hour for one hour, she realizes that if she continues at that same average rate she will be an hour late for her flight. She then travels at a faster speed for the rest of the trip, and arrives 30 minutes before her flight departs. She drove 45 miles in all. What was her speed for the rest of the trip?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is the smallest possible value of $b$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate

$$\lim _{n \rightarrow \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^{2}+j^{2}}$$

The original answer is in \frac{m\pi}{n} format. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3 Let $n$ be a given positive integer, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, let $m$ be the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$. Find the maximum value of $m$ under the condition $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a house, a hall is 20 m long, some meters wide, and 5 m high. Its interior has to be covered with mat. The total expenditure for covering the hall with mat at a cost of Rs. 20 per square meter is Rs. 19000. What is the width of the hall?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( n \) such that the series \( 1\cdot(0)!+3\cdot(1!)+7\cdot(2!)+13\cdot(3!)+21\cdot(4!)+\cdots \) up to \( n \) terms equals \( (4000)4000! \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Markov chain with state space $E = \{1,2,3,4,5\}$ and the transition matrix:

$$ P = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\ 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $$

Assuming $X_1 = 1$, calculate the expected number of transitions between successive visits to state 1, given that state 5 is never visited.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the second homotopy group \( \pi_2(X) \) for the space \( X = D^2 \cup_f S^1 \), where \( f : S^1 \to S^1 \) is a degree \( m \) map.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a random variable Y that is a mixture of a standard uniform random variable X and a degenerate random variable that always takes the value 1. Let p be a positive real number. Find the variance of Y^p as p approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Riley has 64 cubes with dimensions \(1 \times 1 \times 1\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \(4 \times 4 \times 4\) cube. Riley determines the total of the numbers on the outside of the \(4 \times 4 \times 4\) cube. How many different possibilities are there for this total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $S$  be the set of triples  $(a,b,c)$  of non-negative integers with  $a+b+c$  even. The value of the sum
    \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
    can be expressed as  $\frac{m}{n}$  for relative prime positive integers  $m$  and  $n$ . Compute  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( I = \iint_{[0,1]^2} \frac{dx \, dy}{(1+x^2+y^2)^{3/2}}. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pyramid has a base in the shape of a parallelogram with adjacent sides measuring 9 cm and 10 cm, and one of the diagonals is 11 cm. The opposite lateral edges of the pyramid are equal, and each of the longer edges measures 10.5 cm. Calculate the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $0<k<4$, 直线 $l_{1}: k x-2 y-2 k+8=0$ 和直线 $l_{2}: 2 x+k^{2} y-4 k^{2}-4=0$ 与两坐标轴围成一个四边形。求使这个四边形面积最小的 $k$ 值。原始答案的形式为$\frac{m}{n}$，请给出$m+n$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an upper bound for the number of solutions \( N_{\sqrt{2}}(1) \) to the inequality \( \left \lvert \sqrt{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \) for \( \sqrt{2} \) with \( \epsilon=1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 100 x 100 grid, the numbers 1 to 10000 are placed such that each number appears exactly once, and numbers differing by 1 are in adjacent cells. Determine the maximum possible value of the minimum distance S between the centers of any two cells containing numbers that differ by exactly 5000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $g(n)$ be defined as:

$$g(n)=\frac{\sum^{n}_{k=0}\cos\left(\frac{k+1}{n+2}\pi\right)\cos\left(\frac{k+2}{n+2}\pi\right)}{\sum^{n}_{k=0}\cos^2\left(\frac{k+1}{n+2}\pi\right)}$$

Evaluate $\lim_{n\to\infty} g(n)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If ( 1 / 3 ) ^ 32 ( 1 / 125 ) ^ k = 1 / 27 ^ 32, then k =
Please reason step by step, and put your final answer within \boxed{}.


Context:
One side of a rectangular prism is 15 m, one of its diagonals is 17 m, and its height is h m. Find the volume of the prism in cubic meters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are four people in a room. For every two people, there is a $50 \%$ chance that they are friends. Two people are connected if they are friends, or a third person is friends with both of them, or they have different friends who are friends of each other. What is the probability that every pair of people in this room is connected?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Lebesgue measure of a generalized Cantor set formed by removing the middle fifths from the unit interval.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A person travels equal distances with varying speeds and conditions. They start by walking at a speed of 2 km/h on a flat terrain, then switch to riding a bicycle at a speed of 6 km/h uphill with a 5% incline, and finally skateboard at a speed of 6 km/h downhill with a 10% decline and facing a 2 km/h headwind. The person takes a total time of 11 minutes to travel the entire route. Determine the total distance traveled?
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. [6] Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares, She can move the pieces in two ways:
- If a token has free square in front of it, then she can move this token one square to the right
- If the square immediately to the right of a token is occupied by the other token, then she can "leapfrog" the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token.
If a token reaches the 6th square, then it cannot move forward any more, and Rachel must move the other one until it reaches the 5 th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5 th square and the 6 th square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Within a cube structure consisting of 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of the cube (totaling 27 points), determine the total number of groups of three collinear points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $x_0=1$ , and let  $\delta$  be some constant satisfying  $0<\delta<1$ . Iteratively, for  $n=0,1,2,\dots$ , a point  $x_{n+1}$  is chosen uniformly form the interval  $[0,x_n]$ . Let  $Z$  be the smallest value of  $n$  for which  $x_n<\delta$ . Find the expected value of  $Z$ , as a function of  $\delta$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \(\int_{\mathcal{C}(0,3)} \frac{e^{2/z}}{z-2}\,dz\) where \(\mathcal{C}(0,3)\) is the circle centered at the origin with radius 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_0^1\int_0^\infty ye^{-xy}\sin x\,dx\,dy$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the series $\frac{1}{\sin 8^\circ} + \frac{1}{\sin 16^\circ} + \frac{1}{\sin 32^\circ} + \ldots + \frac{1}{\sin 4096^\circ} + \frac{1}{\sin 8192^\circ} = \frac{1}{\sin \alpha}$, where $\alpha \in (0, 90^\circ)$, find the value of $\alpha$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \( 6 \ln(x^2+1) = e^x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 7 points \( A_{1}, A_{2}, \cdots, A_{7} \) on a plane, where no three points are collinear and their convex hull is the triangle \( \triangle A_{1} A_{2} A_{3} \), determine the number of convex quadrilaterals that can be formed by selecting 4 of these points as vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set \( S = \{1, 2, 3, \ldots, 2022\} \) and a permutation \( A = \{a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\} \) of \( S \). It is known that for any \(1 \leq n, m \leq 2022\), the sum \( a_{n} + a_{m} \) is divisible by the greatest common divisor (GCD) of \( n \) and \( m \). Determine the number of possible permutations \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence for the Taylor series representation of the function \( f(x) = x^{0.5} \) centered at 6, given by \( \sum_{n=0}^\infty c_n(x-6)^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangular pyramid with base \( \triangle ABC \), point \( P \) lies inside the triangle such that \( PA^2 = PB^2 + PC^2 \). Find the measure of \( \angle BPC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_0^\infty \log\left(x + \frac{1}{x}\right) \frac{1}{1+x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \int \int \int (x-y) \, dV \) over the solid defined by the limits: \( y \in [0,2] \), \( z \in [-x^2+1, x^2-1] \), and bounded by the parabolic cylinders \( z = x^2 - 1 \) and \( z = 1 - x^2 \) in the xy-plane, forming a square with sides parallel to the coordinate axes and centered at \((0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{C_1} \frac{dz}{z} \) where \( C_1 \) is a contour from \( z=0 \) to \( z=R e^{2\pi i} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The budget of the Petrov family consists of the following sources of income:

- Parents' salary after income tax deduction: 56000 rubles;
- Grandmother's pension: 14300 rubles;
- Son's scholarship: 2500 rubles.

