Context:
There are 3 stops between city A and city B for a train. How many types of tickets need to be prepared for this train?
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Context:
Find the minimum value of the function $$f(a,b)= 1 + a^3 + b^3 - (a + a^2 b + b^2)$$ where $a,b$ are real numbers.
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Context:
A quadrilateral inscribed in a circle has side lengths $a, b, c,$ and $d$. Calculate the ratio of the lengths of the diagonals of this quadrilateral.
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Context:
Using the central limit theorem, determine the minimum number of 100W bulbs needed to ensure that an advertising board is lit for at least 20,000 hours with a probability of at least 0.9. Assume that the life of each bulb follows an exponential distribution with an average life of 600 hours.
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Context:
Find \( f'(8.23) \) where \( f(x) = 23|x| - 37\lfloor x \rfloor + 58\{x\} + 88\arccos(\sin x) - 40\max(x,0) \). Note: For a real number \( x \), \( \{x\} = x - \lfloor x \rfloor \) denotes the fractional part of \( x \).
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Context:
A large spherical model of Earth is to be cut with planar cuts. If each cut intersects the sphere, what is the maximum number of pieces the sphere can be divided into after 6 cuts?
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Context:
Evaluate the limit as a definite integral: $$\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}\frac{e^{2\pi i n/N}}{1-e^{2\pi i n/N}}$$
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Context:
Determine the cardinality of the set \( S = \{ (x, y, z) \mid x^2 + y^2 + z^2 = 2^{2018}, xyz \in \mathbb{Z} \} \).
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Context:
On the board, two sums are written:

\[1+22+333+4444+55555+666666+7777777+88888888+999999999\]

\[9+98+987+9876+98765+987654+9876543+98765432+987654321\]

Determine which one is greater (or if they are equal).
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Context:
Let $X \sim \mathrm{Unif }[0,1]$ and $Y\sim \mathrm{Gamma}(2,r)$, with $r>0$, be two independent random variables. Find the covariance $\mathrm{Cov}(U,V)$ where $U = X$ and $V = XY$. Use the fact that $V \sim \textit{Exp}(r)$.
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Context:
Compute the fundamental group of the quotient space $G/G_x$, where $G$ is a compact, connected, and simply connected Lie group, $\mathfrak{g}$ is its Lie algebra, and $x \in \mathfrak{g}^*$. Here, $G_x$ denotes the isotropy group of $G$. What is $\pi_1(G/G_x)$?
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Context:
Through point \( A \) located on a circle, a diameter \( AB \) and a chord \( AC \) are drawn, where \( AC = 8 \) and \( \angle BAC = 30^\circ \).

Find the length of the chord \( CM \), which is perpendicular to \( AB \).
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Context:
Determine the accumulation points of the sequence \( \{x_n\} \) where \( x_n = n\sin(n) \) for \( n \in \mathbb{N} \).
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Context:
Evaluate the double integral \( \iint_R x \, dx \, dy \) using the change of variables \( x = \sqrt{u+v} \) and \( y = v \), where the region \( R \) is in the first quadrant of the \( xy \)-plane and is bounded by \( y = 0 \), \( y = 16 \), \( y = x^2 \), and \( y = x^2 - 9 \).
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Context:
Let $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form\[0.abcabcabc\ldots=0.\overline{abc},\]where the digits $a$, $b$, and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required?
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Context:
Determine the values of the parameter $x$ for which the infinite series \( \sum_{n=1}^\infty \frac{x^n}{\ln(n+1)} \) converges.
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Context:
Evaluate the expression \( \frac{\partial(\frac{d(x(t))}{dt})}{\partial x(t)} \).
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Context:
Calculate the Gaussian curvature of the surface defined by the set \( S = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \} \) with the first fundamental form given by \( ds^2 = \frac{4}{(1-x^2-y^2)^2}(dx^2 + dy^2) \). Use the formula for Gaussian curvature \( K = \frac{LN - M^2}{EG - F^2} \), where \( E = G = \frac{4}{(1-x^2-y^2)^2} \) and \( F = 0 \).
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Context:
Let $f(x)$ be an irreducible quartic polynomial over $\mathbb{Q}$. Let $L$ be the splitting field of $f(x)$. Suppose $\mathbb{Q}(\alpha) \cap \mathbb{Q}(\beta)=\mathbb{Q}$ for any two distinct roots $\alpha$ and $\beta$ of $f(x)$. Determine the Galois group of $f(x)$ over $\mathbb{Q}$.
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Context:
Compute the limit: \[ \lim_{n \to \infty} \sqrt[n]{ \int_{0}^{1} (1+x^n)^n \, dx} \]
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Context:
Evaluate the integral \( \int^{4}_{0}\int_{0}^{\sqrt{4x-x^2}}\sqrt{x^2+y^2}\, \text{dy dx} \).
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Context:
Evaluate the triple integral \( \iiint\limits_G (x^2 - z^2) \, dx \, dy \, dz \), where the region \( G \) is bounded by the planes \( y = -x \), \( z = x \), \( z = y \), and \( z = 1 \).
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Context:
Evaluate the integral \( \iint_R y\, dx\, dy \) where \( R \) is the region bounded by the \( x \)-axis and the parabolas \( y^2 = 4-4x \) and \( y^2=4+4x \). Use the transformation \( x = u^2 - v^2 \) and \( y = 2uv \) to perform the evaluation.
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Context:
Find the upper bound for the integral \( \left\lvert\int_\gamma \frac{\sin(z)}{z} \,dz\right\rvert \) where \(\gamma\) is a path that traces the circle of radius 2 centered at the origin once.
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Context:
Calculate the value of the infinite sum \( \sum_{j=0}^\infty \frac{\binom{2j}{j}}{2^{2j}(1-4j^2)} \) without using Riemann sums or integrals.
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Context:
Calculate the value of the infinite series \( \sum_{k=1}^{\infty} \frac{(2k-1)(-3)^{k-2}}{2^{k-1}} \).
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Context:
Calculate the volume of the region defined by the inequality \((4x+4y+2z)^2+(4x+2y+4z)^2+(2x+4y+4z)^2 \leq 1\) using the change of variables \(u=4x+4y+2z\), \(v=4x+2y+4z\), \(w=2x+4y+4z\). The Jacobian determinant of the transformation is 40. Express the volume as an integral in the new variables \(u, v, w\) and find the value of the integral.
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Context:
Determine the smallest positive integer \( m \) for which \( m^3 - 90 \) is divisible by \( m + 9 \).
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Context:
Twenty ants live on the faces of an icosahedron, one ant on each side, where the icosahedron have each side with length 1. Each ant moves in a counterclockwise direction on each face, along the side/edges. The speed of each ant must be no less than 1 always. Also, if two ants meet, they should meet at the vertex of the icosahedron. If five ants meet at the same time at a vertex, we call that a collision. Can the ants move forever, in a way that no collision occurs?
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Context:
Determine the matrix representation of the isomorphism \( \varphi: \text{Hom}(V,W) \to M_{m \times n} \), where \( B = (v_1, \ldots, v_n) \) and \( C = (w_1, \ldots, w_m) \) are ordered bases for the finite-dimensional vector spaces \( V \) and \( W \), respectively. Here, \( [T]_B^C \) denotes the coordinate matrix of a linear transformation \( T: V \to W \) with respect to bases \( B \) and \( C \).
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Context:
Find the limit of \( \frac{{{k-1}\choose n}}{{{n+k-1}\choose n}} \) as \( n, k \to \infty \) such that \( \frac{k}{n^2} \to a > 0 \).
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Context:
14. 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:
Let $F \subseteq E$ be a field extension. Let $\alpha \in E$ be a root of an irreducible polynomial $f(x)$ over $F$ with multiplicity 3. If $\beta$ is any other root of $f(x)$ in $E$, what is the multiplicity of $\beta$?
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Context:
Right triangles \(T_1\) and \(T_2\), have areas of \(1\) and \(2\), respectively. A side of \(T_1\) is congruent to a side of \(T_2\), and a different side of \(T_1\) is congruent to a different side of \(T_2\). Find the square of the product of the lengths of the other (third) sides of \(T_1\) and \(T_2\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Mona plays a multiplayer video game on the weekends. She joins groups with four other players to complete objectives in the game. The past weekend, she joined some groups and grouped with 33 unique players. One of the groups included two players she had grouped with before that weekend, and another group included one person she had grouped with before. How many groups did Mona join that weekend?
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Context:
Evaluate the integral \( \int_0^1 \left( \left\lfloor \frac{2}{x} \right\rfloor - 2 \left\lfloor \frac{1}{x} \right\rfloor \right) \, dx \).
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Context:
11. (16 points) A and B are playing a game on a $20 \times 15$ chessboard. At the beginning, a queen is placed on a square of the chessboard except the top-right corner; starting with A, the two players take turns to move the queen. Each move can be a straight or diagonal move of several squares, but only to the right, up, or up-right; the player who moves the queen to the top-right corner square wins. How many starting squares on this chessboard allow A to have a winning strategy?
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Context:
Consider the quotient ring \(S=\mathbb{R}[x_1,x_2,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)\). Determine the least degree of a polynomial \(p\in S\) that satisfies the idempotent property \(p^2=tp\) modulo the ideals \((x_i)\) and \((x_i-1)\) for all \(i=1,2,\dots,n\). Assume that finding the result for \(t=1\) is sufficient, as \(p^2=tp\Rightarrow \left(\frac{p}{t}\right)^2=\frac{p}{t}\Rightarrow q^2=q\text{ for some }q\in S.\) What is the least degree that such a polynomial \(p\) can have?
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Context:
If \( i^{2} = -1 \), then find the sum \( S=\cos 45^{\circ}+i \cos 135^{\circ}+\cdots+i^{n} \cos (45+90n)^{\circ}+\cdots+i^{40} \cos 3645^{\circ} \).
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Context:
Let $\alpha, \beta, \gamma \in \mathbf{R}$ satisfy the equation $\sin \alpha \cdot \cos \beta + |\cos \alpha \cdot \sin \beta| = \sin \alpha \cdot |\cos \alpha| + |\sin \beta| \cdot \cos \beta$. Then, find the minimum value of $(\tan \gamma - \sin \alpha)^2 + (\cot \gamma - \cos \beta)^2$.
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Context:
Solve the equation \( \ln(x) = \sin(x) \) for \( x \).
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Context:
Given the Stooge sort algorithm, which runs in \( O(n^\alpha) \) where \( \alpha \) is minimal, determine the value of \( (243/32)^\alpha \).
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Context:
10.6  Let the insphere of a pyramid  $SABC$  touch the faces  $SAB, SBC, SCA$  at  $D, E, F$  respectively. Find all the possible values of the sum of the angles  $SDA, SEB, SFC$ .
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Context:
Evaluate the sum \( S_n = \sum_{k=n}^{2n-1} \binom{2n-1}{k} x^k (1-x)^{2n-1-k} \) and express it in a closed form involving special functions, where \(|x| \leq 1\).
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Context:
If \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 + x^2 - x + 1 = 0 \), find the value of \( \prod \left( \frac{1}{\alpha^3} + \frac{1}{\beta^3} - \frac{1}{\gamma^3} \right) \).
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Context:
Determine all values of $a$ at which the function \( h(x) \) is differentiable, where \( h(x) = x^{3}-x^{2} \) if \( x \in \mathbb{Q} \) and \( h(x) = 2x-2x^{2} \) otherwise.
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Context:
A partially silvered mirror covers the square area with vertices at \((\pm 1, \pm 1)\). The fraction of incident light it reflects at \((x,y)\) is given by \(\frac{(x-y)^2}{4}\). Assuming a uniform intensity of incident light, calculate the total fraction of light reflected by the mirror.
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Context:
There are twice as many cows in Devonshire as there are hearts on a standard deck of 52 playing cards. If there are 4 hearts on a card, the total cost of the cows when they are sold at a certain price each is $83200. What is the price of each cow?
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Context:
A burger at Ricky C's weighs 120 grams, of which 30 grams are filler.  What percent of the burger is not filler?
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Context:
Determine the smallest integer $A$ such that there exist consecutive integers, including $A$, which sum to 2014. An integer $A$ with this property is called "yummy."
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Context:
What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7?
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Context:
For a positive integer  $n$ , define  $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer  $k$  for which  $7?9?=5?k?$ .
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Context:
8 friends visited a book stall and brought 8 books. Everyone decided to contribute equally to the total bill of $600. If one of the friends had a coupon for 10% off the total bill and another friend had a coupon for 5% off the remaining amount, and if each friend still contributed equally after both coupons were applied to the bill, how much did each friend pay?
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Context:
On an \( 8 \times  8 \) chessboard, 6 black rooks and k white rooks are placed on different cells so that each rook only attacks rooks of
 the opposite color. Compute the maximum possible value of \( k \) .
 (Two rooks attack each other if they are in the same row or column and no rooks are between them.)
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Context:
In a complete graph $K_{12}$, how many distinct paths of length exactly 7 exist between vertices A and B, assuming no repeated vertices along the path?
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Context:
Determine the norm in which the inequality \( ||g*f-g*s|| \leq C ||g|| \text{ } ||f-s|| \) holds, given that \( g \in L^p(\mathbb{R}) \) has compact support and \( 1\leq p<\infty \).
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Context:
Let the set $A=\{(x,y)|\frac{m}{2}\leqslant {(x-2)}^{2}+{y}^{2}\leqslant {m}^{2},x,y\in \mathbb{R}\}$, $B=\{(x,y)|2m\leqslant x+y\leqslant 2m+1,x,y\in \mathbb{R}\}$. If $A\cap B\neq \varnothing$, then the range of the real number $m$ is \_\_\_\_\_\_.
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Context:
Given that the correlation between random variables X and Y is \( \rho \), and Y and Z are independent, with all three variables being standard normal, determine the correlation between X and Z.
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Context:
For a positive integer $n$, let $S_n$ be the total sum of the intervals of $x$ such that $\sin 4n x \geq \sin x$ in $0 \leq x \leq \frac{\pi}{2}$. Find $\lim_{n\to\infty} S_n$. \(60\) points.
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Context:
Find a nontrivial unit polynomial \( p(x) \) in \( \mathbb{Z}_4[x] \) such that \( \deg p(x) > 1 \).
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Context:
Calculate the sum of the series:

