Context:
At a race track, a maximum of 6 horses can race together at a time. There is a total of 30 horses. Races not only depend on speed, but also rely on track conditions, unpredictable horse behavior during each race, and asymmetric track layouts, where each horse encounters unique challenges with varying difficulty levels. Track conditions and layouts change for each race. What is the minimum number w of races we need to conduct to confidently determine the top 3 fastest horses under different conditions and asymmetric layouts?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A technician makes a round-trip to and from a certain service center by the same route. On the way to the center, the technician encounters a traffic jam that causes a 15% increase in travel time and a construction detour that causes an additional 10% increase in travel time. If the technician completes the drive to the center and then completes 20% of the drive back from the center without any delays, determine the completion percentage of the round-trip, accounting for the delays experienced on the way to the center.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola y = 3 - x^2 and the line of symmetry x + y = 0, find the distance |AB| between two distinct points A and B on the parabola that are symmetric with respect to the line.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dima is driving on a straight highway from point A to point B. From point B, a traffic jam is extending towards Dima at a speed of \( v \) km/h. The speed of the car in the traffic jam is 10 km/h, and outside the traffic jam it is 60 km/h. The navigator in the car continuously shows the remaining time for Dima to reach point B based on the current length of the traffic jam. At a certain moment (before reaching the traffic jam), Dima noticed that the navigator showed the same time as it did a few minutes ago. Find \( v \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of angles $\theta$ between 0 and $2 \pi$, other than integer multiples of $\pi / 2$, such that the quantities $\sin \theta, \cos \theta$, and $\tan \theta$ form a geometric sequence in some order.
Please reason step by step, and put your final answer within \boxed{}.


Context:
# Task 7. (14 points)

In a certain company, no two employees have jobs of the same difficulty, and no two employees receive the same salary. On April 1, each employee made two statements:

(a) There will not be 12 employees with more difficult jobs.

(b) At least 30 employees have a higher salary. How many employees are in the company if some employees told the truth twice, and the rest lied twice.

#
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 5 Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9, \cdots, n\}$. Take 3 numbers from $A$ and 2 numbers from $B$ to form a set $A_{i}(i=1,2, \cdots, 20)$ with 5 elements, such that $\left|A_{i} \cap A_{j}\right| \leqslant 2,1 \leqslant i<j \leqslant 20$, find the minimum value of $n$. (2002 China National Training Team for IMO)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
给正方体的 8 个顶点染上 $k$ 个红点, $8-k$ 个蓝点 $(1 \leqslant k$ $<8)$. 凡两端为红色的棱记上数字 $\frac{-1+\sqrt{3} i}{2}$, 凡两端为蓝色的棱记上数字 $\frac{-1-\sqrt{3} i}{2}$, 凡两端异色的棱记上数字 1 , 这 12 个数字之积的所有可取值为 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
在坐标平面上有两个区域 $M$ 和 $N$。$M$ 是由 $y \geqslant 0, y \leqslant x$ 和 $y \leqslant 2-x$ 这三个不等式确定的，$N$ 是随 $t$ 变化的区域，它由不等式 $t \leqslant x \leqslant t+1$ 所确定，$t$ 的取值范围是 $0 \leqslant t \leqslant 1$。设 $M$ 和 $N$ 的公共面积是函数 $f(t)$。原始答案的形式是 $-at^{2}+bt+\frac{c}{d}$，请给出 $a+b+c+d$ 的值，其中 $\frac{c}{d}$ 是最简分数。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\frac{12}5\sqrt2$. What is the volume of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Barney's grocery store sold out all of its items at the beginning of the pandemic, so they ordered 4458 items to restock the shelves. They sold some items that day, and have 575 items in the storeroom. They have 3,472 items left in the whole store. How many items did they sell that day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. Find the area inside the circle, but outside the square. The original answer is in the form \frac{k\pi}{m} - \frac{\sqrt{n}}{p}, please give the value of k + m + n + p.
Please reason step by step, and put your final answer within \boxed{}.


Context:
如果 $$ax^{2} + bx + c > 0 (a ≠ 0)$$ 的解集为 {$x | x < -2 或 x > 4$}, 设 $$f(x) = ax^{2} + bx + c$$, 试比较 $f(-1), f(2), f(5)$ 的大小.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect the diagonal $AC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设正整数 $m, n \leqslant 2021$ ，并且 $\frac{m}{n+1}<\sqrt{2}<$ $\frac{m+1}{n}$, 则这样的数组 $(m, n)$ 的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
如果在区间 $[1,2]$ 上函数 $f(x)=x^{2}+p x+q$ 与 $g(x)=x+\frac{1}{x^{2}}$ 在同一点取相同的最小值，那么 $f(x)$ 在该区间上的最大值是多少？答案的格式为 \(a - \frac{b}{c} \sqrt[3]{d} + \sqrt[3]{e}\)，请计算a + b + c + d + e的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A savings card's password consists of 6 digits, each of which can be chosen from 0 to 9. When someone tries to withdraw money from an ATM, they forget the last digit of the password. If they randomly press the last digit, calculate the probability of pressing the correct digit in no more than 2 attempts. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
2. Let $n$ be a three-digit positive integer that does not contain the digit 0. If any permutation of the units, tens, and hundreds digits of $n$ does not form a three-digit number that is a multiple of 4, find the number of such $n$.
(54th Ukrainian Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.


Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+d e f$ 是奇数的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f(999)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximal possible number of integers you can choose from the set \( \{1, 2, \ldots, 100\} \) such that no product of any non-empty subset of these numbers is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two points on the surface of a sphere, and a circle $k$ which passes through exactly one of these points, how many circles are there on the sphere that pass through both points and are tangent to $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[ s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The *rank* of a rational number  $q$  is the unique  $k$  for which  $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$ , where each  $a_i$  is the smallest positive integer  $q$  such that  $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$ .  Let  $q$  be the largest rational number less than  $\frac{1}{4}$  with rank  $3$ , and suppose the expression for  $q$  is  $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$ .  Find the ordered triple  $(a_1,a_2,a_3)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain number, when increased by 20%, equals twice the result of 20 less than the number decreased by 20%. What is that number? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the mass percentage of Ca in a mixture containing 20 g of Calcium carbonate (CaCO3) and 10 g of Magnesium chloride (MgCl2), assuming that the mixture is anhydrous and there are no impurities present.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $f(x) = \frac{ax+b}{cx+d}$ ($c \neq 0$), the center of symmetry for the graph of $f(x)$ is $(-\frac{d}{c}, \frac{a}{c})$. It is known that $f(x) = \frac{2-2x}{2x-1}$ and the general term of the sequence $\{a_n\}$ is $a_n = f(\frac{n}{2017})$ for $n \in \mathbb{N}$. Calculate the sum of the first 2017 terms of this sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, and the final number displayed on the computer was 12. What is the smallest possible initial number on the computer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two numbers a and b with 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1 are chosen at random. The number c is defined by c = 2a + 2b. The numbers a, b, and c are each rounded to the nearest integer to give A, B, and C, respectively. Find the probability that 2A + 2B = C. Express your answer as a fraction in lowest terms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular 200-sided polygon \( A_{1} A_{2} \cdots A_{200} \), connect the diagonals \( A_{i} A_{i+9} \) for \( i=1, 2, \cdots, 200 \), where \( A_{i+200}=A_{i} \) for \( i=1,2, \cdots, 9 \). How many different intersection points are there inside the regular 200-sided polygon formed by these 200 diagonals?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are line segments of lengths 1, 2, 3, ..., 9, each of a kind. How many different ways can you select several of them to form a square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five numbers are interconnected. The first number is less than the second number by 40%, the third number is 25% more than the first number. The fourth number is 20% less than the second number but 30% more than the third number. The fifth number is 50% more than the fourth number. What is the percentage difference between the second and fifth numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kevin starts with the vectors \((1,0)\) and \((0,1)\) and at each time step, he replaces one of the vectors with their sum. Find the cotangent of the minimum possible angle between the vectors after 8 time steps.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the end-of-term math exam of a certain school, the math scores follow a normal distribution $N\left(76,16\right)$. The exam scores are divided into four levels $A$, $B$, $C$, and $D$ in the proportions of $15\%$, $35\%$, $35\%$, and $15\%$ respectively. Students with scores greater than or equal to $83$ points are classified as level $A$. What should be the score range for level $B$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
We consider a chessboard of size $8 \times 8$ where the squares are alternately colored white and black. An infernal rook is a piece that can attack the squares of its color located on its row, as well as the squares of the other color located in its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The following list shows every number for which more than half of its digits are the digit $2$, in increasing order:
\[
2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots
\]
If the $n$th term in the list is $2022$, what is $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two water droplets fall freely one after another from a $300 \mathrm{~m}$ high cliff. The first droplet has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts falling.