The average monthly expenses of the family include:

- Payment for communal services: 9800 rubles;
- Food expenses: 21000 rubles;
- Transport expenses: 3200 rubles;
- Leisure: 5200 rubles;
- Other expenses: 15000 rubles.

10 percent of the remaining amount is transferred to a deposit for forming a financial safety cushion. Determine the amount that the Petrov family can save monthly for an upcoming vacation.

In the answer, indicate only the number without units of measurement!
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct permutations of the letters in the word 'LYCANTHROPIES' such that the letter 'C' is not adjacent to 'A', 'A' is not adjacent to 'N', and 'N' is not adjacent to 'T'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $e_{60}=17$, $_{10}P_{50}=0.8$, and the $10$-year curtate temporary life expectancy at age $50$ is $9.2$, find $e_{50}$ using the recursive formula $e_x = P_x(1 + e_{x+1})$. Calculate $e_{50}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left| x_{k} - x_{k+1} \right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} | y_{k} - y_{k+1} | \). (2001 Shanghai Mathematics Competition)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint _{ C }^{ }{ \left( { e }^{ x }\sin { y } \right) dx+\left( { e }^{ x }\cos { y+x } \right) dy } \), where \(C\) is the arc of the circumference \( x^2 + y^2 = 1 \) in the first quadrant, traversed in the counterclockwise direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two swimmers start at opposite sides of a 90-foot swimming pool. One swims at 3 feet per second, while the other swims at 2 feet per second. They swim back and forth for 12 minutes without taking any time to turn around. How many times do they meet? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest natural number $m$ such that for each non-negative real numbers $a_1 \ge a_2 \ge \ldots \ge a_{2014} \ge 0$, the following inequality holds: $$\frac{a_1 + a_2 + \ldots + a_m}{m} \ge \sqrt{\frac{a_1^2 + a_2^2 + \ldots + a_{2014}^2}{2014}}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let x be the number of crayons Denise gives to each of her 30 friends, such that the ratio x/210 represents the proportion of crayons each friend receives. If Denise gives away 70% of her crayons among her 30 friends, what is the value of x that corresponds to the number of crayons each friend gets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数$f\left( x \right)=\frac{x{{e}^{x}}+x+2}{{{ex}}+1}+\sin x$，则$f\left( -5 \right)+f\left( -4 \right)+f\left( -3 \right)+f\left( -2 \right)+f\left( -1 \right)+f\left( 0 \right)+f\left( 1 \right)+f\left( 2 \right)+f\left( 3 \right)+f\left( 4 \right)+f\left( 5 \right)$的值是________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = ax^3 - 3x^2 + 1, if f(-a), f(a), and f(3a) form an arithmetic sequence with a non-zero common difference, calculate the number of tangents that can be drawn from the origin to the curve y = f(x).
Please reason step by step, and put your final answer within \boxed{}.


Context:
I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are there for me to choose a license plate with these restrictions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a basketball game, a certain team played a total of 8 games and scored 25, 29, 30, 32, 37, 38, 40, 42 points respectively. What is the 65th percentile of this data set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the integral as \( R \) approaches infinity: \[ \lim_{R \to \infty} \frac{1}{2\pi i}\int_{-R}^{R}\frac{1}{t-(a-bi)}dt \] where \( a \) and \( b \) are real constants and \( b > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$-{{8}^{\frac{2}{3}}}+{{\log }_{3}}\sqrt{27}+2\lg 5+\lg 4+{{7{{\log 7}}2}}=$______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Randomly divide the positive integers $1, 2, 3, 4, 5, 6, 7$ into two groups, ensuring that each group contains at least one number. Calculate the probability that the sum of the numbers in each group is equal. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of an isosceles trapezoid, given that its height is \( h \) and its lateral side is visible from the center of the circumscribed circle at an angle of \( 60^{\circ} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Revise the expression to accurately bound the probability $P(X > t\alpha)$, where $X$ follows a Poisson distribution with parameter $\alpha = N/M$, using Markov's inequality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
From a point A, the bearing to the base of a 60m high tower is 53°, and the angle of elevation to the top of the tower is 16°. From a point B, the bearing to the base of the tower is 300°, and the angle of elevation to the top is 20°. Calculate the distance between points A and B.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct cyclic subgroups of order 12 in the group \( \mathbb{Z}_6 \times \mathbb{Z}_{10}^\times \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle \(ABC\) with side length \(a\), points \(M, N, P, Q\) are located such that \(MA + AN = PC + CQ = a\). Determine the measure of angle \(NOQ\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{0}^{1} f(x) \, dx \) for a function \( f \), where \( f \) is continuous on \([0,1]\), \( f(1) = 0 \), \( \int_{0}^{1} [f'(x)]^2 \, dx = 7 \), and \( \int_{0}^{1} x^2 f(x) \, dx = \frac{1}{3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_1, x_2, ... , x_n$ be a real numbers such that
1) $1 \le x_1, x_2, ... , x_n \le 160$
2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$
Find the largest possible $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $H(w)$, determine an upper bound on the Lipschitz constant of this function, assuming that $\|x_i\| \leq 1$ for all $i$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $x$ at which the Bessel equation $x^2y'' + xy' + (x^2 - n^2)y = 0$ has a regular singularity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest possible value of n such that for any permutation of n points in the plane, the resulting broken line does not intersect itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A random glass breaks at an exponentially distributed time, with a mean of 24 months. Calculate the probability that, out of 500 glasses, at most 210 break within a year.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compare the numbers: \(9^{8^{8^9}}\) and \(8^{9^{9^8}}\). Determine which is larger.
Please reason step by step, and put your final answer within \boxed{}.