$$\sum_{n=2}^{\infty}\frac{3n-5}{n(n^2-1)}.$$
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Context:
Calculate the Fourier transform \( \mathscr{F}(f_a) \) of the function \( f_a(x) = \frac{a}{\pi(x^2+a^2)} \) for \( a > 0 \), given that \( \mathscr{F}(\mathscr{F}(f))(x) = 2\pi f(-x) \) and \( \mathscr{F}(e^{-a|x|}) = \frac{2a}{a+\xi^2} \).
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Context:
Calculate the total surface area of the two cylinders given by the equations \(y^2 + z^2 = 1\) and \(x^2 + y^2 = 1\).
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Context:
Using the method of Lagrange multipliers, determine the minimum value of the function \( f(x,y,z) = x^2y^2z^2 \) subject to the constraint \( x^2 + y^2 + z^2 = r^2 \). Given the critical point \((\pm \frac{r}{\sqrt{3}}, \pm \frac{r}{\sqrt{3}}, \pm \frac{r}{\sqrt{3}})\), identify the minimum value of the function.
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Context:
Given that there are $3$ red cubes, $4$ blue cubes, $2$ green cubes, and $2$ yellow cubes, determine the total number of different towers with a height of $10$ cubes that can be built, with the condition that the tower must always have a yellow cube at the top.
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Context:
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(-2,4)$, calculate the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$.
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Context:
将一枚均匀的股子独立投郑三次, 所得的点数依次记为 $x, y, z$, 则事件 " $\mathrm{C}_{7}^{x}<\mathrm{C}_{7}^{y}<\mathrm{C}_{7}^{2}$ "发生的概率为 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Find the number of positive integer solutions to the equation \(x+y+z+w=20\) such that \(x, y, z,\) and \(w\) are all different.
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Context:
Danny is bringing his sailboat out to the ocean. When he uses his 24 square foot sail, he goes at a speed determined by the wind speed (w). His speed with the larger sail is 50w MPH. Similarly, when he uses his 12 square foot sail, his speed is 20w MPH. The wind speed is expected to increase linearly throughout the day, starting with no wind in the morning (w=0) and reaching a maximum wind speed of 1 by the end of the day.

