Context:
13th Iberoamerican 1998 Problem A3 Find the smallest number n such that given any n distinct numbers from {1, 2, 3, ... , 999}, one can choose four different numbers a, b, c, d such that a + 2b + 3c = d. Solution
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Context:
For the function $f(x) = \cos(x)$, find the degree of the Taylor polynomial needed to approximate $\cos(1/10)$ with an error less than $10^{-5}$ using the Taylor-Lagrange inequality.
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Context:
Find a formula for $\sum_{k=0}^{n}(-1)^k2^ks(n,k)$ using the exponential generating function of the Stirling numbers of the first kind.
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Context:
Compute the number of ordered quadruples $(a, b, c, d)$ of distinct positive integers such that $\binom{\binom{a}{b}}{\binom{c}{d}} = 21$.
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Context:
Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane such that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What is the largest possible number of these angles that can be equal to $90^\circ$?
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Context:
An integer is called <i>parity-monotonic</i> if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there?
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Context:
The plane is divided by lines into square cells with side length $a$. The cells are numbered using pairs of integers $(m, n)$ (where the cell immediately to the right has the number $(m+1, n)$, and the cell immediately above has the number $(m, n+1)$). Find the probability that a coin of radius $r < a/2$ randomly tossed onto the plane will completely fall into a cell for which $m+n$ is divisible by three.
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Context:
Find the greatest number which leaves the same remainder when it divides 1260, 2310, 30030, and 72930, which are all divisible by the first five prime numbers (2, 3, 5, 7, and 11) and their respective combinations.
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Context:
Evaluate the triple integral \( \int \int \int_{\Omega} xy^{2}z^{2} \, dx \, dy \, dz \) where \( \Omega \) is the region bounded by the surfaces \( z=xy \), \( y=x \), \( x=1 \), and \( z=0 \).
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Context:
Calculate the sum $\frac{1}{2.7.12} + \frac{1}{7.12.17} + ... + \frac{1}{1997.2002.2007}$.
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Context:
Select 3 distinct numbers from the set {1, 2, ..., 10} such that they do not form an arithmetic sequence. How many such selections are possible? (Answer with a number).
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Context:
Given an infinite number of uniquely labeled pairs of dice, you roll the pair labeled A to determine the number of times to roll pair B. The sum of pair B's rolls determines the number of times to roll pair C, and so on. For a probability p between 0 and 1 and a target sum s, determine the minimum number of pairs of dice you need to roll to have at least a p probability of reaching a total sum of s or greater.
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Context:
Given the sets A and B as follows:
\[
A = \left\{x^2 + x + 1, -x, -x-1\right\}, \quad B = \left\{-y, -\frac{y}{2}, y+1\right\}
\]
and knowing that A = B, find the value of x^2 + y^2.
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Context:
已知平面上有 100 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。
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Context:
In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in January will be sunny only if the weather changed exactly one year ago on that day. In 2015, January in Anchuria featured a variety of sunny and rainy days. In which year will the weather in January first change in exactly the same pattern as it did in January 2015?
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Context:
Determine the value of $t$ for which the polynomial $f(x) = x^3 - tx - 1$ is irreducible over $\mathbb{Q}[x]$. Provide your answer as a single value of $t$.
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Context:
Given two geometric sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) with positive terms and common ratios \(p\) and \(q\) respectively, where \(p > q\) and \(p \neq 1, q \neq 1\), define \(c_{n} = a_{n} + b_{n}\). Let \(S_{n}\) be the sum of the first \(n\) terms of the sequence \(\{c_{n}\}\). Find \(\lim _{n \rightarrow \infty} \frac{S_{n}}{S_{n-1}}\).
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Context:
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?
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Context:
Determine the number of ordered triples of complex numbers \((x, y, z)\) such that \(x^2 - y^3\), \(y^2 - z^3\), and \(z^2 - x^3\) are rational numbers, and \(x^3(x^2+1) + y^3(y^2+1) + z^3(z^2+1) = 2[(x^2)y^3 + (y^2)z^3 + (z^2)x^3]\).
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Context:
Evaluate the triple integral: \[ \int_{0}^{3} \int_{0}^{\frac{6-2x}{3}} \int_{0}^{\frac{6-3y-2x}{2}} xyz \, dz \, dy \, dx \]
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Context:
Fill the numbers $1$ through $9$ into a $3 \times 3$ grid such that no number is repeated in any cell. Define two cells that share a common edge as a "number pair." Find the maximum number of number pairs where one number is a multiple of the other.
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Context:
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular $12$-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
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Context:
A hotel has 100 rooms and charges guests for their rooms in advance. The probability that any booked guest shows up by 10pm is \( p = \frac{19}{20} \), and guests show up independently. Let \( X \) denote the number of booked guests who show up before 10pm.

The hotel earns $50 in profit for every guest and no-show, but must pay $60 in compensation to each guest who is unable to stay due to overbooking. Assuming unlimited demand for pre-booked rooms, how many rooms should the hotel sell to maximize expected profit?
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Context:
Evaluate the integral \( \int_0^2\int_1^3\min(x,y)\,dx\,dy \) given that \( x < y \).
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Context:
Determine the asymptotic upper bound for the Dirichlet divisor summatory function \( D(N) = \sum_{i=1}^N \left\lfloor\frac{N}{i}\right\rfloor \).
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Context:
Given the plane point sets A = {(x, y) | (y-x)(y-1/x) >= 0} and B = {(x, y) | (x-1)^2 + (y-1)^2 <= 1}, calculate the area of the plane figure represented by A ∩ B. Express your answer in terms of π.
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Context:
在$\vartriangle ABC$中，$AB=\sqrt{2},AC=2,\angle BAC=105{}^\circ $，点$\textit{D}$满足$\overrightarrow{AD}=x\overrightarrow{AB}+y\overrightarrow{AC}$且$x+y=1(x,y\in R)$，则当$|\overrightarrow{AD}|$最小时，$\frac{x}{y}$的值为________.
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Context:
Evaluate the limit: \[ \lim_{x \to 0} \log_{\cos(3x)}\cos(2ix) \] where \( i = \sqrt{-1} \). Note: The use of L'Hospital's rule and expansion is prohibited.
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Context:
A solution contains 8 parts of water for every some parts of lemonade syrup. To make the solution contain 20% lemonade syrup, 2.1428571428571423 parts of the solution should be removed and replaced with water. How many parts of lemonade syrup are there in the original solution?
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Context:
Evaluate the limit \( \lim_{n\to\infty}\int_{-\pi/4}^{\pi/4}\frac{n\cos(x)}{n^2x^2+1}\,dx \).
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Context:
Find the Lanczos derivative of the function \(f(x) = |x|\) at \(x = 0\).
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Context:
Three workers are digging a hole. They take turns working, with each one working as long as it takes the other two to dig half the hole. Working in this way, they finished digging the hole. How many times faster would they have finished the job if they had worked simultaneously?
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Context:
A tech company has successfully developed a new product and decides to borrow 2 million yuan from a bank for production. The contract stipulates that the loan, along with an 8% interest, is to be repaid in a lump sum at the end of two years. After the product enters the market, due to strong sales, the company is able to repay the loan principal and interest, with an additional surplus of 720,000 yuan. If the company's capital increases by the same percentage each year during the production period, find this percentage.
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Context:
Solve the inequality $n \leq 8\lg n$ and find the largest integer $n$ that satisfies it.
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Context:
A train 120 m long running at a certain speed takes 13.884603517432893 seconds to cross a bridge 150 m in length. What is the speed of the train?
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Context:
Given two normally distributed variables, $X$ and $Y$, with means $E(X)=E(Y)=0$, variances $Var(X)=2$ and $Var(Y)=8$, and correlation $Corr(X,Y)=-\frac{1}{2}$, determine the smallest possible value of $E(X^5Y^3)$. Provide your answer as a single numerical value.
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Context:
4. What is the greatest length that a closed, non-self-intersecting broken line can have, which runs along the grid lines of a $6 \times 10$ cell field?
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Context:
A student multiplied a number by 3/5 instead of another fraction. The percentage error is 64.00000000000001%. What fraction should the student have multiplied the number by?
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Context:
In a military exercise, there is a defensive line with 20 posts. Five different new weapons need to be distributed among the posts. The first and last posts should not be equipped with the new weapons. Additionally, among every set of 5 consecutive posts, at least one post should be equipped with a new weapon, and no two adjacent posts should be equipped simultaneously. How many possible arrangements are there to equip the posts with the new weapons?
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Context:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
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Context:
Four small panda bears and five bigger panda bears eat 25 pounds and a certain amount of fresh bamboo sprouts every day, respectively. The pandas eat 2100 pounds of bamboo in a week. How many pounds of bamboo do the bigger pandas eat each day?
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Context:
The general term formula for the sequence $\frac {3}{5}$, $\frac {1}{2}$, $\frac {5}{11}$, $\frac {3}{7}$, $\frac {7}{17}$, ... is __________.
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Context:
Find the sum of the series $\sum f_n$, where $f_n(x)=\frac{nx}{(1+n^2x^2)}-\frac{(n-1)x}{[1+(n-1)^2x^2]}$ for $x\in[0,1]$, using telescoping series.
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Context:
3.281. $\sin \left(\frac{5}{2} \pi+4 \alpha\right)-\sin ^{6}\left(\frac{5}{2} \pi+2 \alpha\right)+\cos ^{6}\left(\frac{7}{2} \pi-2 \alpha\right)$.
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Context:
Professor Smith is an art history professor specializing in the miniature-painting genre in England. She has a collection of 48 miniature paintings. Each painting measures 4 inches by 6 inches. She wants to arrange these paintings in a special exhibit where each painting is placed in a frame with an additional border of 2 inches on all sides.

