Context:
A train 100 m long running at the speed of 65 km/hr takes 13.568145317605362 seconds to cross a bridge of a certain length. What is the length of the bridge in meters?
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Context:
Given a set of sample data with $8$ numbers, the average is $8$, and the variance is $12$. Two unknown numbers are added to this set of sample data to form a new set of sample data. It is known that the average of the new sample data is $9$. Find the minimum value of the variance of the new sample data.
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Context:
Determine the points at which the partial derivative $f_x$ is continuous for the function \( f(x,y) = \frac{x^2 - y^2}{(x^2 + y^2)^{1/3}} \) when \((x,y) \neq (0,0)\), and \( f(x,y) = 0 \) when \((x,y) = (0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Select 5 different letters from the word "equation" to arrange in a row, where "qu" must be together and in the same order. Find the number of different arrangements. Express your answer as a single integer.
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Context:
Each cell of a $10 \times 10$ table contains a natural number. Then, a cell is shaded if the number in that cell is less than one of its neighbors but greater than another neighbor. (Two cells are considered neighbors if they share a side.) As a result, only two cells remain unshaded. What is the minimal possible sum of the numbers in these two unshaded cells?
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Context:
Evaluate the triple integral \( \iiint_B (x^2+y^2+z^2)^2 \, dV \) where \( B = \{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq 1, z \leq 0 \} \).
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Context:
Outside the right-angled triangle $ABC$, squares $ACDE$ and $BCFG$ are constructed on its legs $AC$ and $BC$. The extension of the median $CM$ of the triangle $ABC$ intersects the line $DF$ at point $N$. Find the segment $CN$ if $AC = 4$ and $BC = 1$.
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Context:
In triangle \( \triangle ABC \), the sides are given as \( AB = 7 \), \( BC = 8 \), and \( AC = 9 \). A circle is drawn such that it is tangent to side \( AC \) and the extensions of sides \( AB \) and \( BC \). Let \( D \) be the center of this circle. Calculate \( BD^2 \).
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Context:
Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that is reached according to the original planned time. What is the time of arrival at point \( B \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive real solutions to the equation \( \left(\dfrac{\ln (x)}{x}\right)^2=\pi^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A $39$-tuple of real numbers $(x_1, x_2, \ldots, x_{39})$ satisfies the conditions:
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1, x_2, \ldots, x_{39})$ satisfying these conditions. If this ratio is $\frac{a}{b}$ for coprime positive integers $a$ and $b$, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the probability density function of a continuous random variable $X$:
\[ f_{X}(x) = \begin{cases}\frac{x+1}{a+1} & -1 \leq x < a \\
1 & x=a \\
\frac{x-1}{a-1} & a < x \leq 1 \\
0 & \text{otherwise}\end{cases}\]
Calculate the expected value $E(X)$. Consider the implications of the probability being 1 at $x=a$.
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Context:
Evaluate the double integral \( \iint _R \frac{y^2}{x} \, dx \, dy \) over the region \( R \) in the \( xy \)-plane bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer \( n \) ,define a function \( {f}_{n}\left( x\right) \) at an interval \( \left\lbrack  {0,n + 1}\right\rbrack \) as
 \( {f}_{n}\left( x\right)  = {\left( \mathop{\sum }\limits_{{i = 1}}^{n}\left| x - i\right| \right) }^{2} - \mathop{\sum }\limits_{{i = 1}}^{n}{\left( x - i\right) }^{2}. \)
 Let \( {a}_{n} \) be the minimum value of \( {f}_{n}\left( x\right) \) . Find the value of
 \( \mathop{\sum }\limits_{{n = 1}}^{{11}}{\left( -1\right) }^{n + 1}{a}_{n} \)
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Context:
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer.
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Context:
$\mathrm{Natural~}a,b,c\mathrm{are}>100\mathrm{and}(a,b,c)=1.c|a+b,a|b+c\text{Find minimal }b$
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Context:
Calculate the work done by a particle moving along the spiral curve given by $x = a\cos t$, $y = a\sin t$, $z = bt$ for $0 \le t \le 4\pi$, against the force field $\mathbf{F}(x,y,z) = xy\,\mathbf{i} + yz\,\mathbf{j} + xz\,\mathbf{k}$.
