Context:
The lines \(x=\frac{1}{4}y+a\) and \(y=\frac{1}{4}x+b\) intersect at the point \((1,2)\). The original answer is in the form \(\frac{k}{m}\). Find the value of \(k+m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$.
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Context:
In rectangle $JKLM$, points $N$ and $P$ bisect $\overline{JK}$ and points $Q$ and $R$ bisect $\overline{LM}$. Rectangle $JKLM$ has dimensions $JK = 6$ and $JL = 4$. Find the area of quadrilateral $STUV$ formed by connecting midpoints of sides in the following manner: $S$ is the midpoint of $\overline{JN}$, $T$ is the midpoint of $\overline{KP}$, $U$ is the midpoint of $\overline{LQ}$, and $V$ is the midpoint of $\overline{MR}$.
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Context:
A company purchases 400 tons of a certain type of goods annually. Each purchase is of $x$ tons, and the freight cost is 40,000 yuan per shipment. The annual total storage cost is 4$x$ million yuan. To minimize the sum of the annual freight cost and the total storage cost, find the value of $x$.
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Context:
A robot is standing on the bottom left vertex $(0,0)$ of a $5 \times 5$ grid and wants to go to $(5,5)$, only moving to the right $(a,b) \mapsto (a+1,b)$ or upward $(a,b) \mapsto (a,b+1)$. However, this robot is not programmed perfectly and sometimes takes the upper-left diagonal path $(a,b) \mapsto (a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0 \leq a,b \leq 5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
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Context:
Consider the set \( S = \{A \in \mathbb{Q}^{k \times k} : A^6 = I, A^n \neq I \text{ for all } 0 < n < 6\} \). How many distinct orbits does \( S \) have under conjugation by \( GL_k(\mathbb{Q}) \)?
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Context:
Find the value of $x$ such that $\gcd(a + x, 3a + x - 1) = 2x + 1$, where $a$ is a positive integer.
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Context:
26. A sphere passes through the vertices of one face of a cube and touches the sides of the opposite face of the cube. Find the ratio of the volumes of the sphere and the cube.
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Context:
How many independent sets of size 3 does the Petersen graph have?
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Context:
Evaluate the integral: $$\int_0^{\frac{\pi}{2}}\frac{\tan{x}}{\tan(\frac{x}{2})}\,dx$$
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Context:
Consider the polynomial
\[P(x) = x^3 + 3x^2 + 6x + 10.\]
Let its three roots be \(a\), \(b\), and \(c\). Define \(Q(x)\) to be the monic cubic polynomial with roots \(ab\), \(bc\), and \(ca\). Compute \(|Q(1)|\).
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Context:
The area of a triangle \(ABC\) is \(40 \text{ cm}^2\). Points \(D, E\) and \(F\) are on sides \(AB, BC\) and \(CA\) respectively. If \(AD = 3 \text{ cm}, DB = 5 \text{ cm}\), and the area of triangle \(ABE\) is equal to the area of quadrilateral \(DBEF\), find the area of triangle \(AEC\) in \(\text{cm}^2\). Express your answer as a whole number.
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Context:
Evaluate the integral \( \int_{0}^{2\pi} e^{\cos(\varphi)} \cos(\varphi - \sin(\varphi)) \, d\varphi \) using the method of residues.
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Context:
Define $ n!!$ to be $ n(n-2)(n-4) \cdots3 \cdot1$ for $ n$ odd and $   n\left( {n - 2}\right) \left( {n - 4}\right) \ldots 4 \cdot  2  $ for $ n$ even. When $ \sum\limits_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $\frac{ab}{10}$.
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Context:
Find the positive integer $n$ that satisfi\fes the equation \( {n}^{2} - \left\lfloor  \sqrt{n}\right\rfloor   = {2018} \)

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Context:
Consider the function \( S(n) \) defined for a positive integer \( n \) as follows:
\[
  S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, .
\]
(All logarithms are base 10.) Determine how many integers \( n \) from 1 to 2011 (inclusive) satisfy the condition \( S(S(n)) = n \).
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Context:
Determine the smallest number \( n \) such that the set of integers can be partitioned into \( n \) non-overlapping subsets, none of which contain an infinite arithmetic progression. Is it possible to achieve this with \( n = 2 \)?
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Context:
Given an odd function $f(x)$ that is twice differentiable, with $f(-1)=f(2)=f(3)=-1$, express $f(x)$ in the form $f(x)=a_{2n+1}x^{2n+1}+a_{2n-1}x^{2n-1}+\\cdots+a_{1}x$, where $a_{2n+1}, a_{2n-1}, \\ldots, a_{1}$ are constants. How many real solutions does the differential equation $(f'(x))^2+f''(x)=0$ have?
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Context:
Let $f:(0,5)\to\mathbb{R}-\mathbb{Q}$ be a continuous function such that $f(2)=\pi$. Determine whether $f(\pi)=\pi$.
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Context:
On an $8 \times 8$ board, two identical pieces are placed on cells a1 and c3. Peter and Vasya take turns playing, with Peter starting first. On their turn, a player chooses one of the pieces and moves it either vertically up or horizontally to the right by any number of cells. The player who makes a move to cell h8 wins. Determine which player can always ensure a win regardless of the opponent's moves.
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Context:
A horse runs along a circular path with a speed of 20 km/hr. A fence is positioned along the tangent to the circle at the point where the horse starts. A lantern is placed at the center of the circle, casting the horse's shadow on the fence. Calculate the speed of the shadow along the fence in km/hr when the horse has covered \(\frac{1}{8}\) of the circle.
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Context:
For $\theta<\pi$, find the maximum real number $\theta$ such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0, \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1
$$
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Context:
Among the $3 \times 3$ invertible matrices with entries from the field $\mathbb{Z}/3\mathbb{Z}$, how many matrices are similar to the matrix \( \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)?
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Context:
The real numbers \(x\) and \(y\) satisfy the following system of equations: \[ \begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2} \end{cases} \] Find the numerical value of the ratio \(\frac{y}{x}\).
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Context:
Determine whether the limit \( \lim_{x \to \infty} \frac{\ln(x)^a}{\ln(\ln(x))^{a+b}} \) diverges to infinity for any positive real number \(a\) and integer \(b \geq 1\).
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Context:
Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$  Enter all possible values of $k,$ separated by commas.
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Context:
Determine the values of \( b > 0 \) for which the infinite series \( \sum_{n=0}^\infty \left(\frac{n}{b(n-3)}\right)^n \) converges.
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Context:
Find a formula for \( S(n) \), which represents the number of ways to express the natural number \( n \) as a sum of natural numbers.
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Context:
Determine the minimum number of interior-disjoint almost-squares required to tile an L-shaped region, where the aspect ratio of each almost-square is between 1/2 and 2, and n is an integer.
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Context:
Determine whether the limit \( \lim_{m,n\to\infty} a_{mn} \) exists for the double sequence given by \[ a_{mn} = \frac{m^2n^2}{m^2+n^2}\left(1-\cos\left(\frac{1}{m}\right)\cos\left(\frac{1}{n}\right)\right). \]
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Context:
On a manifold with a metric tensor, let \( \Omega \) be the volume form. Is it true that the Hodge star operator applied to \( \Omega \) results in 1, i.e., \( *\Omega = 1 \)?
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Context:
If the unit vectors $e_{1}$ and $e_{2}$ have an angle of $\dfrac{\pi}{3}$ between them, and the vector $a=e_{1}+\lambda e_{2} (\lambda\in\mathbb{R})$, with $|a|= \dfrac{ \sqrt{3}}{2}$, then $\lambda=$ _______.
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Context:
The field shown has been planted uniformly with wheat. [asy]
draw((0,0)--(1/2,sqrt(3)/2)--(3/2,sqrt(3)/2)--(2,0)--(0,0),linewidth(0.8));
label("$60^\circ$",(0.06,0.1),E);
label("$120^\circ$",(1/2-0.05,sqrt(3)/2-0.1),E);
label("$120^\circ$",(3/2+0.05,sqrt(3)/2-0.1),W);
label("$60^\circ$",(2-0.05,0.1),W);
label("100 m",(1,sqrt(3)/2),N);
label("100 m",(1.75,sqrt(3)/4+0.1),E);
[/asy] At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. What is the fraction of the crop that is brought to the longest side?
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Context:
Find the number of non-isomorphic groups of order 36 that have a normal subgroup of order 4.
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Context:
Given $a \gt 2b \gt 0$, then the minimum value of $\frac{{a}^{4}+1}{b(a-2b)}$ is ____.
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Context:
A tetrahedron has all its faces triangles with sides $13,14,15$. What is its volume?
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Context:
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.
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Context:
In a convex quadrilateral $ABCD$, points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that \( \frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1 \). Another convex quadrilateral $E_1F_1G_1H_1$ is such that $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ respectively, with $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that \( \frac{E_1A}{AH_1} = a \), find \( \frac{F_1C}{CG_1} \) in terms of \( a \).
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Context:
It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?
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Context:
Given an $8\times 8$ chess board, in how many ways can we select $56$ squares on the board while satisfying both of the following requirements:
(a) All black squares are selected.
(b) Exactly seven squares are selected in each column and in each row.
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Context:
Bob is attempting to feed his pets. He feeds his pets as follows:
- No dog receives more food than any other dog
- No cat receives any less food than any other cat
- No hamster receives any less food than any other hamster
- The cats, as a group, receive exactly 1.5 times as much food as a single dog
- The hamsters, as a group, receive exactly 0.25 times as much food as a single dog