If she arranges the framed paintings in a single row along a gallery wall, how long will the row be in inches?
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Context:
If the curve $|y| = 2^x + 1$ and the line $y = b$ have no common points, then the range of values for $b$ is ______.
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Context:
Given the same function $F: \mathbb{R} \rightarrow \mathbb{R}$ defined by $F(x-y) = F(x)F(y)$, determine all possible values of $F(2)$ if $F(1) = 3$.
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Context:
若正整数 $n$ 使得方程 $x^{3}+y^{3}=z^{n}$ 有正整数解 $(x, y, z)$, 则称 $n$ 为 "好数". 那么, 不超过 2019 的好数有 $\qquad$个.
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Context:
Find all solutions to the equation

\[ x^{4} = y^{2} + 2y + 2 \]

where \( x \) and \( y \) are integers.
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Context:
In triangle \( ABC \), points \( A_1 \) and \( A_2 \) are on side \( BC \) such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) are collinear. Determine the length of \( B_1B_2 \).
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Context:
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important. For example, $f(4) = 6$ because $4$ can be written as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1$

Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.
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Context:
Let $X_t$ be a Brownian motion. Calculate the probability that $X_2 > 1$ given that $X_1 = 0$. Express your answer as a single probability value.
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Context:
Inside a circle of radius \( R \), there is a point \( A \). Two perpendicular lines are drawn through this point. Then, these lines are rotated by an angle \( \varphi \) relative to point \( A \). The chords cut by the circle from these lines sweep out a figure in the shape of a cross centered at point \( A \) during the rotation. Find the area of the cross.
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Context:
A non-abelian group $G$ of order $p^3$ has $p^2+p+1$ subgroups of order $p$. Determine the number of subgroups of order $p^2$ in $G$. \( \text{(Express your answer as a single integer.)} \)
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Context:
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,...,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,...,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
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Context:
Given that the ABC stock pays no dividend and is currently priced at $S_0 = 10, with the stock price at expiry $T$ going up to $uS_0$ with probability $p$ and down to $dS_0$ with probability $1 - p$, where $d < 1 < u$, and the interest rate is zero, determine the fair value of a European call option with a strike price of $7$. Assume a European put option with a strike price of $9 is priced at $14/9$ and a European put option with a strike price of $8 is priced at $8/9$. What is the fair value of the call option?
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Context:
Determine the ARIMA parameters \((p,d,q)\) for the process \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\), where \(X_t\) is an \(ARMA(1,2)\) process given by \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\) and \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\).
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Context:
Find all positive integers \(n\) such that \(n\) divides \(a^{25} - a\) for all positive integers \(a\).
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Context:
Determine the value of \(a \in \mathbb{R}\) such that the function \(f : \mathbb{R}^3 \to \mathbb{R}\) defined by \[ f(x,y,z)=\begin{cases} \frac{z}{x}+y & \text{if } x \neq 0 \\ a & \text{if } x=0 \end{cases} \] is continuous at the origin.
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Context:
A container is composed of an upright hollow frustum and a hollow cylinder, each with a base radius of $12 \,\text{cm}$ and a height of $20 \,\text{cm}$. When finely granulated sand is poured into this container, it fills the frustum and partially fills the cylinder, with the sand height in the cylindrical section measuring $5 \,\text{cm}$. If this container is then inverted, what will be the height of the sand in $\text{cm}$?
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Context:
Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\]
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Context:
Find the surface area of a sphere inscribed in a pyramid, whose base is a triangle with sides measuring 13, 14, and 15 cm, given that the apex of the pyramid is 5 cm away from each side of the base.
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Context:
What is the remainder when 71 multiplied by a certain number is divided by 9, if the remainder is known to be 8?
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Context:
Within the range of 0° to 360°, find the angle(s) with the same terminal side as -120°.
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Context:
Sixty-four white 1x1x1 cubes are used to form a 4x4x4 cube, which is then painted red on each of its six faces. This large cube is then broken up into its 64 unit cubes. Each unit cube is given a score as follows:

| Exact number of faces painted red | Score |
| --- | ---: |
| 3 | 3 |
| 2 | 2 |
| 1 | 1 |
| 0 | -7 |

Calculate the total score for the 4x4x4 cube.
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Context:
Bob is building raised beds for his vegetable garden. Each bed is 2 feet high, 2 feet wide, and 8 feet long. The sides are going to be built of 1-foot wide planks. Bob needs a certain number of 8-foot long planks to construct the raised beds. If he needs 50 8-foot long planks in total, how many raised beds is he constructing?
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Context:
If there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+2 x_{2}+\cdots+ n x_{n}=2009$ and $x_{1}+x_{2}+\cdots+x_{n}=0$, where each $x_{i}= \pm 7$ for $i=1,2, \cdots, n$, determine the minimum value of $n$.
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Context:
Mark went to a store where he spent one-half of his money, and then spent $14 more. He then went to another store where he spent one-third of his starting money, and then spent some more. When he had no money left, he had $180 when he entered the first store. How much more did he spend in the second store after spending one-third of his starting money?
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Context:
Find the number of odd numbers in the \(n\)-th row of Pascal's triangle.
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Context:
Given the sequence \( a_n \) defined by \( a_1 = 1111 \), \( a_2 = 1212 \), \( a_3 = 1313 \), and the recursive relation \( a_{n+3} = | a_{n+2} - a_{n+1} | + | a_{n+1} - a_n | \), find the value of \( a_{1414} \).
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Context:
Determine the values of \( x \) for which the series \( \sum_{n=1}^\infty \frac{1}{x^{\ln n}} \) converges.
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Context:
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}$. Compute the sum:
\[ \sum_{i=1}^{1000} g(i). \]
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Context:
Consider 9 distinct apples and 7 distinct bananas to be placed into 6 identical bins. Determine the number of ways to configure the bins such that exactly one bin contains 5 apples, exactly one bin contains 3 bananas, and the remaining 4 bins each contain 1 apple and 1 banana. Provide your answer as a single number.
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Context:
Evaluate the integral \( \int_{\mathbb{R}^3} x_3^2 e^{-\lVert x \rVert_2} \lambda_3(dx) \), where \( x = (x_1, x_2, x_3) \).
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Context:
A safe has a lock with 3 wheels, each of which can be set in 8 different positions. Due to a defect, the safe will open if any two of the three wheels are in the correct position. What is the minimum number of combinations that must be tried to ensure the safe can be opened, assuming the correct combination is unknown?
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Context:
Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, the long axis $AB$ is divided into $2016$ equal parts. From left to right, these points are sequentially named $M_{1}$, $M_{2}$, $…$, $M_{2015}$. A line with a slope of $k(k\neq 0)$ is drawn through point $M_{1}$, intersecting the ellipse $C$ at points $P_{1}$ and $P_{2}$, with $P_{1}$ located above the $x$-axis. Similarly, a line with a slope of $k(k\neq 0)$ is drawn through point $M_{2}$, intersecting the ellipse $C$ at points $P_{3}$ and $P_{4}$, with $P_{3}$ located above the $x$-axis. This process is repeated until a line with a slope of $k(k\neq 0)$ is drawn through point $M_{2015}$, intersecting the ellipse $C$ at points $P_{4029}$ and $P_{4030}$, with $P_{4029}$ located above the $x$-axis. Find the product of the slopes of the $4030$ lines $AP_{1}$, $AP_{2}$, $…$, $AP_{4030}$.
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Context:
How many ordered pairs \((\alpha_1, \alpha_2)\) of permutations in the symmetric group \(S_n\) commute, i.e., satisfy \(\alpha_1 \circ \alpha_2 = \alpha_2 \circ \alpha_1\), where \(\alpha_1, \alpha_2 \in S_n\)?
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Context:
To guard a certain target, someone needs to be on duty day and night, thus there are day shifts and night shifts. A guard can be scheduled for a single day shift, or a single night shift, or a continuous 24-hour shift. After completing a day shift, a night shift, or a 24-hour shift, the guard must rest for at least 1 day, 1.5 days, or 2.5 days respectively. If each shift requires 1 person, what is the minimum number of people needed to participate in the duty rotation?
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Context:
Given a function f such that f(f(n)) + f(n) = 2n + 3, and f(2013) = 2014, what is the value of f(0)?
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Context:
8.4. The shorter leg $AC$ of the right triangle $ABC$ has a length of $b$. A point $D$ is chosen on the hypotenuse $AB$ such that $BD = BC$. A point $E$ is taken on the leg $BC$ such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.