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Context:
Compute the exterior derivative \(d\beta\) of the two-form \(\beta = z(y^2-x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz\) in three-dimensional real space.
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Context:
Find the function \( F(x) = \lim_{y \to 0} f(x,y) \) where \( f: \mathbb{R}^2 \to \mathbb{R} \) is defined as follows:
\[
f(x,y) = \begin{cases} \dfrac{e^{-x^2}-e^{-y^2}}{x^2+y^2} &\text{if } (x,y) \neq (0,0), \\
 k &\text{if } (x,y) = (0,0). \end{cases}
\]
Determine the value of \( k \) such that \( F(x) \) is continuous at \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the functional derivative of the function \( g(x) = x(1) - 1 \) with respect to \( x \), where \( x \) is a function and both \( x \) and \( g(x) \) are differentiable and continuous at all points.
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Context:
For \( n \in \mathbb{N} \), consider non-negative valued functions \( f \) on \( \{1,2, \cdots , n\} \) satisfying \( f(i) \geqslant f(j) \) for \( i>j \) and \( \sum_{i=1}^{n} (i+ f(i))=2023. \) Choose \( n \) such that \( \sum_{i=1}^{n} f(i) \) is maximized. How many such functions exist in that case?
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Context:
Determine the interval of convergence for the series \(\sum_{n=1}^{\infty}\frac{((n+1)x)^n}{n^{n+1}}\). Specifically, evaluate the convergence at the endpoints of the interval.
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Context:
Given the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$, let the left vertex be $M$, the top vertex be $N$, and the right focus be $F$. If $\overrightarrow{NM} \cdot \overrightarrow{NF}=0$, then find the eccentricity of the ellipse. Express your answer as a single value.
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Context:
John is a hack author. He writes a book every certain period of time. He has been writing for 20 years and has earned an average of $30,000 per book. He has made $3,600,000 writing. How often does he write a book?
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Context:
Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\overline{PQ}$ is tangent to segments $\overline{PR}$ and $\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what?
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Context:
2B. Find $x$ for which the function $y=\left(x-x_{1}\right)^{2}+\left(x-x_{2}\right)^{2}+\ldots+\left(x-x_{1996}\right)^{2}$ takes the smallest value.
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Context:
Evaluate the integral \( \iiint_E z \,{\rm d}v \), where \( E \) is the region between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 3 \) in the first octant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 4-tuples \((a,b,c,d) \in \mathbb{N}^{4}\) such that:

\[\mathrm{lcm}(a,b,c) = \mathrm{lcm}(b,c,d)=\mathrm{lcm}(c,d,a)= \mathrm{lcm}(d,a,b)=2^{5}3^{4}5^{3}7^{2}.\]
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Context:
Evaluate the integral \( \int_{-\pi}^{\pi}\ln(4e^2+e^2+4e^2\cos(\theta))d\theta \) when \( a = 2e \).
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Context:
There are 7 boys and 13 girls standing in a line. Let \( S \) be the number of positions where boys and girls are adjacent to each other. Calculate the average value of \( S \) for all possible arrangements of these 20 people. Express your answer as a single integer.
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Context:
Find the largest real $C$ such that for all pairwise distinct positive real $a_{1}, a_{2}, \ldots, a_{2019}$ the following inequality holds $$\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2018}}{\left|a_{2019}-a_{1}\right|}+\frac{a_{2019}}{\left|a_{1}-a_{2}\right|}>C$$
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Context:
Consider a convex polygon with $4n-1$ sides, where $n$ is a positive integer greater than 1. The diagonals of this polygon are divided into $k$ subsets \( S_{1}, S_{2}, \cdots, S_{k} \) such that for every pair of different indices $i$ and $j$, there exist diagonals in subset $S_{i}$ that intersect with diagonals in subset $S_{j}$ at an interior point. What is the maximum possible value of $k$?
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Context:
6 barrels of water have an average capacity of 12 gallons. The capacities of the barrels must be between 8 and 16 gallons, inclusive. What is the maximum possible capacity of the smallest barrel if the median is 14 gallons and the capacity of the largest barrel is a prime number?