Context:
an empty fuel tank is filled with brand z gasoline . when the tank is 3 / 4 empty , it is filled with brand y gasoline . when the tank is half empty again , it is filled with brand z gasoline . when the tank is half empty again , it is filled with brand y gasoline . at this time , what percent of the gasoline in the tank is brand z ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the radius $AO$ of a circle with center $O$, a point $M$ is chosen. On one side of $AO$, points $B$ and $C$ are chosen on the circle such that $\angle AMB = \angle OMC = \alpha$. Find the length of $BC$ if the radius of the circle is 21 and $\sin \alpha = \frac{\sqrt{40}}{7}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Some men can check exam papers in 8 days working 5 hours regularly. The total hours when 2 men will check the double of the papers in 20 days is 8. How many men were initially checking the exam papers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A straight line is drawn across an 8 by 8 checkerboard. What is the greatest number of 1 by 1 squares through which this line could pass?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a rotating round table, there are 8 white teacups and 7 black teacups. Fifteen dwarves wearing hats (8 white hats and 7 black hats) are sitting around the table. Each dwarf picks a teacup of the same color as their hat and places it in front of them. After this, the table is rotated randomly. What is the maximum number of teacups that can be guaranteed to match the color of the dwarf's hat after the table is rotated? (The dwarves are allowed to choose their seating, but they do not know how the table will be rotated.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Farm tax is levied on the 45% of the cultivated land. The tax department collected a certain amount through the farm tax from the village of Mr. Willam. Mr. Willam paid only $480 as farm tax. The percentage of total land of Mr. Willam over the total taxable land of the village is 27.77777777777778%. How much money did the tax department collect from the village?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be critical if for each $i,j \in S$ at least one of the four points $(i,j), (i + 1,j), (i,j + 1)$, and $(i + 1, j + 1)$ $(99 + 1 \equiv 0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $a \gt 0$, if the inequality $8{e}^{ax}(ln2+\frac{ax}{2})-xlnx⩾0$ holds for any $x∈[\frac{1}{e},+∞)$, then the minimum value of $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}: ax-y+a=0$ and $l_{2}: (2a-3)x+ay-a=0$ are parallel, determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $5^{2018}$ has 1411 digits and starts with 3 (the leftmost non-zero digit is 3), for how many integers $1 \leq n \leq 2017$ does $5^n$ start with 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line?
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Given the set of lattice points $S=\{(x, y) \mid x, y \in\{0,1,2,3,4,5\}\}$, find the number of triangles with vertices in $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product of the digits of a 5-digit number is 180. How many such numbers exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the number of derangements in a random permutation of $n$ elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Helga went shopping and visited 6 stores.

At the first store, she tried on 7 pairs of shoes and looked at 4 handbags.

At the second store, she tried on twice the number of shoes she did at the first store and also checked out 6 more bags than the first store.

At the third store, she did not try on any shoes, but she did buy a scarf and a pair of earrings.

At the fourth store, she tried on the same number of shoes as the total number of bags she looked at in the first and second stores combined.

At the fifth store, she tried on half the number of shoes she did at the fourth store but also checked out 8 bags.

Finally, at the sixth store, she tried on the square root of the total pairs of shoes she tried on at the second and fifth stores combined, and also looked at 5 more bags than the total number of bags she had checked out previously.

What is the total number of pairs of shoes and bags Helga tried on or looked at before buying her items?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $A 、 B 、 C$ 分别是复数 $Z_{0}=a \mathrm{i}, Z_{1}=\frac{1}{2}+b \mathrm{i}, Z_{2}=1+c \mathrm{i}$ (其中 $a, b, c$ 都是实数) 对应的不共线的三点. 曲线

$$
Z=Z_{0} \cos ^{4} t+2 Z_{1} \cos ^{2} t \sin ^{2} t+Z_{2} \sin ^{4} t \quad(t \in \mathrm{R})
$$

与 $\triangle A B C$ 中平行于 $A C$ 的中位线只有一个公共点, 求出此点.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求具有下述性质的最小正数 $c$ ：对任意整数 $n \geq 4$,以及集合 $A \subseteq\{1,2, \cdots, n\}$, 若 $|A|>c n$, 则存在函数 $f: A \rightarrow\{1,-1\}$, 满足
$$
\left|\sum_{a \in A} f(a) \cdot a\right| \leq 1
$$
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:
Six horizontal lines and five vertical lines are drawn in a plane. If a specific point, say (3, 4), exists in the coordinate plane, in how many ways can four lines be chosen such that a rectangular region enclosing the point (3, 4) is formed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Initially, Aslı distributes  $1000$  balls to  $30$  boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest value of positive integer $ x$ , such that the number

  $ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$

 is a perfect square .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company hires 4 types of workers with varying levels of expertise and pay rates to work on a project. Type A workers earn Rs. 200 per day, Type B workers earn Rs. 250 per day, Type C workers earn Rs. 300 per day, and Type D workers earn Rs. 350 per day.

A team of 4 men, consisting of one worker of each type, work together 10 hours a day to finish the project and are paid their full rates if they work at least 8 hours a day. For every hour worked beyond 8 hours, they receive overtime pay equal to 1.5 times their standard hourly rate.

Later, the company decides to redistribute the workload. They assign 9 men to the project, with 3 workers of Type A, 2 workers of Type B, 3 workers of Type C, and 1 worker of Type D. This new team works 6 hours a day for the entire week.

Calculate how much the company would have to pay the new team per week including overtime pay if applicable.

Remember there are 7 working days in a week, and overtime pay applies only when a worker works more than 8 hours a day.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}\), where \(\vec{F}(x,y,z) = y^2\hat{i} + 2xy\hat{j} + xz^2\hat{k}\), and \(S\) is the upward-facing surface of the paraboloid \(z = x^2 + y^2\) in the first quadrant, bounded by the planes \(x=0\), \(y=0\), and \(z=1\). Use Stokes' theorem to find the boundary curve \(C\) and evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $A, B$ 分别为椭圆 $C: \frac{x^{2}}{2}+y^{2}=1$ 的左、右顶点, $P$ 是椭圆在第一象限内一点,满足 $|P A|=\lambda|P B|$ 且 $\angle P B A=2 \angle P A B$, 求 $\lambda$ 的值。原始答案的形式为 $\frac{\sqrt{m}}{n}$，请给出 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given condition p: $(x + 1)^2 > 4$, and condition q: $x > a$, and $\neg p$ is a sufficient but not necessary condition for $\neg q$, determine the range of values for $a$. Express your answer as an inequality involving $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
200 participants arrived at the international table tennis championship. The matches are elimination matches, meaning each match involves two players; the loser is eliminated from the tournament, and the winner remains. Find the maximum possible number of participants who have won at least three matches.
Please reason step by step, and put your final answer within \boxed{}.