---

Translation:
8.4. The shorter leg $AC$ of the right triangle $ABC$ has a length of $b$. A point $D$ is chosen on the hypotenuse $AB$ such that $BD = BC$. A point $E$ is taken on the leg $BC$ such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are five boxes \( A, B, C, D, \) and \( E \) with 9, 5, 3, 2, and 1 balls respectively. The first student identifies the box with the fewest balls and then puts one ball from each of the other boxes into this box. The second student then identifies the box with the fewest balls and repeats this process, and so on. After the 199th student has completed their turn, how many balls are in each of the five boxes \( A, B, C, D, \) and \( E \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a polynomial  $P(x,y,z)$  in three variables with integer coefficients such that for any real numbers  $a,b,c,$    $$ P(a,b,c)=0 \Leftrightarrow a=b=c. $$  Find the largest integer  $r$  such that for all such polynomials  $P(x,y,z)$  and integers  $m,n,$    $$ m^r\mid P(n,n+m,n+2m). $$  [i]Proposed by Ma Zhao Yu
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Context:
Evaluate the integral \( \int_{0}^{\pi}\ln\left(\sin\left(x\right)\right)\cos\left(4x\right)\,\mathrm{d}x. \)
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Context:
Determine whether the function \(|x|^{-r}\) is a tempered distribution. Specifically, find if there exist constants \(C\) and \(\alpha\) such that for all \(\phi \in \mathcal{S}(\mathbb{R})\), the inequality \(|\int_{\mathbb{R}} \frac{1}{|x|^r} \phi(x) \, dx| \leq C \|\phi\|_{(\alpha, \beta)}\) holds for some \(r > 0\).
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Context:
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\ln a-b$, with $a$ and $b$ positive integers. Please provide the value of $a + b$.
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Context:
Let $G=(V,E)$ be a planar, $3$-regular connected graph with $\chi'(G)=3$. $G$ also satisfy the following property:
For every two colors, Every path of length $6$ with those two alternating colors, is a cycle.

Find $\vert V \vert$
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Context:
In 4 weeks, Prudence has a varying sleep schedule influenced by her events and activities.

Week 1:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she sleeps for 9 hours. On Saturday, she sleeps 9 hours. She takes a 1-hour nap on Saturdays and Sundays. On Tuesday and Thursday, she takes a 30-minute power nap after work. However, on Wednesday night, she wakes up for 1 hour due to a thunderstorm.

Week 2:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she attends a late-night event, reducing her sleep by 2 hours, resulting in only 7 hours of sleep. On Saturday, she sleeps 8 hours. Her napping schedule remains the same. Additionally, on Monday night, she wakes up for 30 minutes to check on a noise outside.

Week 3:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she sleeps for 9 hours. On Saturday, she sleeps for 9 hours again. Her napping schedule remains the same. On Wednesday, she attends a late-evening yoga class and sleeps 30 minutes less that night.

Week 4:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she attends a late-night event again, resulting in 7 hours of sleep. On Saturday, she sleeps 10 hours. Her napping schedule remains the same. On Thursday, she stays awake for an extra hour to finish a project, reducing her sleep to 5 hours that night.

With these specific events and activities affecting Prudence's sleep schedule, how much sleep does she get in 4 weeks?
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Context:
Two circles are given by the equations $x^2+y^2-2mx=0$ and $x^2+(y-2)^2=1$. The circles are externally tangent. Find the value of the real number $m$. Express your answer as a numerical value, including the ± symbol if necessary.
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Context:
In a connected graph $G=(V,E)$ with $n$ vertices, where all edges form $3$-cycles, the graph has exactly $n-2$ $3$-cycles and a minimum degree of $2$. Every vertex is part of at least one $3$-cycle. Determine the maximum number of vertices with a degree of $2$ in $G$ as a function of $n$. Provide your answer as a function of $n$.
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Context:
Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \).
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Context:
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability  $\frac{1}{2}$ . What is the expected value of the number of games they will play?
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Context:
Find the values of $n$ for which the differential of the square map $P:SO(n) \to SO(n)$ is not an isomorphism.
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Context:
Given a square pyramid \(O-ABCD\) where \(OA = \sqrt{3}\) and \(BC = 2\), if a sphere is centered at \(O\) with a radius of 1, what is the volume of the intersection between the sphere and the square pyramid?
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Context:
What is the average length in letters of a word in this question?

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Context:
A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coordinates of the vertices of $Q_3$.
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Context:
Consider the following increasing sequence $1, 3, 5, 7, 9, \ldots$ of all positive integers consisting only of odd digits. Find the $2017^{th}$ term of the above sequence.
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Context:
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school.  He takes the bus home and arrives at 4:00 p.m.  How many minutes has he spent on the bus?
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Context:
A positive integer is called Tiranian if it can be written as $x^2+6xy+y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?

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Context:
Evaluate the asymptotic behavior of the integral \( \int_0^{at} \frac{e^{-u}}{(t-\frac{u}{a})^\beta} \, \mathrm{d}u \) as \( t \to \infty \), where \( \beta < 1 \).
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Context:
6-27 The function $f(k)$ is defined on $N$ and takes values in $N$, and it is a strictly increasing function (if for any $x_{1}, x_{2} \in A$, when $x_{1}<x_{2}$, we have $f\left(x_{1}\right)<f\left(x_{2}\right)$, then $f(x)$ is called a strictly increasing function on $A$), and it satisfies the condition $f(f(k))=3k$. Try to find: $f(1)+f(9)+f(96)$.

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Context:
Evaluate the line integral \( \oint_C y^2 \,dx + x \,dy \) using Green's Theorem, where the curve \( C \) is parameterized by \( r(t) = 2\cos^3(t) \mathbf{i} + 2\sin^3(t) \mathbf{j} \) for \( t \in [0,2\pi] \). Determine the region \( D \) enclosed by \( C \) as part of your solution.
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Context:
Let $(X,\mathscr{M},\mu)$ be a measure space and $f\colon X \to [0,+\infty[$ be a measurable function. Consider the graph of $f$, defined as ${\rm gr}(f) = \{ (x,f(x)) \mid x \in X \}$. Does the product measure $(\mu \times \mathfrak{m})({\rm gr}(f))$ always equal zero, even if the domain space is not $\sigma$-finite? Here, $\mathfrak{m}$ denotes the Lebesgue measure on $[0,+\infty[$.
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Context:
设有序数组 A=(a_1,a_2,\cdots,a_{10}) 同时满足以下 4 个条件：(1) a_1,a_2,\cdots,a_{10} 是 1,2,\cdots,10 的一个排列；(2) a_1<a_2,a_3<a_4,a_5<a_6,a_7<a_8,a_9<a_{10}；(3) a_2>a_3,a_4>a_5,a_6>a_7,a_8>a_9；(4) 不存在 1\leq i<j<k\leq 10，使得 a_i<a_k<a_j.求这样的有序数组 A 的个数.
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Context:
Determine the time at which the solution of the differential equation \( u_t + uu_x = 0 \) with the initial condition \( u(x,0) = \cos{(\pi x)} \) becomes multi-valued.
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Context:
If $x^2 + bx + b + 3 = 0$ has roots of the form $\frac{-b \pm \sqrt{5}}{2}$, where $b > 0 $, then $b = m+\sqrt{n}$ for positive integers $m,n$. Find $m + n$.
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Context:
Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$.

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Context:
In a $C^*$-algebra, if an element $a$ satisfies $b^*a^*ab = 0$ for all elements $b$ in the algebra, does it necessarily follow that $a = 0$?
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Context:
In trapezoid $K L M N$, the bases $K N$ and $L M$ are equal to 12 and 3, respectively. From point $Q$, lying on side $M N$, a perpendicular $Q P$ is dropped to side $K L$. It is known that $P$ is the midpoint of side $K L$, $P M=4$, and that the area of quadrilateral $P L M Q$ is four times smaller than the area of quadrilateral $P K N Q$.