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Context:
Let the tangent line at point A $(x_0, y_1)$ of the curve $y=(ax-1)e^x$ be $l_1$, and the tangent line at point B $(x_0, y_2)$ of the curve $y=(1-x)e^{-x}$ be $l_2$. If there exists $x_0 \in [0, \frac{3}{2}]$ such that $l_1 \perp l_2$, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum \( S = \sum_{k=1}^{2015} (-1)^{\frac{k(k+1)}{2}} k \) by considering the residue of \( k \) modulo 4.
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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  $x_{1006} = 2011$ .

*Author: Alex Zhu*
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Context:
Let \( {a}_{1},{a}_{2},\ldots ,{a}_{300} \) be nonnegative real numbers,with
$$
\sum_{i=1}^{300} \alpha_i = 1.
$$
 Find the maximum possible value of \( \mathop{\sum }\limits_{{i \neq  j,i \mid  j}}{a}_{i}{a}_{j} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjacent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?
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Context:
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \(3\) and \(4\) units. In the corner where those sides meet at a right angle, he leaves a small unplanted square \(S\) so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from \(S\) to the hypotenuse is \(2\) units. What fraction of the field is planted? The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the matrix that corresponds to reflecting over the $x$-axis.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}}


Context:
Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. Determine the speed of the left ant when it first reaches the line $y=\frac{1}{2}$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parallelogram $ABCD$ with an area of $240$, points $E$ and $H$ are the midpoints of sides $AD$ and $AB$, respectively. On side $BC$, point $G$ is such that $BG = 2GC$, and on side $CD$, point $F$ is such that $DF = 3FC$. Point $K$ is on side $AC$ such that the area of triangle $EKF$ is $33$. Determine the area of triangle $HKG$. \( \text{Provide your answer in square units.} \)
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Context:
In a rectangle $ABCD$, the sides $AB = 3$ and $BC = 4$ are given. Point $K$ is at distances $\sqrt{10}$, 2, and 3 from points $A$, $B$, and $C$, respectively. Find the angle between lines $CK$ and $BD$.
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Context:
Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \(x\) such that \(\gcd(50 + x^2, 4x + 2) = 6\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to insert + and - signs between the numbers 0, 1, 2, ..., 12 such that the resulting expression evaluates to a multiple of 5.
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Context:
Given a natural number \( n \), find the number of distinct triplets of natural numbers whose sum is \( 6n \).
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Context:
What is the result of the sum of three Cantor sets, denoted as \( C + C + C \), where \( C \) is the usual Cantor set obtained from the interval [0,1]? Express your answer as a set.
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Context:
Determine the values of the real parameter \( \alpha > 0 \) for which the function \( \frac{1}{|x|^{\alpha} + |y|^{\alpha} +1} \) is in \( L^2(\mathbb{R}^2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{0}^{\infty}\frac{\sin\left(x^{2}\right) + \cos\left(x^{2}\right) - 1} {x^{2}}\,\mathrm{d}x. \]
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Context:
The school invites 6 out of 10 students to participate in a study seminar, among which students A and B cannot attend at the same time. Calculate the number of different ways to invite the students.
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Context:
Given the numbers \(\log _{\sqrt{x+34}}(2x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2x+23}}(-x-4)\). For which values of \(x\) are two of these numbers equal and the third one greater by 1?
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Context:
For each positive integer \( n \), define \( A_{n} = \frac{20^{n} + 11^{n}}{n!} \), where \( n! = 1 \times 2 \times \cdots \times n \). Find the value of \( n \) that maximizes \( A_{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of subsets of $\{1,3,5,7,9,11,13,15,17,19\}$ where the elements in the subset add to $49$.
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Context:
Let \( s(n) \) denote the sum of the digits (in base ten) of a positive integer \( n \). Compute the number of positive integers \( n \) of at most \( 10^4 \) that satisfy
\[ s(11n) = 2s(n). \]
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Context:
Determine the sum of absolute values for the complex roots of \( 20x^8 + 7ix^7 - 7ix + 20 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have?
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Context:
Consider the rectangular hyperbola \(xy = c^2\). Normals at points \(P, Q, R,\) and \(S\) on the curve are concurrent and meet at point \(O(h, k)\). Find the value of \(OP^2 + OQ^2 + OR^2 + OS^2\).