Context:
From noon till midnight, Clever Cat sleeps under the oak tree and from midnight till noon he is awake telling stories. A poster on the tree above him says "Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time". For how many hours a day does the poster tell the truth?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_n= \frac {1}{n}\sin \frac {n\pi}{25}$, and $S_n=a_1+a_2+\ldots+a_n$. Find the number of positive terms among $S_1, S_2, \ldots, S_{100}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The graph of a power function passes through the point $(2, \frac{1}{4})$. The interval where it is monotonically increasing is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. [30] Consider the following two-player game. Player 1 starts with a number, $N$. He then subtracts a proper divisor of $N$ from $N$ and gives the result to player 2 (a proper divisor of $N$ is a positive divisor of $N$ that is not equal to 1 or $N$ ). Player 2 does the same thing with the number she gets from player 1 , and gives the result back to player 1. The two players continue until a player is given a prime number, at which point that player loses. For how many values of $N$ between 2 and 100 inclusive does player 1 have a winning strategy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = (x-1)3 - $\frac{1}{x-1}$ and g(x) = -x + m, if the sum of the x-coordinates of the intersection points of their graphs is 2, find the value of m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A B C$ be a triangle such that $A B=13, B C=14, C A=15$ and let $E, F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $A E F$ be $\omega$. We draw three lines, tangent to the circumcircle of triangle $A E F$ at $A, E$, and $F$. Compute the area of the triangle these three lines determine.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $a_{k}$ 为 $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$ 中整数项的个数. 则 $\sum\limits_{k=1}^{2018} a_{k}=$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=f′( \frac {π}{4})\cos x+\sin x$, find the value of $f( \frac {π}{4})$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval that contains the solution to the equation $2^{x-1}+x=5$. Express your answer as an interval in standard notation (e.g., (a, b)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to assign each integer from $1$ to $11$ one of the letters $L$, $M$, or $T$ such that no two consecutive multiples of any positive integer $n$ are assigned the same letter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define an odd function $f(x)$ on $\mathbb{R}$ that is monotonically increasing on $(0, +\infty)$ and satisfies $f(1) = 0$. Then, the solution set of the inequality $f(x) > 0$ is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The highway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. It is 54 kilometers from Lishan Town to the county town. At 8:30 AM, a bus left from Lishan Town to the county town and arrived at 9:15 AM. After a 15-minute stop, it continued to the provincial capital and arrived at 11:00 AM. Another bus left from the provincial capital to Lishan Town on the same morning at 8:50 AM, traveling at 60 kilometers per hour. What is the time when the two buses meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the quadrilateral \(ABCD\), points \(E\) and \(F\) are on \(BC\) and \(CD\) respectively, with \(\frac{DF}{FC} = 1\) and \(\frac{CE}{EB} = 2\). If the area of \(\triangle ADF\) is \(m\), and the area of the quadrilateral \(AECF\) is \(n\) (with \(n > m\)), then what is the area of the quadrilateral \(ABCD\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral using the residue theorem: $$\int_{-\infty}^\infty \frac{x\cos\left(-\frac{\pi}{2}x\right)}{x^2-2x+5} \,dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知抛物线 $\Omega: y=x^{2}$, 动线段 A B 在直线 $y=\sqrt{3} x-3$ 上 ( B 在 A 右侧), 且 $|A B|=2 \sqrt{3}$. 过 A 作 $\Omega$的切线, 取左边的切点为 M. 过 B 作 $\Omega$ 的切线, 取右边的切点为 N. 当 $M N / / A B$ 时, 求点 A 的横坐标.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the spherical triangle defined by the points \((0, 0, 1)\), \((0, 1, 0)\), and \(\left(\dfrac{1}{\sqrt{2}}, 0, \dfrac{1}{\sqrt{2}}\right)\) using the spherical Gauss-Bonnet formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are given  $n \ge 2$  distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of  $2$ . For every  $n$  find the largest possible number of elegant pairs.

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


Context:
Find the minimum value of \((s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}\) as \(s\) and \(t\) range over all real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Roberto is raising chickens for eggs. He buys 4 chickens for $20 each. They cost $1 in total a week to feed and each produces 3 eggs a week that will last forever. He used to buy 1 dozen eggs a week and spent a certain amount per dozen. After 81 weeks, the chickens will be cheaper than buying his eggs. How much did he spend per dozen eggs before buying the chickens?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $$\lim_{n\to \infty} \left[\left(1-\frac{1}{n}\right)^n-\left(1-\frac{1}{n}\right)\right]^{-n}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在黑板上写有97个数：$\frac { 4 8 } { k } ( k = 1 ,2 , \cdots , 9 7 )$，每次可任意选择两个数$a$，$b$，将它们擦去，用$2ab-a-b+1$代替，这样经过96次操作后，黑板上还剩下一个数，问：这个数是多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. In the阶梯教室 (tiered classroom) installed with connected desks, 6 people sit in one row, and candidates can only exit the examination room from the two ends of the desk. The time when candidates finish their exams varies. If a candidate sitting inside needs to submit the paper first, they will disturb others. The number of people disturbing others to submit their papers in a row is regarded as a random variable $X$, then the mathematical expectation of $X$ equals $\qquad$ (Su Chun provided the question)

Note: "阶梯教室" is translated as "tiered classroom" to maintain the meaning in English.
Please reason step by step, and put your final answer within \boxed{}.


Context:
a scale 7 ft . 6 inches long is divided into 5 equal parts . find the length of each part .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Perpendiculars $AP$ and $AK$ are drawn from vertex $A$ to the angle bisectors of the external angles $\angle ABC$ and $\angle ACB$ of triangle $ABC$. Find the length of segment $PK$ if the perimeter of triangle $ABC$ is $p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( z = \sqrt{i} - \sqrt{-i} \) and find its value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all values of $a$ and $p$ for the pairs that satisfy the condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all the rational roots of
\[2x^3 - 9x^2 - 32x - 21 = 0.\]Enter all the rational roots, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
$\begin{aligned} \lim_{A\to+\infty}\frac{1}{A}\int_{1}^{A}A^{\frac{1}{x}}dx. \end{aligned}$

Proposed by Jan Šustek, University of Ostrava
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the odd function $f(x)$ is also a periodic function, and the smallest positive period of $f(x)$ is $\pi$, when $x \in \left(0, \frac{\pi}{2}\right)$, $f(x) = 2\sin x$. Find the value of $f\left(\frac{11\pi}{6}\)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is $(a^3+b^3)\div(a^2-ab+b^2+c)$ for $a=7$, $b=6$, and $c=1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知在三棱锥 $D-A B C$ 中, $\angle A C B=\angle A B D=90^{\circ}, C A=C B, \angle B A D=30^{\circ}$.若点 $C$ 在平面 $A B D$ 上的射影恰好在 $A D$ 上, 则二面角 $C-A B-D$ 的平面角的正弦值大小为 \frac{\sqrt{m}}{n}，请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $g(x)$ be a function piecewise defined as \[g(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
2x-41 & x>0.
\end{array}
\right.\] If $a$ is negative, find $a$ so that $g(g(g(10.5)))=g(g(g(a)))$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a city, there are 10,000 bicycles with all possible numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered does not contain the digit 8?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The weather forecast says that in the next three days, the probability of rain each day is $0.4$. Someone used a computer to generate random integers between $0$ and $9$, using $1, 2, 3, 4$ to represent rain, and $5, 6, 7, 8, 9, 0$ to represent no rain. They generated $3$ random numbers as a group, producing $20$ groups of random numbers as follows: $027556488730113537989907966191925271932812458569683431257393$. Based on this, the probability of having at least two rainy days in these three days is approximately \_\_\_\_\_\_
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that f(x) is an even function defined on R, and when x < 0, f(x) + x * f'(x) < 0, and f(-4) = 0, find the solution set of the inequality x * f(x) > 0. Express your answer as an interval or union of intervals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the genus of the field extension $K = \mathbb{C}(T)(\sqrt{T^n + 1})$ over $\mathbb{C}(T)$, where $n$ is an even integer greater than or equal to 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of terms required in the alternating series representation of \( \pi \), given by \( \pi = 4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1} \), to approximate \( \pi \) accurately up to 10 decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the expected value of the number of balls remaining in an urn containing $a$ red and $a$ blue balls if balls are drawn without replacement until all blue balls are drawn?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matrix \( A \in \mathbb{R}^{5 \times 5} \) and a scalar \( \alpha \in \mathbb{R} \), consider the linear system \( Ax = \begin{pmatrix} 1 \\ -1 \\ a^2+a \\ 1 \\ -1 \end{pmatrix} \). The general solution is given by \( \begin{pmatrix} t+u+1 \\ s+t-u \\ s-t+1 \\ -s-t+u \\ -s-u-1 \end{pmatrix} \). Additionally, the vectors \( \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ a^2-3 \end{pmatrix} \) and \( \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\ a^2-2 \end{pmatrix} \) are in the column space of \( A \). It is known that \( \alpha = 0 \) or \( \alpha = -1 \) satisfies these conditions. Determine which value of \( \alpha \) is still valid if \( A^2 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The segment \( AB = 8 \) intersects the plane \( \alpha \) at an angle of \( 30^\circ \) and is divided by this plane in the ratio \( 1:3 \). Find the radius of the sphere passing through points \( A \) and \( B \) and intersecting the plane \( \alpha \) in a circle of the smallest radius.
Please reason step by step, and put your final answer within \boxed{}.