Find the length of segment $P N$.
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Context:
Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$.
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Context:
Positions A, B, and C are our artillery positions. A is 6km due east of B, and C is 4km from B at a bearing of 30° west of north. Position P is an enemy artillery position. At a certain moment, position A detects a signal from the enemy position, and 4 seconds later, positions B and C detect this signal simultaneously. The signal travels at a speed of 1km per second. If A fires at position P, what is the bearing of the fire?
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Context:
Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$, and line $DB$ intersects the circle $\Gamma_1$ for the second time at point $E$. The perpendicular bisectors of segments $CD$ and $EF$ intersect at point $P$. Knowing that $CA = 12$ and $PE = 5$, find $AP$.
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Context:
Find the sum: \( S = 19 \cdot 20 \cdot 21 + 20 \cdot 21 \cdot 22 + \cdots + 1999 \cdot 2000 \cdot 2001 \).
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Context:
Every member of the school's entertainment team can either sing or dance, and at least one of these. It is known that there are 2 people who can sing, 5 people who can dance. If we select 2 people, let $\xi$ be the number of people selected who can both sing and dance. Given $P(\xi > 0) = \frac{7}{10}$, the total number of people in the entertainment team is ______.
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Context:
Arrange the numbers 1, 2, 3, 4, 5, 6 in a row, and denote the i-th number as a_i (i=1,2...6). If a_1!= 1, a_3!= 3, a_5!= 5, and a_1 < a_3 < a_5, calculate the total number of different arrangements. Express your answer as a single integer.
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Context:
Calculate the geometric mean of 1 and 9. Express your answer in the form ±n, where n is a whole number.
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Context:
Given that the binomial coefficient of only the fourth term in the expansion of (1+2x)^n is the largest, the constant term in the expansion of the polynomial (x^2+1)(x+1/x)^n is \_\_\_\_\_\_.
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Context:
Find all generators of the subgroup <21> in the group $Z_{24}$. That is, determine all elements $m \in Z_{24}$ such that $<m> = <21>$.
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Context:
For the equation log 3 n + log x n, the 3 digit number n that makes it a whole number is 216. What is the value of x?
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Context:
Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors as 2013. Compute the least possible value of $|A - B|$.
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Context:
Let proposition $p: f(x)= \sqrt{x^{2}+mx+1}$ have a domain of $\mathbb{R}$; proposition $q$: the equation $\frac{x^{2}}{m}+\frac{y^{2}}{2}=1$ represents an ellipse with foci on the $y$-axis. If "p and q" is a false proposition, and "p or q" is a true proposition, find the range of the real number $m$.
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Context:
Compute the probability $$P(T_{1} < T_{-3} < T_3)$$ for a standard Brownian motion $W_t$, where $T_a$ denotes the hitting time of level $a$.
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Context:
Find the total number of positive four-digit integers \( N \) satisfying both of the following properties:
(i) \( N \) is divisible by 7, and
(ii) when the first and last digits of \( N \) are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting integer need not be a four-digit number.)
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Context:
Calculate the mass of the region in the first quadrant where the tile lies between $1 < xy < 3$ and $1 < x^2 - y^2 < 3$, given the density function $\sigma(x, y) = (x^2 + y^2)(x^4 + y^4)$.
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Context:
In quadrilateral \(ABCD\), it is given that \(\angle DAB = 150^\circ\), \(\angle DAC + \angle ABD = 120^\circ\), and \(\angle DBC - \angle ABD = 60^\circ\). Find \(\angle BDC\).
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Context:
If $z\bar{z}'=-1$, do $z$ and $z'$ correspond to opposite points on the Riemann sphere? Answer 'yes' or 'no'.
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Context:
设 a_1,a_2,\cdots,a_{21} 为 1,2,\cdots,21 的排列，满足|a_{20}-a_{21}|\geq|a_{19}-a_{21}|\geq\cdots\geq|a_1-a_{21}|.这样的排列的个数为__________.
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Context:
The negation of the proposition "_x_∈**R**, _x_2 - _x_ + 3 > 0" is
Please reason step by step, and put your final answer within \boxed{}.

\boxed{_x_∈\mathbb{R}, _x_2 - _x_ + 3 \leq 0}


Context:
In a tournament bracket, a player must be defeated three times to be eliminated. If there are 512 contestants entering the tournament, what is the greatest number of games that could be played?
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Context:
Determine the sign of the derivative of the function \( f(x) = \frac{x+1}{x} + \left(\frac{2x}{x+1}\right)^{\frac{\ln(6/5)}{\ln(31/25)}} \) for \( x > 10^{500} \).
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Context:
A cylindrical, closable container has a diameter of 82 units and a length of 225 units. How many spheres with a diameter of 38 units can be placed inside, and then the container can be closed?
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Context:
Determine whether a symmetric real-valued matrix with a positive determinant is necessarily a covariance matrix. Answer "yes" or "no."
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Context:
Determine the interval of monotonic increase for the function $y=3x-x^{3}$.
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Context:
Calculate the circulation of the vector $\mathbf{a}=\sqrt{1+x^{2}+y^{2}} \mathbf{i}+y[x y+\ln \left(x+\sqrt{1+x^{2}+y^{2}}\right)]$ around the circle $x^{2}+y^{2}=R^{2}$.
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Context:
A box has a square base of side $x$ and height $h$. Find the dimensions $x$ and $h$ when the volume is 12 and the surface area is as small as possible.
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Context:
Evaluate the integral of $g(z)=\cot z$ around a contour that encircles the poles at $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ exactly once.
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Context:
In a non-commutative ring $R$ with unity $1$, but without assuming the uniqueness of left or right inverses, determine if every non-unit element $x$ must be contained in some maximal left ideal. Answer with 'yes' or 'no'.
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Context:
Given the recursive sequence defined by a_0 = 134, a_1 = 150, and a_{k+1} = a_{k-1} - k/a_k for k = 1, 2,..., n-1, find the value of n for which a_n = 0.
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Context:
Evaluate the Cauchy principal value of the integral \( I = \oint \frac{dz}{(z - z_1)(z - z_2)} \) where the poles \( z_1 = e^{-ik} \) and \( z_2 = e^{ik} \) lie on the unit circle contour. Determine if there are conditions under which this integral is not zero.
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Context:
8. What is the least number of odd numbers that can be chosen from the set of all odd numbers lying between 16 and 2016, so that no one of the chosen numbers is divisible by any other?

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Context:
Given Hilbert spaces $H_1$ and $H_2$ with dimensions $n$ and $m$ respectively, consider the map $$\varphi:S_1(H_1)\hat{\otimes}S_1(H_2)\to S_1(H_1\otimes_2 H_2),$$ defined by $$\varphi( A \otimes B) (\xi_1\otimes\xi_2) = (A\xi_1 )\otimes (B\xi_2) \quad(\xi_1\in H_1, \xi_2\in H_2).$$ Is it true that $$\Vert\varphi^{-1}\Vert=\min\{m, n\}?$$ Here, $\hat{\otimes}$ denotes the projective tensor product of Banach spaces.
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Context:
8. A positive integer $x \in$ $\{1,2, \cdots, 2016\}$ is generated with equal probability. Then the probability that the sum of the digits of $x$ in binary is no more than 8 is $\qquad$
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Context:
Consider the analogue of the Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g = T$ in $\mathbb{F}_q[T]$, and let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a non-trivial homomorphism. Define $$L_c(s, \chi) = \sum_f \chi(f) \cdot c^{\text{deg}(f)} \cdot \#(\mathbb{F}_q[T]/(f))^{-s}$$ where $f$ ranges over all monic polynomials in $\mathbb{F}_q[T]$ that are coprime to $g$, and $\text{deg}(f)$ denotes the degree of $f$. Does $L_c(s, \chi)$ necessarily equal 1?
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Context:
Find the minimum value of $$f(x)=\frac{\tan (x-\frac{\pi}{6})}{\tan x}, \qquad x\in \left(\frac{\pi}{3},\pi\right).$$
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Context:
如果$f\left( x \right)=1-{{\log }_{x}}2+{{\log {{x}^{2}}}}9-{{\log }_{{{x3}}}}64$，则使$f\left( x \right)<0$的$x$的取值范围是______.
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Context:
There are $63$ houses at distances of $1, 2, 3, \ldots, 63$ km from the North Pole, respectively. Santa Claus wants to distribute vaccines to each house. To do so, he will have his $63$ assistants, elves named $E_1, E_2, \ldots, E_{63}$, deliver the vaccine to each house; each elf will deliver the vaccine to exactly one house and never return. Suppose that elf $E_n$ takes $n$ minutes to travel $1$ km for each $n = 1, 2, \ldots, 63$, and that all elves leave the North Pole simultaneously. What is the minimum amount of time to complete the delivery?
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Context:
In a ten-mile race First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?

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Context:
## Task 2.