How many millimeters apart will the two droplets be at the moment the first one reaches the base of the cliff? (The result should be calculated to an accuracy of $\frac{1}{10} \mathrm{~mm}$. Air resistance, etc., should not be taken into account.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
a certain debt will be paid in 52 installments from january 1 to december 31 of a certain year . each of the first 25 payments is to be $ 500 ; each of the remaining payments is to be $ 100 more than each of the first 25 payments . what is the average ( arithmetic mean ) payment that will be made on the debt for the year ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The ancient Chinese theory of the "Five Elements" posits that there are five kinds of properties in materials: metal, wood, earth, water, and fire. According to the theory, metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal. Considering these properties, how many different arrangements of the five elements in a row are possible such that no two consecutive materials have a conquering relationship? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
James buys a weight vest for some amount. He then buys 200 pounds of weight plates for $1.2 per pound. A 200-pound weight vest would cost $700 but there is a $100 discount. He saves $110 with his vest. How much did the weight vest cost?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence $\left\{\frac{I_{2n}}{I_{2n+1}}\right\}_{n=1}^{\infty}$, where $I_n=\int_0^{\frac{\pi}{2}}\sin^n x \, dx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the complex numbers \( z_1 \) and \( z_2 \) that satisfy \(\left|z_1\right| = \left|z_1 - 2 z_2\right|\) and \( z_1 \overline{z_2} = \sqrt{3}(1 - \mathrm{i}) \), where \(\mathrm{i}\) is the imaginary unit, find \(\frac{z_1}{z_2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Player A and player B are playing Go. If player A plays as black first, the probability of player A winning is $\frac{4}{5}$; if player B plays as black first, the probability of player B winning is $\frac{2}{3}$. Assuming each game is independent and there are no draws, in a two-game match where the loser of the first game plays first in the second game, and the choice of who plays as black in the first game is equally likely between player A and player B, the probability of player A winning the first game and player B winning the second game is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How to evaluate the integral $$\int_\Omega xye^{x^2-y^2}dxdy$$ over the domain $\Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each side of a triangle is divided into three equal parts. The division points serve as the vertices of two triangles, whose intersection is a hexagon. Find the area of this hexagon if the area of the given triangle is $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four pieces of lumber are placed in parallel positions, perpendicular to line M:

- Piece W is 5m long
- Piece X is 3m long and its left end is 3m from line M
- Piece Y is 5m long and is 2m from line M
- Piece Z is 4m long and is 1.5m from line M

A single cut, perpendicular to the pieces of lumber, is made along the dotted line L. The total length of lumber on each side of L is the same. Calculate the length, in meters, of the part of piece W to the left of the cut. Express your answer as a decimal number with two places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere. Calculate the surface area of this sphere. Express your answer in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Real numbers  $x,y,z$  satisfy   $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0 $$   and none of them lies in the open interval  $(-1,1)$ . Find the maximum value of  $x+y+z$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知实系数多项式 f(x)=x^3+bx^2+cx+d 有复根 z_1, z_2, z_3，且满足 f(2)\geq f(-3)+45. 则 \displaystyle\prod_{k=1}^3(z_k^2+z_k+1) 的最小值为__________.（树懒）
Please reason step by step, and put your final answer within \boxed{}.


Context:
A $3 \times 3$ grid is to be painted with three colors: red, green, and blue, such that:
1. No two squares that share an edge are the same color.
2. No two corner squares on the same edge of the grid have the same color.

For example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? Note that rotations and reflections are considered distinct colorings.
Please reason step by step, and put your final answer within \boxed{}.


Context:
10 white and 20 black chips are arranged in a circle. It is allowed to swap any two chips that have exactly three chips between them. Two arrangements of chips (in these 30 positions) are called equivalent if one can be transformed into the other by several such swaps. How many non-equivalent arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man spends 1/4 of his salary on food, 1/8 of his salary on house rent, 3/10 of his salary on clothes, and 1/6 of his salary on transportation expenses. The man still has $35,000 left with him. Find his total salary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he computes:

\[ 11134640 - 1000000 = 10134640 \]

What is the largest number that always must divide his result?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ring homomorphisms are there from $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lolita can drink some boxes of milk during weekdays. During Saturdays, she can drink twice the number of boxes of milk than on the weekdays, and during Sundays, she can drink thrice the number of boxes of milk than on the weekdays. She drinks 30 boxes of milk per week. How many boxes of milk does she drink on weekdays?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Brielle has a wall clock in his home that rings every 3 hours. If it will first ring at a certain time in the morning, it will ring 8 times in a day. At what time does the clock first ring?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 2. Pasha wrote the numbers $4,5,6, \ldots, 14$ on the board, and then erased one or several of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value of the sum of the remaining numbers on the board?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p=101$. The sum \[\sum_{k=1}^{10}\frac{1}{\binom{p}{k}}\] can be written as a fraction of the form $\dfrac{a}{p!}$, where $a$ is a positive integer. Compute $a \pmod{p}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two equal-armed balances are shown. The first balance has 3 circles balancing 5 squares, and the second balance has 2 circles balancing 3 squares plus 1 circle. How many squares would it take to balance one circle? Express your answer as a single number of squares.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The lengths of the sides of a trapezoid are 2, 10, 10, and 20. Find the length of the segment connecting the midpoints of the diagonals.

(It is known that these points are distinct). If there are multiple possible answers, list them in ascending order separated by commas or semicolons.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Every second, Andrea writes down a random digit uniformly chosen from the set $\{1,2,3,4\}$. She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? Provide your answer as a simplified expression in the form of \(\frac{a+b\sqrt{c}}{d}\). Please find the value of a + b + c + d.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. What digit is most likely to be the units digit of the **sum** of Jack's integer and Jill's integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a small pond, there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, where $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$, it will be eaten by a patiently waiting snake. If the frog reaches pad $10$, it will exit the pond, never to return. Find the probability that the frog will escape being eaten by the snake. The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate: $-1^{2}\times \left(-2024\right)^{0}+(\frac{1}{3})^{-1}-|-5|+\left(-4\right)^{2023}\times \left(-0.25\right)^{2022}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a board with size $25 \times 25$. Some $1 \times 1$ squares are marked, so that for each $13 \times 13$ and $4 \times 4$ sub-boards, there are at least $\frac{1}{2}$ of the squares in the sub-board that are marked. Find the least possible number of marked squares in the entire board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知$A = \{ x \vert x ^ { 2 } - 4 x + 3 < 0 , x \in R \} , B = \{ x \vert 2 ^ { 1 - x } + a \leq 0 , x ^ { 2 } - 2 ( a + 7 ) x + 5$≤0,x∈R}，若A⊆B，则实数a的取值范围是 .
Please reason step by step, and put your final answer within \boxed{}.


Context:
现有长为150cm的铁丝，要截成n(n>2)小段，每段的长为不小于1（cm）的整数．如果其中任意3小段都不能拼成三角形，则n的最大值为
Please reason step by step, and put your final answer within \boxed{}.


Context:
The equation  $$ (x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016) $$  is written on the board, with  $2016$  linear factors on each side. What is the least possible value of  $k$  for which it is possible to erase exactly  $k$  of these  $4032$  linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Joe likes to play video games and he normally spends an average of $50 a month on video games. However, the cost of the games sometimes varies from $40 to $60 per game. Joe also sells his games for an average of $30 each once he is done playing them, but the resale price varies between $20 and $40. In addition to his video game expenses, Joe has a monthly subscription to an online gaming platform, which costs him $15 per month. If he starts with $240, how many months can he continue purchasing and reselling games with variable pricing and pay for his gaming subscription expenses before he is out of money?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $p:(x-m)^{2} \lt 9$, $q:\log _{4}(x+3) \lt 1$, if $\neg q$ is a necessary but not sufficient condition for $\neg p$, then the range of $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Susan made a block with small cubes of 5 cubic cm volume to make a block 7 small cubes long, 7 small cubes wide, and 6 small cubes deep. She realized that she has used more small cubes than she really needed. How many fewer cubes could she have glued together to look like a block with the same dimensions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A, B, C, and D obtained the top four positions (without ties) in the school, and they made the following statements:
A: "I am neither first nor second."
B: "My position is adjacent to C's position."
C: "I am neither second nor third."
D: "My position is adjacent to B's position."