Context:
P.J. starts with \(m=500\) and chooses a positive integer \(n\) with \(1 \leq n \leq 499\). He applies the following algorithm to \(m\) and \(n\): P.J. sets \(r\) equal to the remainder when \(m\) is divided by \(n\). If \(r=0\), P.J. sets \(s=0\). If \(r>0\), P.J. sets \(s\) equal to the remainder when \(n\) is divided by \(r\). If \(s=0\), P.J. sets \(t=0\). If \(s>0\), P.J. sets \(t\) equal to the remainder when \(r\) is divided by \(s\). For how many of the positive integers \(n\) with \(1 \leq n \leq 499\) does P.J.'s algorithm give \(1 \leq r \leq 15\) and \(2 \leq s \leq 9\) and \(t=0\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A papaya tree will grow 2 feet in the first year. In the second year, it will grow some percentage more than the first year. In the third year, the tree will grow 50% more than in the second year. In the fourth year, it will grow twice as much as the third year. In the fifth year, it will grow half as much as the fourth year. When the tree is 5 years old, it is 23 feet tall. What is the percentage increase in growth from the first year to the second year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
对于平面 $\alpha$, 若多面体的各个顶点到平面 $\alpha$ 的距离均相等, 则称平面 $\alpha$ 为多面体的“中位面”.
四面体有多少个互不相同的中位面?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a positive constant \( c \) such that the normalized sum \( \frac{S_n}{n^c} \) converges in distribution to a random variable \( A \), where \( S_n = X_1 + X_2 + \ldots + X_n \) and the characteristic function of the iid random variables \( X_i \) is given by \( \phi(t) = 1 - \sqrt{|t|(2-|t|)} \) for \( t \in [-1,1] \) and \( \phi(t) = 0 \) elsewhere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the norm \( \|\psi\| \) of the functional \( \psi: L^1[-1,1] \to \mathbb{C} \) defined by \( \psi(g)=\int_{-1}^{0}x^2g(1-x)\,\mathrm{d}x. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ideals of order 8 in the factor ring \( R = \mathbb{F}_2[x, y]/\langle x^2, y^2 \rangle \) without using the brute force method.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Xiaoming's mother went to the store to buy soap. The unit prices of brand $A$ and brand $B$ soaps are 6 yuan and 9 yuan respectively. Xiaoming's mother can buy 2 more pieces of brand $A$ soap than brand $B$ soap with all the money she has, and there is no money left. How much money did Xiaoming's mother bring?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_{-\infty}^{\infty} [F(x+10) - F(x)] \, dx \), where \( F \) is a distribution function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We have 21 pieces of type $\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while following this rule. What is the smallest $k$ such that there exists a maximal arrangement of $k$ pieces of type $\Gamma$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $f(x),$ defined for $0 \le x \le 1,$ has the following properties:

(i) $f(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $f(x) \le f(y).$
(iii) $f(1 - x) = 1 - f(x)$ for all $0 \le x \le 1.$
(iv) $f \left( \frac{x}{3} \right) = \frac{f(x)}{2}$ for $0 \le x \le 1.$

Find $f \left( \frac{2}{7} \right).$The answer is in the form rac{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:
In a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A **binary palindrome** is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). Leading zeroes are not permitted in the standard representation. For example, 2015 is a binary palindrome because in base 2 it is \(11111011111\). How many positive integers less than 2015 are binary palindromes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of a regular triangular prism if the side of its base is \(a\) and the lateral surface area is equal to the sum of the areas of the bases.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man is rowing a flat bottom canoe down a river which varies in width. In still water, he can row at a speed of 24 kmph. He realizes that it takes him thrice as long to row up the river as it does to row down the river. On top of that, he has to deal with cross-currents that alter his speed by a factor of 15%. Assuming the cross-currents affect his speed equally in both directions and that the speed of the stream changes linearly with the width, calculate the rate of the stream in a section with double the width of the narrowest section.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be a Lebesgue integrable function on $[0,1]$ with $\lim_{x\rightarrow 1}f(x) = c$, where $c$ is a constant. Determine the value of $\int_0^1 nx^n f(x) \, dx$. \(n \in \mathbb{N}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the nine-sided regular polygon $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1,A_2,\dots,A_9\}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Peter knows that the distance an airplane travels between two countries is 7019 km, while the distance between Spain and Germany is 1615 km. Peter took a flight from Spain, which made a stopover in Germany, while being there, he receives a call from his brother asking him how many kilometers he has to fly to get to the other country and then return to Spain. What is the distance between Spain and the other country?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \(\lim_{x \to 0} \left(\frac{1 + \sin x}{1 - \cos x}\right)^{\frac{1}{x}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the standard notation or terminology for the set \( S = \{ a \in A \mid \sigma(a) = a \} \) if \( A \) is an algebra over a field \( F \) and \( \sigma: A \rightarrow A \) is an involution of the second kind?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the integer value of $x$ that maximizes the function \( f(x) = a\left(1-\dfrac{a}{x}\right)\left(1-\dfrac{a}{bx}\right)^{n-2} \) given the constraints: \( a \in \{1, 1/4, 1/2, 3/4\} \), \( 50 \leq n \leq 500 \), \( 5 \leq b \leq 44 \), and \( 5 \leq x \leq 64 \). Use the derived formula \( x = \dfrac{a(n-1)}{n+b-2} \) as a starting point, and ensure that the final value of \( x \) is an integer within the specified range.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a farm, there are ducks, cows, and spiders. The total number of legs is 72 more than twice the number of heads. Additionally, there are twice as many spiders as ducks, and the total number of animals does not exceed 40. Find the total number of cows in the farm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of $a^x$ using the limit definition: \( \frac{\mathrm{d}}{\mathrm{d}x}a^x = \lim_{h\to0}\frac{a^{x+h}-a^{x}}{h} \). Specifically, determine \( \lim_{h\to0}\frac{a^{h}-1}{h} \) for \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 240 residents in the village of Sosnovka. Some people are acquainted with each other, and some are not. It is known that any five residents can be seated at a round table so that each of them is acquainted with both of their neighbors. What is the smallest number of pairs of acquainted residents in Sosnovka?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum possible value of the expression