Let $N$ be a natural number. We call a staircase a part of a square plate of dimensions $N \times N$ that consists of the first $K$ fields in the $K$-th row for $K=1,2, \ldots, N$. In how many ways can the staircase be cut into rectangles of different areas that consist of the fields of the given plate?
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Context:
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
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Context:
Evaluate \( \lim_{n \to \infty} \int_0^1 \sqrt{\frac{1}{x} + n^2 x^{2n}} \, dx \).
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Context:
It takes Roque two hours to walk to work and one hour to ride his bike to work. However, on Wednesdays, his walking route winds through a park, which adds an extra 30 minutes to his walking time. On Fridays, Roque usually bikes through a busy road that adds 15 more minutes to his bike ride. Additionally, when it's raining, it takes him 20% longer to walk to and from work. Given that Roque walks to and from work on Mondays, Wednesdays, and Thursdays, rides his bike on Tuesdays and Fridays, and it rained on Monday and Thursday this week, how many hours in total did he take to get to and from work this week with walking and biking?
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Context:
In acute triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The area of triangle $ABC$ is $S=\frac{{\sqrt{3}}}{{12}}({{a^2}+{b^2}-{c^2}})$. If $24\left(bc-a\right)=b\tan B$, then the minimum value of $c$ is ____.
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Context:
Work base 3. (so each digit is 0,1,2)

A good number of size $n$ is a number in which there are no consecutive $1$'s and no consecutive $2$'s. How many good 10-digit numbers are there?
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Context:
$1 \cdot 13$ Let the positive integer $M$ in its $n$-ary representation have all distinct digits, and each digit, except the leftmost one, differs from some digit to its left by $\pm 1$. Find the number of all such positive integers $M$. (The answer should be given as an explicit function of $n$ in its simplest form)

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Context:
In a unit cube \(ABCDA_1B_1C_1D_1\), eight planes \(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\) and \(D_1AB_1\) intersect the cube. What is the volume of the part that contains the center of the cube?
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Context:
A rectangle with a perimeter of 100 cm was divided into 70 identical smaller rectangles by six vertical cuts and nine horizontal cuts. What is the perimeter of each smaller rectangle if the total length of all cuts equals 405 cm?
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Context:
In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. Find the perimeter of the triangle. The original answer is in the form $k + m\sqrt{n}$, please give the value of $k + m + n$.
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Context:
For real numbers  $x$  and  $y$  we define  $M(x, y)$  to be the maximum of the three numbers  $xy$ ,  $(x- 1)(y - 1)$ , and  $x + y - 2xy$ . Determine the smallest possible value of  $M(x, y)$  where  $x$  and  $y$  range over all real numbers satisfying  $0 \le x, y \le 1$ .
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Context:
Determine whether the cardinality of the superstructure \(\mathbb{V}\) over \(\mathbb{R}\) and the set of finite subsets \(\mathcal{P}_{fin}(\mathbb{V})\) is \(\beth_\omega\).
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Context:
For $n \in \Bbb N$, let $g_n(x)=\displaystyle x^{\frac{1}{n}}$. Find the value of the following expression:

$$\lim_{n \to \infty} \sum_{a=2}^\infty \frac{g_n\left(\int_0^1(1-x^a)^n dx\right)}{a!}$$
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Context:
Determine whether the sequence \(1, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \ldots, \frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \ldots, \frac{n+1}{n}, \ldots\) has a set of subsequential limits equal to the interval \([0,1]\).
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Context:
Consider the matrix product \(A^iB^{ij}C^{jk}D^k\), where repeated indices are summed. Given that \(B^{ij}\) is symmetric and \(C^{jk}\) is skew-symmetric, determine whether this product vanishes.
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Context:
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$
Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and
$$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$
Determine the largest possible number of elements in $A$.
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Context:
Find the smallest real number \(a\) such that for any positive integer number \(n > 2\) and any arrangement of the numbers from 1 to \(n\) on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than \(a\).
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Context:
How many positive odd integers greater than 1 and less than $150$ are square-free?
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Context:
在 \(\triangle A B C\) 中, 已知 \(\angle C=90^{\circ}, \angle B=30^{\circ}\), \(A C=2, M\) 是 \(A B\) 的中点，将 \(\triangle A C M\) 沿 \(C M\) 翻折，使 \(A 、 B\) 两点间的距离为 \(2 \sqrt{2}\) 。求三棱锥 \(A-B C M\) 的体积 \(V\)。答案的形式为 \(\frac{k \sqrt{m}}{n}\)，请给出k+m+n的值。
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Context:
Given real numbers \( x_{1}, x_{2}, \cdots, x_{2018} \) such that for any \( 1 \leqslant i < j \leqslant 2018 \), we have \( x_{i} + x_{j} \geqslant (-1)^{i+j} \). Find the minimum value of \( \sum_{i=1}^{2018} i x_{i} \).
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Context:
Let  $ABCD$  be a convex quadrilateral with  $\angle ABD = \angle BCD$ ,  $AD = 1000$ ,  $BD = 2000$ ,  $BC = 2001$ , and  $DC = 1999$ . Point  $E$  is chosen on segment  $DB$  such that  $\angle ABD = \angle ECD$ . Find  $AE$ .
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Context:
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. Calculate the minimum number of moves required.
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Context:
A truncated cone with a height of 8 contains a sphere O1 with a radius of 2. The center of sphere O1 is on the axis of the truncated cone, and sphere O1 is tangent to the upper base and the lateral surface of the truncated cone. Another sphere O2 with a radius of 3 can be placed inside the truncated cone such that sphere O2 is tangent to sphere O1, the lower base, and the lateral surface of the truncated cone. How many more spheres with a radius of 3 can be placed inside the truncated cone besides sphere O2? Express your answer as a whole number.
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Context:
Tim has some 10-dollar bills, 11 5-dollar bills, and 17 one-dollar bills. If Tim needs to pay exactly $128, and he uses at least 16 bills, how many 10-dollar bills does he have?
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Context:
For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \leq p(n)$.
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Context:
Given that positive numbers \( a, b, \) and \( c \) satisfy \( a^{2} + b^{2} + 2c^{2} = 1 \). What is the maximum value of \( \sqrt{2}ab + 2bc + 7ac \)?
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Context:
If  $a, b, c$  are real numbers such that  $a+b+c=6$  and  $ab+bc+ca = 9$ , find the sum of all possible values of the expression  $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$ .
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Context:
How many elements does the set $A=\{x \in \mathbb{N} | \exists n \in \mathbb{N}: x \equiv 7^n + 5^n \ (\text{mod } 35), 35 > x \ge 0\}$ have?
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Context:
Trapezoid \(ABCD\) has one base \(AB\) that is twice as long as the other base \(DC\). \(E\) is the intersection point of the two diagonals. If the length of the diagonal \(AC\) is 11 units, calculate the length of \(EC\). Express your answer as a mixed number or decimal, including units.
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Context:
There is a square of checkered paper measuring $102 \times 102$ squares and a connected figure of unknown shape, consisting of 101 cells. What is the largest number of such figures that can be cut from this square with a guarantee? A figure made up of cells is called connected if any two of its cells can be connected by a chain of its cells in which any two adjacent cells have a common side.
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Context:
Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another.
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Context:
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. Find the value of \[f(2) + f(3) + f(4) + f(5)+ f(6).\]
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Context:
Find the least possible number of factors for integer $B$, given that number $A$ has $24$ factors and the product $A \cdot B$ has $105$ factors.
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Context:
How many 5-digit numbers beginning with $2$ are there that have exactly three identical digits which are not $2$?
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Context:
In a plane, there are n circles, each pair of which intersects at two points, and no three circles have a common point. Given that f(1) = 2, f(2) = 4, and f(3) = 8, where f(n) represents the number of regions into which these n circles divide the plane, find the expression for f(n).
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Context:
Recall that the sum of the angles of a triangle is 180 degrees. In triangle $ABC$, angle $A$ is a right angle. Let $BM$ be the median of the triangle and $D$ be the midpoint of $BM$. It turns out that $\angle ABD = \angle ACD$. What are the measures of these angles?
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Context:
A competition consists of $n$ true/false questions. After analyzing the answers from eight contestants, it was found that for any pair of questions, exactly two contestants answered "T, T"; exactly two contestants answered "F, F"; exactly two contestants answered "T, F"; and exactly two contestants answered "F, T." Find the maximum value of $n$.
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Context:
How many (possibly empty) sets of lattice points $\{P_1, P_2, \ldots, P_M\}$, where each point $P_i = (x_i, y_i)$ for $x_i, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$? An infinite slope, e.g., $P_i$ is vertically above $P_j$, does not count as positive.
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Context:
From vertex \( B \) of parallelogram \( ABCD \), its heights \( BK \) and \( BH \) are drawn. It is known that \( KH = a \) and \( BD = b \). Find the distance from point \( B \) to the intersection of the heights of triangle \( BKH \).
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Context:
# Problem 2. (2 points)

In a positive non-constant geometric progression, the arithmetic mean of the second, seventh, and ninth terms is equal to some term of this progression. What is the minimum possible number of this term?
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Context:
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
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Context:
Find the mass of the cylindrical surface \(x^{2} + y^{2} = R^{2}\), enclosed between the planes \(z = 0\) and \(z = H\), if at every point the surface density is inversely proportional to the square of its distance to the origin of coordinates.
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\boxed{2k \pi \arctan \left( \frac{H}{R} \right)}