During a 10-hour sailing period, Danny must decide which sail to use and plans to travel a total distance of 200 miles. How many hours faster can he complete the journey with the larger sail compared to the smaller sail, given the increasing wind speed throughout the day?
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Context:
Given the triangle $\triangle ABC$ is an isosceles right triangle with $\angle ABC = 90^\circ$, and sides $\overline{AB}$ and $\overline{BC}$ are each tangent to a circle at points $B$ and $C$ respectively, with the circle having center $O$ and located inside the triangle, calculate the fraction of the area of $\triangle ABC$ that lies outside the circle.
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Context:
On the lateral sides $AB$ and $CD$ of trapezoid $ABCD$, it is possible to select points $K$ and $L$ respectively, such that the segment $KL$ is not parallel to the bases and is divided into three equal parts by the diagonals. Find the ratio of the bases of the trapezoid.
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Context:
Compute the integral \( \iint_D (x^5y + xy^5) \, dA \) where \( D = \{ (x, y) : 0 < x, 0 < y, x^2 + y^2 < 4, x^2 - y^2 > 1 \} \), using the substitution \( u = x^2 + y^2 \) and \( v = x^2 - y^2 \). Assume \( x > 0 \) in the domain \( D \).
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Context:
A semicircle is cut out of paper with diameter \( AB \). A point \( M \) is marked on \( AB \), and two smaller semicircles with diameters \( AM \) and \( MB \) are cut out from the original semicircle. The area of the remaining figure is \( 16 \pi^3 \). Find the length of the chord passing through point \( M \) and perpendicular to \( AB \) that lies inside the remaining figure. Round your answer to two decimal places if necessary.
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Context:
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$.
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Context:
Suppose that the error, in grams, of a balance has the density function \(f(x)=\frac{1}{4}e^{\frac{-|x|}{2}}\) for \(-\infty<x<\infty\), and that 100 items are weighed independently. Use the Central Limit Theorem to approximate the probability that the absolute difference between the true total weight and the measured total weight is more than 50 grams.
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Context:
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$
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Context:
Calculate the line integral \( \int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1} \) where \( A \) is the line \(|x|+|y|=4\), travelling clockwise and making one rotation. Use Green's theorem to find the value of the integral.
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Context:
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 12.59899208063355 seconds. If the shorter train is 160 m long, how long is the longer train?
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Context:
Find the positive integer $n$ such that:
\[
\underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1
\]
where $f(n)$ denotes the $n$th positive integer which is not a perfect square.
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Context:
How many ways are there to distribute 6 copies of one book, 7 copies of a second book, and 11 copies of a third book between two teachers if each teacher receives exactly 12 books and each teacher gets at least 2 copies of each book?
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Context:
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
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Context:
Given the set \( P = \{ x \mid x = 7^3 + a \times 7^2 + b \times 7 + c \} \), where \( a \), \( b \), and \( c \) are positive integers not exceeding 6, if \( x_1, x_2, \ldots, x_n \) are elements of set \( P \) forming an arithmetic sequence, find the maximum value of \( n \).
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Context:
Determine the number of points on the elliptic curve given by \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\).
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Context:
Determine the number of solutions to the equation \(x^4 - 1 = e^x\).
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Context:
In a soap company, a soap is manufactured with some parts. For making one soap, you will get 1 part as scrap. At the end of the day, they have 251 such scraps. From that, they can manufacture 25 more soaps. How many parts are needed to manufacture one soap?
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Context:
Compute the limit: $$ \lim_{n \to \infty}\int_{0}^{\pi} \frac{\sin\left(x\right)}{1 + 3\cos^{2}\left(nx\right)}\,\mathrm{d}x $$
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Context:
In a group of 21 people, each pair of people can communicate using a unique radio frequency. However, in every group of three people, exactly two of them do not communicate with each other. Determine the maximum number of unique frequencies required for this group.
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Context:
Find the sum of the residues of the rational function \( h(z) = \frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111} \) at its poles in the complex plane.
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Context:
Let \((X_{n})_{n \in \mathbb{N}}\) be a sequence of independent random variables, each with a Bernoulli distribution with parameter \(\frac{1}{2}\). Define the random variable \(Z_{n}\) for each \(n \in \mathbb{N}\) as follows:
\[ Z_{n} = \sum_{k=1}^{n} \frac{X_{k}}{2^k} \]
Find the limit in probability of the sequence \((Z_{n})_{n \in \mathbb{N}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points)
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Context:
Calculate the expected value \( \mathbb{E}(e^{M_n}) \), where \( M_n = \max(X_1, X_2, \ldots, X_n) \) and \( X_1, \ldots, X_n \) are iid random variables each with distribution \( \mathcal{Exp}(1) \). The cumulative distribution function of \( M_n \) is given by \( F_{M_n}(x) = (1 - e^{-x})^n \) for \( x \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $a=(2,1)$, $b=(-3,2)$, and vector $c$ that satisfies $c⊥(a+b)$, and $b/!(c-a)$, find $c=$\_\_\_\_\_\_\_\_.
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Context:
实数 $a 、 b$ 使得方程
$x^{3}-a x^{2}+b x-a=0\quad①$
有三个正实根. 求 $\frac{2 a^{3}-3 a b+3 a}{b+1}$ 的最小值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Compute $(-1 + i \sqrt{3})^8.$  Enter the answer in rectangular form.
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Context:
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the odd part of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m> 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.
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Context:
A computer program evaluates expressions without parentheses in the following way:

1) First, it performs multiplications and divisions from left to right one by one.
2) Then, it performs additions and subtractions from left to right.

For example, the value of the expression $1-2 / 3-4$ is $-3 \frac{2}{3}$. How many different results can we get if in the following expression each $*$ is independently replaced by one of the operators $+$, $-$, $/$, $\times$?

$1 * 1 * 1 * 1 * 1 * 1 * 1 * 1$
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Context:
Determine the cardinality of the set of real-valued continuous functions \( f \) defined on \([0,1]\) such that \( f(x) \) is rational whenever \( x \) is rational.
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Context:
Given that \( y = f(x) \) is a monotonous function defined on \(\mathbb{R}\), with real numbers \( x_{1} < x_{2} \), \(\lambda \neq -1\), \(\alpha = \frac{x_{1} + \lambda x_{2}}{1 + \lambda}\), and \(\beta = \frac{x_{2} + \lambda x_{1}}{1 + \lambda}\). If \(\left| f(x_{1}) - f(x_{2}) \right| < |f(\alpha) - f(\beta)|\), find the range of values for \(\lambda\).
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Context:
If the width of a rectangle is increased by 3 cm and the height is decreased by 3 cm, its area does not change. What would happen to the area if, instead, the width of the original rectangle is decreased by 4 cm and the height is increased by 4 cm?
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Context:
A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?
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Context:
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.
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Context:
Consider the partial differential equation (PDE): $$xu_x - yu_y = 0.$$ If two $\mathcal{C}^1$ functions, $u$ and $v$, both solve this equation and satisfy the condition $u(1, y) = v(1, y)$ for all $y \in \mathbb{R}$, determine the largest subset $A \subset \mathbb{R}^2$ on which $u$ and $v$ are equal.
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Context:
If $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 4$, and $(\overrightarrow{a} + \overrightarrow{b})$ is perpendicular to $\overrightarrow{a}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.
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Context:
Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?
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Context:
Find the asymptotic upper bound for the recurrence relation: \( T(n) = (T(n-1))^2 \) with the initial condition \( T(n) = n \) for \( n \leq 2 \).
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Context:
A vector in 110-dimensional Euclidean space $\mathbf{R}^{110}$ makes equal acute angles with the positive directions of the 110 coordinate axes. What is the approximate measure of that angle in degrees?
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Context:
Consider the bilinear form \(f(A, B) = n\operatorname{Trace}(AB) - \operatorname{Trace}(A)\operatorname{Trace}(B)\) defined on the space of \(n \times n\) complex matrices, \(M_n(\mathbb{C})\). Determine the dimension of the orthogonal complement \(U^\perp\), which consists of all matrices \(A\) such that \(f(A, B) = 0\) for every \(B \in M_n(\mathbb{C})\). Find \(\dim(U^\perp)\).
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Context:
Let \( \overline{\rm AB} \) be a diameter of a circle \( \omega \) and let \( C \) be a point on \( \omega \), different from \( A \) and \( B \). The perpendicular from \( C \) intersects \( \overline{\rm AB} \) at \( D \) and \( \omega \) at \( E (\ne C) \). The circle with center \( C \) and radius \( \rm CD \) intersects \( \omega \) at \( P \) and \( Q \). If the perimeter of \( \triangle{PEQ} \) is \( p \), find the length of \( \overline{\rm PQ} \).
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Context:
Find the sum of the series: $$ \sum_{n=1}^\infty \frac{\sqrt{n-\sqrt{n}}}{n^{1/n + 3/2}} $$
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Context:
Calculate the curl of the radial unit vector $e_r$ in spherical coordinates.
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Context:
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$?
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Context:
Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$:

(a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and

(b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n}$$
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Context:
In \(\mathbb{R}^4\), consider the surface \(S\) consisting of all points that are 1 unit away from the origin. Given five points \(P_1, P_2, \ldots, P_5\) lying on \(S\), determine the maximum angle formed between any two points and the origin, denoted as \(\angle P_nOP_k\) for distinct natural numbers \(n\) and \(k\) from 1 to 5, where \(O\) is the origin.
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Context:
In a regular tetrahedron, the center of the circumscribed sphere coincides with the center of the inscribed sphere. Determine the angle between two adjacent edges of the tetrahedron.
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Context:
Find the value of \(c\) such that the areas of the regions bounded by the parabolas \(y = cx^2\) and \(y = 1 - x^2\) in the first quadrant are equal.
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Context:
Consider the graph $G$ obtained by deleting any edge from the complete bipartite graph $K_{7,8}$. Determine the number of spanning trees in the complement graph $\overline{G}$ of $G$. Provide your answer as a single integer.
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Context:
Calculate the line integral \( \int_l (e^x \sin y - 7y) \, dx + (e^x \cos y - 7) \, dy \) where \( l \) is the upper section of the circle \( x^2 + y^2 = 16x \) from point \( A(16,0) \) to point \( B(0,0) \), using Green's theorem.
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Context:
$\prod\limits_{k=1}^{6}\left(x^{5}-5 x+k\right)=0$ 有 $\qquad$个实根。
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Context:
Let $\{e_i\}$ be a basis of a $\mathbb{C}$-linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix of $h$ for $\{e_i\}$, where $h_{ij}=h(e_i,e_j)$. Let $g\in GL(E)$ be a complex automorphism and consider the new basis $\{e'_i\}=\{ge_i\}$. Denote by $H'=\{h'_{ij}\}$ the corresponding matrix. Find the relation between $H$ and $H'$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the entropy of a discrete random variable with the probability distribution given by \( p(x) = \frac{1}{2^x} \) for \( x = 1, 2, 3, \ldots \).
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Context:
Consider the set \( A = \{1, 2, 3, 4, 5\} \). A function \( f: A \to A \) is chosen at random from the set of all nonempty relations on \( A \). What is the probability that \( f \) is a bijective function such that \( f(x) \neq x \) for all \( x \in A \) and \( f(1) \neq 2 \)?
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Context:
Evaluate the integral \( \int_{0}^{1} x^{m-1} e^{-x} \, dx \).
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Context:
In triangle ABC, where AB = 4024, AC = 4024, and BC = 2012, reflect the line AC over line AB to intersect the circumcircle of triangle ABC at point D (D ≠ A). Determine the length of segment CD.
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Context:
A has 2 dollars and B has 3 dollars. They toss a coin with the probability of heads being \( \frac{1}{3} \). If the coin lands on heads, A gives 1 dollar to B. If it lands on tails, B gives 1 dollar to A. What is the probability that B will win all the money? Provide your answer as a fraction.
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Context:
Given that the domain of the function \( f(x) = \frac{1}{\lg \left(2^x + 4 \cdot 2^{-x} - a \right)} \) is \(\mathbf{R}\), find the range of values for the real number \( a \).
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Context:
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of real numbers $x$ that satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$.
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Context:
Given that the variables $x$, $y$, and $z$ follow Gaussian distributions with means $\mu_x = \mu_y = \mu_z = 1,000,000$ and standard deviations $\sigma_x = \sigma_y = \sigma_z = 200,000$, calculate the probability that the function \[ f(x, y, z) = \frac{x}{1.1} + \frac{y}{1.1^2} + \frac{z}{1.1^3} - 2,000,000 > 0. \]
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Context:
Find a function \( g(\theta) \) such that the quantity \( Q(\theta, W) = g(\theta) W \) follows a chi-squared distribution with \( k \) degrees of freedom, where \( W(X_1, \dots, X_n) = \sum_{i=1}^n \ln\left(\frac{X_i}{8}\right) \) and \( X_1, \dots, X_n \) are i.i.d. with probability density function \( f(x;\theta) = \theta \frac{8^\theta}{x^{\theta +1}} \) for \( x \geq 8 \) and \( \theta > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kelly booked a three week vacation to visit relatives. On the first day, she spent 8 hours traveling by plane, including a 3-hour layover. Due to time zone changes, this took up two calendar days. The next 5 days, she spent at her Grandparents' house. The next day, she spent 10 hours traveling by train, but this journey involved an unexpected 2-hour delay. Then, she spends the next 5 days at her brother's house. After that, she spent 1 day traveling by car and 1 day traveling by bus to her sister's house, crossing two time zones, which added an extra day to her trip. She spent several days at her sister's house, and then spent 1 day traveling by bus and 1 day traveling by car back home without additional layovers. How many days did she spend at her sister's house?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of $n$ is the quotient ring $\mathbb{C}[x_1,\dots,x_n]/(\sum_{i=1}^n x_i^2)$ reduced, meaning it contains no nilpotent elements?
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} \int_0^{2 \pi} \frac{1}{x+\sin^n x+ \cos^n x} \, dx \]
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Context:
In a large cube composed of 64 unit cubes arranged in a 4x4x4 grid, select 16 unit cubes to be painted red such that each vertical column of 4 cubes contains exactly one red cube. How many different ways can the 16 red unit cubes be selected?
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Context:
Given a matrix with its LU decomposition, if the last corner entry is $A(5,5) = 11$ and the last pivot of $A$ is $U(5,5) = 4$, determine the value of $A(5,5)$ that would make the matrix singular.
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Context:
What is the largest number of points that can be arranged on a given circle such that in all the triangles they form, none of the angles is less than $25^\circ$?
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Context:
Determine the maximum possible length of the graph of a function \( f: [0,1] \to [0,1] \) that satisfies the following conditions:

- \( f(0)=0 \)
- \( f(1)=1 \)
- \( f \) is increasing.
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Context:
In triangle $ABC$, with the sides opposite angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, if $2b\cos B - c\cos A = a\cos C$, then the magnitude of angle $B$ is __________.
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Context:
Determine the number of triples of natural numbers \((a, b, c)\) that satisfy the following system of equations:

\[\begin{cases}
\gcd(a, b, c) = 15 \\
\text{lcm}(a, b, c) = 3^{15} \cdot 5^{18}
\end{cases}\]
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Context:
Calculate the flux of the vector field \( F = (3x, 4y, -z) \) through the surface \( S \) defined by the equations \( z^2 = x^2 + y^2 \) and \( z = x^2 + y^2 \).
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Context:
What is the largest integer that cannot be expressed as $2017a + 2018b$ where $a$ and $b$ are positive integers?
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Context:
Determine the number of points on the elliptic curve $E$ given by $y^2 = x^3 + bx + c$ over the integers modulo $221 = 13 \cdot 17$. There exist points $P$ and $Q$ on $E$ such that $11P = \mathcal{O}$ and $7Q = \mathcal{O}$. What is the number of points on $E$?
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Context:
Mary just arrived at the beach. She has 4 times as many towels as Frances does. The total weight of their towels is 60 pounds. If Mary has 24 towels, how much do Frances's towels weigh in ounces?
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Context:
Evaluate the limit: $$\lim_{n\rightarrow \infty}\frac{\sqrt[n^2]{(1!)^{\alpha} \cdot (2!)^{\alpha} \cdot (3!)^{\alpha} \cdots (n!)^{\alpha}}}{n^q}$$ where $\alpha$ is a positive constant.
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Context:
What is the smallest value of \( k \) such that one can mark \( k \) cells on an \( 8 \times 9 \) board, so that no matter how a four-cell figure is placed on the board (allowing for rotations and reflections), it always covers at least one of the marked cells?
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Context:
Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values?
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Context:
A regular octagon has sides of length 2 cm. When rectangles PQTU and RSVW are shaded, four small triangles inside the octagon remain unshaded. Find the total area of these four triangles. Express your answer in square centimeters (cm²) as a whole number.
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Context:
Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. The equation satisfied by $t$ is of the form \(\left(\frac{k}{m}+\frac{1}{n}\right)(t-p)=q\). Please find the value of \(k + m + n + p + q\).
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Context:
Find all non-negative integer solutions to \(5a \cdot 7b + 4 = 3^c\).
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Context:
 Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland?

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Context:
Given a graph, determine the number of spanning trees that include the edges QS and RS.
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Context:
Given an $n$-dimensional vector space $V$ over a field $F$ with basis $\mathcal{B} = \{\mathbf{v_1}, \ldots, \mathbf{v_n}\}$, let $\mathcal{B}^{*} = \{\phi_1, \ldots, \phi_n\}$ be the dual basis for $V^{*}$. If $\psi : V \to V$ is a linear transformation and $A$ is the matrix representing $\psi$ relative to $\mathcal{B}$, with $\psi^t : V^{*} \to V^{*}$ denoting the transpose of $\psi$ and $B$ being the matrix representing $\psi^t$ relative to $\mathcal{B}^{*}$, derive the relationship between matrices $A$ and $B$. What is the relationship between $A$ and $B$?
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Context:
Determine if the function \( f(x) = \begin{cases} x^2 & \text{if } x \in \mathbb{Q}, \\ x^3 & \text{if } x \notin \mathbb{Q} \end{cases} \) is differentiable at \( x = 0 \).
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Context:
Alice and Bob play a game. Bob starts by picking a set \( S \) consisting of \( M \) vectors of length \( n \) with entries either 0 or 1. Alice picks a sequence of numbers \( y_1 \leq y_2 \leq \cdots \leq y_n \) from the interval \( [0, 1] \) and a choice of real numbers \( x_1, x_2 \ldots, x_n \in \mathbb{R} \). Bob wins if he can pick a vector \( (z_1, z_2, \ldots, z_n) \in S \) such that

\[
\sum_{i=1}^{n} x_i y_i \leq \sum_{i=1}^{n} x_i z_i,
\]

otherwise Alice wins. Determine the minimum value of \( M \) so that Bob can guarantee a win.

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Context:
How many distinct cellular automata rules are there that use the Moore neighborhood on a two-dimensional Cartesian lattice if each site can be in one of eight states?
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Context:
Raashan, Sylvia, and Ted play the following game. Each starts with \$1. A bell rings every 15 seconds, at which time each of the players who currently has money simultaneously chooses one of the other two players independently and at random and gives \$1 to that player. What is the probability that after the bell has rung 2019 times, each player will have \$1?

For example, Raashan and Ted may each decide to give \$1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have \$1, Sylvia will have \$2, and Ted will have \$1, and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their \$1 to, and the holdings will be the same at the end of the second round.

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Context:
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?
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Context:
Find the area of the region defined by the inequality \( \left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2 \).
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Context:
Determine whether there exists an element $\beta$ in the field $\mathbb{Q}(y_1, \ldots, y_5)$ such that $\beta^2$ is in the field $\mathbb{Q}(f_1, \ldots, f_5)$, where $f_i$ are the elementary symmetric polynomials in $y_1, \ldots, y_5$, but $\beta$ is not in $\mathbb{Q}(f_1, \ldots, f_5)$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Problem A3

Find the smallest number $\mathrm{n}$ such that given any $\mathrm{n}$ distinct numbers from $\{1,2,3, \ldots, 999\}$, one can choose four different numbers $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ such that $\mathrm{a}+2 \mathrm{~b}+3 \mathrm{c}=\mathrm{d}$.

