Context:
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that
$\frac{x^2}{a^2} + \frac{y^2}{a^2-16} = \frac{(x-20)^2}{b^2-1} + \frac{(y-11)^2}{b^2} = 1.$
 Find the least possible value of $a+b.$
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Context:
A stock investment increased by $15\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year?
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Context:
Arrange three identical black balls and three identical white balls from left to right into a row. A "valid arrangement" is defined as an arrangement where, starting from any position (including this position) to the right, up to the last ball, the number of black balls is greater than or equal to the number of white balls. Calculate the probability of having a "valid arrangement". Express your answer as a simplified fraction.
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Context:
Determine the order of convergence of the modified Newton-Raphson method given by the iteration formula: \[ x_{n+1} = x_n - m \frac{f(x_n)}{f'(x_n)} \] where \( m \) is the multiplicity of the root.
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Context:
Determine the dimension of the vector space \( \mathcal{L}_F(K) \) over \( K \), where \( K/F \) is a finite-dimensional field extension with \([K:F]=n\).
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Context:
Determine the limit of the function \( \cos\left(e^{x^{1/3}+\sin x}\right) \) as \( x \to \pm \infty \).
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Context:
Determine the set of discontinuities \( D_f \) for the function \( f: \mathbb{R} \to \mathbb{R} \) defined by:
\[
f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \frac{1}{n} & x=r_n \in \mathbb{Q} \end{cases}
\] where \( \mathbb{Q} = \{r_1, r_2, \ldots\} \) is the set of rational numbers.
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Context:
$\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. Find the angle $\angle EDF$
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Context:
Initially, a natural number  $n$  is written on the blackboard. Then, at each minute, *Neymar* chooses a divisor  $d>1$  of  $n$ , erases  $n$ , and writes  $n+d$ . If the initial number on the board is  $2022$ , what is the largest composite number that *Neymar* will never be able to write on the blackboard?
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Context:
Company P had 15 percent more employees in December than it had in January. If Company P had 426.09 employees in January, how many employees did it have in December?
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Context:
Determine the optimal reserve price \( r \) that maximizes the seller's expected revenue in a second-price sealed-bid auction with two buyers, where each buyer's valuation is independently drawn from \( \{0, 1, 2\} \) with equal probability.
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Context:
给定点 $A(-2,2)$, 已知 $B$ 是椭圆 $\frac{x^2}{25}+\frac{y^2}{16}=1$ 的动点, $F$ 是左焦点, 当 $|A B|+\frac{5}{3}|B F|$ 取最小值时, 则点 $B$ 的坐标 $\qquad$ .
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Context:
Given the matrix \(K_{3\times3} = \begin{bmatrix} 1&1&1 \\ 3&2&1 \\ 1&2&1 \end{bmatrix}\) with \(|K| = 2\), find the determinant of the matrix expression \(|2K^3 - 2K^4|\).
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Context:
Let $Q$ be a point on or inside the boundary of a square $ABCD$. Determine the minimum value of the sum $\angle QAB + \angle QBC + \angle QCD + \angle QDA$. Provide your answer in radians.
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Context:
In a circle, $AB$ is a diameter, and $CD$ is a chord parallel to $AB$. Point $E$ is the intersection of lines $AC$ and $BD$, and $\angle AED = \beta$, but now $\beta$ is such that $\angle AED + \angle BED = 180^\circ - 2\beta$. The ratio of the area of triangle $\triangle CDE$ to that of triangle $\triangle ABE$ is required.
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Context:
Mary chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
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Context:
A restricted path of length $n$ is a path of length $n$ such that for all $i$ between 1 and $n-2$ inclusive, if the $i$th step is upward, the $i+1$st step must be rightward. Find the number of restricted paths that start at $(0,0)$ and end at $(7,3)$.
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Context:
A triangular piece of paper with an area of 1 is folded along a line parallel to one of its sides and pressed flat. What is the minimum possible area of the resulting figure?
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Context:
Find the simple interest on a certain amount at 25/4% per annum for the period from 4th Feb 2005 to 18th April 2005, if the interest earned is Rs. 25. What is the principal amount?
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Context:
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:

Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.

Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.
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Context:
Find the limit of the following expression as x approaches 0: $$\lim_{x \rightarrow 0} \frac{e^{2x}-e^{-2x}-2x}{x^2}$$
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Context:
Given that the domain of the function $f(x)$ is $\mathbb{R}$ and $$\frac{f'(x)}{2} - f(x) > 2,$$ if $f(0) = -1$, then the solution set of the inequality $$\frac{f(x) + 2}{e^{2x}} > 1$$ is __________.
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Context:
$A=\{1,2,3, \cdots, 15\}, B=\{1,2,3,4,5\}, f$ 是 $A$ 到 $B$ 的映射，若满足 $f(x)=f(y)$ ，则称有序数对 $(x, y)$ 为"好对"，那么"好对"的个数的最小值是多少？
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Context:
In a group of order 21, there is a conjugacy class of order 3. Determine the order of an element in this conjugacy class.
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Context:
Calculate the volume of the tetrahedron with vertices at $A_{1}(14, 4, 5)$, $A_{2}(-5, -3, 2)$, $A_{3}(-2, -6, -3)$, and $A_{4}(-2, 2, -1)$. The height is dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$. Provide your answer as a single numerical value.
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Context:
Edward starts in his house,which is at(0,0)and needs to go point(6,4),which is coordinate for his school. However,there is a
 park that shaped as a square and has coordinates \( \left( {2,1}\right) ,\left( {2,3}\right) ,\left( {4,1}\right) \) ,and(4,3). There is no road for him to walk inside the park but
 there is a road for him to walk around the perimeter of the square. How many different shortest road routes are there from
 Edward's house to his school?
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Context:
Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX \cdot TY$.
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Context:
A number when divided by 1423 leaves 215 as the remainder. When the same number is divided by 109, and then subtracted by the square of their quotient, what will the remainder be?
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Context:
Determine the total number of spanning trees in the undirected labelled graph $Q_3$, which is a 3-dimensional hypercube.
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Context:
Find the value of $k$ for which the integral \( \int_{0}^{\infty} \frac{x^k}{2+4x+3x^2+5x^3+3x^4+4x^5+2x^6} \, dx \) is minimized.
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Context:
The series \(\sum a_n x^n\) and \(\sum b_n x^n\) have radii of convergence of 1 and 2, respectively. Determine the radius of convergence \(R\) of the series \(\sum a_n b_n x^n\).
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Context:
Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.
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Context:
Calculate the definite integral:

$$
\int_{\arcsin (1 / \sqrt{37})}^{\pi / 4} \frac{6 \tan x \, dx}{3 \sin 2x + 5 \cos^2 x}
$$
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Context:
Solve the system of equations:

\[
x^3 - y = 6,
\]

\[
y^3 - z = 6,
\]

\[
z^3 - x = 6.
\]
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Context:
Calculate the expected number of coin tosses needed to achieve either $n$ consecutive heads or $n$ consecutive tails, excluding the first toss.
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Context:
Find the derivative of the matrix product $\mathbf{A}\mathbf{B}\mathbf{c}$ with respect to the matrix $\mathbf{B}$, where $\mathbf{A}$ is an $n\times m$ matrix, $\mathbf{B}$ is an $m\times k$ matrix, and $\mathbf{c}$ is a $k\times 1$ vector.
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Context:
In the convex quadrilateral \(ABCD\), point \(K\) is the midpoint of side \(BC\), and \(S_{ABCD} = 2S_{AKD}\).

Find the length of the median \(KE\) of triangle \(AKD\), if \(AB = a\) and \(CD = b\).
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Context:
In the context of the theorem on Formal Functions (Hartshorne, III.11.1), consider a projective morphism $f: X \to Y$ between noetherian schemes, with a coherent sheaf $\mathcal{F}$ on $X$ and a point $y \in Y$. The natural isomorphism $R^i f_\ast (\mathcal{F})_y^\hat{} \to \varprojlim H^i(X_n, \mathcal{F}_n)$ holds for all $i \geq 0$. The sequence $0 \rightarrow \mathcal{I}^n \mathcal{F} \rightarrow \mathcal{F} \rightarrow \mathcal{F}_n \rightarrow 0$ defines $\mathcal{F}_n$. Determine the starting point of the inverse limit: does it start at $n = 1$ or $n = 0$?
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Context:
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?
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Context:
Let \( n \) be a positive integer. Chef Kao has \( n \) different flavors of ice cream. He wants to serve one small cup and one large cup for
 each flavor. He arranges the \( {2n} \) ice cream cups into two rows of \( n \) cups on a tray. He wants the tray to be colorful,so he
 arranges the ice cream cups with the following conditions:
 - each row contains all ice cream flavors, and
 - each column has different sizes of ice cream cup.

 Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.
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Context:
Find the interval containing the solution to the equation $2^{x-1}+x=5$. Express your answer as an interval in the form (a, b).
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Context:
Let \( g : \mathbb{R}^{3} \to \mathbb{R}^{3} \) be a function defined by \( g(x,y,z) = (3y+4z, 2x-3z, x+3y) \). Define the set \( S = \{(x,y,z) \in \mathbb{R}^{3} : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \} \).

If \[ \int\int\int_{g(S)} (2x+y-2z) \, dx \, dy \, dz = \alpha \int\int\int_{S} z \, dx \, dy \, dz, \] find the value of \( \alpha \).
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Context:
Determine the set of real values of \(\alpha\) for which the improper integral \(\int_0^\infty x^\alpha \cos x\, dx\) converges.
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Context:
Find the third derivative of the given function.

\[
y=\frac{\ln (x-1)}{\sqrt{x-1}}, \quad y^{\prime \prime \prime}=?
\]
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Context:
Determine the line integral of the vector field $\vec{F}(x,y,z) = (3y, 5x, \frac{2x+3}{x^2 + y^2})$ over the intersection curve of the cone $x^2 + y^2 - z^2 = 0$ and the cylinder $(x-1)^2 + y^2 = 4$, using a parameterization without applying Stoke's theorem.
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Context:
Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups.
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Context:
The increasing interval of the function $f(x) = (x - 1)^2 - 2$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a polynomial equation \( z^{4} + a z^{3} + b z^{2} + c z + d = 0 \) with real coefficients, where all roots lie on the unit circle in the complex plane, find the sum of the reciprocals of these roots. Express your answer as a single value.
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Context:
The sector of a circle has a certain perimeter and a central angle of 180 degrees. Its radius is 28.000000000000004 cm. What is the perimeter of the sector?
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Context:
Find the limit:

$$\lim_{x \to -\infty} \frac{\sqrt[n]{x+1} + \sqrt[n]{x+2} - 2\sqrt[n]{x-3}}{\sqrt[n]{x-1} + \sqrt[n]{x-2} - 2\sqrt[n]{x+3}} \quad \text{for} \quad n \in \mathbb{N}, \quad n > 2, \quad \text{odd}$$
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Context:
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
$abc=70,$
$cde=71,$
$efg=72.$
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Context:
The set \( S \) is given by \( S = \{1, 2, 3, 4, 5, 6\} \). A non-empty subset \( T \) of \( S \) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \( T \)?
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Context:
Find the equation of the projection of the surface $x^2+y^2+z^2-yz=1$ onto the $xoy$ plane.
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Context:
Find the distributional limit \( \lim_{\varepsilon\rightarrow 0^+}f_{\varepsilon} \) in \( D'(\mathbb{R}) \), where \( f_\varepsilon(x) \) is defined as follows:

\[
f_\varepsilon(x) = \begin{cases}
\frac{1}{\varepsilon^3}, & x \in (0,\varepsilon) \cup (2\varepsilon,3\varepsilon), \\
\frac{-2}{\varepsilon^3}, & x \in (\varepsilon,2\varepsilon), \\
0, & x \notin (0,3\varepsilon).
\end{cases}
\]
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Context:
Find the equation of the line joining the points of contact of the tangents from the line \(2x-3y = c\) to the curve \(x^2+2y^2=2\). Given that \(c = \pm \sqrt{17}\), determine the equation of the line joining these points of contact.
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Context:
Evaluate the following integral:

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$
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Context:
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then:
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Context:
Evaluate the limit $$\lim_{x\rightarrow 0} \frac{e - (1+x)^{\frac{1}{x}}}{\tan x}$$ using L'Hopital's rule, without relying on series expansion.
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Context:
Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\]
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Context:
Given the function
\[ f(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x \]
find the range of the function \( f(x) \) over the interval \( x \in [-1, 1] \).
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Context:
Evaluate the line integral \( \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz \) where \( C \) is the curve parametrized by \( r(t)=(t-1)i+e^{t^4}j+(t^2+1)k \) for \( 0\leq t\leq 1 \).
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Context:
The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 15 wooden blocks of weights $F_{2}, F_{3}, \ldots, F_{16}$, compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
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Context:
Given that $\sin(\theta + 3\pi) = -\frac{2}{3}$, find the value of $\frac{\tan(-5\pi - \theta) \cdot \cos(\theta - 2\pi) \cdot \sin(-3\pi - \theta)}{\tan(\frac{7\pi}{2} + \theta) \cdot \sin(-4\pi + \theta) \cdot \cot(-\theta - \frac{\pi}{2})} + 2 \tan(6\pi - \theta) \cdot \cos(-\pi + \theta)$.
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Context:
In the arithmetic sequence $\{a_{n}\}$, $\frac{a_{1010}}{a_{1009}} < -1$. If its first $n$ terms' sum $S_{n}$ has a maximum value, then the maximum positive integer value of $n$ that makes $S_{n} > 0$ is $\_\_\_\_\_\_\_\_\_\_.$
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Context:
Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Given $f(x)= \frac{2^{x}}{2(\ln 2-1)x}$, find $f'(1)$.
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Context:
3. [30] Let $A B C$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle O I A=90^{\circ}$. Given that $A I=97$ and $B C=144$, compute the area of $\triangle A B C$.
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Context:
In $\triangle ABC$, if $\overrightarrow{AB} \cdot \overrightarrow{AC} = \overrightarrow{AB} \cdot \overrightarrow{CB} = 4$, then the length of edge $AB$ is _____.
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Context:
已知 $\operatorname{Re}(z) \geq 1$, 求 $\left|\frac{1}{z}+1+\mathrm{i}\right|_{\text {min }}$ 的值。原始答案为 $\frac{\sqrt{m}-n}{k}$ 的格式，请给出 m + n + k 的值。
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Context:
Evaluate the complex integral \( K = \frac{i}{4-\pi} \int \frac{1}{z\cos(z)}\,dz \) over the circle centered at the origin with radius 4. Determine the value of \( K \).
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Context:
Given 6 points in space such that no 4 points are coplanar, find the smallest positive integer \( n \) such that whenever 10 line segments are drawn connecting these points, it is always possible to find \( n \) distinct pairs of triangles, each pair sharing a common edge.
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Context:
Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ with $N \subset M$. Given that $P_M$ and $P_N$ denote the orthogonal projections onto $M$ and $N$, respectively, determine the subspace of $H$ on which the operator $P_M - P_N$ is an orthogonal projection.
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Context:
Shapovalov A.V.

Inside a rectangular sheet of paper, $n$ rectangular holes with sides parallel to the edges of the sheet have been cut out. What is the smallest number of rectangular pieces into which this perforated sheet can be guaranteed to be cut? (The holes do not overlap and do not touch.)
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Context:
Calculate the probability that $X + Y \geq 2$, where $X$ is a binomial random variable with parameters $n = 10$ and $p = 0.5$, and $Y$ is an exponential random variable with rate $\lambda = \frac{1}{5}$. Assume $X$ and $Y$ are independent.
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Context:
Find the distance between the local maximum and local minimum of a cubic polynomial \( P(x) \) of degree 3, given that \( P(-1) = 10 \), \( P(1) = -6 \), there is a maximum at \( x = -1 \), and the derivative \( P'(x) \) has a minimum at \( x = 1 \).
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Context:
If $f(x) = (m-2)x^2 + mx + 4$ ($x \in \mathbb{R}$) is an even function, find the interval where $f(x)$ is monotonically decreasing.
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Context:
Vasya was driving from Sosnovka to Petrovka. Along the way, he saw a sign indicating "70 km to Petrovka". After traveling another 20 km, Vasya saw a sign indicating "130 km to Sosnovka". What is the distance (in kilometers) from Sosnovka to Petrovka? Express your answer as a whole number.
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Context:
Find the limit of the following: $$\lim\limits_{n \to \infty}\frac{3^{-n^2}}{\sum\limits_{k=n+2}^{\infty} 3^{-k^2}}$$
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Context:
Let \( f \in (\mathbb{R}^{3})^* \) be a linear functional defined as \( f(x_1, x_2, x_3) = x_1 - 3x_2 + x_3 \). Consider a basis \( \mathcal{A} = \{\alpha_1, \alpha_2, \alpha_3\} \) for the linear space \( \mathbb{R}^3 \) and its dual basis \( \mathcal{A}^* = \{g_1, g_2, g_3\} \subset (\mathbb{R}^{3})^* \), where \( g_1(x_1, x_2, x_3) = x_1 - x_3 \) and \( g_2(x_1, x_2, x_3) = x_1 - 2x_2 \). If the coefficient of \( f \) in the dual basis representation is 1 for \( g_3 \), determine the vector \( \alpha_3 \).
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Context:
A classroom has the shape of a regular hexagon with a side length of 3 m. A snore-o-meter is installed in each corner, measuring the number of sleeping students within a distance of up to 3 m. How many sleeping students are there in total in the classroom if the sum of the snore-o-meter readings is 7?
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Context:
Given that the area of a triangle with vertices $(x_1, x_1^2)$, $(x_2, x_2^2)$, and $(x_3, x_3^2)$ is 2 square units, and the area of another triangle with vertices $(y_1, y_1^2)$, $(y_2, y_2^2)$, and $(y_3, y_3^2)$ is 10 square units, calculate the value of the determinant:
$$
\begin{vmatrix}
(1+x_1y_1)^2 & (1+x_2y_1)^2 & (1+x_3y_1)^2 \\
(1+x_1y_2)^2 & (1+x_2y_2)^2 & (1+x_3y_2)^2 \\
(1+x_1y_3)^2 & (1+x_2y_3)^2 & (1+x_3y_3)^2 \\
\end{vmatrix}
$$
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Context:
In triangle $ABC$, the area is $5$ and $BC = 10$. Points $E$ and $F$ are the midpoints of sides $AC$ and $AB$, respectively. Lines $BE$ and $CF$ intersect at point $G$. Given that quadrilateral $AEGF$ is cyclic, find the value of $AB^2 + AC^2$. \(\text{(Proposed by Ray Li)}\)
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Context:
Given a cube with edge length \( a \). Let \( N \) be a point on the diagonal of a side face, and \( M \) be a point on a circle located in the plane of the base, with the center at the center of the base and radius \( \frac{5}{12}a \). Find the minimum value of \( |MN| \).
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Context:
For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. Determine the number of three-digit positive integers $m$ for which $m + t(t(m))$ is odd.
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Context:
Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \cdots - 59 + 63 - 67 + 71$.
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Context:
A train 700 m long can cross an electric pole in 40 sec. What is the speed of the train?
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Context:
Find the maximum value of $T$ with $x, y, z \in \Bbb{R_+}$
$$T=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$$
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Context:
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.
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Context:
Suppose the random variable Y has a moment generating function given by \( M(t) = \frac{e^{ct}}{1-dt^2} \) for \(-1 < t < 1\). Given that the mean and variance of Y are 5 and 4, respectively, find the value of \( c + d \).
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Context:
Point $I$ is the center of the inscribed circle of triangle $ABC$, $M$ is the midpoint of the side $AC$, and $W$ is the midpoint of the arc $AB$ of the circumcircle, not containing $C$. It turns out that $\angle AIM = 90^\circ$. In what ratio does point $I$ divide the segment $CW$?
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Context:
Example. Evaluate the definite integral