$$
27abc + a\sqrt{a^2 + 2bc} + b\sqrt{b^2 + 2ca} + c\sqrt{c^2 + 2ab}
$$

where \(a, b, c\) are positive real numbers such that \(a + b + c = \frac{1}{\sqrt{3}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The eight-digit number 757AB384 is divisible by 357. Get B.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A river 42 feet across is to be used for a boat race. Each boat is 3 feet wide and must have at least 2 feet of space between them or the riverbank. There is also a 2 mph current flowing downstream that affects the boats' speed. If each boat's individual speed without any current is 6 mph, calculate the minimum time it takes for all boats to reach the finish line located 1 mile downstream.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such that $\tfrac{1}{E_n}
> 2017$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A glazier has 15 windows of various sizes and shapes, each requiring one pane of glass. The panes have been prepared to match the windows exactly, but the glazier is unaware of this. He selects panes by finding one that is large enough for each window, either exactly fitting or one that can be cut to fit. If no suitable pane is found, he moves to the next window. It is not allowed to assemble a pane from multiple parts. What is the maximum number of windows that can remain unglazed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Chloe wants to purchase a jacket that costs $\$45.50$. She checks her purse and finds she has four $\$10$ bills, ten quarters, and some nickels and dimes. What is the minimum number of dimes that she must have if she has 15 nickels?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $R$ be an equivalence relation on $P(\Bbb Z)$. Determine the number of elements in the equivalence class of a set $S$ that contains exactly $k$ elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two table tennis players, \(A\) and \(B\), are preparing to play 7 matches against each other. Player \(A\) is offered 100 times the money if he wins all 7 matches. Given that past experience shows \(A\) has twice the chance of winning 4 matches and losing 3, as opposed to winning 5 matches and losing 2, determine the fair multiplier instead of 100 times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of permutations \( \alpha \in S_9 \) such that \( \alpha \omega \alpha^{-1} = \gamma \), where \( \omega = (13624)(587)(9) \) and \( \gamma = (15862)(394)(7) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F$ be a focus of the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$. If there exists a point $P$ on $C$ such that the midpoint of the line segment $PF$ is exactly one endpoint of its imaginary axis, then the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $K$ such that all roots of the polynomial \( m^3 + m^2 + 2m + K = 0 \) are negative.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A grocery store sells three different types of fruit: apples, oranges, and bananas, with varying quantities in each box. The markup on a box of apples is 10 percent of the cost, the markup on a box of oranges is 15 percent of the cost, and the markup on a box of bananas is 20 percent of the cost. The costs of the boxes are $30 for apples, $40 for oranges, and $50 for bananas. The box of apples contains 25 apples, the box of oranges contains 20 oranges, and the box of bananas contains 15 bananas.

Calculate the weighted average markup percentage of the total selling price for all three boxes of fruits, considering the quantity of each fruit type. (Note: Markup = Selling Price - Cost)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $\{a\_n\}$ consists of positive numbers and satisfies $a\_1=1$, $\sqrt { \frac {1}{ a\_{ n }^{ 2 }}+3}= \sqrt { \frac {1}{ a\_{ n+1 }^{ 2 }}}$. Let $b\_n= \frac {1}{ a\_n^{ 2 } a\_{ n+1 }^{ 2 }}$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is denoted as $S\_n$. If $S\_n < t$ holds true for any $n∈N^{*}$, then the range of the real number $t$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal number \( M \) of telephone connections needed to ensure that no more than 1% of calls from city A to city B are blocked. Assume city A has 2000 phones, each user needs a connection for 2 minutes on average during 10 working hours, and calls are random. Use a Gaussian distribution with a mean of \( 6.6667 \) connections. Ignore impossible cases of less than 0 or more than 2000 calls.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The volume of a truncated cone circumscribed around a sphere is twice the volume of the inscribed sphere. By how many times is the radius of the base circle of the truncated cone the radius of the top circle's radius?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \sqrt[n]{\frac{(2n - 1)!}{n!}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the series \( \sum_{n=0}^\infty \frac{x^n}{n\#} \) converges, where \( n\# \) is the product of all primes less than or equal to \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{n=1}^{\infty}\frac{n^n}{(H_n)^2n!}\) where \(H_n=\sum\limits_{k=1}^n\frac1k\) is the \(n\)th harmonic number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{\gamma}\omega \) on the curve \( \gamma(t)=(t\cos(2\pi t), t\sin(2\pi t), t^2) \), where \( t \in [0,1] \), and the curve is contained in the paraboloid \( S \) defined by \( S = \{(x,y,z) \mid z = x^2 + y^2\} \). The differential form is \( \omega = ydx + xdy + (x^2 + y^2)dz \). Evaluate the integral and find the value of \( \int_{\gamma} \omega \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)$ be a polynomial of degree 3 with real coefficients and leading coefficient 1. Let the roots of $P(x)$ be $a, b, c$. Given that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a, b, c, d$ such that $\gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In rectangle $ABCD$, where $AB = 6$ and $BC = 3$, a point $P$ is chosen on $AB$ such that $\angle APD = 2 \angle CPB$. Determine the length of $AP$. \( \text{Express your answer as a decimal.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train of length 400 m moving at a speed of 72 kmph takes 69.99440044796417 seconds to cross another train of a certain length, moving at 36 kmph in the same direction. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是。原始答案为 \frac{a}{b} 格式，请给出 a + b 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which natural numbers \( n \), does there exist a bijection \( f : \mathbb{N} \to \mathbb{Q}^n \) and a norm \( \|\cdot\| \) on \( \mathbb{R}^n \) such that \( \|f(k+1)-f(k)\|=1 \) for all \( k \in \mathbb{N} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A king has eight sons, and each night he sends three of them to guard the golden apples from the Firebird. No two sons agree to guard together more than once. What is the maximum number of nights this arrangement can continue?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 13 weights arranged in a row on a table, ordered by mass (the lightest on the left, the heaviest on the right). It is known that the mass of each weight is an integer number of grams, the masses of any two adjacent weights differ by no more than 5 grams, and the total mass of the weights does not exceed 2019 grams. Find the maximum possible mass of the heaviest weight under these conditions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a drawer, there are 5 white socks, 3 brown socks, and 4 blue socks. How many ways can you choose a pair of socks such that the socks are of different colors and at least one of them is blue?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If five pairwise coprime distinct integers \( a_{1}, a_{2}, \cdots, a_{5} \) are randomly selected from \( 1, 2, \cdots, n \) and there is always at least one prime number among them, find the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Natalia is participating in a week-long cycling competition. On Monday, she rode 40 kilometers and achieved a 5% bonus for her speed. On Tuesday, she rode 50 kilometers and earned a 3% bonus. On Wednesday, she rode 50% fewer kilometers than she did on Tuesday and gained a 7% bonus for her speed. On Thursday, she rode as many kilometers as the sum of the distances covered on Monday and Wednesday, and earned a 4% bonus for her speed.