Context:
An inspector rejects 0.07% of the meters as defective. He examined a certain number of meters and found 2 defective meters. How many meters did he examine?
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Context:
Determine the values of $c$ for which the intersection of the cone $K$ and the plane $E_c$ in $\mathbb{R}^3$ is a circle or an ellipse.
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Context:
Determine the smallest integer \( N \) such that the inequality \( \sum_{k=0}^{17} \binom{N}{k} \times 0.1^k \times 0.9^{N-k} < 0.004 \) holds.
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Context:
设正整数 $a, b, c, d$ 满足 $a>b>c>d$, 且$$a b+b c+c a+d^{2} \mid(a+b)(b+c)(c+a)$$求 $a b+b c+c a+d^{2}$ 的素因子个数的最小值. (素因子的个数计重数, 如 12 的素因子个数为 3 )。
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Context:
Determine the limit of the sequence defined by successive compositions of the natural logarithm function, starting with a positive number greater than 1. Does this sequence converge to zero as the number of compositions approaches infinity?
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Context:
Bill buys a stock that decreases by $20\%$ on the first day, and then  on the second day the stock increases by $30\%$ of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days?
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Context:
Determine whether the following sum diverges to infinity:
\[
\sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\log x )^{100} < m \leq x} \left(\frac{m}{n}\right) \frac{\mu^2(2  m )}{m},
\]
where \(\mu\) is the M"obius function and \(\left(\frac{m}{n}\right)\) is the Jacobi quadratic symbol for positive squarefree odd integers \(n,m\).
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Context:
Calculate the Littlewood-Richardson coefficient \(C^{(2,1)}_{(1), (1,1)}\).
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Context:
Determine whether the following determinant is non-zero:
\[
D = \begin{vmatrix}
1 & 2 & 3 & \cdots & 2008 & 2009 & 2010 & 2011 \\
2^2 & 3^2 & 4^2 & \cdots & 2009^2 & 2010^2 & 2011^2 & 2012^2 \\
3^3 & 4^3 & 5^3 & \cdots & 2010^3 & 2011^3 & 2012^3 & 2012^3 \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
k^k & (k+1)^k & \cdots & 2011^k & 2012^k & \cdots & 2012^k & 2012^k \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
2010^{2010} & 2011^{2010} & 2012^{2010} & \cdots & 2012^{2010} & 2012^{2010} & 2012^{2010} & 2012^{2010} \\
2011^{2011} & 2012^{2011} & 2012^{2011} & \cdots & 2012^{2011} & 2012^{2011} & 2012^{2011} & 2012^{2011}
\end{vmatrix}
\]
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Context:
Find all integers $m$ such that $\frac{2m^2+7m-9}{m^2+m+1}$ is integer
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Context:
Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. Calculate the number of possible ordered pairs $(f, m)$. Express your answer as a single integer.
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Context:
Find a real sequence \((u_j)_{j \in \mathbb{N}}\) such that the series \(\sum_{j=1}^\infty u_j\) converges and \(\sum_{j=0}^\infty (2^j+1)\exp(-u_j2^j) < \infty.\)
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Context:
Determine the volume of a sphere inscribed in a regular pyramid, where the height is \( h \) and the dihedral angle at the base is \( 60^{\circ} \).
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Context:
Find the value of the integral $I_n=\int_{0}^{1}(1-x^2)^ndx$ for any $n \in \mathbb N$.
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Context:
Find the infimum of the set \( \{ \lambda \in \mathbb{R} : x^2y'' - 3xy' - \lambda y = 0, \lim_{x \to \infty} y(x) = 0 \} \). Assume \( x > 0 \) and use the substitution \( x = e^t \) to solve the differential equation.
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Context:
Determine whether \( \lim \sup_{n\to\infty}\frac{m_n}{n} = 1 \), where \( m_n \) is the largest integer such that there exists a subset \( T \subseteq \{0,1\}^n \) with \(|T|=n\) and for distinct \( x, y \in T \), the Hamming distance \( d^H_n(x,y) \geq m_n \).
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Context:
Calculate the correlation coefficient \( \text{Corr}(N, M) \) where \( N \) and \( M \) are Poisson random variables with parameter 1, defined on the intervals \((0,2)\) and \((1,3)\) respectively.
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Context:
From some time in the morning to 4 pm, it rained for 4 hours. It did not rain for 5 hours. At what time did the period start?
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Context:
Michelle is drawing segments in the plane. She begins from the origin facing up the $y$-axis and draws a segment of length $1$. Now, she rotates her direction by $120^\circ$, with equal probability clockwise or counterclockwise, and draws another segment of length $1$ beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?
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Context:
In town X, some percent of the population are employed, and 80 percent of the population are employed males. If 33.33333333333333 percent of the employed people in town X are females, what percent of the population are employed?
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Context:
整数［$\frac { 1 0 ^ { 8 3 } } { 1 0 ^ { 3 1 } + 3 }$］的末两位数字是__ .（先写十位数字，后写个位数字；共中[x]表示不超过x的最大整数）.
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Context:
An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$?
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Context:
Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\pi$.
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Context:
Determine the number of solutions to the equation \( 3^a - 7^b - 1 = 0 \) where \( a \) and \( b \) are real numbers.
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Context:
Given vectors $\overrightarrow{a}=(\cos{35}^\circ,\sin{35}^\circ)$ and $\overrightarrow{b}=(\cos{65}^\circ,\sin{65}^\circ)$, find the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$.
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Context:
An engineer is designing an engine. Each cycle, it ignites a negligible amount of fuel, releasing $ 2000 \, \text{J} $ of energy into the cubic decimeter of air, which we assume here is gaseous nitrogen at $ 20^\circ \, \text{C} $ at $ 1 \, \text{atm} $ in the engine in a process which we can regard as instantaneous and isochoric. It then expands adiabatically until its pressure once again reaches $ 1 \, \text{atm} $, and shrinks isobarically until it reaches its initial state. What is the efficiency of this engine?

[i]Problem proposed by B. Dejean[/i]
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Context:
On the eve of the Spring Festival, a convenience store sold a batch of goods that were purchased at 12 yuan per piece at a price of 20 yuan per piece, selling 240 pieces per day. After a period of sales, it was found that if the price per piece was increased by 0.5 yuan, then 10 fewer pieces would be sold per day; if the price per piece was decreased by 0.5 yuan, then 20 more pieces would be sold per day. In order to achieve a daily sales profit of 1980 yuan for the goods, how much should each piece be priced?
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Context:
Let $m \in \mathbb{R}$, the line $x + my = 0$ intersects with the line $mx - y - 2m + 4 = 0$ at point $P(x, y)$. Find the maximum distance from point $P$ to the line $l: (x-1)\cos\theta + (y-2)\sin\theta = 3$.
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Context:
Evaluate the convergence and find the sum of the series:
\[ \sum_{k=0}^{\infty} \frac{\left(\frac{1}{2}\right)^{2 k+4} 4\left(\begin{array}{c}
2 k+3 \\
k
\end{array}\right)}{k+4} \]
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Context:
Determine the highest order of differentiability of the function $f(x)$ if the function $g(x) = \int_0^x f(y) (y - x)^4 \, dy$ is continuously differentiable six times.
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Context:
What is a necessary and sufficient condition for an $n \times n$ matrix $A$ over $\mathbb{C}$ such that $A^m \to 0$ as $m \to \infty$?
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Context:
Find the smallest natural number \( N \) such that \( \frac{(N!)^2}{(2N - 1)!} \leq \frac{1}{25000} \).
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Context:
Find the other $x$-coordinate of the intersection between the curves $y = \log x$ and $y = \sqrt{1-x^2}$. What is this $x$-coordinate?
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Context:
Consider the two points \( A(4,1) \) and \( B(2,5) \). For each point \( C \) with positive integer coordinates, we define \( d_C \) to be the shortest distance needed to travel from \( A \) to \( C \) to \( B \) moving only horizontally and/or vertically. For example, for the point \( C(3,4) \), we compute \( d_C \) as follows:

- To get from \( A \) to \( C \) moving only horizontally and/or vertically, we can move 1 unit to the left then 3 units up for a total distance of \( 1+3=4 \).
- The shortest path from \( C \) to \( B \) moving only horizontally and/or vertically is to move 1 unit left and then 1 unit up (or 1 unit up then 1 unit left) for a total distance of \( 1+1=2 \).

Thus, for \( C(3,4) \), we have that \( d_C = 4 + 2 = 6 \).

The positive integer \( N \) has the property that there are exactly 2023 points \( C(x, y) \) with \( x > 0 \) and \( y > 0 \) and \( d_C = N \). What is the value of \( N \)?
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Context:
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are known. A circle with center \(O\) is constructed using side \(AC\) as the diameter, intersecting side \(BC\) at point \(K\). It turns out that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\).
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Context:
Prankster Pete and Good Neighbor George visit a street of $2021$ houses (each with individual mailboxes) on alternate nights, such that Prankster Pete visits on night $1$ and Good Neighbor George visits on night $2$, and so on. On each night $n$ that Prankster Pete visits, he drops a packet of glitter in the mailbox of every $n^{th}$ house. On each night $m$ that Good Neighbor George visits, he checks the mailbox of every $m^{th}$ house, and if there is a packet of glitter there, he takes it home and uses it to complete his art project. After the $2021^{th}$ night, Prankster Pete becomes enraged that none of the houses have yet checked their mail. He then picks three mailboxes at random and takes out a single packet of glitter to dump on George’s head, but notices that all of the mailboxes he visited had an odd number of glitter packets before he took one. In how many ways could he have picked these three glitter packets? Assume that each of these three was from a different house, and that he can only visit houses in increasing numerical order.