$$
\int_{0}^{2} \frac{4 \sqrt{2-x}-\sqrt{2+x}}{(\sqrt{x+2}+4 \sqrt{2-x})(x+2)^{2}} d x
$$
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Context:
There is a repeating decimal $0.2\dot{5}8\dot{7}$. What is the sum of the digits in the 1st, 99th, 199th, and 299th decimal places?
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Context:
Suppose that $k \geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining $k$ items from the original are placed in the even positions of the new list, in the same order as they appeared in the original list. For example, an in-shuffle performed on the list $P Q R S T U$ gives the new list $P S Q T R U$. A second in-shuffle now gives the list $P T S R Q U$. Ping has a list of the 66 integers from 1 to 66, arranged in increasing order. He performs 1000 in-shuffles on this list, recording the new list each time. In how many of these 1001 lists is the number 47 in the 24th position?
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Context:
Consider a finite population from which $X_1, X_2, X_3, \ldots$ are sampled without replacement. The central limit theorem suggests that for i.i.d. random variables, the distribution of the sample mean approaches a normal distribution as the sample size $n$ increases. However, when sampling without replacement, independence does not hold, but the approximation to normality can still be valid when $n$ is small relative to the population size. Determine the value of $n$ for which the probability distribution of the sample mean is closest to the normal distribution described by the central limit theorem.
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Context:
数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=1$，${{a}_{n+1}}=-a_{n}^{2}+c{{an}}-1$，若$\left\{ {{a}_{n}} \right\}$单调递减，则实数$c$的取值范围是__________.
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Context:
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 - n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n = 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.
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Context:
Let $\textbf{Z}_n$ be the set $\{0, 1, \ldots, n - 1\}$ with addition mod $n$. Consider subsets $S_n$ of $\textbf{Z}_n$ such that $(S_n + k) \cap S_n$ is nonempty for every $k$ in $\textbf{Z}_n$. Let $f(n)$ denote the minimal number of elements in such a subset. Determine the value of $$\lim_{n \to \infty} {\ln f(n) \over \ln n}.$$
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Context:
Polina wrote on the first page of her notebook \( n \) different positive
 integers. On the second page she wrote all pairwise sums of the
 numbers from the first page, and on the third - absolute values of
 pairwise differences of number from the second page. After that
 she kept doing same operations,i.e. on the page \( {2k} \) she wrote all
 pairwise sums of numbers from page \( {2k} - 1 \) , and on the page
 \( {2k} + 1 \) absolute values of differences of numbers from page \( {2k} \)
 At some moment Polina noticed that there exists a number \( M \)
 such that, no matter how long she does her operations, on every
 page there are always at most \( M \) distinct numbers.
 What is the biggest \( n \) for which it is possible?
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Context:
Determine the index of the subgroup $A \ast (ba)B(ba)^{-1}$ in the free product $A \ast B$, where $A$ and $B$ are non-trivial groups, and $a \in A$, $b \in B$ are non-identity elements.
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Context:
\begin{quote}
Ted quite likes haikus,
poems with five-seven-five,
but Ted knows few words.

He knows $2n$ words
that contain $n$ syllables
for every int $n$.

Ted can only write
$N$ distinct haikus. Find $N$.
Take mod one hundred.
\end{quote}

Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$.
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Context:
Let \(x_1, x_2, \ldots, x_n\) be real numbers in arithmetic sequence, which satisfy \(|x_i| < 1\) for \(i = 1, 2, \dots, n,\) and \[|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of \(n\)?
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Context:
Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms.
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Context:
Determine the maximal size of a set of positive integers with the following properties:

$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

$2.$ No digit occurs more than once in the same integer.

$3.$ The digits in each integer are in increasing order.

$4.$ Any two integers have at least one digit in common (possibly at different positions).

$5.$ There is no digit which appears in all the integers.
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Context:
Let\[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\]Compute $\Big|\lfloor 2008 X\rfloor\Big|$.
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Context:
What is the largest possible sum of the interior angles of a triangle on a torus? Consider the geometry of the torus and the placement of the triangle to determine this maximum sum.
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Context:
Determine the number of elements in the set $D_n = \{ a^ib^j \mid \text{ order}(a)=n, \text{ order}(b)=2, a^ib = ba^{-i} \}$ that have order $n$.
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Context:
The number of people who purchased book A is twice the number of people who purchased book B. The number of people who purchased both books A and B is 500, which is twice the number of people who purchased only book B. Additionally, there is a third book, book C. The number of people who purchased only book C is equal to the number of people who purchased both books A and B. The total number of people who purchased any of these 3 books is 2,500. Given these constraints, what is the number of people x who purchased only book A?
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Context:
## Task Condition

Find the derivative.

$y=\operatorname{arctg} \frac{\operatorname{tg} \frac{x}{2}+1}{2}$
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Context:
Calculate the limit of the following expression as $n$ approaches infinity, using the asymptotic expansion of the Barnes G-function:

$$\lim_{n\to\infty} \frac{G(n+\frac{11}{8})G(n+\frac{13}{8})^{2}G(n+\frac{15}{8})}{G(n+\frac{5}{2})^{2}G(n+\frac{7}{4})^{2}}$$
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Context:
The product of two positive integers plus their sum is 71. The integers are relatively prime, and each is less than 15. What is the sum of the two integers?
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Context:
Divide the $n$ positive integers $1, 2, 3, \ldots, n$ ($n \in \mathbb{N}^*$) into two groups, such that no two numbers in each group sum up to a perfect square, and there are no identical numbers in the two groups. What is the maximum value of $n$?
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Context:
A staircase has four steps. Calculate the total number of possible ways to climb the staircase, considering different step sequences as distinct. Express your answer as a single integer.
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Context:
There exist positive integers $k$ and $3\nmid m$ for which
$$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times  54\times 55}.$$Find the value $k$.
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Context:
Find all natural numbers \( n \) such that
$$
\min_{k \in \mathbf{N}}\left(k^{2}+\left\lfloor \frac{n}{k^{2}} \right\rfloor \right) = 1991 \quad (n \in \mathbf{N}).
$$
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Context:
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
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Context:
Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(2) = Q(4) = Q(6) = Q(8) = b\]and
\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\]
What is the smallest possible value of $b$?
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Context:
Three people are playing table tennis, where the player who loses a game gives up their place to the player who did not participate in that game. In the end, it turned out that the first player played 10 games, and the second player played 21 games. How many games did the third player play?
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Context:
Compute the definite integral:

$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$
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Context:
Given a positive integer \( n \geq 3 \) and a sequence of \( n \) different real numbers \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) whose sum is positive. If a permutation \(\left(b_{1}, b_{2}, \cdots, b_{n}\right)\) satisfies \( b_{1}+b_{2}+\cdots+b_{k} > 0 \) for every \( k = 1, 2, \cdots, n \), then this permutation is called good. Determine the minimum number of such good permutations.
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Context:
A parliament consists of 30 members. Each pair of members are either friends or enemies, and each member has exactly 6 enemies. Every set of 3 members forms a 3-member committee. A committee is called a "good committee" if all three members are either mutual friends or mutual enemies. Determine the number of all good committees.
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Context:
Find the area enclosed by the parametric curve given by \( r(t) = (16\sin^3(t), 13\cos(t) - 5\cos(2t) - 2\cos(3t) - \cos(4t)) \) for \( 0 \le t \le 2\pi \). Use the integral \( \int_0^{2\pi} y \cdot x' \, dt \) and the hint: \( \int_0^{2\pi} \sin(nx)\sin(mx) \, dx = \int_0^{2\pi} \cos(nx)\cos(mx) \, dx = \begin{cases} \pi  & \text{for } m = n \\ 0 & \text{otherwise} \end{cases} \).
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Context:
Let  $ABC$  be a triangle with incenter  $I$ . Let  $M_b$  and  $M_a$  be the midpoints of  $AC$  and  $BC$ , respectively. Let  $B'$  be the point of intersection of lines  $M_bI$  and  $BC$ , and let  $A'$  be the point of intersection of lines  $M_aI$  and  $AC$ .

If triangles  $ABC$  and  $A'B'C$  have the same area, what are the possible values of  $\angle ACB$ ?
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Context:
The cost price of an article is some percentage of the marked price. After allowing a discount of 25%, the gain percent is 17.1875%. What is the percentage of the cost price to the marked price?
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Context:
Vasya has a triangle with vertices labeled 1, 2, and 3. Rotating the triangle clockwise around vertex 1 by its angle 15 times returns it to its original position. Similarly, rotating around vertex 2 by its angle 6 times also returns it to its original position. Determine the minimum number of times, \( n \), that Vasya must rotate the triangle around vertex 3 by its angle to return it to its original position.
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Context:
Evaluate the limit: \[ \lim_{n\rightarrow \infty}P\left(\sum_{i=1}^n X_i \leq \frac{n}{2}+ n^{3/4} \right) \] where \(X_1, X_2, \ldots, X_n\) are i.i.d. random variables with a uniform distribution on \([0, 1]\).
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Context:
Determine the points at which the function \( f(x,y) = \begin{cases} xy\sin\left(\frac{1}{x^2+y^2}\right), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \) is differentiable.
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Context:
The price of an article has undergone three successive discounts: first by 15%, then by 20%, and finally by 25%. After these three discounts are applied sequentially, the final cost of the article is Rs. 250. What was the original cost of the article?
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Context:
Find the relationship between the initial conditions \(x(0) = x_0\) and \(\dot{x}(0) = y_0\) that ensures the solution \(x(t) = Ae^{k_1t} + Be^{-k_2t}\) of the differential equation approaches zero as \(t \to \infty\).
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Context:
In acute triangle $\Delta ABC$, $O$ is the circumcenter, $\angle A=45^\circ$. If $\frac{\cos B}{\sin C}\overset{\rightarrow}{AB}+\frac{\cos C}{\sin B}\overset{\rightarrow}{AC}=2m\overset{\rightarrow}{AO}$, find the value of $m$.
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Context:
Let \(Q\) be a set of permutations of the numbers \(1, 2, \ldots, 100\) such that for any two numbers \(a\) and \(b\) with \(1 \leq a, b \leq 100\), the number \(a\) appears to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Determine the largest possible number of permutations in the set \(Q\).
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Context:
Several schools took part in a tournament. Each player played one match against each player from a different school and did not play anyone from the same school. The total number of boys taking part differed from the total number of girls by 1. The total number of matches with both players of the same sex differed by at most one from the total number of matches with players of opposite sex. What is the largest number of schools that could have sent an odd number of players to the tournament?
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Context:
Evaluate the integral \( \int_0^\pi e^{ix} \sqrt{1 - e^{2ix}} \, dx \).
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Context:
A train crosses a platform in 40 seconds while it crosses a signal pole in 20 seconds. The length of the platform is the same as the length of the train. What is the length of the train?
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Context:
You have a Lego block of length 10 and three additional Lego blocks of lengths 3, 4, and 5. These shorter blocks are randomly placed on the block of length 10 such that their boundaries are within the boundaries of the block of length 10. The position of a short block is defined by the left-most knob it covers on the 10-block. The block of length 3 can be placed in positions 1 to 8, the block of length 4 in positions 1 to 7, and the block of length 5 in positions 1 to 6, all with equal probability. The blocks are placed independently. Let $X$ be the number of knobs on the large block that are covered by all three shorter blocks. Calculate the expected value $E[X]$. \(X \in \{0, 1, 2, 3\}\).
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Context:
Find the optimal solution to the optimization problem:
\[\begin{aligned}
&\min_{\mathbf{x} \in \mathbb{R}^n} && \|\mathbf{B}\mathbf{x}\|_1 \\
&\ \ \text{s.t.} && \|\mathbf{x}\|_2 \leq 1
\end{aligned}\]
where \(\mathbf{B} \in \mathbb{R}^{n \times n}\).
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Context:
25. Let $\dot{P}$ be a 40 -sided convex polygon. Find the number of triangles $S$ formed by the vertices of $P$ such that any two vertices of $S$ are separated by at least two other vertices of $P$.
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Context:
1. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. What is the smaller part of the volume separated from it by a plane passing through the midpoint of edge $B C$, the center of face $C D D_{1} C_{1}$, and point $X$ on edge $A A_{1}$ such that $A X: A_{1} X=1: 3 ?$
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Context:
Barbara has 9 candies. She buys 18 more and shares them with her 6 friends. Each friend receives an equal amount of candies. If each friend also receives some of Barbara's original candies, how many candies does each friend get?
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Context:
Find a vector potential $G$ such that $\nabla \times G = (x, -y, 0)$. Provide your answer as a vector $G$.
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Context:
From a point P on the surface of a sphere with radius R, draw three mutually perpendicular chords PA, PB, and PC. Find the value of $PA^2 + PB^2 + PC^2$.
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Context:
Consider the stochastic differential equation \(dZ_t = aZ_t dt + \sigma Z_t dB_t\), where \(B_t\) is a Brownian motion, \(a = 0.03\), and \(\sigma = 0.3\). Given that \(Z_0 = 100\), find the probability \(P(Z_1 > 103.5)\).
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Context:
Jason is counting the number of cars that pass by his window. He notices that there are twice as many blue cars as there are red cars. Furthermore, there are three times as many red cars as there are orange cars. He also notices that there are half as many yellow cars as there are orange cars, but three times as many yellow cars as there are purple cars. Lastly, there are five times as many green cars as there are purple cars. If Jason counted a total of 987 cars, how many of them were purple?
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Context:
Equilateral $\\triangle ABC$ has side length $\\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\\triangle ABC$, with $BD_1 = BD_2=\\sqrt{11}$. Find $\\sum^4_{k=1}(CE_k)^2$.
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Context:
Compute
$$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$
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Context:
Given that the focus of the parabola $y^{2}=8x$ is $F$, two perpendicular lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Let these lines intersect the parabola at points $A$, $B$ and $D$, $E$, respectively. The minimum area of quadrilateral $ADBE$ is ____.
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Context:
Compute the limit: \[ \lim_{x \to \infty} \int_{0}^{\pi} \frac{\sin(t)}{1+\cos^{2}(xt)} \, \mathrm{d}t. \]
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Context:
Two distinct similar rhombi share a diagonal. The smaller rhombus has an area of $1$, and the larger rhombus has an area of $9$. Compute the side length of the larger rhombus.
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Context:
Define $$p(x)= \sum_{i=1}^{n+1}2^{i}\cdot\frac{\prod_{j=1,j\neq i}^{n+1}(x-j)}{\prod_{j=1,j\neq i}^{n+1}(i-j)}.$$ Find $p(n+2)$.
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Context:
Lucy was born on Wednesday, December 1st, 2004. That Wednesday was the first day of her life. Her parents held a party for her on the $1000$th day of her life. On which day of the week was the party?
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Context:
Solve the equation \(x = e^{-1/x}\) for \(x\) analytically.
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Context:
Compute the cross-product term \(d(x_1)d(x_2)\) for the following independent Wiener processes:
\[ dx_1 = a_1 dt + b_1 dw_1(t) \]
\[ dx_2 = a_2 dt + b_2 dw_2(t) \]
where \(w_1\) and \(w_2\) are independent Wiener processes.
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Context:
Determine whether there exists a subfield of the real numbers that is both uncountable and locally compact.
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Context:
Given a cube, on each edge, write a number, either $1$ or $-1$. For each face of the cube, multiply the four numbers on the edges of this face and write the product on this face. Finally, add all the eighteen numbers that were written on the edges and faces of the cube.