If Bob has 9 dogs, 6 cats, and 10 hamsters, approximately what percentage of the food bag does a cat receive?
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Context:
Compute the total area of the infinitely many triangles in the figure, where each triangle has a height of \( \frac{6}{7} \). The \( x \)-values for the bases of the triangles, from left to right, are \( \frac{27}{64} \), \( \frac{9}{16} \), \( \frac{3}{4} \), \( 1 \), and so on.
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Context:
Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers?
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Context:
Given a strict order $(P, \prec)$ and a sequence of distinct elements $a_1, a_2, \ldots, a_n$ from $P$, determine whether there exists a permutation $\sigma$ of $\{1, 2, \ldots, n\}$ such that $a_{\sigma(n)} \not\prec a_{\sigma(m)}$ for all $n < m$. Answer 'yes' or 'no'.
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Context:
On the board we write a series of  $n$  numbers, where  $n \geq 40$ , and each one of them is equal to either  $1$  or  $-1$ , such that the following conditions both hold:

(i) The sum of every  $40$  consecutive numbers is equal to  $0$ .
(ii) The sum of every  $42$  consecutive numbers is not equal to  $0$ .

We denote by  $S_n$  the sum of the  $n$  numbers of the board. Find the maximum possible value of  $S_n$  for all possible values of  $n$ .
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Context:
Calculate the value of the following expression: \[ \frac{\int_0^1 \frac{dt}{(1-t^4)^{1/2}}}{\int_0^1 \frac{dt}{(1+t^4)^{1/2}}}. \]
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Context:
Given real numbers $a$ and $b$ satisfying $b(e^{a}-1)+a=e^{b}-\ln b$, then the range of $b^{2}e^{n}-2e^{b}$ is ____.
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Context:
Find the real value of $a$ for which the definite integral \( f(a) = \int_{0}^{\infty} \frac{x^a}{2x^6 + 4x^5 + 3x^4 + 5x^3 + 3x^2 + 4x + 2} \, dx \) is minimized.
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Context:
Use the fourth order Runge-Kutta method with step size \( h = 1 \) to approximate \( \sqrt{2} \) from the initial value problem given in the previous question. Provide your answer rounded to five decimal places.
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Context:
Find the limit of the series as \( x \) approaches 0: \[ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} \]
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Context:
Evaluate the limit:
$$\lim_{x\to 0} \frac{\sin(\sin(\sin x)) - x}{x^3}$$
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Context:
In triangle \(ABC\), the angle bisectors \(BK\) and \(CL\) are drawn. Point \(N\) is marked on segment \(BK\) such that \(LN \parallel AC\) and \(NK = LN\). Determine the measure of angle \(ABC\).
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Context:
Determine how many different ways there are to assign the elements of the set \( M = \{1, 2, 3, 4, 5\} \) into three ordered sets \( A, B,\) and \( C \) such that the following conditions are satisfied: each element must belong to at least one of the sets, the intersection of all three sets is empty, and the intersection of any two sets is not empty. (i.e., \( A \cup B \cup C = M, A \cap B \cap C = \varnothing \), and \( A \cap B \neq \varnothing, B \cap C \neq \varnothing, C \cap A \neq \varnothing \))
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Context:
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?
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Context:
Let $a=256$. Find the unique real number $x>a^2$ such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
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Context:
\section*{Problem 3 - 131033}

Consider a four-sided pyramid with a square base. The vertices of this base are the points \(A, B, C\), and \(D\). The apex of the pyramid is \(S\). All eight edges have the length \(a\).

\(E\) and \(F\) are the midpoints of the edges \(S B\) and \(S C\), respectively. A plane through the points \(A, E, F\), and \(D\) divides the pyramid into two sub-bodies.

Calculate the ratio of the volumes of these two sub-bodies!
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Context:
Solve the inequality \(\sqrt{x^{2}+x-56}-\sqrt{x^{2}+25 x+136}<8 \sqrt{\frac{x-7}{x+8}}\) and find the sum of its integer solutions within the interval \([-25, 25]\).
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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:
Ant Amelia starts on the number line at \(0\) and crawls in the following manner. For \(n=1,2,3,\) Amelia chooses a time duration \(t_n\) and an increment \(x_n\) independently and uniformly at random from the interval \((0,1).\) During the \(n\)th step of the process, Amelia moves \(x_n\) units in the positive direction, using up \(t_n\) minutes. If the total elapsed time has exceeded \(1\) minute during the \(n\)th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most \(3\) steps in all. What is the probability that Amelia’s position when she stops will be greater than \(1\)? The answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
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Context:
Vasya wrote down 11 natural numbers in one notebook. Petya wrote down the greatest common divisors of each pair of numbers recorded in Vasya's notebook in another notebook. It turned out that every number written in one of the two notebooks is also in the other notebook. What is the maximum number of distinct numbers that could have been written in Vasya's notebook?
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Context:
We define: \( a @ b = a \times (a+1) \times \ldots \times (a+b-1) \). Given \( x @ y @ 2 = 420 \), determine \( y @ x = (\quad) \)
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Context:
find the value of 18396 x 9999 = m ?
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Context:
Find the joint density function $g(u, v)$ for the transformed random variables $U = \exp(X) + 2\exp(Y)$ and $V = 2\exp(X^2) + \exp(Y^2)$, where $X$ and $Y$ are independent Exponential random variables with mean 1. Then, evaluate $g(3e, 3e)$. Provide your answer as a numerical value.
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Context:
In a group of $n$ people, each person initially knows $k$ unique pieces of gossip. What is the minimum number of phone calls required for everyone in the group to know all the gossip?
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Context:
Find the degree and a basis for the field extension \( \mathbb{Q}(\sqrt{2},\sqrt[3]{3}) \) over \( \mathbb{Q}(\sqrt{2}) \).
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Context:
Determine whether the following limit exists: \[\lim_{m \to \infty} \left(\frac{N}{m} \right)^{\frac{N}{m-N}}-\left(\frac{N}{m} \right)^{\frac{m}{m-N}}\]
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Context:
Given the probability density function (pdf) \( f(x) = 10x^9 \) for \( 0 < x < 1 \), and assuming independence, let \( Y = \max(X_1, X_2, \ldots, X_8) \). Find the cumulative distribution function (CDF) for \( Y \) and calculate the probability \( P(0.9999 < Y < 1) \).
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Context:
	 Determine the smallest positive integer  $n \ge 3$  for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where  $A$  denotes the result when the numbers  $2^{10}$ ,  $2^{20}$ ,  $\dots$ ,  $2^{10n}$  are written in decimal notation and concatenated (for example, if  $n=2$  we have  $A = 10241048576$ ).

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Context:
How many ways are there to color every integer either red or blue such that \(n\) and \(n+7\) are the same color for all integers \(n\), and there does not exist an integer \(k\) such that \(k, k+1\), and \(2k\) are all the same color?
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Context:
What is the probability that it takes exactly $n$ random numbers uniformly distributed in $(0,1)$ to exceed $1$ when summed together?
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Context:
Two chords, measuring 9 and 17, are drawn from a single point on a circle. Find the radius of the circle if the distance between the midpoints of these chords is 5.
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Context:
Consider a sequence of one-to-one conformal mappings $\{f_n\}$, each mapping the open unit disk $\mathbb{D}$ onto the right half-plane $\mathbb{A} = \{z \in \mathbb{C} : \mathfrak{R} z > 0\}$. Assume that $f_n$ converges uniformly to a function $f$ on compact subsets of $\mathbb{D}$, but $f$ is not one-to-one. Determine the real part of $f(0)$. \( \mathfrak{R} f(0) = \boxed{?} \)
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Context:
7. Let $R S T U V$ be a regular pentagon. Construct an equilateral triangle $P R S$ with point $P$ inside the pentagon. Find the measure (in degrees) of angle $P T V$.
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Context:
Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), we call \( \{a, b\} \) and \( \{c, d\} \) a "balanced pair" of the set \( M \). How many balanced pairs are there in the set \( M \)?
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Context:
Determine if 11 is a quadratic residue modulo 839 using quadratic reciprocity.
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Context:
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{3}}{2}$, a line is drawn through a point $M$ on the ellipse to intersect the ellipse at points $A$ and $B$, with slopes $k_1$ and $k_2$ respectively. If points $A$ and $B$ are symmetric with respect to the origin, find the value of $k_1 \cdot k_2$.
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Context:
Let $X_n$ be a random variable with an exponential distribution with mean $n$ for each $n \geq 1$. Define $Y_n = X_n - 10 \left[\frac{X_n}{10}\right]$, where $[u]$ denotes the greatest integer less than or equal to $u$. Determine the distribution function of $Y_n$ and find the limit of this distribution function as $n \to \infty$.
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Context:
On a chessboard, $n$ white rooks and $n$ black rooks are arranged such that rooks of different colors do not attack each other. Find the maximum possible value of $n$.
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Context:
Evaluate the change in $z$, denoted as $\Delta z$, when moving from the point $(0,0)$ to the point $(1,1)$ along the curve $y = x^2$. Given the differential $dz$ as:

$$dz = xy\,dx + xy\,dy$$

Express $dz$ in terms of $d(xy(x+y))$ and use the path to determine $\Delta z$. Assume $f(x, y) = \frac{xy(x+y)}{2}$. What is $\Delta z$?
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Context:
Given positive numbers $a$ and $b$ satisfying $a+2b=6$, find the minimum value of $\frac{1}{a+2}+\frac{2}{b+1}$. Express your answer as a fraction in simplest form.
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Context:
In the Cartesian coordinate system, define $d(P, Q) = |x_1 - x_2| + |y_1 - y_2|$ as the "polyline distance" between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$. Then, the minimum "polyline distance" between a point on the circle $x^2 + y^2 = 1$ and a point on the line $2x + y - 2 \sqrt{5} = 0$ is __________.
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Context:
Find an expression for the coefficients of the polynomial given by the falling factorial $x^{\underline n} = x (x-1) \dots (x-n+1)$, which is a polynomial of degree $n$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{(-1)^{n - k} \left[{n \atop k}\right]}


Context:
Circle \(\omega\) has a radius of 5 and is centered at \(O\). Point \(A\) lies outside \(\omega\) such that \(OA = 13\). The two tangents to \(\omega\) passing through \(A\) are drawn, and points \(B\) and \(C\) are chosen on them (one on each tangent), such that line \(BC\) is tangent to \(\omega\) and \(\omega\) lies outside triangle \(ABC\). Compute \(AB + AC\) given that \(BC = 7\).
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Context:
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
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Context:
Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \(\gcd(n,2^n - 1) = 3.\)
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Context:
Determine $w^2+x^2+y^2+z^2$ if

$ \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 $
$ \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 $
$ \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 $
$ \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 $
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Context:
The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \geq 1$.
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Context:
Compute the dimension of the Zariski Tangent space of the affine scheme \(\operatorname{Spec} A\), where \(A = \mathbb{Z}[2i] \cong \mathbb{Z}[x]/(x^2 + 4)\), at the point corresponding to the maximal ideal \((2, x)\).
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Context:
Suppose the sequence \((x_n)_{n \ge 1}\) converges to \(x \in \mathbb{R}\). Does the sequence \(\frac{1}{n}\{x_{mn+1}+x_{mn+2}+ \cdots +x_{2mn-1}+x_{2mn}\}\), where \(m \in \mathbb{Z^+}\), converge?
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Context:
Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$.
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Context:
Evaluate the expression \( \frac {(x + \frac{1}{x})^6 - (x^6 + \frac{1}{x^6}) - 2} {(x + \frac{1}{x})^3 - (x^3 + \frac{1}{x^3})} \) and determine its minimum value.
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Context:
Evaluate the integral $$\int_0^3 \frac{x^3}{1-x^4}\, dx.$$
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Context:
Olya wrote fractions of the form \( \frac{1}{n} \) on cards, where \( n \) includes all possible divisors of \( 6^{100} \). She then arranged these cards in some order. After that, she wrote on the board the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, and so on, finally the sum of the numbers on all the cards. Each sum Olya wrote on the board as an irreducible fraction. What is the minimum number of different denominators that could appear on the board?
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Context:
Let $X$ be a connected oriented triangulation (polyhedron) space, homeomorphic to a geometric realization of an oriented simplicial complex $S$ with dimension $n$, where the boundary $\partial S$ of $S$ is 0. Is the homology group $H_n(X, \mathbb{Z})$ isomorphic to $\mathbb{Z}$?
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Context:
Calculate the discriminant of the number field \(\mathbb{Q}(\sqrt{-3})\) using the basis \(\{1, \sqrt{-3}\}\) and the complex embeddings \(\{\sigma_1, \sigma_2\}\), where \(\sigma_1\) is the identity map and \(\sigma_2\) is the conjugate map. Verify your result.
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Context:
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$?
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Context:
Find all the solutions to
\[\sqrt[3]{15x - 1} + \sqrt[3]{13x + 1} = 4 \sqrt[3]{x}.\]Enter all the solutions, separated by commas.
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Context:
The circle inscribed in a right trapezoid divides its larger lateral side into segments of lengths 1 and 4. Find the area of the trapezoid.
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Context:
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$. Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.

[i]Proposed by Michael Ren[/i]
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Context:
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
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Context:
Suppose S_n is the sum of the first n terms of a geometric sequence {a_n} with common ratio q > 0. Compare the products S_{n+1}a_n and S_na_{n+1}. Express your answer as a comparison (e.g., >, <, =).
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Context:
Find the sum of all $x$ from $2$ to $1000$ inclusive such that$$\prod\limits_{n=2}^x \log_{n^n}(n+1)^{n+2}$$is an integer.
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Context:
Find the number of vertical asymptotes in the graph of
\[y = \frac{(x + 8) (x + 5)^2 (x + 1)^3 x^5 (x - 3)^2}{(x + 7) (x + 5)^2 (x + 1) x (x - 3)^3 (x - 4)}.\]
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Context:
A circle is inscribed in trapezoid $ABCD$. The extensions of the lateral sides of the trapezoid $AD$ and $BC$ beyond points $D$ and $C$ intersect at point $E$. The perimeter of triangle $DCE$ and the base of the trapezoid $AB$ are 60 and 20 respectively. The angle $ADC$ is $\beta$. Find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{30 \sin \beta}{4 + \cos \beta}}


Context:
Determine the set of all accumulation points of the sequence of sets in \(\mathbb{R}^2\) defined by \(K_n = \left\{(x,y) \in \mathbb{R}^2: \left\|(x,y) - \left(\frac{1}{n},\frac{1}{n}\right)\right\|_\infty = \frac{1}{n}, xy \neq 0\right\}\), where \(\| \cdot \|_\infty\) denotes the infinity norm, and \(K = \bigcup\limits_{n\in\mathbb{N}} K_n\).
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Context:
There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students.

The task is to find the smallest number $k$ such that the desired arrangement can be achieved with no more than $k$ swaps.

Illustration of a swap:
$$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$$
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Context:
Task B-2.4. A square and an equilateral triangle are inscribed in a circle of radius 1 so that they share one vertex. Determine the area of the common part of the square and the triangle.
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Context:
Determine the dimension of the symplectic group \( \text{Sp}(2n, F) \) where \( F \) is the algebraic closure of a finite field.
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Context:
During a commercial break in the Super Bowl, there were three commercials with varying lengths: one 5-minute commercial, one 6-minute commercial, and one 7-minute commercial. Additionally, there were eleven 2-minute commercials, but two of them overlapped with a 3-minute interruption causing them to restart after the interruption ends. How many minutes was the commercial break?
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Context:
Let  $S$  be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant  $r$ , such that there exists one way to colour all the points in  $S$  with three colous so that the distance between any two points with same colour is less than  $r$ .
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Context:
Two robots start at position X=0 on the X-axis. Each turn, each robot moves one unit forward or backward with equal probability (0.5), independently of each other and of previous turns. After 450 turns, let D be the distance between the two robots. Calculate approximately the probability that D is less than or equal to 6, using the Central Limit Theorem.
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Context:
Calculate the volumes of the solids formed by rotating the figures bounded by the graphs of the functions about the $Oy$ axis.