It is known that A, B, C, and D respectively obtained the places $A, B, C, D$. Determine the four-digit number $\overrightarrow{\mathrm{ABCD}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在一个 10$\times10$ 的方格表中有一个由4$n$个 1 x 1 的小方格组成的图形，它既可被$n$
个“田”型的图形覆盖，也可被$n$个“田”或
“毌”型(可以旋转)的图形覆盖.求正整数$n$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a point \( P \) inside \( \triangle ABC \), perpendiculars are drawn from \( P \) to \( BC, CA, \) and \( AB \) with feet \( D, E, \) and \( F \) respectively. Semicircles are constructed externally on diameters \( AF, BF, BD, CD, CE, \) and \( AE \). These six semicircles have areas denoted \( S_1, S_2, S_3, S_4, S_5, \) and \( S_6 \). Given that \( S_5 - S_6 = 2 \) and \( S_1 - S_2 = 1 \), find \( S_4 - S_3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=x^{3}+ax^{2}+bx+a$ has an extremum at $x=1$ with the value $10$, find $f(2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each edge of a triangular pyramid (tetrahedron) is equal to \( a \). Find the maximum area that the orthogonal projection of this pyramid on a horizontal plane can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
函数$f(x)={{\log }_{a}}({{x}^{2}}-4x+3)$（$a>0$，$a\ne 1$）在区间$[m,+\infty )$上存在反函数，则实数$m$的取值范围是____
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the mass percentage of I in Aluminum iodide.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the perimeter of $\triangle ABC$ is 12, and its sides are $a$, $b$, and $c$, with $b$ being the longest side, find the range of possible values for $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A river boat travels at a constant speed from point A to point B. Along the riverbank, there is a road. The boat captain observes that every 30 minutes, a bus overtakes the boat from behind, and every 10 minutes, a bus approaches from the opposite direction. Assuming that the buses depart from points A and B uniformly and travel at a constant speed, calculate the interval time (in minutes) between each bus departure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
when 242 is divided by a certain divisor the remainder obtained is 6 . when 698 is divided by the same divisor the remainder obtained is 13 . however , when the sum of the two numbers 242 and 698 is divided by the divisor , the remainder obtained is 5 . what is the value of the divisor ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An island has  $10$  cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\int_{0}^{2\pi}(1+\cos x)^n\sin nx dx$$ using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $$\int_0^\infty \dfrac {\log{x}}{(x^2+1)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mr. Grey's house was worth $100,000. During a booming market, he sold the house to Mr. Brown at a profit of 12%. After one year, the market experienced a slight decline, and Mr. Brown invested 5% of the purchasing price on improvements. He then sold the house to his friend Mr. Green with a 4% profit. Mr. Green rented out the house for a year, during which the market faced another decline of 3%. Mr. Green then decided to sell the house to Ms. White, who bought it with a 10% loss. After six months, the market started recovering and experienced an 8% increase in property prices. Ms. White then decided to sell the house back to Mr. Brown with a 5% profit. How much was Mr. Brown's final purchase price after buying the house back from Ms. White?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A=\{x|x=sin\frac{2π}{2023}+sin\frac{4π}{2023}+sin\frac{6π}{2023}+…+sin\frac{2kπ}{2023},k∈Z,k＞0\}$. Find the number of distinct elements in set $A$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a bag containing 8 white balls and 2 red balls. Each time a ball is randomly taken out, and then a white ball is put back into the bag. What is the probability that all red balls have been taken out exactly after the 4th draw?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When \((1+x)^{38}\) is expanded in ascending powers of \(x\), \(N_{1}\) of the coefficients leave a remainder of 1 when divided by 3, while \(N_{2}\) of the coefficients leave a remainder of 2 when divided by 3. Find \(N_{1} - N_{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Naomi has a broken calculator. All it can do is either add one to the previous answer, or square the previous answer. (It performs the operations correctly.) Naomi starts with 2 on the screen. In how many ways can she obtain an answer of 1000 ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The structure consists of two weightless rods \(AB\) and \(CD\), a weightless pulley, and a load with mass \(m = 3\) kg, which is suspended at point \(G\). All cords are weightless and inextensible. The rod is connected to the hinge at point \(A\), which allows the rod to rotate within the plane of the drawing. The lengths are \(CG = \frac{1}{4} CD\) and \(AE = \frac{1}{4} AB\), and point \(F\) is the midpoint of rod \(AB\). The acceleration due to gravity is \(g = 10 \ \mathrm{m} / \mathrm{s}^2\). Determine the tension force \(T_{0}\) in the cord attached to rod \(AB\) at point \(E\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If we also want to know whether the counterfeit coin is lighter or heavier than the others in the previous problem, what is the maximum number of coins that can be determined with three weighings?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An elliptic curve $E: y^2 = x^3 + bx + c$ is defined over the integers modulo $221 = 13 \cdot 17$. Points $P$ and $Q$ on $E$ satisfy $11P = \mathcal{O}$ and $7Q = \mathcal{O}$, where $\mathcal{O}$ is the point at infinity. Determine the number of points on $E$, denoted as $\sharp E$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系内, $M \in\left\{(x, y) \left\lvert\, \frac{x^{2}}{200}+\frac{y^{2}}{8} \leq 1\right.\right\}, A(2,1)$, 若 $\Delta O M A$ 的面积不超过 3, 则满足条件的整点 $M$ 个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Linda was gathering eggs from her chicken coops. When she was done, she had three times as many white eggs as brown eggs, and she had an equal number of small, medium, and large eggs. Additionally, she collected 6 green eggs which were all medium-sized. On her way back to the house, she stumbled, dropping her egg basket. Upon inspecting the damage, she found that all 20 of the brown eggs survived - 10 large, 5 small, and 5 medium, but now she only had 35 eggs left. How many eggs did Linda break?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the circumcircle of $\triangle ABC$ has a radius of $1$, with its center at $O$, and $3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \overrightarrow{0}$, find the measure of $\angle AOB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum of all positive real numbers \(x \leq 5\) satisfying \(x=\frac{\left\lceil x^{2}\right\rceil+\lceil x\rceil \cdot\lfloor x\rfloor}{\lceil x\rceil+\lfloor x\rfloor}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \( f \) is a function such that \( f(a+b) = \frac{1}{f(a)} + \frac{1}{f(b)} \), find all possible values of \( f(2011) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an $n \times 6$ rectangular grid with $6n$ unit squares, each unit square is filled with either a 0 or a 1. A filling method is called an "N-type filling" if it ensures that there is no rectangular subgrid whose four corners have the same number. Otherwise, the filling method is called a "Y-type filling." Find the smallest positive integer $n$ such that no matter how the numbers are filled, the filling method will always be a "Y-type filling."
Please reason step by step, and put your final answer within \boxed{}.


Context:
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,…,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? The standard answer format is $a^{b} c^{d}$. Please give the value of a + b + c + d.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, if $\angle B=30^\circ$, $AB=2 \sqrt {3}$, $AC=2$, find the area of $\triangle ABC$\_\_\_\_\_\_．
Please reason step by step, and put your final answer within \boxed{}.


Context:
在空间直角坐标系 $x y z$ 中，先以 z 轴为旋转轴旋转 $\frac{\pi}{3}$ ，再以 x 轴为旋转轴旋转 $\frac{\pi}{3}$ ，两次操作后等效于绕某一轴旋转，旋转角为 $\theta(\theta \in(0, \pi])$ ，则
$\left[\tan ^{2} \theta\right]=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical.
Please reason step by step, and put your final answer within \boxed{}.