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Context:
Determine the number of points of discontinuity for the function \( f(\theta) = \int_{-1}^{1} \frac{\sin\theta \, dx}{1-2x\cos\theta +x^2} \) where \( \theta \in [0,2\pi] \).
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Context:
Find the extremal function $y(x)$ that satisfies the integral \( \int^{3}_{0}(x(y')^3 -3y(y')^2)dx \) with the boundary conditions $y(3) = 1$ and $y(0) = 7$.
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Context:
Calculate the cardinality of the stabilizer \( \operatorname{Stab}_{G}(\langle e_1, e_3 \rangle) \) for the action of \( G=GL(V) \) on \( X=\{W\subset V:\text{sub-vector space,}\dim =2\} \), where \( V=\Bbb F_2^3 \).
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Context:
In a ten-story building, you live on the ground floor, and each of your nine friends lives on a different floor. You randomly draw the numbers 1 through 9 from a hat, determining the order in which you visit your friends' floors. If the distance between each floor is 4 meters, calculate the average number of meters you travel by elevator, moving from one floor to the next in the order drawn.
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Context:
已知关于$x$的不等式$$(2ax-\ln x)\left(x^{2}-(a+1)x+1\right)\geqslant0$$ 对于任意$x\in(0,+\infty)$恒成立。则实数$a$的取值范围是_______。标准答案格式为$\left[\frac{a}{be},c\right]$，计算a+b+c的值。
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Context:
Find the limit of the sequence \((z_n)\) defined by \[ z_n = \frac{(1+2)^{p+1}+(1+2+3)^{p+1}+\cdots+(1+2+\cdots+n)^{p+1}}{n^{p+2}} \] using the Stolz-Cesaro theorem.
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Context:
Find the inverse of the element \( \beta \) in the finite field \( GF(81) \), where \( \beta \) satisfies the equation \( \beta^4 + \beta^2 + 3 = 0 \).
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Context:
There is exactly one real number \( x \) with the property that both \( x^{2} - t x + 36 = 0 \) and \( x^{2} - 8 x + t = 0 \).

What is the value of \( x \) ?
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Context:
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is $24$ and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?
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Context:
As shown in the diagram, in square $ABCD$, extend $BA$ to $G$ such that $AG = BD$. What is the measure of $\angle BCG$ in degrees?
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Context:
On a sheet of paper, a grid of \( n \) horizontal and \( n \) vertical lines is drawn. How many distinct closed \( 2n \)-segment broken lines can be drawn along the grid lines such that each broken line passes through all horizontal and all vertical lines at least once?
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Context:
Given propositions $p$ and $q$:
- Proposition $p$: The function $f(x)=x^{2}+ax-2$ has exactly one root in the interval $[-1,1]$.
- Proposition $q$: The inequality $x^{2}+3(a+1)x+2 \leqslant 0$ holds true for all $x$ in the interval $[\frac{1}{2}, \frac{3}{2}]$.
If the compound proposition "$p$ and $q$" is false, find the range of values for the real number $a$.
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Context:
Find the monotonic increasing interval of the function $f(x)=( \frac {1}{2})^{ \sqrt {x-x^{2}}}$. Express your answer as an interval using standard interval notation.
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Context:
A cube with a volume of 1 cubic centimeter (cm3) was cut by the number of vertexes from a cube with one edge 3 centimeters (cm) long. Find the volume of the remaining part.
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Context:
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$
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Context:
Determine the appropriate limits for the parameter $t$ in the parametrization $(a\cos t, a\sin t, b-b\cos t)$ to describe the curve of intersection between the cylinder $x^2+y^2=a^2$ and the plane $\frac{x}{a}+\frac{z}{b}=1$.
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Context:
Find a set of numbers \( S = \{s_1, s_2, \ldots, s_n\} \) such that the sum of the greatest common divisors of all subsets of \( S \) equals the sum of the elements of \( S \).
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Context:
Given the infinite series \( \alpha = \frac{5}{2!\cdot 3} + \frac{5\cdot 7}{3!\cdot 3^2} + \frac{5\cdot 7 \cdot 9}{4!\cdot 3^3} + \cdots \), find the value of \( \alpha^2 + 4\alpha \).