$$
y=\arccos \frac{x}{5}, \quad y=\arccos \frac{x}{3}, \quad y=0
$$
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Context:
Given that the function f(x) = ax + sin(x) has extreme points, find the range of the real number a. Express your answer as an inequality.
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Context:
A man sold oranges at the rate of 12 a rupee and experienced a certain loss. To gain 44%, he must sell them at the rate of 7.5 a rupee. What was the percentage of his loss?
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Context:
Let \( \Omega \) be a bounded open set in \( \mathbb{R}^n \), and let \( f, g \in L^2(\Omega) \) with constants \( C_1, C_2 > 0 \). Given the inequality:
\[ \int_{\Omega} fg \, dx \leq C_1 + C_2 \|f\|_2^2 \]
find an estimate for \( \|g\|_2 \).
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Context:
Suppose $f(x)$ is such that \( \int_{-\infty}^\infty e^{tx} f(x) \, dx = \arcsin(t - \sqrt{1/2}) \) for all \( t \) where the right-side expression is defined. Compute \( \int_{-\infty}^\infty x f(x) \, dx \).
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Context:
Find the expected value of a random variable X with the following probability density function:

$$f(x) = \frac{1}{\lambda} \left(1 + \frac{\xi}{\lambda}(x-\mu)\right)^{-(1 + \frac{1}{\xi})}$$

where $\lambda > 0$, $\xi \neq 0$, and $\xi < 1$.
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Context:
Three married couples arrange a party. They arrive at the party one at a time, the couples not necessarily arriving together. They all, upon arriving, shake the hand of everyone already there, except their own spouse. When everyone has arrived, someone asks all the others how many hands they shook upon arriving, and gets five different answers. How many hands did he himself shake upon arriving?
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Context:
3. Given is $\triangle A B C$ where $\measuredangle C$ is obtuse. Points $D \in A C, E \in A B$, and $F \in B C$ are marked on its sides such that quadrilateral $C D E F$ is a rhombus. If $A E=140, B E=84$, and $r=15 \sqrt{3}$, where $r$ is the radius of the circle inscribed in the rhombus $C D E F$, calculate the area of $\triangle A B C$.
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Context:
Given the Markov chain with transition matrix \( \begin{pmatrix} 0 & 0 & 0.2 & 0.8 & 0 \\ 0 & 0 & 0 & 0.9 & 0.1 \\ 0.6 & 0 & 0 & 0 & 0.4 \\ 0.2 & 0.8 & 0 & 0 & 0 \\ 0 & 0.9 & 0.1 & 0 & 0 \end{pmatrix} \), calculate the expected number of visits to state B before reaching state A, starting from state C.
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Context:
Given the function $y=x^3$ and the graph of $y=2^x+1$ intersect at the point $(x_0, y_0)$, determine the interval that $x_0$ falls into. Express your answer in standard interval notation (e.g., (a, b)).
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Context:
Evaluate the integral \( \int_{S^2} \sin(\theta) \, d\theta \wedge d\phi \).
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Context:
A table tennis team has 4 players, among whom player A is not suitable for doubles matches. The team's coach needs to arrange for 3 players to participate in 5 matches, with the third match being a doubles match and the rest being singles matches. Each player plays two matches, and no player plays consecutive matches in the first two or last two matches. Calculate the total number of ways the coach can arrange for the players to participate in the matches.
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Context:
A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction.
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Context:
Triangle  $ABC$  with  $\measuredangle A = 90^\circ$  has incenter  $I$ . A circle passing through  $A$  with center  $I$  is drawn, intersecting  $\overline{BC}$  at  $E$  and  $F$  such that  $BE < BF$ . If  $\tfrac{BE}{EF} = \tfrac{2}{3}$ , then  $\tfrac{CF}{FE} = \tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
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Context:
Find the largest positive integer  $N$  so that the number of integers in the set  $\{1,2,\dots,N\}$  which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
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Context:
If $\mathcal{A}$ is an antichain in $2^{\mathbb{N}}$, measurable with respect to the measure $\mu$ generated from cylindrical sets, does it follow that $\mu(\mathcal{A}) = 0$?
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Context:
Let $R$ be a commutative ring with identity, and let $P$ be a prime principal ideal of $R$. Suppose there exists a proper principal ideal $I$ of $R$ such that $R \supsetneq I \supsetneq P$. Determine if $P^2 = P$.
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Context:
Given that all vertices of a regular triangular prism $ABC-A_1B_1C_1$ are on the surface of a sphere with radius 1, when the volume of the regular triangular pyramid is at its maximum, the height of the regular triangular pyramid is ______.
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Context:
Determine the values of \( k \) for which the second partials test is inconclusive for the function \( f(x, y) \) at the point \((0, 0)\).
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Context:
A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find $k$.
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Context:
Let $A \in M_n$ be a doubly stochastic matrix. Is the largest singular value of $A$, denoted as $\sigma_1(A)$, equal to 1?
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Context:
$ABC$ is a triangle with $AB = 33$, $AC = 21$, and $BC = m$, where $m$ is an integer. There are points $D$ and $E$ on sides $AB$ and $AC$ respectively such that $AD = DE = EC = n$, where $n$ is also an integer. Find the value of $m$.
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Context:
How many ways are there to put five beads on a necklace if there are eight distinct beads to choose from, and rotations and reflections of the necklace are considered the same?
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Context:
Calculate the value of the sum \( \sum_{r=0}^5 \binom{32}{6r} \).
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Context:
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
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Context:
Determine the number of field homomorphisms from a finite field with 9 elements to a finite field with 4 elements.
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Context:
Two pipes A and B separately fill a cistern in some minutes and 5 minutes respectively, and a third pipe C can empty at a rate of 14 L/min. If all the pipes are open when the cistern is full, it is emptied in 1 hour. The cistern holds 39.99999999999999 litres. How many minutes does it take for pipe A to fill the cistern?
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Context:
Compute the nearest integer to $$100 \sum_{n=1}^{\infty} 3^{n} \sin ^{3}\left(\frac{\pi}{3^{n}}\right)$$
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Context:
A long rope is folded in half, then folded in half again along the middle of the folded rope, and this process is repeated for a total of 6 folds. After the 6th fold, the rope is cut along the middle. How many segments will the rope be cut into? Express your answer as a whole number.
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Context:
Assume that $X_1, X_2, X_3, X_4$ are independent and identically distributed random variables such that $P(X_1=0)=0.3$, $P(X_1=1)=0.1$, and $X_1$ has a density $f(x)=0.6$ for $x$ in $(0,1)$. Calculate $P(X_1 + X_2 + X_3 + X_4 \leq 1)$. Provide your answer as a probability value.
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Context:
Determine whether the set \( S = \mathbb{C} \setminus \{z = x + iy \in \mathbb{C} | \, \text{Re}(z) = 0 \text{ and } \text{Im}(z) > 0\} \) forms a star domain. If it does, identify the point at which it is centered.
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Context:
In how many ways can you divide $n$ students into 4 distinct groups such that each student is in exactly one group and each group is assigned to a unique project?
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Context:
Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}} $$
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Context:
If $x+2 y-3 z=7$ and $2 x-y+2 z=6$, determine $8 x+y$.
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Context:
今有茭草八千五百六十八束，欲令撒星更落一形垛之，问底子几何？（撒星更落一形的放法是第一层1束，第二层(1+4)束，第三层(1+4+10)束，···，第n层$( 1 + 4 + 1 0 + \cdots + C _ { n + 2 } ^ { 3 } )$束.）
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Context:
The right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (with $a>0$, $b>0$) is $F$, and $B$ is a point on the left branch of the hyperbola. The line segment $BF$ intersects with one asymptote of the hyperbola at point $A$, and it is given that $(\vec{OF} - \vec{OB}) \cdot \vec{OA} = 0$ and $2\vec{OA} = \vec{OB} + \vec{OF}$ (where $O$ is the origin). Find the eccentricity $e$ of the hyperbola.
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Context:
How many ordered triples $\left(a, b, c\right)$ of positive integers are there such that at least two of $a, b, c$ are prime and $abc = 11(a + b + c)$?
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Context:
Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $.
$$\sum\limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty} \frac{\cos(\alpha n +\beta  m)}{2^n3^m}$$can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.
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Context:
Use the Cauchy-Frobenius lemma to find the number of orbits of the action of $D_8$, the dihedral group of $8$ elements, on the set of $k$-colorings of the edges of a square.
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Context:
Dividing $x^{1951}-1$ by $P(x)=x^4+x^3+2x^2+x+1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.
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Context:
Determine the necessary and sufficient condition for the convergence of the integral \( \int_0^\infty \frac{x^2+1-\cos x}{(x^2+1)x^p} \, dx \), where \( p \) is a real number. Is the condition \( 1 < p \le 2 \) correct?
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Context:
Solve the equation
$
3^x -5^y  = z^2.
$
in positive integers.
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Context:
Lillian collects 205 candies. Lillian's father gives her 2 times the amount of candies she collected. If Lillian shares the candies equally with her 7 friends, how many candies will each friend receive?
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Context:
Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\tfrac{p}{q}$ for relatively prime positive integers $p, q$. What is $p + q$?