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Context:
Given that there are 5 balls in a pocket, among which there are 2 black balls and 3 white balls, calculate the probability that two randomly drawn balls of the same color are both white.
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Context:
Consider the line $a+bt$ where $a,b>0$. Let $B(t)$ be Brownian motion and let $\tau=\inf\{t>0:B(t)=a+bt\}$ be the first hitting time of that line, with the understanding that $\tau=\infty$ if the line is never hit. Compute the probability that Brownian motion hits that line, i.e., $P(\tau<\infty)$. Express your answer in terms of $a$ and $b$.
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Context:
When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school?
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Context:
In pentagon $ABCDE$, $BC=CD=DE=2$ units, $\angle E$ is a right angle and $m \angle B = m \angle C = m \angle D = 135^\circ$.  The length of segment $AE$ can be expressed in simplest radical form as $a+2\sqrt{b}$ units.  What is the value of $a+b$?
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Context:
Let \( f(m, n) = 3m + n + (m + n)^2 \). Find \( \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} 2^{-f(m, n)} \).
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Context:
Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$.
For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
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Context:
In a trapezoid \(ABCD\), the diagonals \(AC\) and \(BD\) are mutually perpendicular, and \(\angle BAC = \angle CDB\). The extensions of the lateral sides \(AB\) and \(DC\) intersect at point \(K\), forming the angle \( \angle AKD \), which is equal to \(30^\circ\). Find the area of triangle \(AKD\) if the area of the trapezoid is \(P\).
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^{2\pi}\frac{n (t+1) \sin (t)}{t (n t+1)^2}\,dt \]
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Context:
Let's call a natural number "suitable" if it is the smallest among all natural numbers with the same sum of digits as it has. Find all suitable numbers that are exact squares of natural numbers.
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Context:
The solution to the equation $$\sqrt {x^{2}+6x+10}+ \sqrt {x^{2}-6x+10}=8$$ is \_\_\_\_\_\_.
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Context:
4 men can check exam papers in 8 days working 5 hours regularly . what is the total hours when 2 men will check the double of the papers in 20 days ?
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Context:
Find the number of values of $x$ such that the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.
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Context:
Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realized that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?
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Context:
A positive integer is said to be 'good' if each digit is 1 or 2 and there are neither four consecutive 1's nor three consecutive 2's. Let \( a_{n} \) denote the number of \( n \)-digit positive integers that are 'good'. Calculate the value of \(\frac{a_{10} - a_{8} - a_{5}}{a_{7} + a_{6}}\).
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Context:
As shown in the figure, point $E$ is on the side $AB$ of rectangle $ABCD$ and point $F$ is on the side $BC$. Connect $CE$ and $DF$, which intersect at point $G$. If the area of $\triangle CGF$ is 2, the area of $\triangle EGF$ is 3, and the area of the rectangle is 30, find the area of $\triangle BEF$.
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Context:
A line is drawn through the midpoints of any two edges of the triangular prism $ABC-A_1B_1C_1$. How many such lines are parallel to the plane $ABBA_1$?
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Context:
Compute the length of the segment tangent from the origin to the circle that passes through the points $(4,5)$, $(8,10)$, and $(10,25)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若 $0<x<2$, 则函数 $y=x(4-2x)$ 的最大值为 $\sqrt{2}$.
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Context:
Evaluate the integral using the Cauchy integral formula, given that the integrand has a removable singularity within the contour.
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Context:
For $n \geq 3$, consider the set $V = \mathbb{C} \setminus \{1, 2, ..., n\}$. Determine if the automorphism group $Aut(V)$ is finite, and if so, find an upper bound for its order. Specifically, verify whether $|Aut(V)| \leq n!$.
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Context:
Determine the annihilator of the tensor $e_1 \wedge e_2 + e_3 \wedge e_4$ in the vector space $V = \langle e_1, e_2, e_3, e_4 \rangle$.
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Context:
Given \(x, y, z \in (0, 2]\), find the maximum value of the expression

\[
A = \frac{\left(x^{3} - 6\right) \sqrt[3]{x + 6} + \left(y^{3} - 6\right) \sqrt[3]{y + 6} + \left(z^{3} - 6\right) \sqrt[3]{z + 6}}{x^{2} + y^{2} + z^{2}}
\]
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Context:
Determine the radius of convergence of the function \( \frac{\sinh(z)}{z} \) about the point \( z = \pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values of $b$ for which the random variables $X$ and $Y$ are dependent, given their joint density function $f_{X,Y}(x,y)$. Provide your answer as an interval or union of intervals.
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Context:
Calculate the surface area of the portion of the plane \(2x + y + 2z = 16\) that lies within the region bounded by \(x = 0\), \(y = 0\), and \(x^2 + y^2 = 64\).
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Context:
Find the non-zero function \( q(x) \) that is orthogonal to all polynomials of degree less than or equal to \( n \) with respect to the weight function \( w(x) = \sqrt{1-x^2} \) over the interval \([-1, 1]\).
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Context:
Given $2$ red and $2$ white balls, a total of $4$ balls are randomly arranged in a row. The probability that balls of the same color are adjacent to each other is $\_\_\_\_\_\_$.
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Context:
Determine the amortized time complexity for a series of n operations on a disjoint set data structure, given that all union operations are performed before any find operations.
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Context:
Determine the Galois group of the polynomial \( f(x) = x^6 - 3x^3 + 2 \) over the field of rational numbers \( \mathbb{Q} \).
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Context:
Coach $F$ of country $F$ plans to have players $A_{1}, A_{2}, \cdots, A_{7}$ participate in three practice matches, each lasting 90 minutes. At any given moment, only one player is on the field. The total playing time for $A_{1}, A_{2}, A_{3}, A_{4}$ must be divisible by 7, and the total playing time for $A_{5}, A_{6}, A_{7}$ must be divisible by 13. How many different possible allocations of total playing time are there, considering these conditions?
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Context:
In the right triangle \(ABC\), a circle is constructed with \(AC\) as its diameter. The circle intersects side \(AB\) at point \(E\). Point \(G\) is on side \(BC\) such that segment \(AG\) intersects the circle at point \(F\). Given that segments \(EF\) and \(AC\) are parallel, \(BG = 2CG\), and \(AC = 2\sqrt{3}\), find the length of \(GF\).
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Context:
How many integers $n$ have the property that there exist positive integers $a, b, c$ such that $a^n + b^n = c^n$? Provide your answer as a single integer.
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Context:
Determine the condition that must hold between two real numbers $a$ and $b$ such that the equation \( \left(\frac{1+iz}{1-iz}\right)^n = a+ib \) has only real solutions for $z$. What is this condition?
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Context:
Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\sin nx}{(1+2^x)\sin x} \, dx \) for \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{\pi}{2} \left(1 - (-1)^n\right)}