On Friday, she increased her ride distance by 20% compared to the distance she covered on Thursday, and earned a 6% speed bonus. On Saturday, she covered 75% of the distance she covered on Friday but faced a challenging elevation gain, resulting in a 2% bonus. Finally, on Sunday she managed to ride as many kilometers as the difference between the distances she covered on Saturday and Wednesday, and due to an easier elevation, earned a 10% bonus for her speed.

Considering the base distances ridden each day, the bonuses earned for speed, and additional weekend rides with fluctuating elevation gains, how many kilometers did Natalia effectively ride in total during the week-long competition, including those earned from her speed bonuses?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\omega\) be a complex number such that \(\omega^{2013} = 1\) and \(\omega^m \neq 1\) for \(m = 1, 2, \ldots, 2012\). Determine the number of ordered pairs of integers \((a, b)\) with \(1 \le a, b \le 2013\) such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is an algebraic integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{n\to \infty} \frac{a_n}{b_n} \) given that \( a_n \to \infty \), \( b_n \to \infty \), and \( a_n + b_n \) is bounded.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \rightarrow 0} -2\int_{0}^{x^{2}} \cos t^{2} \, dt \] and express the result in terms of a Maclaurin series expansion up to the order of $x^2$. Ignore terms of order $o(x^4)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all polynomials  $P$  with real coefficients such that   $$ \frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x) $$   holds for all nonzero real numbers  $x,y,z$  satisfying  $2xyz=x+y+z$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? Provide your answer as a simplified fraction in the form \frac{k}{m}, and find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $p$ 为质数, $\sqrt{p}$ 的小数部分为 $x, \frac{1}{x}$ 的小数部分为 $\frac{\sqrt{p}-31}{75}$. 求所有满足条件的质数 $p$ 的值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \] where \( \delta_{a} \) is the Dirac distribution at the point \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the direction vector of line $l$ is $\overrightarrow{e}=(-1,\sqrt{3})$, calculate the inclination angle of line $l$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\prod_{k=1}^{6}\left(x^{5}-5 x+k\right)=0$ 有 $\qquad$个实根.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ted loves creating haikus, which are Japanese three-line poems with syllable patterns of 5, 7, and 5 respectively. However, his vocabulary is limited. Specifically, for integers $1 \le n \le 7$, he knows $2n$ words that each contain $n$ syllables. Words cannot cross between lines but may be repeated.

If Ted can create $N$ distinct haikus, find the remainder when $N$ is divided by $100$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
定义函数 A(x)=\ds{m=1}{\infty}\ds{n=0}{\infty}\df{m!n!}{(m+n+1)!}x^m，计算 [100A(\frac{2}{3})].
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$.

For example, if $n = 117$, it has six divisors: $1, 3, 9, 13, 39, 117$. Two of these divisors, namely $3$ and $13$, have unit digits equal to $3$. Hence, for $n = 117$, $p = 33.33\ldots$.

Find the maximum possible value of $p$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a circle of length 2013, there are 2013 marked points dividing it into equal arcs. Each marked point has a chip. We call the distance between two points the length of the smaller arc between them. For what largest $n$ can the chips be rearranged so that there is again a chip at each marked point, and the distance between any two chips, initially no more than $n$ apart, increases?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number \( a \) is a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). For which natural values of \( n \) does the equality \( a^{4} + a^{3} = a^{n} + 1 \) hold?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the residue of \( \cot^2(z) \) at each of its poles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty} \left(\prod_{k=1}^{k=n} \prod_{r=0}^{r=k}\frac {n-k+r}{r}\right)^{1/((n)(n+1))}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (with masses of 1 kg, 2 kg, 3 kg, ..., up to 20 kg), balancing a watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on paper the mass of the weights he used. What is the minimum number of different numbers that could appear in his records, given that the mass of each weight is an integer in kilograms?
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task A-2.2.

A 2021-digit number is written. Each two-digit number formed by two consecutive digits of this number (without changing the order) is divisible by 17 or 23. The unit digit of this number is 7. What is its first digit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f:[0,1] \rightarrow \mathbb{R} \) be a continuous function such that \( \int_0^1 f(x) \, dx = 0 \). Compute the limit:
\[ \lim _{t\to 0} \frac{1}{t} \int_0^1 (f(x+t)-f(x))x \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim _{x\to \infty }\left(\frac{\left((3x)!\right)}{x^{2x}(x!)^2}\right)^{\frac{1}{x}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of a plane that is invariant under the transformation given by the matrix \( \begin{pmatrix}4&-23&17\\ 11&-43&30\\ 15&-54&37\end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a polynomial \(P(x)\) of degree \(n\) with algebraic coefficients, where each coefficient has a minimal polynomial of degree \(\le D\) over \(\mathbb{Q}\), determine an upper bound on the degree of the minimal polynomials over \(\mathbb{Q}\) for the roots of \(P(x)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ x$ and $ y$ be positive integers such that $ 7x^5=11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a +b+c+  d$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Determine the smallest possible value of the total number of intersection points of these circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 3 Let $P_{m}=\left\{A_{1}, A_{2}, \cdots, A_{m}\right\}$ be a partition of an $n$-element set $X$ (i.e., $A_{1}, A_{2}, \cdots, A_{m}$ is a partition of $X$). If one of the $A_{i}$ is further divided into two sets, this results in a partition $P_{m+1}$ of $X$, which consists of $m+1$ sets. $P_{m+1}$ is called a refinement of $P_{m}$. If
$$
P_{1}, P_{2}, \cdots, P_{n}
$$