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Context:
In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
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Context:
Let $(X, \mathcal{U})$ be a compact Hausdorff uniform space where $\bigcap \{U: U\in \mathcal{U}\}= \Delta_X$. Determine whether there exists a sequence $\{U_n\}_{n\in\mathbb{N}}$ such that $U_{n+1}\subseteq U_n$ and $\bigcap_{n\in\mathbb{N}}U_n=\Delta_X$. Answer 'yes' or 'no'.
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Context:
Let $f$ be an entire function of order $r = 1$. Determine the order of the derivative function $f'$. Is it finite or not?
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Context:
Let $X$ be a smooth projective variety, and $D$ a Cartier divisor on $X$ inducing a surjective morphism $f: X \rightarrow C$, where $C$ is a curve. Can we conclude that the self-intersection number $D^2 = 0$?
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Context:
Calculate the probability $P(1.75 < \bar{X} < 2.5)$, where $\bar{X}$ is the sample mean of $36$ independent random variables, each following a chi-squared distribution with $2$ degrees of freedom.
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Context:
Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probability that Bob wins can be written as a fraction \( \frac{a}{b} \) in lowest terms. What is \( a + b \)?
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Context:
ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD.
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Context:
A square is divided into $n^{2}$ equal smaller squares. It is known that a certain polyline passes through the centers of all these smaller squares (the polyline may intersect itself). What is the minimum number of segments in this polyline?
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Context:
Find the magnitude of the product of all complex numbers  $c$  such that the recurrence defined by  $x_1 = 1$ ,  $x_2 = c^2 - 4c + 7$ , and  $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$  also satisfies  $x_{1006} = 2011$ .
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Context:
Given a regular tetrahedron with all four vertices lying on the surface of a sphere with radius 1, where the three vertices of the base are on a great circle of the sphere, find the volume of the regular tetrahedron.
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Context:
A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \).
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Context:
Find the center of gravity of a hemispherical shell of radius $a$, assuming the density is $1$ and it is placed so that its base is on the $x-y$ plane. Use the method of slicing parallel to the $x-y$ axis.
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Context:
Given the numbers 1, 3, 5 and 2, 4, 6, calculate the total number of different three-digit numbers that can be formed when arranging these numbers on three cards.
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Context:
An expression arranged according to a certain rule: $-\frac{b}{a}$, $\frac{3b}{{a}^{2}}$, $\frac{5b}{{a}^{3}}$, $\frac{7b}{{a}^{4}}$, $\ldots \ldots $ The $n$th expression is ______.
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Context:
A circle with area $40$ is tangent to a circle with area $10$. Let $R$ be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
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Context:
Let $A \in \mathbb{R}^{n \times n}$ be a matrix where the column vectors $k_i \in \mathbb{R}^n$ for $i = 1, 2, \ldots, n$ satisfy the condition $k_i = (i+2)k_{i+2}$ for $i = 1, 2, \ldots, n-2$. Given that $n > 3$, choose one eigenvalue of $A$ and determine the smallest possible dimension of the eigenspace corresponding to the chosen eigenvalue.
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Context:
A person initially has an unknown amount of money. They spend 1/3 of it on clothes, then 1/5 of the remaining on food, 15% of what's left on utilities, 1/8 of the remaining on entertainment, and finally 20% of what's still left on travel. After all these expenses, they have $3,500 left. How much money did they have in the beginning?
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Context:
Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of measurable functions from a finite measure space $(E, \mathcal{E}, m)$ to $\mathbb{R}$ that converges in measure to a measurable function $f:E \to \mathbb{R}$. For a given $\eta >0$, determine if the following limit holds:
\[
\lim_{n \to \infty} m\left(\{x \in E : |f_n(x)-f(x)||f_n(x)| > \eta\}\right) = 0?
\]
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Context:
Suppose you have a function \( g : [0, \infty) \to [0, \infty) \) such that \( \lim_{x \to +\infty} (2g(2x) - g(x)) = 0 \). Additionally, for every \( M > 0 \), the restriction \( g|_{[0,M]} \) is bounded. Does it follow that \( \lim_{x \to +\infty} g(x) = 0 \)?
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Context:
Given the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$, use Horner's Rule to calculate the value of $v_4$ when $x = -4$.
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Context:
sue ' s trail mix is 30 % nuts and 70 % dried fruit . jane ' s trail mix is 60 % nuts , 30 % chocolate chips and 10 % pretzels. Tom's trail mix is 40% nuts, 50% dried fruit, and 10% granola. If the combined mixture of sue, jane and tom's trails mix contains 45 % nuts , what percent of the combined mixture is dried fruit ?
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Context:
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 3$, $|\overrightarrow{b}| = 2\sqrt{3}$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -9$, find the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$.
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Context:
Given the function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$, if it is a power function and is an increasing function on the interval $(0, +\infty)$, determine the real number $m$.
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Context:
Two players play alternately on a 5 x 5 board. The first player always enters a 1 into an empty square and the second player always enters a 0 into an empty square. When the board is full, the sum of the numbers in each of the nine 3 x 3 squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make if the second player plays so as to minimize this score?
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Context:
What is the limit superior of:

$$ \limsup_{x\rightarrow 10} 1*\mathbb{I}_{x<10}(x)+2*\mathbb{I}_{x=10}(x)+\frac{1}{2}*\mathbb{I}_{x>10}(x) ? $$
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Context:
ABC is a triangle. D is the midpoint of BC. E is a point on the side AC such that BE = 2AD. BE and AD meet at F and ∠FAE = 60°. Find ∠FEA.
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Context:
Evaluate the following limit:

$$\lim\limits_{N \to \infty}{\oint_{C_N}{\frac{z}{\exp(z)-1}}\cdot\frac{dz}{z^{2\cdot k+2}}},$$

where $C_N=\{z\in \mathbb C : |z|=2\pi(N+\frac{1}{2}) \}$ and $k\in \mathbb N$ is fixed.
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Context:
Professor Artie, a professor of art history, is organizing an art exhibition to demonstrate the concept of artistic subjectivity and interpretation. She has selected 5 different pieces of art for the exhibition, each representing a unique interpretation of a historical event. To highlight the diversity of interpretations, she decides to display each piece in 3 different rooms, each room offering a different lighting setup that influences how the art is perceived.

Professor Artie wants to ensure that each piece is viewed 7 times over the course of the exhibition by different groups of students. If she plans to have 2 tours per room each day, how many days will the exhibition last if she wants each piece to be viewed the desired number of times?
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Context:
已知 $[x]$ 为高斯函数，求满足 $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=x$ 的解的组数。
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Context:
There are three boxes of stones. Each hour, Sisyphus moves a stone from one box to another. For each transfer of a stone, he receives from Zeus a number of coins equal to the number of stones in the box from which the stone is drawn minus the number of stones in the recipient box, with the stone Sisyphus just carried not counted. If this number is negative, Sisyphus pays the corresponding amount (and can pay later if he is broke).

After 1000 years, all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
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Context:
The quantities of rice in warehouses A and B are the same. Dad, Mom, and Yangyang each need 10 days, 12 days, and 15 days respectively to transport all the rice from one warehouse on their own. Dad and Mom start transporting the rice from warehouses A and B respectively at the same time. Yangyang first helps Mom and then helps Dad. As a result, they finish transporting all the rice from both warehouses at the same time. How many days did Yangyang help Mom?
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Context:
The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$.  There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$.  Find $n-10m$.
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Context:
Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \(\frac{\text{gcd}(m,n)}{\text{lcm}(m,n)}\) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \(\frac{r}{s}\), compute $r + s$.
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Context:
A line passing through point $P(-\sqrt{3},-1)$ and point $A(-2,0)$ intersects with the circle $x^2+y^2=1$ at a common point. The range of the slope angle of line $l$ is __________.
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Context:
How many positive integers are within $810$ of exactly $\lfloor \sqrt{810} \rfloor$ perfect squares?

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Context:
Determine whether the equation $x(3y-5)=y^2+1$ has any integral solutions. If so, provide one such solution.
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Context:
Let $\vartriangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\vartriangle AB'C'$ with $AB' = 7\sqrt6$, $B'C' = 15\sqrt6$, $AC' = 20\sqrt6$ such that $\angle BAB' = \frac{5\pi}{12}$ . The lines $BB'$  and $CC'$  intersect at point $D$. Let $O$ be the circumcenter of $\vartriangle BCD$, and let $O' $ be the circumcenter of $\vartriangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{ d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$
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Context:
Given that for \( |x| \leqslant 1 \), the inequality
\[
2px^2 + qx - p + 1 \geqslant 0
\]
always holds, find the maximum value of \( p + q \).
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Context:
棋盘上标有第 $0,1,2, \cdots, 100$ 站, 棋子开始时位于第 0 站, 棋手抛郑均匀硬币走跳棋游戏, 若掷出正面,棋子向前跳出一站; 若郑出反面, 棋子向前跳出两站, 直到跳到第 99 站（胜利大本营）或第 100 站（失败集中营) 时, 游戏结束, 设棋子跳到第 $n$ 站的概率为 $P_{n}$.
求 $P_{3}$ 的值;
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all polynomials \( p(x) \) such that \( p(x+c) = p(x) + c \) for any constant \( c \).
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Context:
The number  $125$  can be written as a sum of some pairwise coprime integers larger than  $1$ . Determine the largest number of terms that the sum may have.
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Context:
Determine a lower bound for the Hausdorff measure \( \mathcal{H}^s([0,1]) \) for \( s \in [0,1) \) using a covering strategy different from the standard one.
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Context:
A photo archive is arranged in identical albums, with exactly 4 photos on each page. The 81st photo ends up on the 5th page of one album, and the 171st photo ends up on the 3rd page of another. How many photos does each album hold?
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Context:
consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)