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Context:
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1}\frac{n\cos(x)}{1+n^2x^{3/2}}\, dx.$$
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Context:
Avik is holding six identical ropes in his hand, with the midpoints in his fist. The ends of the ropes are lying on either side of his hand. Kamrul randomly selects and ties pairs of rope ends together on one side, and then does the same on the other side. If the probability of forming a loop with all six ropes is expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers, find the value of \(a + b\).
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Context:
Point \( O \) divides segment \( AB \) into segments \( OA = 6 \) and \( OB = 4 \). A circle is drawn with center at point \( O \), and tangents to this circle are drawn from points \( A \) and \( B \), intersecting at point \( M \), with the points of tangency lying on the same side of line \( AB \). Find the radius of the circle if \( OM = 12 \).
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Context:
Given the function $g:\mathbb{R}^3 \to \mathbb{R}^3$ defined by $g(x, y, z) = (3x + 4z, 2x - 3z, x + 3y)$ and the region $S = \{(x, y, z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\}$, find the value of $\alpha$ for which the integral equality holds:

$$\iiint_{g(S)} (2x + y - 2z) \,dxdydz = \alpha \iiint_S z \,dxdydz$$
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Context:
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. We need to arrange seating for 2 people, and these 2 people cannot sit next to each other (even one in the front and one in the back are considered not adjacent). Find the total number of different seating arrangements. Express your answer as a numerical value.
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Context:
What is the minimum number of colors required to paint all the cells of a 6 by 6 square grid such that each row, column, and diagonal of the square has all cells with different colors? Note: A diagonal of the square is defined as all rows of at least two cells going diagonally from one edge of the square to the other at an angle of $45^\circ$ or $135^\circ$ to the horizontal.
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Context:
例7 设$a _ { i }$、$b _ { i } ( i = 1$,2,···,n)是有理数，使得对任意的实数x都有$x ^ { 2 } + x + 4 = \sum _ { i = 1 } ^ { n } ( a _ { i } x + b _ { i } ) ^ { 2 }$，求n的最小可能值．（2006年国家集训队试题）
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Context:
The scoring rules for a soccer match are: 3 points for a win, 1 point for a draw, and 0 points for a loss. How many possible outcomes are there for a team that plays 14 matches and scores a total of 19 points?
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Context:
A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$?
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Context:
Determine the limit points of the set \( S = \left\{ \left(m+\dfrac{1}{4^{|p|}},n+\dfrac{1}{4^{|q|}}\right) : m,n,p,q \in \mathbb{Z} \right\} \).
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Context:
In this figure, \( \angle RFS = \angle FDR \), \( FD = 4 \) inches, \( DR = 6 \) inches, \( FR = 5 \) inches, \( FS = 7\frac{1}{2} \) inches. Find the length of \( RS \), in inches, as a decimal. Multiply your final answer by 100 and provide the result.
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Context:
The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.
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Context:
Find the radius of convergence of the power series $\sum_0^{\infty} n!x^{n^2}$.
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Context:
Calculate the area of the figures bounded by the curves given in polar coordinates.

$$
r=2 \sin 4 \phi
$$
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Context:
Mary picked 122 oranges, Jason picked 105 oranges, Tom picked 85 oranges, and Sarah picked 134 oranges. If there was a 10% increase in the number of oranges picked by each person, how many oranges were picked in total?
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Context:
Mark rolls 5 fair 8-sided dice. What is the probability that at least three of the dice show the same number?
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Context:
Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)?
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Context:
In the park, there are initially 25 oak trees, 40 maple trees, and 20 birch trees. Park workers will plant 73 oak trees, 52 maple trees, and 35 birch trees over the course of 3 days. On the first day, they will plant 40% of the oak trees, 50% of the maple trees, and 30% of the birch trees. On the second day, they will plant 35% of the oak trees, 25% of the maple trees, and 45% of the birch trees. On the third day, they will plant the remaining trees. Additionally, they will remove 15 oak trees, 10 maple trees, and 5 birch trees on the second day. Calculate the total number of oak, maple, and birch trees in the park after the workers have completed their tasks over the 3 days.