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


Context:
Determine whether the product \( p = \prod_{j=2}^\infty \zeta(j)^{-1} \) is greater than zero, assuming the product converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the star-shaped figure below, all side lengths are equal to 3, and the three largest angles of the figure are 210 degrees. The area of the figure can be expressed as \( \frac{a \sqrt{b}}{c} \), where \( a, b, \) and \( c \) are positive integers such that \( a \) and \( c \) are relatively prime and \( b \) is square-free. Compute \( a + b + c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the expression \( \sum_{i=0}^{a-1}\binom{a}{i}\binom{2a-i}{a} \) is even.
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Context:
Determine whether the limit \(\lim_{(u,v)\to(-\infty,\infty)}e^{u+jv}\) exists, where \(u,v\in\mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the kernel of the homomorphism \( \phi : \mathbb{R}[X] \to M_{2 \times 2}(\mathbb{R}) \) defined by \( \phi\left(\sum_{i=0}^d r_i X^i\right) = \sum_{i=0}^d r_i \mathbf{A}^i \), where \( \mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \), without using the Cayley-Hamilton Theorem.
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Context:
Given a pair  $(a_0, b_0)$  of real numbers, we define two sequences  $a_0, a_1, a_2,...$  and  $b_0, b_1, b_2, ...$  of real numbers by  $a_{n+1}= a_n + b_n$  and  $b_{n+1}=a_nb_n$  for all  $n = 0, 1, 2,...$ . Find all pairs  $(a_0, b_0)$  of real numbers such that  $a_{2022}= a_0$  and  $b_{2022}= b_0$ .
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Context:
Evaluate the integral \( \int_{0}^{1}{(\{2x\}-1)(\{3x\}-1)}\,dx \), where \( \{x\} \) denotes the fractional part of \( x \).
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Context:
In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Given a positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$ and $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are relatively prime to $n$. Find all $n$ for which one of the three numbers $n,\tau(n), \varphi(n)$ is the arithmetic mean of the other two.
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Context:
Find the smallest positive number  $\lambda $  , such that for any complex numbers  ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$  ,if   $z_1+z_2+z_3=0$ , then   $$ \left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda . $$
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Context:
Determine if it is possible to express the right-hand side of the identity \( (1+6x^3+9x^4)^3+(1-6x^3+3x-9x^4)^3+(1-9x^3-6x^2)^3=9x^2+9x+3 \) as a perfect cube by adding a suitable term to both sides. If possible, find the term.
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Context:
Solve the system of equations: \[ \begin{cases} x + y^2 = 7 \\ x^2 + y = 11 \end{cases} \]
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Context:
Let $k$ be a field of characteristic zero, $A$ a simplicial commutative $k$-algebra, and $M$ a simplicial $A$-module. Consider the trivial square-zero extension $A \oplus M$ as an $A$-algebra. Is the relative cotangent complex of $A \oplus M$ over $A$ (i.e., the cotangent complex of the map $A \rightarrow A \oplus M$) isomorphic to $M$ in the derived category of $A$-modules?
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Context:
Find the general form of a differentiable function $g(x)$ that satisfies the functional equation $g(x+y) = g(x) + g(y)$ for all real numbers $x$ and $y$. Assume $g(x)$ is differentiable at $x = 0$. What is the general form of $g(x)$?
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Context:
In a game with Odin and Evelyn, there are initially $3k$ empty boxes, where $k$ is a positive integer. Odin goes first and can only write odd numbers in the boxes, while Evelyn can only write even numbers. On each turn, a player can either write a non-negative integer in an empty box or replace a number in a box with a strictly smaller non-negative integer. The game ends when a player cannot move, resulting in the other player's victory, or when there are exactly $k$ boxes with the number $0$. In the latter case, Evelyn wins if all other boxes contain the number $1$, otherwise Odin wins. Determine who has a winning strategy.
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Context:
Determine the type and order of the singularity at \( z=0 \) for the function \( \frac{1}{z(e^z-1)} \).
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Context:
Evaluate the line integral \( \oint_{C} \left(-\frac{x^2y}{(x^2+y^2)^2}\,dx + \frac{x^3}{(x^2+y^2)^2}\,dy\right) \) where \( C \) is the boundary of the region defined by \( 4x^2 + y^2 = 1 \). Consider the implications of the singularity at the origin on the validity of Green's Theorem for this problem.
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Context:
There are $100$ books in a row, numbered from $1$ to $100$ in some order. An operation is choose three books and reorder in any order between them(the others $97$ books stay at the same place). Denote that a book is in correct positionif the book $i$ is in the position $i$. Determine the least integer $m$ such that, for any initial configuration, we can realize $m$ operations and all the books will be in the correct position.
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Context:
Determine the minimum size of a family of subsets of the set \([n]=\{1,2,\ldots,n\}\) such that for any distinct elements \(i,j,k\in[n]\), there exists a subset in the family containing both \(i\) and \(j\) but not \(k\).
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Context:
Find the sum of the series:
\[ 1 + \frac{1\cdot3}{6} + \frac{1\cdot3\cdot5}{6\cdot8} + \cdots \]
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Context:
For which greatest $n$ there exists a convex polyhedron with $n$ faces having the following property: for each face there exists a point outside the polyhedron such that the remaining $n - 1$ faces are seen from this point?
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Context:
Determine the value of \( n \) such that all solutions of the differential equation \( y'' + ny = \cos(nx) \) are unbounded.
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Context:
(2000•上海)圆锥曲线 $$\begin{cases}x=4\sec\theta+1 \\y=3\tan\theta \end{cases}$$ 的焦点坐标是多少？
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Context:
Determine the domain of the operator $|\Delta|^{1/2}$, where $U$ is a bounded open subset of $\mathbb{R}^N$ and $\Delta: H^2(U) \subset L^2(U) \to L^2(U)$ is the Laplace operator.
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Context:
In a regular tetrahedron \( ABCD \), the circumcenter is denoted as \( O \). Let \( E \) be the midpoint of edge \( BC \). Determine the measure of the dihedral angle between the planes \( A-BO-E \).
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Context:
We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.
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Context:
Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.
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Context:
Given a matrix \( B \) and a matrix \( \Psi \) defined as:

\[\Psi=\begin{bmatrix} e_{2}^{T} & \\ e_{2}^{T}B \\ e_{2}^{T}B^2 \\ .\\ .\\ .\\ e_{2}^{T}B^{m-1}\end{bmatrix}\]

where \( e_{2} \) is the standard basis vector with the second entry 1 and others 0. If \( B \) is a block diagonal matrix with at least two \( 2\times 2 \) rotation matrices as blocks, determine if the determinant of \( \Psi \) is zero.
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Context:
Let $C$ be a nonsingular curve of degree $n$ defined over a field $F$. If a line intersects $C$ in $n-1$ points that are defined over $F$, is the $n$th point of intersection also defined over $F$? Answer 'yes' or 'no'.
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Context:
Regular hexagon $NOSAME$ with side length $1$ and square $UDON$ are drawn in the plane such that $UDON$ lies outside of $NOSAME$. Compute $[SAND] + [SEND]$, the sum of the areas of quadrilaterals $SAND$ and $SEND$.

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Context:
Determine the splitting field $F$ of the polynomial $f(x)=x^4+16$ over the real numbers $\mathbb{R}$ and find the degree of the extension $[F:\mathbb{R}]$. Provide your answer as a pair $(F, [F:\mathbb{R}])$.
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Context:
Let $S$ be the set of all point $(x,y)$ in the coordinate plane such that $0 \le x \le \frac{\pi}{2}$ and $0 \le y \le \frac{\pi}{2}$. What is the area of the subset of $S$ for which $$\sin^2x-\sin x \sin y + \sin^2y \le \frac34?$$

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Context:
Gavril threw a ball with a speed of 5 m/s at an angle to the horizon on a sunny day when the sun was directly overhead. After 1 second, the ball hit his abdomen 1 meter below the point of release. Assuming the force exerted by the air is horizontal and constant, and the acceleration due to gravity is \( g = 10 \mathrm{~m/s}^2 \), determine the maximum distance the ball's shadow moved away from Gavril's feet.
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Context:
3. How many spheres exist that touch all the faces of the given triangular pyramid $T$?

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Context:
Evaluate the integral \( \int \cdots \int f(x_1, \cdots, x_n) \prod_{i=1}^{n} \delta(x - x_i) \, dx_i \). Is the result equivalent to \( f(x, \cdots, x) \)?
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Context:
Evaluate the integral \( \displaystyle\int\limits_\gamma \frac{2z+1}{e^{\pi z}-1}dz \) along the specified path \( \gamma \).
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Context:
Example 3 If $x, y, z$ are all positive real numbers, find the maximum value of $\frac{x y z}{(1+5 x)(4 x+3 y)(5 y+6 z)(z+18)}$. (2003 Singapore Mathematical Olympiad)
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Context:
Given triangle $ABC$ with base $AB$ fixed in length and position. As the vertex $C$ moves on a straight line, the intersection point of the three medians moves on:
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Context:
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 28. What is $r$?