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Context:
Given the sample data: $110$, $120$, $120$, $120$, $123$, $123$, $140$, $146$, $150$, $162$, $165$, $174$, $190$, $210$, $235$, $249$, $280$, $318$, $428$, $432$, find the $75$th percentile.
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Context:
Given the function $f\left( x \right)=\log_{\frac{1}{e}}\left( x^{2}+\frac{1}{e} \right)-\left| \frac{x}{e} \right|$, find the range of $x$ that makes $f\left( x+1 \right) < f\left( 2x-1 \right)$. Express your answer in interval notation.
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Context:
函数 $f(x)=\frac{1-2 x}{\left(1-x+x^{2}\right)^{2}}$, 则 $\left[\frac{f^{(2022)}(0)}{2022!}\right]=$ $\qquad$.
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Context:
The coordinates of the point $\left(-2,3\right)$ with respect to the $x$-axis are ____.
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Context:
In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?
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Context:
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $\angle MF_{2}F_{1}$. Find the eccentricity of ellipse $C$.
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Context:
Mrs. Carter's algebra class consists of 48 students. Due to a schedule conflict, 40 students took the Chapter 5 test, averaging 75%, while the remaining 8 students took it the following day, achieving an average score of 82%. What is the new overall mean score of the class on the Chapter 5 test? Express the answer as a percent.
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Context:
Pipe A and Pipe B fill water into a tank of capacity 1000 litres, at a rate of 200 L/min and 50 L/min, respectively. Pipe C drains at a rate of 25 L/min. Pipe D has a varying flow rate, initially at 75 L/min for the first two cycles, then increasing by 25 L/min for each subsequent cycle.

The process is as follows: Pipe A is open for 1 minute and closed, then Pipe B is open for 2 minutes and closed. Pipe C is opened and drains for another 2 minutes. After that, Pipe D is opened for 1 minute. This process is repeated until the tank is filled. How long will it take to fill the tank?
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Context:
Given \((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}\), find \( a_0 + a_1 + 2a_2 + 3a_3 + \cdots + 20a_{20} \).
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Context:
Compute Compute \( \left\lbrack  {\mathop{\sum }\limits_{{k = {2018}}}^{\infty }\frac{{2019}! - {2018}!}{k!}}\right\rbrack \) . (The notation \( \lceil x\rceil \) denotes the least integer \( n \) such that \( n \geq  x \) .)
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Context:
3. Let $O$ be the center of the circumcircle of triangle $ABC$, points $O$ and $B$ lie on opposite sides of line $AC$, $\angle AOC = 60^\circ$. Find the angle $AMC$, where $M$ is the center of the incircle of triangle $ABC$.
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Context:
Find the smallest positive integer $k$ such that there exist positive integers $M,O>1$ satisfying
\[ (O\cdot M\cdot O)^k=(O\cdot M)\cdot \underbrace{(N\cdot O\cdot M)\cdot (N\cdot O\cdot M)\cdot \ldots \cdot (N\cdot O\cdot M)}_{2016\ (N\cdot O\cdot M)\text{s}}, \]where $N=O^M$.
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Context:
设 a_1,a_2,\cdots,a_{2023} 是正整数，满足对任意 1\leq k\leq 2023，均有a_k=\ds{i=1}{2023}|a_k-a_i|.求 a_1+a_2+\cdots+a_{2023} 的最小值.
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Context:
Determine whether the following limit exists as \( z \) approaches \(-3\):
\[ \lim_{z\rightarrow -3}\frac{\log(z+3)}{e^{\cos(z)}}(z+3) \]
where \( \log(z) \) is the complex logarithm.
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Context:
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.
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Context:
In a game, two players, Sam and Joe, each pick a nonnegative integer at random. The probability that the number is in any set $A$ is given by a finitely additive function $\mu$ defined on $2^X$, where $\mu(A) = 0$ for every finite set $A$ and $\mu(X) = 1$, with $X = \mathbb{N}$. The player with the larger number wins. A coin is tossed to determine whose number is revealed first. It lands on heads, so Sam's number is revealed, but Joe's number remains unknown. Based on this information, who is more likely to win?
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Context:
A loaf of sausage is marked with thin transverse rings. If you cut along the red rings, you get 5 pieces; if along the yellow rings, you get 7 pieces; and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors?
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Context:
I-2 Consider a chessboard $n \times n$ where $n>1$ is a positive integer. We select the centers of $2 n-2$ squares. How many selections are there such that no two selected centers lie on a line parallel to one of the diagonals of the chessboard?

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Context:
Determine the asymptotic bounds for the sum \( \sum\limits_{k=1}^n k^k \).
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Context:
Let $\{a_n\}$ be a sequence of real numbers. Is it possible to find a subsequence $\{a_{n_k}\}$ such that the limit of the ratio of consecutive terms, $\lim \dfrac{a_{n_{k+1}}}{a_{n_k}}$, exists and is a member of the set $\{-1, 0, 1, \infty\}$? Answer 'yes' or 'no'.
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Context:
Consider a matrix \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that:
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\). Determine the maximum number of matrices that satisfy these conditions such that no two of them are similar.
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Context:
Given the numbers 0, 1, 2, 3, 4, 5, 6, determine the total number of 3-digit numbers that can be formed from these digits without repetition and divided by 5.
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Context:
Consider a well-defined set $S$ and a subset $A$ of $S$ defined as follows: \[ A = \{x \mid (x \in S) \wedge (x \not\in S)\}. \] Determine whether $A$ represents the empty set $\varnothing$ due to the contradictory predicate.
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Context:
Let $S$ be a subset of $\{1,2,3,\cdots,1989\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest number of elements $S$ can have?
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Context:
Evaluate the integral of \( \log(z) \) over the unit circle in the complex plane.
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Context:
Minh enters the numbers $1$ through $81$ into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by $3$?
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Context:
Let $k$ be a field, and consider the set $S = A_1 \cup \{(0,0)\}$, where $A_1 = \{(x,y) \in k^2: x \neq 0\}$. Determine whether $S$ is open or closed in the affine plane $\mathbb{A}^2$. Provide your answer as either "open", "closed", or "neither".
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Context:
The circle touches the sides $AB$ and $BC$ of triangle $ABC$ at points $D$ and $E$ respectively. Find the height of triangle $ABC$ dropped from point $A$, given that $AB = 5$, $AC = 2$, and points $A$, $D$, $E$, and $C$ lie on the same circle.
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Context:
Let $N = \lceil\delta^{-\gamma}\rceil$, with $0 < \gamma < 1$. Determine whether \( \lim_{\delta \to 0} f(t,\delta,\lceil \delta^{-\gamma}\rceil)=0 \) for all \( t > 0 \), where \( f(t,\delta, N)= e^{-t/\delta} \sum_{k=0}^{N-1}\frac{(t/\delta)^k}{k!} \).
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Context:
Tommy goes for a run around his neighborhood and decides to calculate how many wheels he saw. All the trucks in his neighborhood have 4 wheels, all the cars have 4 wheels, bicycles have 2 wheels, and buses have 6 wheels. However, some vehicles were partially hidden, so Tommy could only see 75% of their wheels. If he saw 12 trucks, 13 cars, 8 bicycles, and 3 buses, and assuming all hidden wheels were distributed evenly among the vehicle types, how many wheels did Tommy see in total?
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Context:
Consider a $(-1)$-curve $E \subseteq S$ inside a non-singular complex projective surface, where $E \cong \mathbb{P}^1_\mathbb{C}$ and $E^2 = -1$. Let $\sigma \in \text{Aut}(\mathbb{C})$ be a field automorphism, and define the fibered product $E^\sigma := E \times_{\text{Spec } \mathbb{C}} \text{Spec } \mathbb{C}$. Is it true that ${(E^\sigma)}^2 = -1$? In other words, is the auto-intersection preserved by the action of $\text{Aut}(\mathbb{C})$ on $E$?
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Context:
Let \(X \in \mathbb{R}^{m}\) and \(W \in \mathbb{R}^{p}\) be random vectors. Determine whether the statement \(E[X|W] = 0\) implies \(E[XW^{T}]=0\).
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Context:
Let $ABCD$ be a convex quadrilateral with $AB=CD=10$, $BC=14$, and $AD=2\sqrt{65}$.  Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of $\triangle APB$ and $\triangle CPD$ equals the sum of the areas of $\triangle BPC$ and $\triangle APD$.  Find the area of quadrilateral $ABCD$.
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Context:
Find the number of solutions, in degrees, to the equation $\sin^{10}x + \cos^{10}x = \frac{29}{16}\cos^4 2x,$ where $0^\circ \le x^\circ \le 2007^\circ.$