Context:
对于非负整数，函数f定义如下：

$f ( 0 ) = 0 , f ( x ) = f ( [ \frac { x } { 1 0 } ] ) + [ \lg \frac { 1 0 } { x - 1 0 [ \frac { x - 1 } { 1 0 } ] } ] ( x > 0 )$.问：f(x)在$0\leq x\leq 1996$中取最大值时，x的值是多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the universal set be U = {1,2,3,4,5,6,7,8,9,10}. Given the conditions:
1. A is a subset of U.
2. If x is in A, then 2x is not in A.
3. If x is not in A, then 2x is not not in A.
Calculate the number of sets A that satisfy all three conditions simultaneously.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain candy store, 22% of the customers are caught sampling the candy and are charged a small fine. However, 10% of the customers who sample the candy are not caught. Additionally, 15% of the customers who are caught sampling candy are repeat offenders, and 35% of the customers who sample the candy end up buying additional items after sampling. What is the total percent of all customers who sample candy, including those who are caught, not caught, repeat offenders, and those who buy additional items?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a real number \(a\) such that there exists exactly one square whose four vertices lie on the curve \(y = x^3 + a x\). Determine the side length of this square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a geometric sequence of real numbers, the sum of the first $2$ terms is $7$, and the sum of the first $6$ terms is $91$. The sum of the first $4$ terms is
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, $H$ is the orthocenter, $BC = a$, the circumradius of $\triangle ABC$ is $R$, and $a = 2AH - 2R$. Find the value of $\sin A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ to the area of triangle $ABC$ can be written in the form $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest natural number  $n$  such there exist natural numbers  $x_{1}, x_{2}, \ldots, x_{n}$  and natural  $a_{1}< a_{2}< \ldots < a_{n-1}$  satisfying the following equations for  $i =1,2,\ldots,n-1$ : \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
8 men, working a certain number of hours a day can complete a work in 24 days. If 12 men work the same number of hours a day, they can complete the same work in 16 days. How many hours a day were the 8 men working?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the maximum value and the minimum value of the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \) sum up to 4, then find \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知${{2}^{2x}}-{{25}}={{2}^{x+2}}$，则$\lg \left( {{x2}}+1 \right)=$__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the expansion of $(x-2{y}^{3}){(x+\frac{1}{y})}^{5}$, the coefficient of $x^{2}$ term is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
令 $z_{k}=\cos \frac{2 k-2}{5} \pi+i \sin \frac{2 k-2}{5} \pi(k=1,2,3,4,5)$ ，若 $a_{i}=\prod_{i \neq j}\left(z_{i}-z_{j}\right)$ ，请计算 $a_{1} a_{2} a_{3} a_{4} a_{5}$ 的值，并将结果表示为一个整数。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons `cos` or `sin` a total of 2001 times? (So you might press `cos` five times, then `sin` six times, and so on with a total of 2001 presses.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
13.364. The base of the power was increased by $k$ times, and the exponent was decreased by the same factor, as a result of which the power itself did not change. Find the base of the power that has such a property.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral $$\int_{|z|=1} \frac{z^7}{10z^8-5z^4+2z^2+iz+1}dz$$ given that the polynomial in the denominator has all its zeros inside the open unit disc.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\alpha$ and $\beta$ are two interior angles of an oblique triangle, if $\frac{{\cos \alpha - \sin \alpha}}{{\cos \alpha + \sin \alpha}} = \cos 2\beta$, then the minimum value of $\tan \alpha + \tan \beta$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $\int_{\gamma}\frac{dz}{z^2+3}$ where $\gamma$ is the upper half of the unit circle $|z|=1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Any six points are taken inside or on a rectangle with dimensions $2 \times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
To estimate the lower probability that the average burning duration of the selected 200 electric lamps differs from the average burning duration of the entire batch in absolute value by less than 5 hours, given that the standard deviation of the burning duration of lamps in each box is less than 7 hours.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive real numbers \(a\), \(b\), and \(c\), compute the maximum value of
\[
\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of solutions to
\[\cos 4x + \cos^2 3x + \cos^3 2x + \cos^4 x = 0\]for $-\pi \le x \le \pi.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Aunt Gemma has six dogs with different sizes and appetites. She feeds them three times a day. The first dog consumes 250 grams of food, the second dog eats 350 grams, the third dog requires 450 grams, the fourth dog needs 550 grams, the fifth dog consumes 300 grams, and the sixth dog requires 400 grams per meal. If Aunt Gemma bought 2 sacks of dog food, each weighing 50 kilograms, and feeds the dogs their specified amounts at each meal, how many days will those sacks of food last?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $I$  be the incenter of  $\triangle ABC$ , and  $O$  be the excenter corresponding to  $B$ . If  $|BI|=12$ ,  $|IO|=18$ , and  $|BC|=15$ , then what is  $|AB|$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
记 M(100,\bR) 是所有 100 阶实方阵的集合，设 S=\{A\in M(100,\bR)\ |\ A^4+A^2=0\}. 称 A,B\in S 是相似的，如果存在可逆的 100 阶实方阵 P 使得 P^{-1}AP=B. 求 S 中互不相似的矩阵个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let positive real numbers $x$ and $y$ satisfy $2x+y=1$. The minimum value of $\frac{8}{{x+1}}+\frac{1}{y}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function y = |log2(x)| with a domain of [1/m, n] (m, n are positive integers) and a range of [0, 2], calculate the number of integer pairs (m, n) that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ with the sides of $\triangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在$\triangle{ABC}$中，中线$\overline{AD}$和$\overline{BE}$在$G$点相交，且$\triangle{AGE}$是等边三角形。那么$\cos(C)$可以写成$\frac{m\sqrt{p}}{n}$的形式，其中$m$和$n$是互质的正整数，$p$是一个不被任何素数的平方整除的正整数。请计算$m+n+p$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given real numbers \( a, b, \) and \( c \) that satisfy
\[ f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \]
for any real number \( x \). What is the maximum value of \( a + b + c \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Initially, all edges of the  $K_{2024}$  are painted with  $13$  different colors. If there exist  $k$  colors such that the subgraph constructed by the edges which are colored with these  $k$  colors is connected no matter how the initial coloring was, find the minimum value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
在边长为 8 的正方形 $A B C D$ 中, $M$ 为边 $B C$ 的中点, $N$ 为边 $D A$ 上一点, 且 $D N=3 N A$. 若对于常数 $m$ ，在正方形 $A B C D$ 的边上恰有六个不同的点 $P$ ，使得 $\overrightarrow{P M} \cdot \overrightarrow{P N}=m$ ，则实数 $m$ 的取值范围是 $($用不等式表示$)$，形如$a < m < b$，请给出$a + b$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x+y+z=30$, $3x+y-z=50$, and $x$, $y$, $z$ are non-negative real numbers, then the range of $M=5x+4y+2z$ is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two distinct squares of the 8 by 8 chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1,2, \ldots, 64$ there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rhombus, where the shorter diagonal is equal to its side of length 1, rotates around a line passing through the end of the longer diagonal and perpendicular to this diagonal. Find the volume of the resulting solid of revolution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在数列 $\left\{a_1, a_2, \cdots\right\}=\{1,3,3,3,5,5,5,5,5, \cdots\}$ 中每个奇数 $k$ 出现 $k$ 次, 设有整数 $p, q, r$存在, 对所有正整数 $n$, 满足 $a_n=p[\sqrt{n+q}]+r$, 其中 $[x]$ 表示不大于 $x$ 的最大整数, 求 $p+q+$ $r$ 的值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function $f$ satisfies the equation $f(xy) = f(x) + f(y)$ for any positive $x$ and $y$, and that $f\left(\frac{1007}{1}\right) = 1$, find the value of $f(2007)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the orthogonal projection of the vector $(1, 1, 1)$ onto the null space of the matrix $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two unbiased coins are tossed repeatedly. Determine the expected number of tosses required until the last five outcomes of both coins are identical. What is this expected number of tosses?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n \) such that \(|\sin x - (x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{9!} - \frac{x^{11}}{11!})| < |\cos(c)/13!|\), for some \( c \in (-0.1, 0.1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An insurance policy covers two employees of a company, reimbursing no more than one loss per employee per year. The policy reimburses the full amount of the loss up to a company-wide maximum of $8000. The probability of an employee incurring a loss is 40%, independent of the other employee. Each loss amount is uniformly distributed between $1000 and $5000. Given that one employee has incurred a loss exceeding $2000, calculate the probability that the total losses will exceed the reimbursements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the generating function for the sequence \( B(n) \) defined by the recurrence relation: \[ B(n) = \sum_{a_1 + \cdots + a_k = n, a_i \geq 1} \frac{1}{k!} \binom{n}{a_1 \cdots a_k} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square truth table of size $n^2$, an algorithm takes two arbitrary rows as input ($a$ and $b$ such that $a < b$) and performs a fixed number of logical operations ($s$) on them. The result is a series of Boolean values. To perform the above operations on all combinations of $a$ and $b$ requires $sn^2$ steps, which means the time complexity is $O(n^2)$. Now consider the case in which the result of each iteration as described above returns some unknown number of TRUE values $c$ such that $c < b - a - 1$ and exactly $c$ additional operations must be performed before starting over with a new $a$ and $b$. Determine the time complexity in the worst case.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} (f(x+1) - f(x)) \, dx \) given that \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a number that is divisible by all integers less than or equal to half of itself, but not divisible by any integer greater than half of itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle ABC, where AB = AC and \( \angle ACB = 72^\circ \), point D is located inside the triangle such that \( \angle DBC = 42^\circ \) and \( \angle DCB = 54^\circ \). If line AD intersects BC at point E, determine the measure of \( \angle AEC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \(f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( a \) such that the four planes given by the equations:
\[
\begin{cases}
α_1 :& x + 2ay + az = a^2 \\
α_2 :& x + ay + az = a \\
α_3 :& x + a^2 z = a^3 \\
α_4 :& ax + ay + a^2 z = a \\
\end{cases}
\]
do not have any point in common.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( \omega(X,Y) \) for the differential form \( \omega = 2x \, dx \wedge dy + y^2 \, dx \wedge dz \) and the vector fields \( X = x^2y \frac{\partial}{\partial y} + x \frac{\partial}{\partial z} \) and \( Y = x \frac{\partial}{\partial y} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using dimensional analysis, find the value of the exponent 'm' in the equation:
\[ \int \frac{x^m \, dx}{\sqrt{a^2 - x^2}} = b^n \sin^{-1} \left[\frac{x}{a}\right] \]
where 'a' and 'b' are constants with dimensions of length.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of isomorphism classes of countable models of the empty theory over one binary relation. In other words, how many countable unlabeled graphs are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \( 2x - e^{-x} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \frac{n! F_n(1)}{\ln n} \] where $F_0(x) = \ln x$ and for $n \geq 0$ and $x > 0$, $F_{n+1}(x) = \int_0^x F_n(t) \, dt$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two independent arrival processes $N_t = X_t + Y_t$, where $X_t$ and $Y_t$ each occur with probability rate $0.5$, find the expected value $E[N_t|X_t = n]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( I = \int \limits_\Gamma x^2 \, dy - 2xy \, dx \), where \( \Gamma \) is the part of the curve \( x^4 - y^4 = 6x^2y \) from point \( A(-4\sqrt{2}, 4) \) to point \( B(0,0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $X \sim N(0,1)$ and $Y \sim N(X, 1)$, where $Y-X$ is independent of $X$, calculate the probability $P(Y<0 \mid X>0)$. This probability is relevant in the context of pricing a Slalom option in a Normal model without drift.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of positive integer solutions \((a, b, c, d)\) for the following system of equations:

\[\sqrt{2020}(\sqrt{a} + \sqrt{b} )= \sqrt{ (c+ 2020)(d+ 2020)}\]
\[\sqrt{2020}(\sqrt{b} + \sqrt{c} )= \sqrt{ (d+ 2020)(a+ 2020})\]
\[\sqrt{2020}(\sqrt{c} + \sqrt{d} )= \sqrt{ (a+ 2020)(b+ 2020)}\]
\[\sqrt{2020}(\sqrt{d} + \sqrt{a} )= \sqrt{ (b+ 2020)(c+ 2020)}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that the field extension \( \mathbb{F}_{3}(\alpha) \) is equal to \( \mathbb{F}_{3^5} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi}\frac{\cos(3x)}{5-4\cos(x)}dx \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\lambda_1+\lambda_2x)^n-x^n}{1-x}\,dx,$$ where $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the function defined by the following recursive relations: \( f(x,0)=f(x-1,1) \), \( f(0,y) = (y+1) \mod 5 \), and \( f(x,y) = f(x-1, f(x,y-1)) \). Find \( f(333,3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Generalized Directional Derivative (Clarke Derivative) of the function \( f(x) = -|x| \) at the point \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of \( \lambda > 0 \) for which the integral \( \int _0 ^\infty \frac{e^{\sin x}\sin 2x}{x^\lambda} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezium $ABCD$, the sides $AB$ and $CD$ are parallel. A semicircle with diameter $AB$ touches $CD$ and passes through the midpoints of diagonals $AC$ and $BD$. Determine the measure of angle $\angle BOE$, where $O$ is the center of the semicircle and $E$ is the intersection point of $AC$ with the semicircle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $H$ be the line at infinity in $\mathbb{CP}^{2}$, and let $P$ and $Q$ be distinct points on $H$. Let $X$ be the blow-up of $\mathbb{CP}^{2}$ at $P$ and $Q$; let $E_{1}$ and $E_{2}$ be the exceptional divisors over $P$ and $Q$, respectively, and let $L$ be the proper transform of $H$. What is the self-intersection number of $L$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the integral over the semicircle as \( R \to \infty \):

\[ \lim_{R\to\infty}\int_{\Gamma_R}\frac{e^{izt}}{\pi\cos(iz)}dz \]

where \( t > 0 \) and \( \Gamma_R \) is the upper semicircle of radius \( R \) in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alison is eating 2401 grains of rice for lunch. She eats th
 rice in a very peculiar manner: every step, if she has only
 one grain of rice remaining, she eats it. Otherwise, she
 nds the smallest positive integer \( d > 1 \) for which she ca
 group the rice into equal groups of size \( d \) with none left
 over. She then groups the rice into groups of size \( d \) ,eats
 one grain from each group, and puts the rice back into a
 single pile. How many steps does it take her to finish all
 her rice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a
 time in random order. Given that the youngest person leaves the meeting sometime before the oldest
 person leaves the meeting, the probability that the third, fourth, and fifth people to leave the meeting do so in order of their ages
 (youngest to oldest) is \( \frac{m}{n} \) ,where \( \mathrm{m} \) and \( \mathrm{n} \) are relatively prime positive integers. Find \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let
\[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\]
where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate
\[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a $2\times2$ invertible matrix with real entries such that $\det(A) = k$. Given that $\det(A + k \operatorname{adj}(A)) = 0$, find the value of $\det(A - k \operatorname{adj}(A))$. \( \operatorname{adj}(A) \) denotes the adjugate of matrix $A$. \( k \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi} e^{ib \sin{\theta}} \, d\theta \) where \( b \) is a constant, using analytic methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of trees on the vertex set \( \{v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8\} \) that have at least two vertices with a degree of at least 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the infinite series \( \sum_{n = 1}^\infty \left( \frac{1}{n} - \frac{1}{n + x} \right) \) for \( x = 0.500 \). Round your answer to 3 decimal places and ensure the precision is at least 12 significant digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the quotient ring \( \mathbb{Z}[\sqrt{-19}]/I \), where \( I=(18+\sqrt{-19}, 7) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $X \sim \text{Binomial}(n, p)$ and $Y \sim \text{Bernoulli}\left(\frac{X}{n}\right)$, find the conditional expectation $E[X \mid Y]$. Express your answer in terms of $n$, $p$, and $Y$. \( \text{Note: } Y \text{ is a Bernoulli random variable whose parameter depends on } X. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square contains \( K \) points (\( K > 2 \)). Determine the minimum number of triangles required to divide the square such that each triangle contains at most one point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a call graph $G = (V, E)$ where each vertex $v \in V$ represents a phone number and each edge $\{v, w\} \in E$ indicates a call between $v$ and $w$, a set $S \subseteq V$ of authorized seeds with $|S| = 300$ is monitored. Calls can be investigated for vertices within a distance of 3 hops from any seed. If each phone number has interacted with exactly 100 other numbers, determine the tightest upper bound on the number of phone numbers that can be queried without a warrant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct squares of elements in the alternating group $A_4$, considering that it has 12 elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a function $f$ on the unit interval $0 \le x \le 1$ by the rule
\[f(x)
= \begin{cases}
		    1-3x &  \text{if } 0 \le x < 1/3 \, ; \\
			  3x-1 &  \text{if } 1/3 \le x < 2/3 \, ; \\
			  3-3x &  \text{if } 2/3 \le x \le 1 \, .
		  \end{cases}\]Determine $f^{(2018)}(1/730)$. Recall that $f^{(n)}$ denotes the $n$th iterate of $f$; for example, $f^{(3)}(1/730) = f(f(f(1/730)))$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
$a_1=2021^{2021}$
$0 \le a_k < k$ for all integers $k \ge 2$
$a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of times the random draw function \( R(n) \), which selects integers between \( 0 \) and \( n-1 \) (inclusive), must be applied starting with \( 10^{100} \) until the integer \( 0 \) is obtained.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of the matrix product $C = AB$ with respect to the matrix $B$, where $A \in \mathbb{R}^{n \times d}$ and $B \in \mathbb{R}^{d \times m}$. Express your answer in terms of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $n$ for which two $n \times n$ matrices are similar if they have the same minimal polynomial and the same characteristic polynomial.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a square matrix $B$ has $k$ distinct nonzero eigenvalues, what is the minimum possible nullity of $B$? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x\to\infty}\left(\frac{\max\{f(x),g(x)\}}{\min\{f^4(x),g^4(x)\}}\right) \) given that \( \lim_{x\to\infty}f(x)=\lim_{x\to\infty}g(x)=\infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum number of positive integers $a$ and $b$ such that every integer greater than a certain threshold can be expressed as a non-negative integer combination of $a$ and $b$. What is this minimum number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sequence of vehicles, two-thirds of all trucks have a car behind them, and one-fifth of all cars have a truck behind them. What proportion of the vehicles are cars?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of \(|z|\) for complex numbers \(z\) satisfying the equation \(|(z+i)^2-3| = 2|z-i|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the pullback of the 2-form $\alpha(x, y) = xy \, dx \, dy$ under the function $f:\mathbb{R}^2 \to \mathbb{R}^2$ given by $f(x, y) = (e^{2x}, xy)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
计算下列实变函数定积分．