What is the smallest possible sum that can be obtained?
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Context:
For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.
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Context:
What relationship must hold between the edges of a cuboid for it to be divided into two congruent cuboids that are similar to the original one?
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Context:
In a village, there are three age groups of workers: young, middle-aged, and old. The work capacities of young and old workers are only 75% of middle-aged workers.

Eight middle-aged men can dig a pit in 20 days. If a middle-aged man works half as much again as a boy, determine the number of days it would take for the following team to dig a similar pit: 2 young men, 2 middle-aged men, 2 old men, 5 young boys, 5 middle-aged boys, and 5 old boys.
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Context:
Determine the dimension of the tangent space at a point where multiple circles intersect in \(\mathbb{R}^2\).
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Context:
Find the extremals for the functional \( J(y) = \int_{0}^{1}\left[\dot{y}^{2}\right]\mathrm{d}t \) subject to the constraints \( y(0) = 1 \), \( y(1) = 4 \), and \( \int_{0}^{1}y\mathrm{d}t = 2 \).
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Context:
Given an element \( \alpha \) in a field with characteristic 2, and its minimal polynomial \( f(x) = x^9 + x + 1 \) with respect to \( GF(2) \), determine the order of \( \alpha \).
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Context:
考察集合$A = \{ 1 , 2 , 3 , 4 , \cdots ,2 ^ { n } \}$，其中，$n \geq 2 .$如果集合$A$中两个元素的和是2的幂，那么，它们之中的一个恰好属于子集$B$.求集合$A$中子集$B$的个数.
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Context:
How many ways are there to partition the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots, 2^{2005}\right\}$ into two non-empty disjoint subsets $A$ and $B$, such that the equation $x^{2} - S(A)x + S(B) = 0$ has integer roots, where $S(M)$ denotes the sum of all elements in the set $M$?
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Context:
Alice, Bob, and Carol play a game in which each of them chooses a real number between \(0\) and \(1.\) The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between \(0\) and \(1,\) and Bob announces that he will choose his number uniformly at random from all the numbers between \(\frac{1}{2}\) and \(\frac{2}{3}.\) Armed with this information, what number should Carol choose to maximize her chance of winning? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Amelia has 27 unit cubes. She selects one and paints one of its faces. She then randomly glues all 27 cubes together to form a $3 \times 3 \times 3$ cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.
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Context:
Consider a sequence of independent random variables $\{X_1, X_2, \ldots, X_N, \ldots\}$, where each $X_i$ is distributed such that $\text{Pr}(X_i = 1) = \text{Pr}(X_i = -1) = \frac{1}{2}$. Define $Y_N$ as the maximum absolute partial sum among the first $N$ terms, given by $Y_N = \max_{1 \leq i \leq N} \left| \sum_{j=1}^{i} X_j \right|$. Determine an upper bound for the limit of the probability $\lim_{N\to\infty}\text{Pr}\left(\frac{Y_N}{N} \leq \alpha\right)$ as $N$ approaches infinity, for $0 < \alpha < 1$. Provide your answer as a single expression or value.
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Context:
In a circle with center $O$ and radius $r$, let $AB$ be a chord smaller than the diameter. The radius of the inscribed circle in the smaller circular segment formed by the radii $AO$ and $BO$ is denoted by $\varrho$. Express $AB$ in terms of $r$ and $\varrho$.
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Context:
How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey?
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Context:
Given real numbers $a$, $b$, $c$ that satisfy $3x+4y-5\leqslant ax+by+c\leqslant 3x+4y+5$ for any real numbers $x$, $y$, find the minimum value of $a+b-c$. Provide your answer as a single integer.
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Context:
Consider a regular pentagon with vertices numbered from 1 to 5 in a clockwise direction. Initially, two ladybugs are placed at vertices 1 and 3. At each time step, each ladybug moves independently to one of its two adjacent vertices with equal probability. Calculate the expected number of time steps for the two ladybugs to meet at the same vertex.
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Context:
4. In space, there are 4 points $A, B, C, D$, satisfying $A B=B C=C D$. If $\angle A B C=\angle B C D=\angle C D A=36^{\circ}$, then the angle formed by line $A C$ and line $B D$ is $\qquad$
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Context:
Determine the number of distinct permutations of an $n$-element list such that no two permutations are equivalent under the condition that a permutation matches another if they can be obtained by flipping one another, preserving the relative order of neighboring elements.
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Context:
Evaluate the integral \( I = \int_{-1}^0 \int_{0}^1 \delta(x-y) \, dx \, dy \), where \( \delta(t) \) is the Dirac delta function defined as the limit of a symmetric Gaussian probability density function.
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Context:
Determine the number of integer sequences $m_1, m_2, \ldots, m_k$ such that $m_1, m_k \in \{1, 2, \ldots, 12\}$, $m_n \in \{1, 2, \ldots, 6\}$ for $1 < n < k$, and $m_1 + 2m_2 + \ldots + 2m_{k-1} + m_k = 24$. Provide your answer as a single integer.
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Context:
Let \(g(x)\) be a function defined as:

\[g(x)=\begin{cases} cx(x-1)+d&;x<1\\ x+2&;1\le x\le3\\ rx^2+sx+2&;x>3 \end{cases}\]

If \(g(x)\) is continuous for all \(x \in \mathbb{R}\) except \(x=1\), \(|g(x)|\) is differentiable everywhere, \(g'(x)\) is continuous at \(x=3\), and \(|c+r+d+s|=m\), find the value of \(m\).
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Context:
一个盒子里有无限可数个球，编号分别为 0,1,2,3,\cdots. 现在从中随机取球，每次取出编号为 k 的球的概率为 \df{e^{-1}}{k!}. 如果连续有放回地取 6 个球，记这 6 个球编号之和为 4 的概率为 p. 则 [10|\ln p|]=__________.
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Context:
Given $n$ zigzag lines, where each zigzag line intersects every other zigzag line exactly twice and no three lines intersect at the same point, determine into how many regions they divide the plane.
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Context:
Farmer James invents a new currency such that for every positive integer $n \leq 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be *nice* if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
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Context:
Determine the number of functors from the category of diagrams with exact rows in an abelian category to the category of six-term exact sequences in the abelian category, where the functors act identically on objects and morphisms except for possibly associating a different connecting morphism to each diagram.
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Context:
The product of four consecutive positive odd numbers ends in 9. Find the two penultimate digits of this product.
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Context:
What is the range of values of \( m \) for which the equation \( m(x^2 + y^2 + 2y + 1) = (x - 2y + 3)^2 \) represents an ellipse in a Cartesian coordinate system?
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Context:
Calculate the integral \( \int_{-1}^1 |z| \, dz \) where the integration contour is the upper semi-circle with unit radius.
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Context:
Calculate the limit of the sum of binomial coefficients given by:
\[ S_{n} = \sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right). \]
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Context:
Determine the smallest integer $n$ such that a square $ABCD$ can be decomposed into $n$ non-overlapping triangles, each having only acute angles.
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Context:
Determine the dimension of the vector space of alternating n-linear functions on the set of n x n matrices over a field F.
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Context:
Evaluate the exterior $2$-form $\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$ on $\mathbb{R}^n$ using the vectors $A = (1, 1, 1, \dots, 1)$ and $B = (-1, 1, -1, \dots, (-1)^n)$. Compute the result.
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Context:
We have some cards that have the same look, but at the back of some of them is written $0$ and for the others $1$.(We can\u0027t see the back of a card so we can\u0027t know what\u0027s the number on it\u0027s back). we have a machine. we give it two cards and it gives us the product of the numbers on the back of the cards. if we have $m$ cards with $0$ on their back and $n$ cards with $1$ on their back, at least how many times we must use the machine to be sure that we get the number $1$? (15 points)
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Context:
A group of 10 children are taken to an amusement park. They all try a variety of rides and attractions with the following costs:

Ferris wheel: $5 per child (6 children)
Roller coaster: $7 per child (4 children)
Merry-go-round: $3 per child (everyone)
Bumper cars: $4 per child (7 children)
Haunted house: $6 per child (5 children)
Log flume: $8 per child (3 children)

During their visit, they also bought some snacks:

3 cones of ice cream: $8 per cone (4 children)
1 hot dog: $6 each (5 children)
1 slice of pizza: $4 each (3 children)
1 large pretzel: $5 each (2 children)
1 cotton candy: $3 each (6 children)
1 soda: $2 each (7 children)