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Context:
Determine if the Mellin transform of the function \( f(t) = e^{-t} \int_0^t e^w g(w)\,dw \) converges in the strip \( a<\Re(z)<b \), given that the Mellin transform of \( g(t) \) converges in the same strip.
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\int_0^n \frac{e^{-t} \sin{\frac{t}{n}} dt}{(1+\frac{t}{n}^2)\arctan{\frac{t}{n}}}.$$
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Context:
If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$ then $x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} = $
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Context:
How many nine-digit numbers composed of eight 3's and one 0 leave a remainder of 1 when divided by 4? Express your answer as a single integer.
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Context:
Let \(\{ x \}\) denote the nearest integer to \(x\), so that \(x - \frac{1}{2} \leq \{ x \} < x + \frac{1}{2}\). Define the sequence \(u_1, u_2, u_3, \ldots\) by \(u_1 = 1\) and \(u_{n+1} = u_n + \{ u_n \sqrt{2} \}\). For example, \(u_2 = 2\), \(u_3 = 5\), \(u_4 = 12\). Find the units digit of \(u_{1985}\).
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Context:
Given the differential equation $ty''(t)+2y'(t)+ty=0$ for $t>0$, with initial conditions $y(0+)=1$ and $y'(0+)=0$, find the value of $Y(1)$, where $Y(s)$ is the Laplace transform of $y(t)$. Assume that $Y(s) = -\tan^{-1}(s) + C$ and determine the constant $C$ to find $Y(1)$. Provide your answer as a single value.
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Context:
Mom gave Vasya money for 30 pencils. It turned out that the pencil factory is running a promotion in the store: in exchange for a receipt for the purchase of a set of 20 pencils, $25\%$ of the cost of the set is refunded, and in exchange for a receipt for the purchase of a set of 5 pencils, $10\%$ is refunded. What is the maximum number of pencils that Vasya can buy?
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Context:
If $D$ is a dense subset (not a subspace) of a Hilbert space $H$, is the orthogonal complement of $D$ equal to $\{ 0 \}$?
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Context:
Determine whether the integral \( \int_0^\infty \frac{\sin ax}{x} \, dx = \frac{\pi}{2} \) holds true when \( a = 0 \).
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Context:
Find the limit \( \mathcal{T} = \lim\limits_{n\to\infty} nt_n \), where \( t_n = \frac{1}{2n+1} - \frac{1}{2n+2} + \frac{1}{2n+3} - \frac{1}{2n+4} + \ldots + \frac{1}{4n-1} - \frac{1}{4n} \). Relate this limit to the Riemann sums for the function \((1+x)^{-2}\).
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Context:
Given an arithmetic sequence \(\{a_n\}\), where the sum of the first \(m\) terms is \(S_m = \frac{m}{n}\) and the sum of the first \(n\) terms is \(S_n = \frac{n}{m}\) with \(n \neq m\), determine whether \(S_{m+n}\) is greater than 4.
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Context:
Jirka had three identical rectangles. First, he placed them next to each other as shown in the picture and obtained a rectangle with a perimeter of $20 \mathrm{~cm}$. He then arranged them differently and obtained a rectangle with a different perimeter. What was the perimeter of this rectangle? (In each arrangement, no two rectangles overlap and there are no gaps between them.)
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Context:
Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).

Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of \( g(3) \). Find \( m \times t \).
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Context:
There are three triangles: an acute triangle, a right triangle, and an obtuse triangle. Sasha took one triangle, and Borya took the remaining two. It turned out that Borya can place one of his triangles against the other (without overlapping) to form a triangle identical to Sasha's. Which triangle did Sasha take?
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Context:
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?

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Context:
Let $a,b \in \mathbb{R}_+$ be real positive numbers with $\frac{1}{2}<a<1$ and let $I=[0,b]$ be a closed real interval. Define the sequence of functions $f_n(x) : I \to \mathbb{R}$ by \[ f_n(x) = { \dfrac{n}{(2n-1)^{a+x}} - \dfrac{n}{(2n)^{a+x}} } \] and the sequence of functions $g_n(x) : I \to \mathbb{R}$ by \[ g_n(x) = \dfrac{n \cdot \ln(2n)}{(2n)^{a+x}} - \dfrac{n \cdot \ln(2n-1)}{(2n-1)^{a+x}}. \] Let $h : I \to \mathbb{R}$ be the function \[ h(x)= \sum_{n=1}^\infty \left| \dfrac{f_n(x)-f_n(0)}{x} - g_n(0) \right|^2. \] Determine whether \[ \lim_{x \to 0^+} h(x)=0. \]
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Context:
Let  $a_0, a_1,\dots, a_{19} \in \mathbb{R}$  and   $$ P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}. $$   If  $P(x)=P(-x)$  for all  $x \in \mathbb{R}$ , and   $$ P(k)=k^2, $$   for  $k=0, 1, 2, \dots, 9$  then find   $$ \lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}. $$
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Context:
5. Determine the number of all coverings of a chessboard $3 \times 10$ by (nonoverlapping) pieces $2 \times 1$ which can be placed both horizontally and vertically.

(Stanislava Sojáková)

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Context:
Find a root of the polynomial \( f(x) = x^4 + 10x^2 + x + 3 \) modulo \( 3^4 \).
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Context:
Which element is the major component in solar cells?
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Context:
There are 62 boxes in total, with small boxes being 5 units per ton and large boxes being 3 units per ton. A truck needs to transport these boxes. If the large boxes are loaded first, after loading all the large boxes, exactly 15 small boxes can still be loaded. If the small boxes are loaded first, after loading all the small boxes, exactly 15 large boxes can still be loaded. How many large boxes are there among these boxes in total?
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Context:
How long does a train 375 m long running at the speed of 78.5 km/hr take to cross a bridge 450 m in length?
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Context:
Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.
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Context:
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

\[
\sqrt[4]{a(b+2c)} + \sqrt[4]{b(c+2d)} + \sqrt[4]{c(d+2a)} + \sqrt[4]{d(a+2b)}
\]
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Context:
In the Cartesian coordinate system, let A, B, and C be three distinct points on the curve $y= \frac {1}{x-1}$. Let D, E, and F respectively be the midpoints of BC, CA, and AB. Then, the circle passing through points D, E, and F always passes through a fixed point \_\_\_\_\_\_.
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Context:
17. [10] How many ways are there to color every integer either red or blue such that $n$ and $n+7$ are the same color for all integers $n$, and there does not exist an integer $k$ such that $k, k+1$, and $2 k$ are all the same color?
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Context:
Let ellipse $C$:$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ have foci $F_{1}(-c,0)$ and $F_{2}(c,0)$. Point $P$ is the intersection point of $C$ and the circle $x^{2}+y^{2}=c^{2}$. The bisector of $\angle PF_{1}F_{2}$ intersects $PF_{2}$ at $Q$. If $|PQ|=\frac{1}{2}|QF_{2}|$, then find the eccentricity of ellipse $C$.
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Context:
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 15x\]for all integers $x.$ Find the smallest possible value of $a.$
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Context:
At 8:10 AM, Feifei walked from home to school. After 3 minutes, the dog started running to catch up with her and caught up with her 200 meters away from home. The dog immediately ran back home after catching up with her, and then turned back to chase Feifei again, catching up with her 400 meters away from home. The dog once again ran back home after catching up with her, and then started chasing Feifei again, finally catching up with her at the school. Feifei reached the school at __ minutes past 8 o'clock.

(Note: Fill in the blank with the appropriate time)
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Context:
Find the sum of all integers $n$ with $2 \le n \le 999$ that have the following property: if $x$ and $y$ are randomly selected without replacement from the set $\{ 1, 2, \dots, n \}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.
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Context:
Find the operator norm of the linear operator $A$ defined by \begin{equation} Ax=(x_1+x_2, x_2+x_3, \ldots) \end{equation} for all $x \in \ell_2$. It has been shown that this operator is bounded with \begin{equation} \sum_{i=0}^\infty|x_i+x_{i+1}|^2 \leq 6\sum_{i=0}^\infty|x_i|^2. \end{equation} Determine the norm of $A$ by considering the supremum of $Ax$ for $|x|=1$.
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Context:
Given that the three internal angles \( A, B, C \) of triangle \( \triangle ABC \) satisfy the equations \( \cos A = \sin B = 2 \tan \frac{C}{2} \), find the value of \( \sin A + \cos A + 2 \tan A \).
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Context:
Find the remainder when \( \frac{3^{2015} + 1}{4} \) is divided by 1000.
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Context:
After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$?
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Context:
Tracy has been collecting novels from her friends to donate to the Children For The Future charity organization. In the first week, she collects a certain number of books on her own, and then two of her friends volunteer to help her in the following weeks. The book collection process for the following weeks is as follows:

- In the second week, Tracy collects twice as many books as she did in the first week, while her first friend collects three times as many books as Tracy did in the first week.
- In the third week, Tracy collects three times as many books as she did in the first week, while her first friend collects four times as many books as Tracy did in the first week, and her second friend starts collecting and gathers half as many books as Tracy did in the first week.
- In the fourth week, Tracy collects four times as many books as she did in the first week, her first friend collects five times as many books as Tracy did in the first week, and her second friend collects as many books as Tracy did in the first week.
- In the fifth week, Tracy collects five times as many books as she did in the first week, her first friend collects six times as many books as Tracy did in the first week, and her second friend collects twice as many books as Tracy did in the first week.
- In the sixth week, Tracy collects six times as many books as she did in the first week, her first friend collects seven times as many books as Tracy did in the first week, and her second friend collects three times as many books as Tracy did in the first week.