$\int _ { - \infty } ^ { \infty } \frac { x \sin m x } { 2 x ^ { 2 } + a ^ { 2 } } d x ( m > 0 , a > 0 )$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring homomorphisms from the ring of real numbers to itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
10 play kabadi, 35 play kho kho only, and some people play both games. In total, there are 45 players. How many people play both games?
Please reason step by step, and put your final answer within \boxed{}.


Context:
You cook a pot of beef stew for you and your roommate. Normally, it lasts 7 days between the two of you with each of you eating equal portions. Now, 3 guests are staying with you and your roommate, and everyone will be eating from the stew. You observe that your roommate and the guests eat 1.5 times more than you do. How many days will the beef stew now last if everyone eats their respective portions daily?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( k \) such that the solutions to the differential equation \( y'' + 2y = \cos(kt) \) are unbounded.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\angle BAC = 60^{\circ}$, $D$ is a point on side $BC$ such that $AD = \sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\triangle ABC$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Milo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Euler characteristic of a Y-shaped pipe, which has three openings that converge together.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On each of the five cards, a number is written. The cards are lying on the table with the numbers facing down. By paying a ruble, you can point to any three cards, and you will be told the sum of the numbers written on them. For what minimum price can you certainly determine the sum of all five numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define the function $f(x)$ on $\mathbb{R}$ such that $f(0)=0$, $f(x)+f(1-x)=1$, $f\left(\frac{x}{5}\right)=\frac{1}{2}f(x)$, and for $0 \leq x_1 < x_2 \leq 1$, $f(x_1) \leq f(x_2)$. Find the value of $f\left(\frac{1}{2007}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use differentials to approximate the percentage decrease in mass of an object moving at a velocity of 0.85c as its velocity decreases to 0.83c, according to Einstein's Special Theory of Relativity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the double integral $$\int_{-1}^{1}\int_{-1}^{1}|x+y|\, {\rm d}x\, {\rm d}y$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Misha has thought of a five-digit number with all distinct digits. Igor wants to guess this number. In each turn, Igor can choose several positions of the number, and Misha will reveal the digits in those positions in random order. What is the minimum number of turns Igor needs to guarantee that he can determine the entire number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest non-negative integer \( n \) such that the ring \( S_n = C^n([0,1], \mathbb{R}) \), consisting of all continuously \( n \)-times differentiable functions on the interval \([0,1]\), is not an integral domain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number of lines that are equidistant from point A (-3, 2) and B (1, 1) and have a distance of 2 is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Count the number  $N$  of all sets  $A:=\{x_1,x_2,x_3,x_4\}$  of non-negative integers satisfying  $$ x_1+x_2+x_3+x_4=36 $$  in at least four different ways.
Please reason step by step, and put your final answer within \boxed{}.


Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{def}$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( I \) be the center of the inscribed circle in triangle \( ABC \). It is given that \( CA + AI = BC \). Determine the value of the ratio \( \frac{\widehat{BAC}}{\widehat{CBA}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set \( A = \{1, 2, 3, 4, \cdots, 2^n\} \), where \( n \geq 2 \). If the sum of two elements in set \( A \) is a power of 2, then exactly one of them belongs to subset \( B \). Determine the number of such subsets \( B \) in set \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with edge length 8 is balanced on one of its vertices on a horizontal table such that the diagonal from this vertex through the interior of the cube to the farthest vertex is vertical. When the sun is directly above the top vertex, the shadow of the cube on the table is a regular hexagon. The area of this shadow can be written in the form \( a \sqrt{b} \), where \( a \) and \( b \) are positive integers and \( b \) is not divisible by any perfect square larger than 1. Find the value of \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
All minions either have $1$ or $2$ eyes, and have one of $4$ possible hairstyles. They are all thin and short, fat and short, or thin and tall. Gru doesn’t want to have any $2$ minions that look exactly alike, so what is the maximum possible amoount of minions can he get?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim\limits_{n \to \infty} \frac{(n+3)(n+1)^{n+1}}{(n+2)^{n+2}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) be a holomorphic function on the unit disk \( \mathbb{D} \) such that \( |f(z)| < 1 \) for all \( |z| < 1 \). Given that \( f(0) = \frac{1}{2} \), find a bound for \( |f'(0)| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation given by \( T(1) = 1 \) and \( T(n) = 3T\left(\sqrt[3]{n}\right) + \log_{2}(\log_{2}n) \) using the master theorem, assuming \( n = 2^k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let BC be the latus rectum of the parabola \(y^2 = 4ax\) with vertex A. Determine the minimum length of the projection of BC on a tangent drawn in the portion BAC.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of tails when tossing a fair coin until you achieve $n$ consecutive heads.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum: $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer $n$ such that the error in approximating $\sin(x)$ using its Taylor series expansion up to the $n$th term is less than $10^{-8}$ for all $x$ in the interval $(-0.5, 0.5)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of elements of order 5 in a group of order 80 using Sylow's theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to win tic-tac-toe in $\mathbb{R}^{n}$? (That is, how many lines pass through three of the lattice points $(a_{1}, \ldots, a_{n})$ in $\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\{1,2,3\}$? Express your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a linear transformation \( T: V \rightarrow W \) where \( V = C^\infty(-\infty, \infty) \), the vector space of real-valued continuous functions with continuous derivatives of all orders on \((-\infty, \infty)\), and \( W = F(-\infty, \infty) \), the vector space of real-valued functions defined on \((-\infty, \infty)\). The kernel of \( T \) should be \( P_3 \), the space of polynomials of degree \( \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the height \( m \) of the horizontal line that divides the volume of the solid obtained by rotating the region bounded by \( y = x^2 \), \( y = 9 \), and the \( y \)-axis, around the vertical line \( x = 3 \), into two equal parts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{q\rightarrow 1} (1-q^{2}){\psi^2(q)}$$ using a q-continued fraction for the Ramanujan theta function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using Taylor series: $$\lim_{h\to0}\frac{f(a-5h^2)-f(a+3h)}{h}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the asymptote of a hyperbola as the line $5x - 4y + 5 = 0$ and a tangent to the hyperbola as the line $4x + 5y - 7 = 0$, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the permutation $X$ such that $\tau \circ X = \sigma$, given the permutations $\tau = (1,3,5)(2,4)$ and $\sigma = (1,3)(2)(4,5)$. Express $X$ in cycle notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Poisson process with rate \(\lambda = 3\), where three customers arrive in the first hour, calculate the probability that the third customer arrives between the 30th and 60th minute, conditional on the total of three arrivals. Express this probability as \(P(X_3 \in [0.5, 1] \mid N(1) = 3)\), knowing that \(X_3 \sim \text{Gamma}(3, 3)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the solid $D$ bounded below by the sphere $x^2 + y^2 + (z - 1)^2 = 1$ and above by the cone $z = \sqrt{x^2 + y^2}$ using cylindrical coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest power of three that divides the largest product of several natural numbers whose sum is 2011.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the expectation \( M_T = \mathbb{E}\left[W_T\int_0^T t \, dW_t \right] \) where \( 0 < t < T \) and \( W = (W_t)_{t \geq 0} \) is a standard Brownian motion starting at 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \( 5^{11469} \mod 1911 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest symmetric group $S_n$ into which the group $G = S_4 \times \mathbb{Z}/5\mathbb{Z}$ can be embedded, given that there exists an element $(\sigma, i) \in G$ with order 15, where $\sigma$ is a four-cycle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A standard deck of 52 cards is distributed equally among 4 players, with each player receiving 13 cards. What is the probability that one of the players receives 13 cards, each with a different rank? Express your answer as a fraction or a decimal rounded to four decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability of drawing at least one repeated ball when drawing 10 times with replacement from a set of 100 balls numbered from 0 to 99.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A) \le n$. Determine a lower bound for the quantity $$\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr),$$ where $I$ is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least value of \(b\) such that at least one permutation of the numbers consisting of one \(1\), two \(2\)'s, three \(3\)'s, ..., nine \(9\)'s can be expressed in the form \(a^b\), where \(a\) and \(b\) are natural numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{-1}^{1} \LegP[n, x] \frac{d}{dx} (\LegP[n+1,x]) \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a number $a$, with $a < 2000$, that is not divisible by 7 and satisfies the congruence $a \equiv 7^{3003} \pmod{m}$, given that $\phi(m) = 1000$. Use Euler's Theorem, which implies $7^{1000} \equiv 1 \pmod{m}$, to solve this without a calculator.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the coefficient of the $x^{\frac{3}{2}}$ term in the expansion of $\left(\sqrt{x} - \frac{a}{x}\right)^6$ is 30, find the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of permutations of the letters ABCDEFGHI such that none of the sequences ABC, DEF, or GHI appear consecutively in the permutation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $r$ for which the sequence $a_n = \dfrac{4^n}{n^3} + (r^2)^n$ and the series $S = \sum_{n=0}^\infty \frac{4^n}{n^3 + (r^2)^n}$ converge.
Please reason step by step, and put your final answer within \boxed{}.