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Context:
There exist two triples of real numbers \((a,b,c)\) such that \(a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}\) are the roots to the cubic equation \(x^3-5x^2-15x+3\) listed in increasing order. Denote those \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\). If \(a_1, b_1,\) and \(c_1\) are the roots to the monic cubic polynomial \(f\) and \(a_2, b_2,\) and \(c_2\) are the roots to the monic cubic polynomial \(g\), find \(f(0)^3+g(0)^3\).
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Context:
Let $k$ be a real number. In the Cartesian coordinate plane $xOy$, consider the following sets of points:
\[ A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \]
\[ B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \]
Determine the value of $k$ such that the intersection of $A$ and $B$ is a single-element set.
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Context:
Compute the integral \( \int_{\gamma} \frac{f(z)}{z^3}dz \), where \( f(z) = az^3 + bz^2 + cz + d \) and \( \gamma : [0, 4\pi] \to \mathbb{C} \), \( \gamma(t) = e^{it} \). Use the Cauchy integral formula to find the result, considering the interval \([0, 4\pi]\).
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Context:
Find $b^2$ if the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide.
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Context:
The lateral edge of a regular quadrilateral pyramid makes an angle \( \alpha \) with the plane of the base. A plane is constructed through a vertex of the base and the midpoint of the opposite lateral edge, parallel to one of the diagonals of the base. Find the angle between this plane and the plane of the base of the pyramid.
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\boxed{\arctan \left( \frac{\tan \alpha}{3} \right)}


Context:
On a circular track, Alphonse is at point \( A \) and Beryl is diametrically opposite at point \( B \). Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but different, speeds. After running for a while they notice that when they pass each other it is always at the same three places on the track. What is the ratio of their speeds?
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Context:
Let $f(x)$ be an even function defined on $\mathbb{R}$. For $x < 0$, $f(x) + x f'(x) < 0$, and $f(-4) = 0$. Determine the solution set of the inequality $x f(x) > 0$.
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Context:
A point \((X, Y)\) is uniformly distributed on the unit square \([0,1]^2\). Let \(\theta\) be the angle between the x-axis and the line segment connecting \((0,0)\) to the point \((X,Y)\). Find the expected value \(E[\theta]\).
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Context:
Alexey plans to buy one of two car brands: "A" costing 900,000 rubles or "B" costing 600,000 rubles. On average, Alexey drives 15,000 km per year. The cost of gasoline is 40 rubles per liter. The cars use the same type of gasoline. The car is planned to be used for 5 years, after which the car of brand "A" can be sold for 500,000 rubles, and the car of brand "B" for 350,000 rubles.

| Car Brand | Fuel Consumption (L/100km) | Annual Insurance Cost (rubles) | Average Annual Maintenance Cost (rubles) |
| :--- | :--- | :--- | :--- |
| "A" | 9 | 35,000 | 25,000 |
| "B" | 10 | 32,000 | 20,000 |

Using the data from the table, answer the question: How much more expensive will it be for Alexey to buy and own the more expensive car?
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Context:
A set of data arranged in ascending order is $1$, $4$, $4$, $x$, $7$, $8$ $(x \neq 7)$. If the median of this data set is $5$ times the mode, then the $60$th percentile of this data set is ______.
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Context:
There is a method of setting a six-digit password: the first 2 digits are the last 2 digits of the birth year, the middle 2 digits are the birth month, and the last 2 digits are the birth date. For example, if someone was born on April 23, 1998, the six-digit password would be 980423. How many such possible passwords are there?
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Context:
Let \( f, g \in L^2 \). Find the limit \( \lim_{t \to \infty} \frac{\|f+tg\|_2-\|f\|_2}{t} \).
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Context:
Evaluate the integral $$\int_{0}^{2\pi} \cos (m \theta)\cos(n \theta) \text{d} \theta$$ where $m$ and $n$ are integers.
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Context:
Consider a game where a number $u$ is drawn from a uniform distribution on $[0,1]$. On one side of a piece of paper, $u$ is written, and on the other side, $u^2$ is written. One side of the paper is randomly drawn, and you win one dollar if you correctly guess whether the observed number is $u$ or $u^2$. Determine the expected value of this game.