How much did they spend altogether?
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Context:
The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys.
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Context:
Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t.
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Context:
What is the quantity equivalent to '2% of 1'?
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Context:
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 100$ such that $i^x+i^y$ is a real number.
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Context:
Given the ellipse \( C: \frac{x^{2}}{3} + y^{2} = 1 \) with the upper vertex as \( A \), a line \( l \) that does not pass through \( A \) intersects the ellipse \( C \) at points \( P \) and \( Q \). Additionally, \( A P \perp A Q \). Find the maximum area of triangle \( \triangle A P Q \).
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Context:
Given vectors $\vec{a}=(m,4)$ and $\vec{b}=(3,-2)$, and $\vec{a}$ is not parallel to $\vec{b}$, find the value of $m$.
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Context:
Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$.
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Context:
Consider the bounded solution $u(x,t)$ of the heat equation \(\frac{\partial u}{\partial t} -\frac{\partial^2 u}{\partial x^2} = 0\), where \(u(x, 0) = \frac{e^{2x} - 1}{e^{2x} + 1}\). Determine \(\lim_{t \to \infty} u(1, t)\).
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Context:
8.4. On a chessboard, the centers of some cells are marked in such a way that no triangle with marked vertices is a right triangle. What is the maximum number of points that could have been marked?
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Context:
What is the minimum number of elementary row or column transformations needed to find the inverse of a given matrix using Gauss-type methods?
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Context:
How many distinct garlands can be formed by arranging all $6n$ flowers of one kind and $3$ flowers of another kind in a circular fashion, considering that the garland is considered the same if it can be rotated or reflected to match another arrangement?
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Context:
6. In the Cartesian coordinate system $x O y$, the point set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly selected from $K$, then the probability that the distance between each pair of these three points is no more than 2 is $\qquad$ .
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Context:
Let $x_1 = \frac{1}{20}$, $x_2 = \frac{1}{13}$, and \[ x_{n+2} = \frac{2x_nx_{n+1}(x_n + x_{n+1})}{x_n^2 + x_{n+1}^2} \] for all integers $n \geq 1$. Evaluate $\sum_{n=1}^\infty \left( \frac{1}{x_n + x_{n+1}} \right)$.
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Context:
Consider a random walk on the set of natural numbers \(\mathbb{N}\), starting at 0. The transition probabilities are given by:

\[ p(0,1)=0.75, \quad p(n,n-1)=0.5, \quad p(n,n+1)=0.5 \quad \text{for } n>0. \]

Calculate the expected time \(\mathbb{E}[T_{50}]\) for the random walk to reach the value 50.
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Context:
Determine the number of orderings of the totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}[\zeta]$, where $\zeta$ is a primitive $p$-th root of unity.
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Context:
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Find the positive integer.
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Context:
A function \(f\left(a \frac{b}{c}\right)\) for a simplified mixed fraction \(a \frac{b}{c}\) returns \(\frac{a + b}{c}\). For instance, \(f\left(2 \frac{5}{7}\right) = 1\) and \(f\left(\frac{4}{5}\right) = \frac{4}{5}\). Find the sum of the three smallest positive rational \(x\) where \(f(x) = \frac{2}{9}\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
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Context:
In a $4 \times 4$ table, we place $k$ blocks such that:

1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we remove a block, there is at least one cell that is not covered.

Find the maximum value of $k$.

Note: The blocks can overlap.
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Context:
In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that:
[LIST]
[*] There are no two numbers written in the first row that are equal to each other.[/*]
[*]  The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*]
[*] The two numbers written in each column are different and they add up to a rational number.[/*]
[/LIST]
Determine the maximum quantity of irrational numbers that can be in the chessboard.
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Context:
How many strings of length $5$ formed from the digits $0$, $1$, $2$, $3$, $4$ are there such that for each $j \in \{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies this condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)
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Context:
设数列$\{ a _ { n } \}$的通项公式为$a _ { n } = n ^ { 3 } - n , n \in N ^ { * }$，将该数列中个位数字为0的项，按从小到大的顺序排列构成数列$\{ b _ { n } \}$，则$b _ { 2 0 1 8 }$被7除所得的余数为
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Context:
Determine the maximal number of real intersections of the unbounded components of two plane cubic curves, $C_1$ and $D_1$, in the projective plane.
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Context:
If $a,b,c,d$ are distinct positive real numbers such that $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $ac=bd$, find the minimum value of $$\dfrac{a}{c}+\dfrac{b}{d}+\dfrac{c}{a}+\dfrac{d}{b}.$$
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Context:
Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \).
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Context:
From the set $\{1,2,3,…,10\}$, select any three different numbers so that these three numbers form a geometric sequence. There are \_\_\_\_\_\_ such geometric sequences.
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Context:
Let \( x \) and \( y \) be acute angles such that \( \sin y = 2005 \cos (x + y) \sin x \). Find the greatest possible value of \( \tan y \).
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Context:
Determine the maximum order of an element in the group $SL_2(\mathbb{Z} / p \mathbb{Z})$ for an odd prime $p$, given that the characteristic polynomial $P_A(x) = x^2 - \text{tr}(A)x + 1$ has no roots in $\mathbb{Z} / p \mathbb{Z}$. Provide your answer as a single integer.
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Context:
Find the range of the function $y=2^{-x^{2}+2x}$. Express your answer as an interval, specifying the endpoints and whether they are included or excluded.
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Context:
Find the angle between the vector \((2,2,1,1)\) and the linear subspace spanned by the vectors \((3,4,-4,-1)\) and \((0,1,-1,2)\).
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Context:
## Problem Statement

Find the derivative.

$$
y=x \ln (\sqrt{1-x}+\sqrt{1+x})+\frac{1}{2}(\arcsin x-x)
$$
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Context:
Find the smallest positive integer \( k \) such that the sum of the coefficients \( a_0 + a_1 + a_2 + \cdots + a_{k-1} \) in the expansion of \((1+x)(1+2x)(1+3x) \cdots (1+kx)\) is divisible by 2005.
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Context:
将有 2023 个顶点的完全图的每条边染色，使得有公共点的任意两条边颜色不同. 定义每个点所连出的边的颜色为一个颜色集，求不同颜色集个数的最小值.
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Context:
Every second Saturday and all Sundays are holidays. In a month of 30 days beginning on a Saturday, how many working days will be there if all Sundays are not holidays?
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Context:
On the extension of side $AC$ of triangle $ABC$, a point $T$ is marked beyond point $A$ such that $\angle BAC = 2 \angle BTC$. Find the area of triangle $ABC$, given that $AB = AC$, $BT = 42$, and $AT = 29$.
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Context:
Find the sum of the coefficients of $x^{20}$ and $x^{21}$ in the power series expansion of $\frac{1}{(1-x^3)^4}$.
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Context:
Among the non-empty subsets of the set \( A = \{1, 2, \cdots, 10\} \), how many subsets have the sum of their elements being a multiple of 10?
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Context:
A <b>palindrome</b> is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $\boxed{1:01}$, $\boxed{4:44}$, and $\boxed{12:21}$. How many times during a $12$-hour period will be palindromes?
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Context:
Evaluate the following limit using L'Hopital's Rule and Leibniz Integral Rule:

$$\lim_{a \to 1} \frac{1}{1-a} \int _{a} ^{1} \frac{\ln(x)}{x-1}\ dx$$
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Context:
Determine the cardinality of the set $F$ of all entire functions $f$ such that $f(x) = |x^k|$ for all $x \in (-1, 1)$, where $k$ is an odd integer $k \geq 1$. Provide your answer as a single number.
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Context:
Two cars start from the opposite places of a main road, 113 km apart. The first car runs for 25 km and takes a right turn and then runs 15 km. It then turns left and then runs for another 25 km and then takes the direction back to reach the main road. In the meantime, due to a minor breakdown, the other car has run only 35 km along the main road. What would be the distance between the two cars at this point?
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Context:
Solve the equation (for positive $x$)
$$x^x=\frac{1}{\sqrt2}$$
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Context:
Let \( \{X_t \mid t\in\mathbb{N}\} \) be an iid sequence of normal random variables on a probability space \((\Omega,\mathcal{F},P)\). Define \(A(\omega) = \{X_t(\omega) \mid t\in\mathbb{N}\}\) for all \(\omega\in\Omega\) and let \(\overline{A(\omega)}\) be its closure. Let \(\ell\) be the Lebesgue measure. Determine the probability \(P(\{\omega \mid \ell(\overline{A(\omega)}) = 0\})\).
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Context:
How can the centroid of 10 points be constructed using a fixed-opening compass in only 38 steps?
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Context:
Determine the asymptotic behavior of the sum \( \sum_{n=1}^N \Omega(n) \) as \( N \to \infty \), where \( \Omega(n) \) is the sum of the exponents in the prime factorization of a positive integer \( n \).
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Context:
 Let $a, b, c$ and $d$ be rational numbers. If it is known that the equation $x^4 + ax^3 + bx^2 + cx + d = 0$ has $4$ real roots, two of which are $\sqrt2$ and $\sqrt{2008}$. The value of $a + b + c + d$ is ...
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Context:
Let  $k$  be a positive integer and  $P$  a point in the plane. We wish to draw lines, none passing through  $P$ , in such a way that any ray starting from  $P$  intersects at least  $k$  of these lines. Determine the smallest number of lines needed.
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Context:
You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let $\theta$ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is $\lfloor 100 \cdot \theta \rfloor$ ?
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Context:
Tony and Maria are training for a race by running all the way up and down a 700 m long ski slope. They each run up the slope at different constant speeds. Coming down the slope, each runs at double his or her uphill speed. Maria reaches the top first, and immediately starts running back down, meeting Tony 70 m from the top. When Maria reaches the bottom, how far behind is Tony? Express your answer in meters as a whole number.
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Context:
Given the function $f(x)= \sqrt {-3x^{2}+ax}- \frac {a}{x}(a > 0).$ If there exists $x\_0$, such that $f(x\_0) \geqslant 0$ holds, then the minimum value of $a$ is $\_\_\_\_\_\_$.
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Context:
Evaluate the contour integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{\cos(z)}{z^7-2z^6}dz \), where \(\gamma\) is the path that traverses the circle centered at \(1/4\) with radius \(1/2\) counter-clockwise.
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Context:
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. The original answer for the perimeter of the hexagon is in the form $k\sqrt{3}$. Please find the value of $k$.
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Context:
Given a triangle inscribed in a circle of radius $\rho$, let $S$ denote the area, $L$ the perimeter, and $R=\sqrt{S}/L$. Assuming the three vertices are uniformly distributed on the circle, what is the maximum value of $R$?
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Context:
A crust is 25% lighter in weight than a white bread, and it is also 20% more expensive. However, the crust is consumed completely, whereas 15% of the bread always dries out. Assuming equal consumption, by what percentage do we spend more if we buy crust instead of bread?
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Context:
Evaluate the limit: $$\lim_{n\rightarrow\infty}\frac{n^{1/2}+n^{1/3}+\cdots+n^{1/n}}{n}.$$
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Context:
100 schoolchildren participated in the Mathematical Olympiad where 4 tasks were given. The first problem was solved by 90 students, the second by 80, the third by 70, and the fourth by 60. No student solved all the problems. Students who solved both the third and fourth questions received an award. How many students were awarded?
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Context:
The equation $g(x) = 0$ has a simple root in the interval $(1,2)$. The function $g(x)$ satisfies the conditions $|g'(x)| \ge 4$ and $|g''(x)| \le 3$. Assuming the Newton-Raphson method converges for all initial approximations in $(1,2)$, determine the maximum number of iterations required to obtain the root correct to 6 decimal places after rounding.
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Context:
The angle between the height of a regular triangular pyramid and a lateral edge is $\alpha (\alpha < \pi / 4)$. In what ratio does the center of the circumscribed sphere divide the height of the pyramid?
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Context:
Given that $X$ and $Y$ are independent standard normal random variables, find the probability that both $X$ and $X+Y$ are positive.
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Context:
Find the preimage of the set \( \{w \mid |w|=1\} \) under the bilinear transformation \( w=\frac{z+2i}{2iz-1} \). In other words, determine the set of points \( z \) that map to the unit circle \( |w|=1 \).
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Context:
the total cost of a vacation was divided among 5 people, with each person contributing a different amount. If the total cost of the vacation had been divided equally among 7 people, the cost per person would have been $40 less. Moreover, if the cost had been divided among 8 people, each person would have paid $60 less. What was the total cost of the vacation?
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Context:
Calculate the line integral \( \int_\gamma f \, dx \) for the vector field \( f(x) = (x_1^2 + 5x_2 + 3x_2x_3, 5x_1 + 3x_1x_3 - 2, 3x_1x_2 - 4x_3) \), where \( \gamma \) is the helix with radius 1 and pitch 1 from \((0, 1, 0)\) to \((0, 1, 2\pi)\).
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Context:
Determine the value of $a$ such that the function \( f(z) = \int_{1}^{z} \left(\frac{1}{w} + \frac{a}{w^3}\right)\cos(w)\,\mathrm{d}w \) is single-valued in the complex plane.
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Context:
Calculate the rank of the coherent sheaf \( \widetilde{k[t]/(t)} \) on \( \text{Spec}(k[t]) \).
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Context:
Let $E(n)$ denote the largest integer $k$ such that $5^k$ is an integral divisor of the product $1^12^23^3\cdots n^n.$ Calculate
$$\lim_{n\to\infty}\frac{E(n)}{n^{2}}$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Jingoistic James wants to teach his kindergarten class how to add in Chimese. The Chimese also use base-10 number system, but have replaced the digits 0-9 with ten of its own characters. For example, the two-digit number 十六 represents $16$. What is the sum of the two-digit numbers 十六 and 六十?
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Context:
The three-digit number $n = CCA$ (in base $10$), where $C \neq A$, is divisible by $14$. How many possible values for $n$ are there?
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Context:
For how many ordered triples $(a, b, c)$ of positive integers are the equations $abc+9=ab+bc+ca$ and $a+b+c=10$ satisfied?
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Context:
Suppose $A$ and $B$ are two non-singular matrices such that $B \neq I$, $A^6 = I$, and $AB^2 = BA$. What is the least value of $k$ for which $B^k = I$?
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Context:
Evaluate the limit: $$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$$
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Context:
In a movie projector, the cross-section of the spotlight is part of an ellipse. The filament is located at the focal point \( F_{2} \), and the distance from the filament to the vertex \( A \) of the mirror is \( |F_{2} A| = 1.5 \) cm. The major axis of the ellipse has a length of \( |B C| = 5.4 \) cm. Calculate the distance the bulb should be placed from the film gate to ensure the strongest light is received.
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Context:
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken - when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
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Context:
Determine a quadratic expression that generates prime numbers for more consecutive integer values of 'n' than the expression \(36n^2 - 810n + 2753\).
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Context:
A sweater nets the merchant a 35% profit on the wholesale cost at which he initially purchased the item when sold at a certain discount. The sweater is marked up by 70.37% from wholesale at its normal retail price. What is the discount percentage at which the sweater is sold?
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Context:
$\begin{array} {l}\overline{AB}\text{is a diameter of a circle. Tangents }\overline{AD}\mathrm{and}\overline{BC}\text{are drawn so that}\overline{AC}\mathrm{and}\overline{BD}\text{intersect in a point on the circle. If} \\ \overline{AD}=a\mathrm{and}\overline{BD}=b,a\neq b,\text{the diameter of the circle is:} \end{array}$
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Context:
Rectangles \(ABCD\), \(DEFG\), and \(CEIH\) have equal areas and integer side lengths. Given that \(BC = 37\), find the length of \(DG\).
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Context:
For the function \( y=\frac{1}{|x|-1} \), find the minimum area of all circles centered at the point symmetric to its intersection with the y-axis about the origin that intersect the graph of the function.
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Context:
A coin is tossed 50 times. Let X represent the number of heads obtained in all 50 tosses, and let Y represent the number of heads obtained in the first 20 tosses. Calculate the conditional expectation E[Y | X = i].
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Context:
To vigorously improve transportation and celebrate the 60th National Day, a region plans to open a bus route between locations A and B on the occasion of the 60th National Day. It is known that A and B are 15 kilometers apart. The planning requirements for the bus are as follows: the distance between two adjacent stops is equal, the interval time before and after each stop for the bus is 3 minutes, ignoring the parking time, and the designed speed of the bus is 60 kilometers per hour. Calculate the minimum number of buses required to be put into operation between locations A and B for a round trip. Express your answer as a whole number.
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Context:
Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions:

1. Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime.
2. All other pairs of the five integers are relatively prime.
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Context:
Find the gradient of the function \( f(C) = \text{trace}(X X^T C - \log(C))\) with respect to \(D\), where \(C\) is parametrized as \( C = (D + U F U^T)^{-1} \).
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Context:
Determine the number of points at which a line in \(\mathbb{CP}^n\) intersects \(\mathbb{CP}^2\). Assume the line is a copy of \(\mathbb{CP}^1\).
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Context:
Find the dimension of the space of global sections of the tangent sheaf of $\mathbf{P}^3$ using the Euler sequence.
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Context:
A basket of 2750 apples is divided equally among a group of apple lovers. If 60 people join the group, each apple lover would receive 12 apples less. How many apples did each person get before 60 people joined the feast?
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Context:
Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form

$\frac{d^{n}}{dx^{n}}f(e^{x})= \displaystyle\sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$).

From the coefficients $a_{kn}$ compose the sequence of polynomials

$P_{n}(x)= \displaystyle\sum_{k=0}^{n} a_{kn}x^{k}$ ($n=0,1,2,\dots$)

and find a closed form for the function

$F(t,x) = \displaystyle\sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$
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Context:
Start by writing the integers  $1, 2, 4, 6$  on the blackboard. At each step, write the smallest positive integer  $n$  that satisfies both of the following properties on the board.

- $n$  is larger than any integer on the board currently.
- $n$  cannot be written as the sum of  $2$  distinct integers on the board.