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Context:
Let $\rho_1$ and $\rho_2$ be two representations of a group $G$, with $V$ and $W$ as the corresponding $FG$-modules. The tensor product $\rho_1 \otimes \rho_2$ is defined by $\rho_1 \otimes \rho_2: G \to GL(V \otimes_F W)$, where $(\rho_1 \otimes \rho_2)(g)(v \otimes w) = (\rho_1(g)(v), \rho_2(g)(w))$. Suppose the $FG$-module $V \otimes_{FG} W$ affords the representation $\rho_3$. Determine if $\rho_3 \sim \rho_1 \otimes \rho_2$. Provide your answer as 'yes' or 'no'.
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Context:
Evaluate the limit using L'Hopital's rule to determine if it diverges: $$\lim_{x \to 2} \frac{\int_1^{(x-2)^5} \cos(t^4)\, dt}{(x-2)^{18}}$$
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Context:
Given triangle $ABC$, reflect vertex $A$ over the perpendicular bisector of side $BC$, then reflect the resulting point over the perpendicular bisector of side $CA$ to obtain point $A'$. Similarly, reflect point $B$ over the perpendicular bisector of side $CA$, and then over the perpendicular bisector of side $AB$ to obtain point $B'$. Finally, reflect point $C$ over the perpendicular bisector of side $AB$, and then over the perpendicular bisector of side $BC$ to obtain point $C'$. If triangle $A'B'C'$ is congruent to triangle $ABC$, does this imply that triangle $ABC$ is equilateral? Answer 'yes' or 'no'.
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Context:
Determine whether the limit \( \lim_{(x,y)\to(0,0)}\frac{x^3y^3}{x^8+y^4} \) exists.
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Context:
Determine whether a countable set can have uncountably many infinite subsets such that the intersection of any two distinct subsets is finite. Answer 'yes' or 'no'.
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Context:
We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?

Proposed by Nathan Xiong
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Context:
Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.
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Context:
Problem 4. For a positive integer $n$, two players A and B play the following game: Given a pile of $s$ stones, the players take turn alternatively with A going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $\mathrm{A}$ and $\mathrm{B}$ play perfectly, for how many values of $s$ the player A cannot win?

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Context:
Determine the number of ordered pairs \((x, y)\) of real numbers such that \(-100\pi \le x \le 100\pi\), \(-100\pi \le y \le 100\pi\), \(x + y = 20.19\), and \(\tan x + \tan y = 20.19\).
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Context:
The Caterpillar owns five different matched pairs of socks. He keeps the ten socks jumbled in random order inside a silk sack. Dressing in the dark, he selects socks, choosing randomly without replacement. If the two socks he puts on his first pair of feet are a mismatched pair and the two socks he puts on his second pair of feet are a mismatched pair, then what is the probability that the pair he selects for his third set of feet is a mismatched pair?

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Context:
Let $U$ be an open disc centered at $0$ with radius $R > 0$ in the complex plane, and let $f: U \setminus \{0\} \rightarrow \mathbb{C}$ be a holomorphic function. Suppose that for some $a \in \mathbb{C}$, the radial limits of $f$ at $0$ exist and are equal to $a$ for any $\theta \in [0, 2\pi)$:
\[
\lim_{r \rightarrow 0} f(r e^{i \theta}) = a,
\]
where $r$ is a positive real number. Determine whether the limit of $f$ at $0$ exists.
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Context:
Calculate the expected number of draws from a standard 52-card deck (with replacement) until you have exactly two Kings and two Jacks. Note that Kings and Jacks are not replaced once drawn, and the deck is shuffled after each draw. If you draw a third King or Jack, it is put back into the deck.
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Context:
How many positive integers divide the number $10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$ ?
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Context:
Suppose a positive sequence $\{a_n\}_{n=1}^\infty$ has a convergent subsequence with $\lim\limits_{j\to\infty}a_{n_j}=\mu$. If for all $n>0$, $|a_{n+1}-a_n|<\frac{1}{n}$, determine whether $\lim\limits_{n\to\infty}a_{n}=\mu$ is true.
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Context:
Given that Figure 1 is the graph of an even function \( f(x) \) and Figure 2 is the graph of an odd function \( g(x) \), determine the sum of the number of real roots of the equations \( f(f(x))=0 \), \( f(g(x))=0 \), \( g(g(x))=0 \), and \( g(f(x))=0 \). Let the number of real roots of these equations be \( a \), \( b \), \( c \), and \( d \) respectively. Find \( a + b + c + d \).
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Context:
Evaluate the integral: \[ \int_0^2 \left((1+x^3)^{1/2} + (x^2+2x)^{1/3}\right) \, dx \]
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Context:
Given a countably infinite set, what is the minimum number of infinite subsets that must be added to a cofinite topology to transform it into a discrete topology?
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Context:
For the function \( p_k(x) = (x + \sqrt{x^2 - 4})^k - (x - \sqrt{x^2 - 4})^k \), where \( x > 2 \) and \( k, j \) are natural numbers with \( k > j \), determine if \( p_k(x) > p_j(x) \) holds true.
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Context:
Find the value of $\beta^*$ that minimizes the Kullback-Leibler divergence when the observed random variables $X_1$ and $X_2$ are independent and normally distributed with means $\mu_1$ and $\mu_2$, and variances $\sigma_1^2$ and $\theta\sigma_1^2$, respectively.
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Context:
Evaluate the principal value of the integral: \[ \text{p.v.} \int_{-\infty}^{\infty} \frac{x \sin 4x}{x^2 - 1} \, dx \]
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Context:
Given a ring $S$ with identity and a specified number of elements, determine the number of ring homomorphisms from $\mathbb{Q}[x]$ to $S$. Provide your answer as a single integer.
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Context:
83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation)
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Context:
Find the remainder when the integer \( {\left(\sqrt{3} + 5\right)}^{103} - {\left(\sqrt{3} - 5\right)}^{103} \) is divided by 9.
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Context:
Find the exact value of the norm \( \|T\| \) for the operator \( T: \ell^2 \to \ell^2 \) defined by:

\[\begin{cases} (Tx)_1 = 0, \\ (Tx)_n = -x_n + \alpha x_{n+1}, \quad n\ge 2 \end{cases} \]

where \( \alpha \in \mathbb{C} \).
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Context:
Let $ABCD$ be a convex quadrilateral with positive integer side lengths, where $\angle{A} = \angle{B} = 120^{\circ}$, $|AD - BC| = 42$, and $CD = 98$. Determine the maximum possible value of $AB$. \( \boxed{} \)
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Context:
Consider the probability space \(([0,1]^2, \mathcal{B}[0,1], Leb_2)\). Let \(X: [0,1]^2 \rightarrow \mathbb{R}\) and \(Y: [0,1]^2 \rightarrow \mathbb{R}\) be random variables defined by \(X(\omega_1, \omega_2) = \omega_1 + \omega_2\) and \(Y(\omega_1, \omega_2) = \omega_2 - \omega_1\). Find the value of the joint cumulative distribution function \(F_{X,Y}(0.58,0.35)\).
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Context:
Given that
 \( \frac{1}{2!{17}!} + \frac{1}{3!{16}!} + \frac{1}{4!{15}!} + \frac{1}{5!{14}!} + \frac{1}{6!{13}!} + \frac{1}{7!{12}!} + \frac{1}{8!{11}!} + \frac{1}{9!{10}!} = \frac{N}{1!{18}!} \)
 find the greatest integer that is less than \( \frac{N}{100} \) .
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Context:
Let $(X_1, X_2, \ldots)$ be an independent sequence of random 2-dimensional vectors, where for each $n$, $X_n$ is uniformly distributed on the square with vertices $[\pm n, \pm n]$. Calculate the probability \( P(\{w: |X_n(w)| \rightarrow \infty \ \text{as} \ n \rightarrow \infty\}) \).
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Context:
Find the largest integer $n$ such that $n! < 100^{n-1}$. Provide your answer as an integer.
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Context:
\n\t Consider a three-person game involving the following three types of fair six-sided dice.\n\\begin{itemize}\n\t\\item Dice of type $A$ have faces labelled $2$, $2$, $4$, $4$, $9$, $9$.\n\t\\item Dice of type $B$ have faces labelled $1$, $1$, $6$, $6$, $8$, $8$.\n\t\\item Dice of type $C$ have faces labelled $3$, $3$, $5$, $5$, $7$, $7$.\n\\end{itemize}\nAll three players simultaneously choose a die (more than one person can choose the same type of die, and the players don\u0027t know one another\u0027s choices) and roll it.  Then the score of a player $P$ is the number of players whose roll is less than $P$\u0027s roll (and hence is either $0$, $1$, or $2$).\nAssuming all three players play optimally, what is the expected score of a particular player?
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Context:
Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.
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Context:
Determine the number of zeros of the polynomial \( z^4 + 3z^2 + z + 1 \) that lie in the right half-plane.
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Context:
$ P$ is the intersection point of diagonals of cyclic $ ABCD$. The circumcenters of $ \triangle APB$ and $ \triangle CPD$ lie on circumcircle of $ ABCD$. If $ AC + BD = 18$, then area of $ ABCD$ is ?
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Context:
Given a distributional equation $u=h$ with $u \in L^2$ and $h\in W^{-1,2}$, determine if $h\in L^2$.
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Context:
Let \( \omega \) be a complex number which satisfies the equation \( i \omega = \frac{u^2 -1}{u^2 +1} \) for some suitable complex number \( u \). Find the value of \( \tan^{-1}{\omega} \).
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Context:
Evaluate the limit: $$\lim_{x \to 0} \left( 1 + \sin \left( \frac{3}{x} \right) \right)^x$$
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Context:
Find a function $f(x)$ such that the $n$-th derivative of $f$ evaluated at $0$ is $n^{n-1}$, where $n$ is a positive integer.
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Context:
In how many ways can 9 black and 9 white rooks be placed on a 6 × 6 chessboard such that no white rook can capture a black rook? A rook can capture another piece if it is in the same row or column as the other piece, with no other pieces between them.
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Context:
Find the limit of the sequence defined by the recurrence relation \(x_{n+1} = \alpha x_n + (1+\alpha) x_{n-1}\) for \(\alpha \in (0,2)\) and all \(n \geq 1\), in terms of \(\alpha\), \(x_0\), and \(x_1\).
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Context:
Given the equation \( \theta+\tan^{-1}\left(\frac{6}{4-x}\right)+\tan^{-1}\left(\frac{3}{x}\right)=\pi \), where \( x \) is a function of time \( t \), find the value of \( \frac{d\theta}{dt} \) when \( x=2 \) and \( \frac{dx}{dt}=4 \) cm/sec.
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Context:
In an exotic country, the National Bank issues coins that can take any value in the interval [0, 1]. Find the smallest constant $ c > 0 $ such that the following holds, no matter the situation in that country:

Any citizen of the exotic country that has a finite number of coins, with a total value of no more than 1000, can split those coins into 100 boxes, such that the total value inside each box is at most $ c $.
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Context:
Identify a topological space whose fundamental group is isomorphic to \(\mathbb{Z}_3\).
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Context:
Determine the number of subfields $K$ of the complex splitting field $L$ of the polynomial $f(x) = x^5 - 4x + 2$ over $\mathbb{Q}$, such that the degree of the extension $(K : \mathbb{Q}) = 2$. Provide your answer as a single integer.
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Context:
Determine the value of \( x \) such that the equation \( e^{e^{10^{10^{2.8}}}} = 10^{10^{10^{10^{x}}}} \) holds true, given that you can only change the topmost number on the right side while keeping all other numbers as 10.
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Context:
Calculate the value of \( \displaystyle \sum ^{2020}_{n=1} n^{2019} \pmod{2019} \).
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Context:
Determine all positive integers $n$ such that $n$ divides $a^{25} - a$ for every positive integer $a$.
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Context:
Calculate the expected value of the largest power of 2 that divides a randomly chosen non-zero positive even integer $E$, expressed as $E = A \cdot 2^n$, where $A$ is an odd number.
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Context:
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum .
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Context:
Find the elements of $U(L)$ that satisfy the grouplike element property for a Lie algebra $L$.
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Context:
Let $F$ be a finite field of order 32. Determine the number of solutions $(x, y) \in F \times F$ for the equation $x^2 + y^2 + xy = 0$. Provide your answer as a single integer.
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Context:
Determine all real numbers \(x\) for which the series \(\sum_{n=1}^\infty \left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin nx}{n}\) converges.
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Context:
Suppose $\alpha$ is a root of the polynomial $x^3 + x^2 + 1$ in the finite field $\mathbb{F}_2$, and let $K = \mathbb{F}_2(\alpha)$. If $f$ is an irreducible polynomial in $K[x]$ of degree 4, and $\beta$ is a root of $f$, consider the splitting field $L$ of $f$ over $K$. Given that $K$ has 8 elements, determine the number of elements in the field $L$.
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Context:
Given a function \(f: \mathbb{R} \to \mathbb{R}\) that satisfies the equation \[f(2xy) + f(f(x + y)) + f(x + y) = xf(y) + yf(x)\] for all real numbers \(x\) and \(y\), determine the sum of all possible absolute values \(|f(48)|\).
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Context:
The sports plane flew along a diamond-shaped route in windy weather. He flew through the first three sides of the rhombus in $a $, $b$ and $c$ hours, respectively. How long did it take him to cover the fourth side of the diamond? (The speed of an aircraft is a vector equal to the sum of two vectors: the aircraft own speed and the wind speed. Wind speed is a constant vector. The aircraft own speed is a vector of constant length).
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Context:
Determine the interval of convergence for the series \(1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots\) using an appropriate convergence test.
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Context:
Determine the maximum number of axes of symmetry that a three-dimensional figure, consisting of three lines, can have if no two of the lines are parallel and no two coincide.
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Context:
Solve the equation \(3^{\log_a x} + 3x^{\log_a 3} = 2\) for \(x\).
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Context:
Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?

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Context:
Determine the number of terms required to approximate the sum of the series \(\sum_{n=0}^\infty \frac{(-1)^n(5)^{2n+1}}{(2n+1)(2n+1)!}\) to within an error of \(10^{-6}\).
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Context:
Find the value of \(\left(\frac{1}{4}\right)!\).
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Context:
Is the conditional variance Var(X|\(\mathcal{P}(\Omega)\)) equal to 0 for any random variable X?
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Context:
Consider the vector space $[0,1]^\mathbb{N}$ of real sequences with values in the interval $[0,1]$ together with the product topology. Determine if the subspace consisting of all monotone sequences with values in $[0,1]$ is connected.
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Context:
Determine the value of $f(0,0)$ for polynomials $f(x,y)$ and $g(x,y)$ in two variables, given that $g(0,0)=6$ and \((x^2+2xy+3y^2)f(x,y)=(4x^2+5xy+6y^2)g(x,y)\) holds for all real numbers $x$ and $y$ satisfying \(x^2y+6y=xy^3+5x^2+2x\).
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Context:
Determine whether the $n$-th singular homology group of an oriented, compact, and connected $n$-dimensional smooth manifold with boundary, denoted as $H_n(M)$, is trivial.
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Context:
Determine the number of ordered pairs of integers \((m, n)\) such that \(mn \geq 0\) and \(m^3 + n^3 + 99mn = 33^3\).
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Context:
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.

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Context:
Suppose $h$, $i$, $o$ are real numbers that satisfy the products $hi = 12$, $ooh = 18$, and $hohoho = 27$. Find the value of the product $ohio$.

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Context:
Consider a continuous and non-decreasing distribution function $F$ with an associated density function $f$. Let $S_1$ be the set where $F(s+\varepsilon)> F(s-\varepsilon)$ for all $\varepsilon>0$, and $S_2$ be the closure of the set where $f(s)>0$. Determine if these two sets are equal, i.e., is $S_1=S_2$?
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Context:
Given a pair (N, S) in the Tonalpohualli calendar, where N is the number and S is the symbol, determine the original day number from 1 to 260 that corresponds to this pair. For example, find the day number that produces the pair (3, 12).
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Context:
Determine whether the series \( \sum\limits_{n=1}^{\infty}\frac{n^{n-1}}{n!e^n} \) converges to 1.
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Context:
Consider the product measure \((\mu \times \nu)\) defined on \(\mathbb{R} \times \widetilde{\mathbb{R}}\), where \(\mu\) is the Lebesgue measure on \(\mathbb{R}\) and \(\nu\) is the counting measure on \(\widetilde{\mathbb{R}}\), the real line with the discrete topology. Let \(E = \{(x, x) \mid x \in [0, 1]\}\). Determine whether \((\mu \times \nu)(E) = 0\) when \(\widetilde{\mathbb{R}}\) is endowed with the discrete topology and the associated discrete \(\sigma\)-algebra.
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Context:
Find the dimension of the symplectic group \(\operatorname{Sp}(2n, \mathbb{R})\), which is the group of invertible matrices \(A\) such that \(\omega = A^T \omega A\), where \(\omega = \begin{pmatrix} 0 & \mathit{Id}_n \\ -\mathit{Id}_n & 0 \end{pmatrix}\).
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Context:
Given a set of points $X$ in $\mathbb{R}^2$, define $g(X)$ as the geometric median of $X$, where $g(X) = \arg \min_{y \in \mathbb{R}^2} \sum_{x_i \in X} ||x_i - y||_2$. For two neighboring sets $X$ and $X'$ that differ by only one element ($X \sim X'$), determine the supremum of the Euclidean distance between their geometric medians, $\sup_{X':X \sim X'} ||g(X) - g(X')||_2$. Find this supremum.
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Context:
Let $f:\mathbb{R}^{3}\to\mathbb{R}^{2}$ be defined by $f(r,s,t)=(r^{3}s+t^{2},rst)$. Let $g:\mathbb{R}^{2}\to\mathbb{R}^{11}$ and $h:\mathbb{R}^{11}\to\mathbb{R}^{4}$ be differentiable functions. Compute the Jacobian matrix $J_{h\circ g\circ f}(0,21,0)$. What is the size of the resulting Jacobian matrix?
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Context:
Calculate the flux of the vector field \( F(x,y,z) = (3z^2y, 2x^2y, 2y^2z) \) through the surface \( S = \{ (x,y,z) \in \mathbb{R}^3 : 2\sqrt{x^2+y^2} \leq z \leq 1+x^2+y^2 \} \). Use the divergence theorem to evaluate the flux.
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Context:
Evaluate the limit: $$\lim_{n\to \infty} \Biggl(\left(1+\dfrac{-n^2+2n-3}{n^3-n^2}\right)^{\dfrac{n^3-n^2}{-n^2+2n-3}}\Biggr)^{\frac{n^4-2n^3+3n^2}{n^3-n^2}}$$
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Context:
Find an element \( \alpha \) such that \( \mathbb{Q}(i,\sqrt[3]{2}) = \mathbb{Q}(\alpha) \).
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Context:
In an infinite and stationary universe, consider the intensity of light from distant objects. If the intensity falls off as $1/d^k$, where $d$ is the distance to the object, determine how the intensity changes as a function of dimensionality.
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Context:
Let $G$ be a finite group of order $2p$, where $p$ is an odd prime. Suppose $G$ has a subgroup $H$ of order $p$ and for every element $g \in G$, $g^2 \in H$. Determine the number of elements of order 2 in the group $G$. \( \boxed{} \)
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Context:
What is the smallest number of subsets needed to partition the set of all integers such that none of the subsets contain an infinite arithmetic progression?
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Context:
Given a Hilbert space $H$ with an orthonormal basis $(e_n)_{n \in \mathbb{N}}$, and a bounded linear operator $T \in \mathbf{B}(H)$ represented by the matrix $A = (\alpha_{nm})$ where $\alpha_{nm} = (Te_n|e_m)$, find the matrix representation of the adjoint operator $T^*$.
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Context:
The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.
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Context:
Determine the values of \( n \) for which a regular hexagon can be divided into \( n \) parallelograms of equal area.
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Context:
In a regular dodecahedron, how many distinct planes can intersect the dodecahedron to form a regular hexagon?
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Context:
Real numbers $x,y,$ and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that $$\vert{x}\vert+\vert{y}\vert+\vert{z}\vert+\vert{x+y+z}\vert=\vert{x+y}\vert+\vert{y+z}\vert+\vert{z+x}\vert?$$
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Context:
Jerry recently returned from a trip to South America where he helped two old factories reduce pollution output by installing more modern scrubber equipment. Factory A previously filtered $80$% of pollutants and Factory $B$ previously filled $72$% of pollutants. After installing the new scrubber system, both factories now filter $99.5$% of pollutants.