Context:
To each pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:

1. $a*(b*c) = (a*b)*c$ for all $a, b, c$.
2. $a*a = 1$ for all $a$.

Solve the equation $x*36 = 216$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What relationship exists between $A$ and $p$ if

$$
A=\frac{x^{2}-3 y^{2}}{3 x^{2}+y^{2}}
$$

and

$$
\frac{p x y}{x^{2}-(2+p) x y+2 p y^{2}}-\frac{y}{x-2 y}=\frac{1}{2} ?
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a+b+c+d=3$ and $a^2+b^2+c^2+d^2=45$, find the value of the expression:
\[ \frac{a^5}{(a-b)(a-c)(a-d)} + \frac{b^5}{(b-a)(b-c)(b-d)} + \frac{c^5}{(c-a)(c-b)(c-d)} + \frac{d^5}{(d-a)(d-b)(d-c)} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions in positive integers to the equation \(n_1 + n_2 + n_3 + n_4 + n_5 = 20\) under the condition that \(n_1 < n_2 < n_3 < n_4 < n_5\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation: \( T(n) = T\left(\frac{n}{4}\right) + T\left(\frac{n}{2}\right) + n^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the best bound constant \( C \) such that \( \sup_{R^3/D}|u| \leq C \cdot \sup_{\partial D}|g| \) for the Helmholtz equation \( \Delta u + k^2 u = 0 \) in the exterior region \( \mathbb{R}^3 \setminus D \), where \( D \) is an open and bounded region in three-dimensional space. Assume Dirichlet boundary condition \( u = g \) on \( \partial D \) and the usual radiation condition at infinity. Consider \( \partial D \) to be smooth and \( g \) continuous on \( \partial D \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of people required to attend 20 meetings, each with 8 attendees, such that no pair of attendees appears more than once across all meetings.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a method for randomly generating a classical Hilbert propositional calculus formula $F$ with $n$ variables. The formula $F$ is generated as follows:
- $F = x_i$ for each $i \leq n$ with probability $\frac{1}{n+2}$.
- $F = \neg F_1$, where $F_1$ is generated independently using the same method.
- $F = F_1 \to F_2$, where $F_1$ and $F_2$ are generated independently using the same method.

Given that the generation process terminates with probability 1, what is the probability that the formula $F$ is a tautology?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all functions \(f(x)\) such that \(D^*f = Df\), where \(D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_{G}(x+z) \,d x d y d z \), where the domain \( G \) is bounded by the planes \( x+y=1, x-y=1, x+z=1, z=0, x=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\sin^3{x}}{x^3}dx \) using contour integration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the closed-form expression for the determinant of the matrix \( \mathbf{G} \), where
\[\mathbf{G} = a\mathbf{I} + b\boldsymbol{ee}^T\]
in which \(a\) and \(b\) are constants, \(\boldsymbol{e}\) is a column vector with all elements being \(1\), and \((\cdot)^T\) denotes the transpose operation. \(\mathbf{G}\) is a \(u\times u\) matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( (u,v) = f(x,y) = (x + y, x^2 - y^2) \), compute the second partial derivative \( \frac{\partial^2 x}{\partial v^2} \) at the point \((u,v) = (2,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest possible size of an $n \times n$ matrix $A$ such that $A^3 = 0$ and $A$ has a Jordan form consisting of exactly 4 blocks.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of points of continuity of the function \( f(x) \) defined as \( f(x) = \tan x \) for rational numbers and \( f(x) = x^2 + 1 \) for irrational numbers within the interval \([0, 6\pi]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of the real number $a$ for which the expression \( a(x_1^2+x_2^2+x_3^2)+2(x_1x_2+x_1x_3+x_2x_3) \) can be expressed as \( \alpha x^2+\beta y^2+\gamma z^2 \), where \( \alpha, \beta, \gamma \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the number of 12-digit words that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if each pair of neighboring digits must differ by exactly $1$. Find the remainder when $A$ is divided by $2008$. \( \text{Provide your answer in the form } \boxed{\text{remainder}}. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $m$ for which the equation \( e^{\vert{x}\vert}=m(x+1) \) has exactly two distinct real solutions.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{(-\infty, -e^2) \cup (1, \infty)}