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Context:
Evaluate the inward flux of the vector field \( F = \langle y, -x, z \rangle \) over the surface \( S \) of the solid bounded by \( z = \sqrt{x^2+y^2} \) and \( z = 3 \).
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Context:
Calculate the total number of button presses required to try all possible 4-digit codes on a 10-key keyboard, assuming you do not need to press enter after each digit and consecutive digits can form different codes.
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Context:
Find the largest possible value of the product \( xyz \) for positive numbers \( x, y, \) and \( z \) such that the inequality \( \sqrt{\frac{x^2+y^2+z^2}{3}} \ge \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \) holds, and the answer is a two-digit number when simplified.
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Context:
Find the four roots of
\[2x^4 + x^3 - 6x^2 + x + 2 = 0.\]Enter the four roots (counting multiplicity), separated by commas.
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Context:
What is the minimum number of points required to color an entire 3D space black using a device that colors all points at an irrational distance from each given point, assuming the initial color of the space is white?
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Context:
Find the Jordan measure of the set \( M = \left\{ \begin{pmatrix} -r +2 t -t\\ 3r+t \\2r+2t \end{pmatrix} : 1\leq r \leq 3, 0 < s < 1, -1 < t \leq 2 \right\} \).
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Context:
Find the real number \( a \) such that the conditional expectation \( E(X | Y = aX) = \frac{2}{9} \), given the joint probability density function for random variables \( X \) and \( Y \) is:
\[
f(x, y) = \begin{cases} 2(x+y) & \text{if } 0 < x < y < 1 \\ 0 & \text{otherwise} \end{cases}
\]
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Context:
Given the formulas for the surface area \( S_n(1) \) and volume \( V_n(1) \) of an \( n \)-sphere with radius 1:
\[ S_n(1) = \frac{n \pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)} \quad \text{and} \quad V_n(1) = \frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)} \]
where \( \Gamma(z) \) is the gamma function defined as:
\[ \Gamma(z) = \int_0^\infty e^{-x} x^{z-1}\, dx \]
Find the integers \( v \) and \( s \) such that \( S_s(1) \) and \( V_v(1) \) are maximum. Determine the value of \( v-s \).
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Context:
Find the Fourier transform of the function \( xe^{-x^2/2} \).
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Context:
Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$.
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Context:
A closed cardboard box is in the shape of a cube. By cutting exactly 7 out of its 12 edges, the box can be unfolded onto a plane in various forms. How many distinct unfoldings can be obtained by cutting exactly 7 edges of the cube?
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Context:
Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$.
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Context:
Determine the cardinality of the set \( H = \{ f : \mathbb{N} \to \mathbb{N} \mid f \text{ is an involution} \} \), where an involution is a function that is its own inverse.
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Context:
正方体旋转群元素最多的两个共轭类的元素个数之和为 $\qquad$.
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Context:
Given the function \( f(n) = k \), where \( k \) is the \( n \)-th digit after the decimal point in the repeating decimal \( 0.\dot{9}1827364\dot{5} \), determine the value of \( \underbrace{f\{f \cdots f[f}_{1996 \text{ times }}(1)]\} \). Express your answer as a single digit.
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Context:
A man's wages were decreased by 40%, then increased by 30%. After that, his wages were decreased by 20% and then increased by 10%. Calculate the overall percent change in his wages.
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Context:
Given the surface parameterization \( g(s, t) = (s^2 + 2t, s + t, e^{st}) \), find \( \frac{\partial y}{\partial z} \) at the point \((3, e)\) near \( g(1, 1) = (3, 2, e) \).
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Context:
Calculate the second type integral
$$
\iint_{\sigma} x \, dy \, dz + y \, dx \, dz + z \, dx \, dy
$$
where $\sigma$ is the outer surface of the sphere $x^{2} + y^{2} + z^{2} = R^{2}$.
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Context:
Find the number of odd positive integers greater than 1 and less than 250 that are twinning, where an integer is twinning if for every prime divisor \( p \) of the integer, \( p - 2 \) divides the integer.