Jerry explains the level of pollution reduction to Michael, "Factory $A$ is the much larger factory. It's four times as large as Factory $B$. Without any filters at all, it would pollute four times as much as Factory $B$. Even with the better pollution filtration system, Factory $A$ was polluting nearly three times as much as Factory B."

Assuming the factories are the same in every way except size and previous percentage of pollution filtered, find $a+b$ where $a/b$ is the ratio in lowest terms of volume of pollutants unfiltered from both factories $\textit{after}$ installation of the new scrubber system to the volume of pollutants unfiltered from both factories $\textit{before}$ installation of the new scrubber system.

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Context:
Determine the cardinality of the set of all sequences in \(\mathbb{R}\) that converge to a real number \(a\).
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Context:
If an infinite series $S$ converges to a number $N$, does the exponential function $e^S$ converge to $e^N$?
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Context:
Find the value of \( c \) such that the sequence \((S^3_n - cn)_{n \in \mathbb{N}_0}\) is a martingale, where \( S_n \) is a random walk.
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Context:
Find the splitting field and its degree over \( \mathbb{Q} \) for the polynomial \( f(x) = x^4 - 2 \).
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Context:
For a positive integer \( n \) we denote by \( d\left( n\right) \) the number of positive divisors of \( n \) and by \( s\left( n\right) \) the sum of these divisors. For
 example, \( d\left( {2018}\right) \) is equal to 4 since2018has four divisors(1,2,1009,2018)and
 \( s\left( {2018}\right)  = 1 + 2 + {1009} + {2018} = {3030} \) .
 Determine all positive integers \( x \) such that \( s\left( x\right)  \cdot  d\left( x\right)  = {96} \) .
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Context:
One writes, initially, the numbers  $1,2,3,\dots,10$  in a board. An operation is to delete the numbers  $a, b$  and write the number  $a+b+\frac{ab}{f(a,b)}$ , where  $f(a, b)$  is the sum of all numbers in the board excluding  $a$  and  $b$ , one will make this until remain two numbers  $x, y$  with  $x\geq y$ . Find the maximum value of  $x$ .
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Context:
Find all real numbers $x$ so that the product $(x + i)((x + 1) + i)((x + 2) + i)$ is pure imaginary.  Enter all the solutions, separated by commas.
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Context:
Let $n\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than $1$ that is a perfect $n^{\mathrm{th}}$ power.
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Context:
Given an orthogonal $3 \times 3$ matrix $R$ represented as:

$$R = \begin{bmatrix}
r_{11}&r_{12}&r_{13} \\
r_{21}&r_{22}&r_{23} \\
r_{31}&r_{32}&r_{33} \\
\end{bmatrix}$$

where the parameters $r_{11}, r_{21}, r_{31}, r_{12}, r_{22}, r_{32}$ are known, determine how many unique solutions exist for the remaining parameters $r_{13}, r_{23}, r_{33}$. If there is a unique solution, describe the method to find it using the orthogonality condition, considering that $R$ represents a rotation.
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Context:
Determine the average number of coin flips required such that the difference between the percentage of heads and tails is less than 0.1% with a probability of at least 99%.
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Context:
A father has a pie made for his two sons. Eating more than half of the pie will give indigestion to anyone. While he is away, the older son helps himself to a piece of the pie. The younger son then comes and has a piece of what is left by the brother. Assume that the size of each of the two pieces eaten by the sons is random and uniformly distributed over what is currently available. What is the expected size of the remaining piece given that no son has indigestion?
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Context:
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not by 100 (such as
 2012). The200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7,2012, a Tuesday. On what
 day of the week was Dickens born?

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Context:
Find the basis and determine the dimension of the tangent space at the identity for the symplectic group $Sp(2n, \mathbb{R})$. Provide your answer in terms of $n$.
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx. \]
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Context:
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that
\[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]
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Context:
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$.
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Context:
What is the perimeter of the boundary of the region consisting of all points which can b
 expressed as \( \left( {{2u} - {3w},v + {4w}}\right) \) with \( 0 \leq  u \leq  1 \) , \( 0 \leq  v \leq  1 \) , and \( 0 \leq  w \leq  1 \) ?

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Context:
Let \( n \geq 2 \) be a given integer. For two permutations \((\alpha_1, \ldots, \alpha_n)\) and \((\beta_1, \ldots, \beta_n)\) of \(1, \ldots, n\), consider the \(n \times n\) matrix \(A = (a_{ij})_{1 \leq i,j \leq n}\) defined by \(a_{ij} = (1 + \alpha_i \beta_j)^{n-1}\). Determine all possible values of \(\det(A)\).
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Context:
公园里的自动旋转喷灌装置的喷灌半径是5米，它的最大喷灌面积是多少平方米．
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Context:
Evaluate the line integral \( \int_{\alpha} x \, dy - y \, dx \) where \( \alpha \) is the boundary of the parallelogram with vertices \((0,0), (a,b), (c,d), (a+c,b+d)\) oriented counterclockwise.
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Context:
Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers.
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Context:
Determine if there is a closed form solution to the equation $ae^a = 1$.
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Context:
The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels.

(Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)

[i]Proposed by Michael Ma[/i]

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Context:
Determine the eigenvalues of the linear operator \( T: R[X]_n \rightarrow R[X]_n \) defined by \( T(f(x)) = f(ax+b) \), where \( R[X]_n \) is the vector space of polynomials of degree \( \leq n \).
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Context:
设数列 $\left\{x_{n}\right\}$ 满足 $x_{1} \in\{5,7\}$ 及当 $k \geqslant 1$时，有 $x_{k+1} \in\left\{5^{x_{k}}, 7^{x_{k}}\right\}$ 。试确定 $x_{2009}$ 的末两位数字的所有可能值，并给出这些可能值的和。
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Context:
In the diagram, all rows, columns, and diagonals have the sum 12. Find the sum of the four corner numbers.
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Context:
已知 2023 可以拆分为几个正整数之和，所有整数中的最大值和最小值相差不超过 1。有多少种可能？
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Context:
Determine the number of onto functions \( f : A \to A \) such that \((f \circ f)(1) = 2\), where \( A = \{1, 2, 3, \ldots, 9\} \).
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Context:
Compute the expected value of $X_{n+1}$, given that $X_n = X_0 e^{\mu S_n}$, where $X_0 > 0$, $S_n$ is a symmetric random walk, and $\mu > 0$.
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Context:
Given a finite set $A$ with $n$ elements, find the smallest number of subsets required to generate the power set of $A$. Provide the general formula for this minimum size.
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Context:
Find all pairs of positive integers \((n, k)\) such that \((n + 1)^k - 1 = n!\).
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