Context:
Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\log(1+b\sin x)}{\sin x}\,\mathrm{d}x \) given that \(|b|<1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the Divergence Theorem to evaluate the surface integral \( \iint\limits_S F\cdot dS \) where \( F(x,y,z)=xi+y^2j+(z+y)k \) and \( S \) is the boundary of the region enclosed by the cylinder \( x^2+y^2=4 \) and the planes \( z=x \) and \( z=8 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sum the series\[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the system of equations:
$$
\begin{align*}
xw^3 + yz^2 + z^3 &= -1 \\
zw^3 - xz^3 + y^2w &= 1
\end{align*}
$$
Find the value of $\frac{\partial z}{\partial x}$ at the point $(x, y, z, w) = (1, -1, -1, 1)$ without eliminating the function $w$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum of the irrational solutions of the equation
$$\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A grasshopper sits on a square of a 5x4 rectangular board. Each minute, it jumps at random to a neighboring square, with 2, 3, or 4 possible moves depending on its position (2 at corners, 3 at edges, 4 at the center), all equally likely. The grasshopper always remains on the board. What is the long-run average probability that the grasshopper will be on a corner square of the board after many jumps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of fleas that can be placed on a $10 \times 10$ board such that each flea jumps to an adjacent cell every minute, maintaining its direction as long as possible, and no two fleas ever occupy the same cell at the same time over the course of an hour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of arrangements of the 10 letters ABCDEFGHIJ such that none of the patterns ABE, BED, or HID appear in the arrangement.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral \( \int \int_S (2z^2 - x^2 - y^2) \, dS \), where \( S \) is the surface defined by the intersection of the cone \( z = \sqrt{x^2 + y^2} \) and the cylinder \( (x-1)^2 + y^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of Hahn-Banach extensions of a non-zero continuous linear functional $f:(c_{00},\|\cdot\|_1) \to \mathbb{C}$ to the space $(\ell^1,\|\cdot\|_1)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the expected duration of a randomly selected long-distance telephone call, given that the duration $Y$ has the following properties: $P(Y=3)=0.2$, $P(Y=6)=0.1$, and otherwise $Y$ has a continuous density function given by \( f(y) = \frac{1}{4}ye^{-y/2} \) for \( y > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on \( \alpha > 0 \) such that the function \[ f(\tau) = \frac{(x(t) - x(\tau))^2}{(t - \tau)^{\alpha + 1}} \] is integrable near \( \tau = t \), where \( x(t) \) is a continuous and differentiable function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equations \( \frac{1}{\omega+a}+\frac{1}{\omega+b}+\frac{1}{\omega+c} = 2\omega^2 \) and \( \frac{1}{\omega^2+a}+\frac{1}{\omega^2+b}+\frac{1}{\omega^2+c} = 2\omega \), where \( \omega \) is a complex cube root of unity, find the value of \( \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Lucas sequence is defined by these conditions: \( {L}_{0} = 2,{L}_{1} = 1 \) ,and \( {L}_{n + 2} = {L}_{n + 1} + {L}_{n} \) for all \( n \geq  0 \) . Determine
 the remainder when \( {L}_{2019}^{2} + {L}_{2020}^{2} \) is divided by \( {L}_{2023} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is 11/4. To the nearest whole percent, what percent of its games did the team lose?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest possible size of a subset $M$ of the set $\{1, 2, \ldots, 2006\}$ such that for any three elements $x, y, z$ in $M$ with $x < y < z$, the sum $x + y$ does not divide $z$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the function $g : \mathbb{R}^3 \to \mathbb{R}$ defined by $g(x,y,z) = x^3 - 2y^2 + z^4$. Determine the values of $w$ for which $g^{-1}(w)$ is a submanifold of $\mathbb{R}^3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the integral: \[ \int_{-\infty}^\infty \Gamma(1+ix)\Gamma(1-ix)\,dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the rational function \( Q(x) \) such that \( x \cdot Q(x+2018) = (x-2018)Q(x) \) for all \( x \neq 2018 \) and \( x \neq 0 \), and \( Q(1) = 1 \), find the value of \( Q(2017) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^1{\frac{(2x^3-3x^2)f'(x)}{f(x)}}\,dx \) given that the function \( f(x) \) is differentiable on \([0, 1]\) and satisfies the conditions \( f(1) = 1 \) and \( f(x)f(1-x) = e^{x^2 - x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct groups of order 2015.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{20}\) in the expansion of \((x^2 + x^4 + x^6 + \dots)^3 (x + x^3 + x^6 + \dots)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((r_k)_{k \in \mathbb{N}}\) be an enumeration of \(\mathbb{Q}\). Define \(U_{1/p} = \bigcup_{k \in \mathbb{N}} \left(r_k - \frac{1/p}{2^k}, r_k + \frac{1/p}{2^k}\right) \subset \mathbb{R}\). Determine the set \(\bigcap_{p \in \mathbb{N}^*} U_{1/p}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the area of triangle $\vartriangle ABC$ with vertices $(a,a^2),(b,b^2),(c,c^2)$ is $\frac{1}{4}$ square units, and the area of triangle $\vartriangle PQR$ with vertices $(p,p^2),(q,q^2),(r,r^2)$ is $3$ square units, determine the value of the determinant:

$$ \begin{vmatrix} (1+ap)^2 & (1+bp)^2 & (1+cp)^2 \\ (1+aq)^2 & (1+bq)^2 & (1+cq)^2 \\ (1+ar)^2 & (1+br)^2 & (1+cr)^2 \\ \end{vmatrix} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer n such that the inequality \( \prod_{k=1}^n \frac{\arctan\left( \frac{3k+1}{\sqrt{8k^3+8k^2+2k}}\right)}{\arccos \left( \frac{1}{\sqrt{2k+1}} \right)}>1000 \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of imaginary roots of the equation \( \frac{A_1^2}{x-a_1} + \frac{A_2^2}{x-a_2} + \frac{A_3^2}{x-a_3} + \cdots + \frac{A_n^2}{x-a_n} = k \), where \(a_1, a_2, \ldots, a_n, A_1, A_2, \ldots, A_n,\) and \(k\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( r \) for which the integral \( \int_{1}^{\infty} \frac{\cos(x)}{x^r} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of curly brackets in the expression \( \wp^5(\varnothing) \), where \( \varnothing \) is the empty set and counts as \( \{ \} \) (1 curly bracket).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Aerith records a time duration in both days:hours:minutes:seconds format and in seconds. Bob subtracts the number of seconds from the concatenated days:hours:minutes:seconds format, ignoring punctuation. What is the largest number that always divides the result of this subtraction?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Evaluate the integral \( \int_{-\infty}^\infty e^{-(x-ti)^2} \, dx \), where \( t \in \mathbb{R} \) and \( i = \sqrt{-1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Fourier transform of the function \( \delta(x)f(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_{|z|=3} \frac{z^{17}}{(z^2+2)^3(z^3+3)^4} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a real-valued function $f(x)$ that satisfies the functional equation \[ f(x-y) = f(x) \cdot f(y) - f(a-x) \cdot f(a+y) \] where $a$ is a constant, and knowing that $f(0) = 1$ and $f(2a) = -1$, find the value of $f(2a+x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_A f \) where \( A = \{(x,y) : 0 \le x + y \le 1, 0 \le x + 3y \le 3\} \subseteq \mathbb{R}^2 \) and \( f(x,y) = \sqrt{x+y} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of linear conditions that must be added to the set \( \mathcal{T} \) to ensure that \(|M_{\gamma_{1}}(Q, \mathcal{T'})| < |M_{\gamma_{2}}(Q, \mathcal{T'})|\) holds for any choice of linear conditions, where \( \mathcal{T'} \) is a subset of \( \mathcal{T} \) defined by these linear conditions. Assume \( 0 < \gamma_1 < \gamma_1 + 1 \leq \gamma_2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum number of integer lattice points contained in a centrally symmetric convex subset of \(\mathbb{R}^N\) with a volume that is not an integer multiple of \(2^N\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the pole in the expansion of \( \frac{z - \sin z}{z^3} \) around \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n} \cdot \sum_{i=1}^n \frac{1}{1+(\frac{i}{n})^i}$$ using the concept of definite integrals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the differential equation for the family of parabolas with a fixed directrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle in the axial section of a cone if a sphere with its center at the vertex of the cone, touching its base, divides the volume of the cone in half.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \arccos\left(\frac{1 + \sqrt{17}}{8}\right)}


Context:
Find a prime number \( p \) such that there exists a sequence of integers \( S_n \) where \( (S_n)^2 \) converges to 21 under the \( p \)-adic norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the vector space \( \mathbb{R}(x) \) over the field \( \mathbb{R}(x^3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((X_1, X_2, \ldots)\) be an independent sequence of random variables, where each \(X_n\) is uniformly distributed on \([0, n]\). Calculate the probability \(P(\{w: X_n(w) \rightarrow \infty \text{ as } n \rightarrow \infty\})\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a Gaussian integer \( \alpha \in \mathbb{Z}[i] \) such that the ideal \((3+5i,1+3i)\) in \( \mathbb{Z}[i] \) is equal to the principal ideal \((\alpha)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $u$ and $v$ are differentiable functions of $r$ and $s$, such that $ru^2+sv^3-uv=2$ and $uv^2-ru+sv=3$. Determine the value of $\frac{\partial u}{\partial r}$ at the point $(r,s,u,v)=(1,2,1,2)$.
Please reason step by step, and put your final answer within \boxed{}.