Find the  $100$ -th integer that you write on the board. Recall that at the beginning, there are already  $4$  integers on the board.
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Context:
Find the smallest integer \( n \) such that \( n! > 10^{100} \).
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Context:
In a convex quadrilateral \(ABCD\), the lengths of sides \(AB\) and \(BC\) are equal, \(DB\) is the bisector of angle \(ADC\), and \(AD : DC = 4 : 3\). Find the cosine of angle \(AKB\), where \(K\) is the intersection point of the diagonals \(AC\) and \(BD\), and \(BK : KD = 1 : 3\).
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Context:
If \( A = 2011^{2011} \) and \( B = (1 \times 2 \times \cdots \times 2011)^2 \), determine whether \( A \) is greater than, equal to, or less than \( B \). Fill in the blank with “ $>$ ”, “＝” or “＜”.
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Context:
Calculate the second-order directional derivative \(D_{\mathbf{v}_0 \mathbf{u}_0}\mathbf{f}(\mathbf{x}_0)\) of the vector-valued function \(\mathbf{f}(\mathbf{x}) = (x_1 + x_2)^2\mathbf{e}_1 + x_1\sin(x_2)\mathbf{e}_2\) at the point \(\mathbf{x}_0 = \mathbf{e}_1 + \pi\mathbf{e}_2\) in the direction of \(\mathbf{u}_0 = (\mathbf{e}_1 + 2\mathbf{e}_2)\) and \(\mathbf{v}_0 = (\mathbf{e}_1 - \mathbf{e}_2)\).
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Context:
Calculate the area of the region enclosed by the curves $xy=4$, $xy=8$, $xy^3=5$, and $xy^3=15$. Provide your answer as a single numerical value.
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Context:
Let  $S$  be a subset of  $\{1, 2, 3, \cdots, 1989 \}$  in which no two members differ by exactly  $4$  or by exactly  $7$ . What is the largest number of elements  $S$  can have?
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Context:
A rectangular paper is folded into two non-congruent parts such that the sum of the perimeters of the two parts is 34 cm when folded along one set of sides, and the sum of the perimeters of the two parts is 38 cm when folded along the other set of sides. What is the area of the paper?
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Context:
Evaluate the limit: $$\lim_{r\to0}\frac{\int_{-r}^{r}\int_{-r}^{r}e^{-x^2-y^2+2xy}dxdy}{\int_{-r}^{r}\int_{-r}^{r}e^{-x^2-y^2}dxdy}$$
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Context:
A woman purchased 3 towels at Rs. 100 each with a 10% discount, 5 towels at Rs. 150 each with a 5% discount, and two towels at a certain rate that she cannot remember. However, she remembers that after taking into account the discounts and a 5% sales tax applied to all the towels, the average price of the towels was Rs. 145. Find the unknown rate of the two towels before the discount and tax.
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Context:
Evaluate the line integral \( \int_C P\,dx + Q\,dy \) where \( P(x,y) = \frac{y+x}{x^2+y^2} \) and \( Q(x,y) = \frac{y-x}{x^2+y^2} \) around the circle \( C: x^2+y^2=a \) oriented clockwise.
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Context:
Use Stokes' Theorem to transform the surface integral \( \int_S \text{rot}\vec{F}\cdot \vec{dS} \) into a line integral and calculate it for \( \vec{F}(x,y,z)=(y,z,x) \), where \( S \) is the part of the paraboloid \( z=1-x^2-y^2 \) with \( z\ge 0 \) and the normal vector has a non-negative \( z \) component. Assume \( \vec{F}(x,y,z) \) is conservative.
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Context:
Find the minimum value of \( \frac{1}{2} x^T P x \) subject to the constraint \( a^T x = b \), where \( P \) is a symmetric negative definite matrix.
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Context:
Let \(ABCD\) be a square with side length 1. Points \(P\) and \(Q\) are located on the plane such that \(Q\) is the circumcenter of \(\triangle BPC\) and \(D\) is the circumcenter of \(\triangle PQA\). Determine the largest possible value of \(PQ^2\), and express your answer in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), where \(a\) and \(b\) are rational numbers.
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Context:
Find the value of $x$ that satisfies the equation $\sin(x) = x + 0.5$. Express your answer to three decimal places.
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Context:
In a store, there are 21 white and 21 purple shirts hanging in a row. Find the smallest $k$ such that, regardless of the initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts, so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively.
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Context:
Find the function $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$, given that $if = -v + iu$ and $\Re[(1+i)f(z)] = e^x (\cos y - \sin y)$. What is $f(z)$?
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Context:
Let \( S \) be a set with \( |S| = 2019 \). Suppose \( A_1, A_2, \ldots, A_n \) are subsets of \( S \) such that the union of any three of them equals \( S \), but the union of any two of them does not equal \( S \). Determine the maximum possible value of \( n \).
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Context:
In \(\triangle ABC\), let \(D\) and \(E\) be the feet of the perpendiculars from \(A\) to the angle bisectors of angles \(B\) and \(C\), respectively. The line through \(D\) and \(E\) intersects \(AC\) at \(X\) and \(AB\) at \(Y\). Determine the fractional part of the area of \(\triangle ABC\) that is occupied by the region \(XYBC\).
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Context:
In the isosceles triangle $ABC$, the angle $A$ at the base is $75^{\circ}$. The angle bisector of angle $A$ intersects the side $BC$ at point $K$. Find the distance from point $K$ to the base $AC$ if $BK = 10$.
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Context:
Find the conditions on $a$ and $b$ such that the function $\psi_{|D}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $\psi_{|D}(x,y) = (x^2 - y^2, x^2 + y^2)$ and restricted to $D = \{(x,y) \in \mathbb{R}^2 \mid x,y>0\}$, has a preimage at the point $(a,b)$. Provide the conditions on $a$ and $b$ for which this is true.
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Context:
Find a function \( f(x) \) that satisfies the functional equation \( f(x \cdot y) = f(x) + f(y) \) for all \( x, y > 0 \).
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Context:
Determine the point at which the differential equation \( \frac{dx}{dt} = x + x^{2020}\cos(t) \), when solved using a Bernoulli substitution, becomes a linear inhomogeneous equation.
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Context:
How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? Provide your answer as an integer.
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Context:
Find the integer closest in value to
\[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}.\]
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Context:
How old is the eldest brother?

Determine this given that the age of the middle brother is the product of the ages of his two brothers, the sum of the ages of all three brothers is 35, and the sum of the decimal logarithms of their ages is 3.
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Context:
A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver.
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Context:
An array $\mathbb{T}$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. Determine the number of independent components in this array.
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Context:
Let \( A \in \mathbb{R}^{2\times 2} \) be a matrix such that \( \det(A) = d \neq 0 \) and \( \det(A + d \cdot \text{Adj}(A)) = 0 \). Evaluate \( \det(A - d \cdot \text{Adj}(A)) \).
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Context:
Find a non-smooth function \( f : \mathbb{R} \to \mathbb{R} \) such that \( f^3 \) is smooth.
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Context:
13.7.9 Given the parabola $y^{2}=2 p x(p>0), A B$ is a chord passing through the focus $F$. If $A B$ makes an angle $\theta\left(0<\theta \leqslant \frac{\pi}{2}\right)$ with the $x$-axis, find $\angle A O B$.

---

The original text has been translated into English while preserving the original formatting and line breaks.
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Context:
In a binomial tree model, let $u=2$ and $d=1/2$. Given that $S_n^3$ is a martingale under risk-neutral probabilities, determine the risk-free interest rate $r$.
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Context:
How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?
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Context:
If  $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$  can be written in the form  $a^b$  for positive integers  $a,b$  with  $a$  squarefree, find  $a+b$ .

*Proposed by Deyuan Li and Andrew Milas*
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Context:
A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours?
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Context:
Four friends make cookies from the same amount of dough with the same thickness. Art's cookies are circles with a radius of 2 inches, and Trisha's cookies are squares with a side length of 4 inches. If Art can make 18 cookies in his batch, determine the number of cookies Trisha will make in one batch.
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Context:
How many whole numbers between 1 and 2000 do not contain the digit 2?
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin^2 x}{x^2} \, dx \) using the principal value and complex analysis techniques.
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Context:
Determine the smallest integer $n$ such that the group $Q_8 \times Q_8$ is isomorphic to a subgroup of the symmetric group $S_n$.
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Context:
How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct?
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Context:
Given that the graph of the function $f(x) = \frac{9^x-a}{3^x}$ is symmetric about the origin, and $g(x) = \lg(10^x+1)+bx$ is an even function, find the value of $a+b$.
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Context:
A parabola is given by the equation $y^{2}=2px (p > 0)$, with a focus at point $F$. A line with a slope angle of $\frac{\pi}{4}$ passes through the focus $F$, intersecting the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through the point $(0, 2)$, calculate the value of $p$. Express your answer as a simplified fraction.
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Context:
Determine the values of \( \beta \) for which the differential equation \( \tau\frac{dx}{dt}=-x+\frac{e^{\beta(x-\frac{1}{2})}}{1+e^{\beta(x-\frac{1}{2})}} \) has two additional fixed points beyond \( x=0.5 \).
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