If they have collected a total of 1,025 books now, how many books did Tracy collect in the first week?
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Context:
In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC = DC = 3$, $OD = 2$, and $AB = 6$, find the radius of the circle. If the radius is \(\sqrt{x}\), what is the value of \(x\)?
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Context:
Determine the minimum number of pairs \((x, R(x))\) required to fully characterize a rotation \(R\) in \(\mathbb{R}^n\).
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Context:
Determine the degree of the splitting field of the polynomial \((x^2+x+1)(x^3+x+1)\) over \(\mathbb{Q}\).
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Context:
Let point $P$ be a moving point on the curve $y=2x^2$. The tangent line to the curve $y=2x^2$ at point $P$ is denoted as $l$. The line passing through point $P$ and perpendicular to line $l$ intersects the curve $y=2x^2$ at another point $Q$. Find the minimum value of $|PQ|$.
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Context:
Find the number of ways to order the integers $1, 2, 3, 4, 5, 6, 7$ from left to right so that each integer has all its divisors, besides itself, appearing to the left of it.
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Context:
V is the pyramidal region defined by the inequalities \( x, y, z \geq 0 \) and \( x + y + z \leq 1 \). Evaluate the integral:

\[ \int_V x y^9 z^8 (1 - x - y - z)^4 \, dx \, dy \, dz. \
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Context:
Find the angle between the non-intersecting diagonals of two adjacent lateral faces of a right quadrilateral prism, given that the plane in which they lie forms an angle $\alpha$ with the base plane.
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Context:
若存在实数 a 及正整数 n 使得 f(x)=\cos2x-a\sin x 在 (0,n\pi) 内恰有 2022 个零点，则满足条件的正整数 n 的值有__________个.
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Context:
Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \notin \mathbf{Q}$, $f(x)=0$; and when $x=\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\left(\frac{7}{8}, \frac{8}{9}\right)$.
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Context:
Compute the limit \( \lim_{n \to \infty} n\left(\frac{{\pi}^2}{6} - \sum_{k=1}^{n} \frac{1}{k^2} \right) \).
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Context:
Determine the minimum number of equally spaced subintervals, N, required such that the difference between the Upper and Lower Riemann sums for the integral \( \int_{1}^{17} \frac{1}{x^3} \, dx \) is less than \( \frac{1}{100} \). The Upper and Lower Riemann sums are given by:

\[ L_{n} = \frac{16}{N} \sum_{i=1}^{N}\frac{1}{(2i+1)^3} \]
\[ U_{n} = \frac{16}{N} \sum_{i=1}^{N}\frac{1}{(2i-1)^3} \]

The difference between them is:

\[ U_{n} - L_{n} = \frac{16}{N} \left[1-\frac{1}{(2N)^3} - \frac{1}{(2N +1)^3}\right] \]

Find the smallest N such that:

\[ \frac{16}{N} \left[1-\frac{1}{(2N)^3} - \frac{1}{(2N +1)^3}\right] < \frac{1}{100} \]
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Context:
The circumferences of Lebanese cedar cones are modeled as Gamma-distributed random variables with a mean of 5.40 cm and a variance of 3.24 cm². A botanist is interested in collecting cones with a circumference greater than 6 cm. Calculate the expected number of cones she needs to inspect to find her third cone with a circumference greater than 6 cm.
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Context:
The length of a red bus is four times the length of an orange car. The length of the orange car is also 3.5 times shorter than the length of a yellow bus. A green truck is parked between the red bus and the yellow bus, and the green truck is as long as two orange cars. There is a row of trees that covers 25% of the view between the red bus and the yellow bus. If the red bus is 48 feet long, and the yellow bus and green truck are parked end to end next to the red bus, what length of the yellow bus may the red bus driver see, considering the trees that partially obstruct the view?
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Context:
如果非负整数 $m$ 及其各位数码之和均为 6 的倍数，则称 $m$ 为 "六合数". 求小于 2012 的非负整数中六合数的个数。
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Context:
Determine the minimum value of \( n \) such that there exist real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) satisfying \( x_{1}+2 x_{2}+\cdots+ n x_{n}=2009 \) and \( x_{1}+x_{2}+\cdots+x_{n}=0 \), where each \( x_{i}= \pm 7 \) for \( i=1,2, \ldots, n \).
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Context:
A train of certain length traveling at 72 kmph takes 13.998880089592832 seconds to cross a bridge of 170 m in length. How long is the train?
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Context:
For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$
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Context:
$3$名男同学、$3$名女学生和$2$位老师站成一排拍照合影，要求$2$位老师必须站正中间，队伍左右两端不能同时是一男学生与一女学生，则总共有__________种排法．
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Context:
Given the universal set $I=\{(x,y)|x\in\mathbb{R},\ y\in\mathbb{R}\}$, set $A=\{(x,y)|\frac{y-3}{x-2}=1\}$, and set $B=\{(x,y)|y=x+1\}$, find $(C_{I}A)\cap B$.
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Context:
Find the mean number of records set by the first $n$ jumpers, where each jumper's height is an independent and identically distributed continuous random variable. What is the behavior of this mean as $n$ approaches infinity?
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Context:
What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$?

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Context:
Let $G$ be the centroid of $\triangle ABC$, and $\frac{\sqrt{7}}{\overrightarrow{GA}}\sin A + 3\overrightarrow{GB}\sin B + 3\sqrt{7}\overrightarrow{GC}\sin C = \overrightarrow{0}$, then the magnitude of angle $B$ is \_\_\_\_\_\_.
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Context:
Dr. Riley, an archaeologist studying ancient artifacts, has discovered a collection of 120 bronze coins buried in an old excavation site. Over the years, corrosion has affected 45% of these coins, making them difficult to analyze and categorize. Dr. Riley plans to clean the corroded coins to restore them for further study.

After cleaning each corroded coin, Dr. Riley finds that 60% of them can be fully restored, while the rest are too damaged to be of any use.

How many of the original 120 bronze coins are successfully restored and usable for further study after the cleaning process?
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Context:
Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle  ADC = \angle  BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?
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Context:
Compute the residues of all singularities of $\tanh(z)$ and evaluate the integral $\int_C \tanh(z) \, dz$, where $C$ is the circle of radius $12$ centered at $z_0=0$. Provide your answer as a single value.
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Context:
Marco visits Ristorante Machiavelli every Thursday for dinner. He has been a regular customer for 5 years. Each visit, he orders his favorite pasta dish for $12 and a dessert for $5. Every third visit, he also orders a special appetizer for an additional $8. How much money has Marco spent on average at the restaurant each year, considering he visits every week without missing a Thursday?
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Context:
A train passenger arrives at a busy railway station and chooses a taxi stand. At stand A, taxis arrive according to a Poisson process with a rate of 2 taxis per minute, and passengers choose this stand with a probability of 0.7. At stand B, taxis arrive according to a Poisson process with a rate of 1 taxi per minute, and passengers choose this stand with a probability of 0.3. After waiting for 3 minutes at the front of the queue without a taxi, what is the probability that the passenger is waiting at taxi stand A?
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Context:
Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.
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Context:
If the inequality $2ae^{2x}-\ln x+\ln a\geqslant 0$ holds for any real number $x \gt 0$, then the minimum value of the real number $a$ is ____.
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Context:
Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?

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Context:
Evaluate the limit: \[ \lim_{(x, y) \to (0, 0)} (1+x^2+y^2)^{\frac{1}{x^2+y^2+x y^2}}. \]
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Context:
A magician has one hundred cards numbered $1$ to $100$. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.

A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.

How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)

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Context:
Given $a<0$ and $-1<b<0$, then the order from smallest to largest of $a$, $ab$, $ab^2$ is __________.
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Context:
Let $X$ be a set such that for every metric space $Y$ containing $X$, the set $X$ is open in $Y$. Must $X$ be the empty set?
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