Context:
In a base \( a \) numeral system, determine the number of digits such that any integer ending in this digit will have all of its positive integer powers also end in the same digit.
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Context:
Find the constant value of $g$ when $\varepsilon$ is small, given the equation:
\[3\sinc\left(-1+ \frac\varepsilon T \right)-3\sinc\left(1+\frac\varepsilon T\right)-\sinc\left(-3+\frac\varepsilon T\right)+\sinc\left(3+\frac\varepsilon T\right) = \frac\varepsilon g\]
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Context:
Teams A and B each have 7 players who compete according to a predetermined sequence in a Go relay match. Both teams start with their first player competing against each other, the loser gets eliminated, and the winner competes against the second player of the losing team, and so on, until one team's players are all eliminated. The remaining team wins. How many possible match sequences are there?
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Context:
设集合 $S=\{1,2,3, \cdots, 10\}, S$ 的子集 $A$ 满足
$$
A \cap\{1,2,3\} \neq \varnothing, \quad A \bigcup\{4,5,6\} \neq S,
$$
这样的子集 $A$ 的个数为 $\qquad$.
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Context:
Find the probability density function (pdf) of the time of the first arrival, \( T_1 \), given that there are exactly two arrivals by time 1 in a Poisson process \((N_t, t \geq 0)\).
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Context:
4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $60^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
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Context:
Determine the number of points at which a line in $\mathbb{CP}^n$, defined as a copy of $\mathbb{CP}^1$, intersects with $\mathbb{CP}^2$. Provide your answer as a single integer.
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Context:
Determine whether the function \( f(x,y) = (1+x)^t(1+y)^{1-t} - x^ty^{1-t} \), where \( t \in [0,1] \), satisfies \( f(x,y) \geq 1 \) for all \( x, y \geq 0 \).
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Context:
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\dfrac{x}{y}$?
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Context:
A block is released from point O on an incline where the x-axis is along the incline. The coefficient of friction varies as \( \mu = kx \) with \( k = \tan\theta \). Given that the acceleration of the block is \( g\sin\theta(1-x) \), find the maximum distance traveled by the block.
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Context:
Six-digit integers will be written using each of the digits $1$ through $6$ exactly once per six-digit integer. How many different positive integers can be written such that all pairs of consecutive digits of each integer are relatively prime? (Note: $1$ is relatively prime to all integers.)
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Context:
Let \( f \) be a differentiable function satisfying \( f(x+y)=(f(x))^{\cos y}\cdot (f(y))^{\cos x} \) for all \( x,y \in \mathbb{R} \). Given that \( f(0)=1 \) and \( f'(0)=\ln 2 \), evaluate the integral \( \int_{\frac{\pi}{2}}^{\frac{17\pi}{2}} \frac{f(x)}{f(x)+f(\pi /2-x)} \, dx \) and find the value of \( k \) such that the integral equals \( k\pi \).
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Context:
Evaluate the integral \( \displaystyle \int_{\partial \mathcal{D}(0,2)} \frac{8z^2-8z+1}{4z^3-8z^2+z-2}dz \) using the residue theorem.
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Context:
A function f(x) defined on R satisfies f(0)=0, f(x)+f(1-x)=1, f(x/5)=1/2f(x), and f(x1) <= f(x2) when 0 <= x1 < x2 <= 1. Find the value of f(1/2007). Express your answer as a fraction in simplest form.
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Context:
Determine whether the iterative scheme \(X_{n+1}=\frac{1}{2}(X_n)(1+\frac{a}{(X_n)^2})\) converges with order 2, given that \(\theta\) is the solution where \(\theta^2 = a\).
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Context:
Consider a sequence of complex numbers lying in a sector with an angular width less than \( \pi \). If the sum of the real parts of the sequence converges for any two non-collinear directions in the sector, does the sum of the moduli of the sequence also converge? Answer 'yes' or 'no'.
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Context:
Find the value of \( k \in \mathbb{R} \) such that \[ \lim_{x \to 0} \frac{1 - \cos(kx)}{x^2} = \int_0^\infty xe^{-x} \, \mathrm{d}x. \]
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Context:
Determine whether the sequence \( \sqrt[n]{n!} \) converges or diverges as \( n \to \infty \).
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Context:
Evaluate the integral \( \int_{-\infty}^0 \frac{dx}{(x-2)\sqrt{1-x}}. \)
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Context:
Suppose $u \in H^1(\mathbb{R}^2) \cap C(\mathbb{R}^2)$. Determine whether $\limsup_{|x| \to \infty} u(x) = 0$.
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Context:
Evaluate the integral: $$\int_{0}^{1}\frac{\sin^{-1}\sqrt x}{x^2-x+1}dx$$
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Context:
How many positive integer solutions does the following system of equations have?

$$
\left\{ \begin{array}{l}
\sqrt{2020}(\sqrt{a} + \sqrt{b}) = \sqrt{(c + 2020)(d + 2020)} \\
\sqrt{2020}(\sqrt{b} + \sqrt{c}) = \sqrt{(d + 2020)(a + 2020)} \\
\sqrt{2020}(\sqrt{c} + \sqrt{d}) = \sqrt{(a + 2020)(b + 2020)} \\
\sqrt{2020}(\sqrt{d} + \sqrt{a}) = \sqrt{(b + 2020)(c + 2020)}
\end{array} \right.
$$

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Context:
The value of $\log_5{3125}$ is between which consecutive integers $c$ and $d$? Find $c+d$.
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Context:
A plane intersects a rhombus through one of its sides, forming angles of \( \alpha \) and \( 2\alpha \) with the diagonals of the rhombus. Determine the acute angle of the rhombus.
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Context:
A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates 2, 0, and 18 in some order. Find the sum of all possible values of $A$.
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Context:
Ned is an explosives specialist racing to deactivate a time bomb on the top floor of a building. The building has a certain number of flights of stairs to the top floor, and Ned can run up one flight of stairs in eleven seconds. The time bomb has 72 seconds left on the timer, and Ned has spent 165 seconds running up the stairs. Ned will have 17 seconds to diffuse the bomb when he reaches it. How many flights of stairs are there in the building?
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Context:
Let  0<a<b.  Evaluate $$\lim_{t \to 0} \left \{ {\int_{0}^{1} [bx+a(1-x)]^{t}dx}\right \} ^{1/t}$$[The  final  answer should  not  involve  any  operations other  than  addition,  subtraction,  multiplication,  division,  and  exponentiation.] When \(a = 1\) and \(b = 2\), the limit can be written as \(\frac{a}{be}\), where both \(a\) and \(b\) are positive integers. Find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a polynomial of degree \( k \) in one variable. If the polynomial vanishes at \( x = a \), and \( k-2 \) derivatives vanish at \( x = a \) and 2 derivatives vanish at \( x = b \), does the polynomial vanish identically? Answer with "yes" or "no".
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Context:
In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. Given that $PQ = 7$, $QR = 6$, and $RS = 8$, find the area of triangle $ABC$. \( \text{Express your answer in square units.} \)
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Context:
Compute the number of subsets  $S$  of  $\{0,1,\dots,14\}$  with the property that for each  $n=0,1,\dots,
6$ , either  $n$  is in  $S$  or both of  $2n+1$  and  $2n+2$  are in  $S$ .

*Proposed by Evan Chen*
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Context:
已知直角梯形的一条腰与一条对角线相等，且互相垂直，则其上底与下底之比为多少？
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Context:
In a cyclic quadrilateral \(ABCD\), \(\angle ADB = 48^{\circ}\) and \(\angle BDC = 56^{\circ}\). Inside triangle \(ABC\), a point \(X\) is marked such that \(\angle BCX = 24^{\circ}\), and the ray \(AX\) is the angle bisector of \(\angle BAC\). Determine the measure of angle \(CBX\).
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Context:
Determine the value of $C^{2}_{100}+C^{97}_{100}$. Express your answer as a combination in the form $C^{r}_{n}$.
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Context:
On an island, there are 2018 residents consisting of knights, liars, and followers. Knights always tell the truth, liars always lie, and followers mimic the majority of the previous responses. Each resident was asked, "Are there more knights than liars on the island?" and responded with "Yes" or "No". There were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents?
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Context:
Determine whether the integral \( \int_{0}^{+\infty} \frac{x \log(x)}{(x^{2}+1)^{2}} \, dx \) converges. If it does, find its value.
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Context:
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin \(i\) is \(2^{-i}\) for \(i = 1, 2, 3, \ldots\). More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins 3, 17, and 10.) What is \(p + q\)?

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Context:
All vertices of a regular tetrahedron \( A B C D \) are located on one side of the plane \( \alpha \). It turns out that the projections of the vertices of the tetrahedron onto the plane \( \alpha \) are the vertices of a certain square. Find the value of \(A B^{2}\), given that the distances from points \( A \) and \( B \) to the plane \( \alpha \) are 17 and 21, respectively.
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Context:
5 设$x _ { 1 } ,$ $x _ { 2 } , \cdots , x _ { 1 9 9 0 }$是1,2，···，1990的一个排列，求$F = \vert \cdots \vert \vert x _ { 1 } -$ $x _ { 2 } \vert - x _ { 3 } \vert - \cdots \vert - x _ { 1 9 9 0 } \vert$的最大值．（第24届全俄数学奥林匹克试题）
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Context:
Solve the system of equations:
 \( \left\{  \begin{array}{l} x + \left\lbrack  y\right\rbrack   + \{ z\}  = {3.9} \\  y + \left\lbrack  z\right\rbrack   + \{ x\}  = {3.5} \\  z + \left\lbrack  x\right\rbrack   + \{ y\}  = 2 \end{array}\right. \)
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Context:
Determine whether the inequality between the harmonic mean and the geometric mean, \( H(f) \leq G(f) \), holds for any positive integrable function \( f \) in the continuous case.
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Context:
Evaluate the integral \( \oint \frac{\mathrm{d}z}{q(z)} \) where \( q(z) \) is a polynomial of degree \( m \geq 2 \) with no repeated roots, and the closed path is large enough to enclose all the roots of \( q(z) \).
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Context:
For how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13$?
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Context:
Evaluate the following limit:

$$\lim_{\epsilon \to 0^+} \left(\int_{2}^{2+\epsilon}＋\int_{3-\epsilon}^{3}\right)\frac{\ln (x-1)}{(x-2)^2} dx$$
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Context:
Given a 6x6 matrix \( A \) with rank 5, determine the rank of the matrix \( B = A^2 + A + I \).
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Context:
A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. The original answer is in the form \frac{k}{m}, where \frac{k}{m} is a fully reduced fraction. Please find the value of k + m.
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Context:
Compute the dimension of the Zariski cotangent space of the variety $V(Y-X^2)$ at the origin in $k^2$, where $k$ is an algebraically closed field. Use the definition of the Zariski cotangent space as the quotient $\mathfrak{m}_{(0,0),V}/\mathfrak{m}_{(0,0),V}^2$, where $\mathfrak{m}_{(0,0),V}$ is the maximal ideal of the local ring $\mathcal{O}_{(0,0),V}$ at the origin.
Please reason step by step, and put your final answer within \boxed{}.