are all partitions of the $n$-element set $X$, and each is a refinement of the previous one (clearly, in this case, $P_{n}$ consists of $n$ sets, while $P_{1}$ consists of only one set, which is $X$), then (7) is called a chain of length $n$. Find the number of chains of length $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the number of ordered 10-tuples of positive integers $(a_1, a_2, \ldots, a_{10})$ such that \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{10}} = 1 \) is even or odd.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation for real and positive parameters $a$, $b$, and $c$:
\[ \sqrt{a+bx}+\sqrt{b+cx}+\sqrt{c+ax}=\sqrt{b-ax}+\sqrt{c-bx}+\sqrt{a-cx} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct countable subsets that can be formed from the set of real numbers \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-archimedean extensions of the absolute value of $\mathbb{Q}$ for the field obtained by adjoining the $n$th root of a prime number $p$ to $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements of order 10 in the symmetric group $S_{10}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the series \( a_n = \frac{1}{\sqrt{n}} \left( 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \) converges or diverges as \( n \to \infty \). If it converges, find its sum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value \( \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right] \) where \( Z \sim N(0, \sigma^2I) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \cdot CD = AB \cdot DA$. What is the largest possible value of $BD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the graph of the function \( y = f(x) \) is symmetric with respect to the point \( (1,1) \) and the line \( x + y = 0 \), if \( f(x) = \log_{2}(x + 1) \) when \( x \in (0,1) \), find the value of \( f\left(\log_{2} 10\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \[
\left\lfloor \frac{2017! + 2014!}{2016! + 2015!}\right\rfloor.
\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the determinant of the principal matrix solution \( \Pi(t,t_0) \) for the system of differential equations \( \dot{x} = A(t)x \) in \( \mathbb{R}^2 \), where the matrix \( A(t) \in GL(\mathbb{R}^2) \) has eigenvalues \( \alpha_1(t) = t \) and \( \alpha_2(t) = 0 \). Calculate \( \det(\Pi(1,0)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability that in a sequence of coin flips, there will be a point where the number of heads is at least twice the number of tails?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(x_1 + x_2 + \cdots + x_{10} = 15\) where \(x_1 \geq 1\) and \(0 \leq x_i \leq 9\) for \(i = 1\) to \(10\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
11. (5 points) Arrange the natural numbers $1,2,3,4$, in ascending order without any gaps, to get: 1234567891011121314. In this sequence of digits, when the first set of 5 consecutive even digits appears, the position of the first (even) digit from the left is the how many-th digit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{n=1}^{\infty} \frac{\ln n}{n(n- \ln n)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the random variables \(X_1, \ldots, X_n\) that are independently and identically distributed (i.i.d.) according to the probability density function (PDF) \(f(x) = \frac{1}{\sigma^{3} \sqrt{2\pi}}x^{2}e^{-\frac{x^2}{2\sigma^2}}\), where \(\sigma > 0\), find the distribution of the sum \(\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \beta \) for which a transcritical bifurcation occurs in the equation \( x_{n+1} = \beta x_{n} (2 - x_{n}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many onto functions exist from a set \( A \) with 6 elements to a set \( B \) with 6 elements, such that the function maps the element 1 in \( A \) to the element 2 in \( B \), and no element in \( A \) maps to itself in \( B \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \(2^x \cdot 3^y = 1 + 5^z\) in positive integers \(x, y,\) and \(z\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic behavior of \( \mu(\mathcal{C} \cap \{A: \|A\|_1 \le r\}) \) as \( r \to \infty \), where \( \|A\|_1 \) denotes the 1-norm of matrix \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Picard group of the product space \( \mathbb{P}^n \times \mathbb{C}^* \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the constant $c$ such that the inequality \( \left\|\begin{pmatrix} x^2 - y^2\\2xy \end{pmatrix}\right\| \leq |c|\left\|\begin{pmatrix} x\\y \end{pmatrix}\right\| \) holds for all $x$ and $y$ in polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( A = \bigcap_{k=1}^{\infty} A_k \), where \( A_k = \{ a \in \mathbb{N}^{\mathbb{N}} \mid \sum_{i=100k}^{100k+99} a_i = 200 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square of side length $1$ and a circle of radius $\frac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A right regular hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$ with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, and $FF'$, each perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. A plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained within the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $x, y, z$ are positive real numbers satisfying $x + 2y + 3z = 10$, find the maximum value of the expression $x^2y^2z^6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
实数 $x, y, z, w$ 满足 $x+y+z+w=1$ ，则 $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ 的最大值是 $\qquad$.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:
Given an increasing sequence $\{a_{n}\}$ where all terms are positive integers, the sum of the first $n$ terms is $S_{n}$. If $a_{1}=3$ and $S_{n}=2023$, calculate the value of $a_{n}$ when $n$ takes its maximum value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the index of the unit circle with respect to the vector field given by the system of differential equations:
\[\begin {cases}
\frac{dx}{dt} = 2x^2-1\\
\frac{dy}{dt} = 2xy
\end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the flux of the vector field \( \vec{F} = \langle x, y, 5 \rangle \) through the surface \( S \), which is the region enclosed by \( x^2 + z^2 = 1 \), \( y = 0 \), and \( x + y = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let 3 < x < 4, 4 < a < x, 6 < y < 8, and 8 < b < y. Given that a^2 + b^2 > x^2 + y^2, find the greatest possible positive integer difference of x and y.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \( n \) people, and it is known that any two of them can call each other at most once. Among them, the total number of calls between any group of \( n-2 \) people is the same and is \( 3^{k} \) times, where \( k \) is a natural number. Find all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals:
Please reason step by step, and put your final answer within \boxed{}.


Context:
A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the three students with strong computer skills. How many different allocation schemes are there? (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the fractional equation $\frac{1-x}{x-2}=\frac{a}{2-x}-2$ has a positive root, then $a=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, find the value of ${\log_{3}{a_1} + \log_{3}{a_5} +...+\log_{3}{a_{10}}}$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn + 13m + 13n - m^2 - n^2$. From the value of this expression alone, he miraculously determines both $m$ and $n$. What is the value of the above expression?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 345^\circ.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the set $\{1,2,\ldots,n\}$, where $n \geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ such that:
\[ \sum_{i=1}^n \alpha(i) \beta(i) = \frac{19}{10} \sum_{i=1}^n \gamma(i)\delta(i) . \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the 2011 Shanghai Spring College Entrance Examination, there were 8 universities recruiting students. If exactly 3 students were admitted by exactly 2 of these universities, then the number of ways this could happen is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a topological space \( E \) with the topology \( \theta = \{ G \subseteq E \mid \text{card}(E \setminus G) < \infty \} \cup \{ \emptyset \} \), where \( \text{card}(E) = \infty \), find the interior \( \mathring{A} \) of a set \( A \) such that \( \text{card}(A) = \infty \) and \( \text{card}(E \setminus A) = \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the nth derivative of the function \( f(x) = \frac{1}{x}e^x - \frac{1}{x^2}e^x + \frac{1}{x^2} \) at \( x = 0 \). Express your answer in the form \( \frac{1}{a(n)} \), where \( a(n) \) is an expression in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alex writes down some distinct integers on a blackboard. For each pair of integers, he writes the positive difference of those on a piece of paper. Find the sum of all $n \leq 2022$ such that it is possible for the numbers on the paper to contain only the positive integers between $1$ and $n$, inclusive, exactly once.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the infinite series: \[ \sum_{n=0}^{\infty} \frac{16n^2 + 20n + 7}{(4n+2)!} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence given by $a_0 = 3$ and such that for $i \ge 1$, we have $ai = 2^{a_{i-1}} + 1$. Let $m$ be the smallest integer such that $a^3_3$ divides $a_m$. Let $m'$ the smallest integer such that $a^3_m$ divides $a_{m'}$ . Find the value of $m'$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 6?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of \(x^2 y + y^2 z + z^2 x\) for real numbers \(x, y, z\) with the conditions that their sum is zero and the sum of their squares is six.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One eighth of three fifths of four sevenths of five eleventh of a number minus one ninth of two thirds of three fourths of five eighths of that same number is 30. What will be 75% of that number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $A = \{1, 2, 3, 5, 8, 13, 21, 34, 55\}$, find how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functions \(f(t)=(\int_0^{\sqrt{t}}e^{-x^2} dx)^2\) and \(g(t)=\int_0^1\frac{e^{-t(1+x^2)}}{1+x^2} dx\) for \(t>0\), find \(f'(t)+g'(t)\).
Please reason step by step, and put your final answer within \boxed{}.