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Context:
Solve the following equation:

$$
\left(5^{x}-2^{x-2}\right)^{2}+2 \log \left(5^{x}+2^{x-2}\right)=x
$$
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Context:
一只青蛙在正方形 $A B C D$ 的四个顶点间跳跃, 每次跳跃总是等可能地跳至与当前所在顶点相邻的两个顶点之一，且各次跳跃是独立的. 若青蛙第一次跳跃前位于顶点 $A$ ，则它第 6 次跳跃后恰好仍位于顶点 $A$ 的概率为 $\qquad$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Find the remainder when \(20^{25^{30^{35}}}\) is divided by 120.
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Context:
Determine the asymptotic behavior of the expression:
\[ \frac{\int_{-n}^{n}ne^{-nx^{2}}dx}{\int_{-n}^{n}e^{-x^{2}}dx} \] as \( n \to \infty \).
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Context:
How do I evaluate $$\int_\Gamma \frac{e^{iz}}{(z^2 + 1)^2}\,dz$$ where $\Gamma$ is the circle $|z| = 3$ positively oriented?
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Context:
Determine the number of linear Boolean functions from \({0, 1}^n\) to \({0, 1}\). A function \(F\) is linear if for all \(x, y \in {0, 1}^n\), \(F(x \oplus y) = F(x) \oplus F(y)\).
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Context:
Given the function $f(x)= \frac{1}{e} \cdot e^{x}+ \frac{a}{2}x^{2}-(a+1)x+a (a > 0)$, where $e$ is the base of the natural logarithm. If the function $y=f(x)$ and $y=f[f(x)]$ have the same range, find the maximum value of the real number $a$. Express your answer as a single real number.
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Context:
Evaluate the double integral \( \iint_E x \, dx \, dy \) over the region \( E = \{(x,y) \mid 0 \le x, 0 \le y \le 1, 1 \le x^2 + y^2 \le 4\} \).
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Context:
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.
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Context:
A pyramid's base is a square with unit side length. One of its side edges is also of unit length and coincides with the height of the pyramid. What is the largest dihedral angle?
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Context:
If the space vectors $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ satisfy $|\overrightarrow{{e_1}}|=|2\overrightarrow{{e_1}}+\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\overrightarrow{{e_1}}$ in the direction of $\overrightarrow{{e_2}}$.
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Context:
Given that \(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\) are 7 distinct prime numbers, and the sum of any two of \(a, b, c\) is 800. Let \(d\) be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of \(d\).
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Context:
There are 28 identical egg containers in the mart. Each egg container has the same number of eggs. When you select the 14th egg from the front, the 20th from the back, the 3rd from the left, and the 2nd from the right in an egg container, find all the number of eggs in the egg containers at this mart.
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Context:
An engineer undertakes a project to build a road 15 km long in 300 days and employs 50 men for the purpose. After a certain number of days, he finds only 2.5 km of the road has been completed. He must employ approximately 75 extra men to finish the work in time. How many days had passed when he realized the progress?
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Context:
In triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to the internal angles $A$, $B$, $C$ respectively, and satisfy $\left(b-a\right)\left(\sin B+\sin A\right)=c(\sqrt{3}\sin B-\sin C)$. Find the value of $A$.
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Context:
In a round of the DES cryptosystem, the input is the 64-bit string $1010101010101010101010101010101010101010101010101010101010101010$, and the round key is the 48-bit string $111111111111111111111111111111111111111111111111$. Compute the 53rd bit of the output for this round.
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Context:
Find the number of distinct values obtained from the expression:

$$ \frac{n^2 - 2}{n^2 - n + 2} $$

where \( n \in \{1, 2, 3, \ldots, 100\} \).
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Context:
Three lattice points are chosen at random with replacement in the interior of the square defined by \(-99 \le x \le 100\) and \(-99 \le y \le 100\). The probability that the area of the triangle formed by these points is an integer can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).
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Context:
Find the closure of \(l^1(\mathbb{N})\) in \(l^2(\mathbb{N})\).
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Context:
Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{\ln(1+nx)}{n x^n} \) converges.
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Context:
A circle passes through the vertices $A$ and $C$ of triangle $ABC$, intersects side $AB$ at point $D$ and side $BC$ at point $E$. Find the angle $CDB$ if $AD=5$, $AC=2\sqrt{7}$, $BE=4$, and $BD:CE=3:2$.
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Context:
Find the smallest integer \( n \) such that \( f_3(n) > T(n+3) \), using the bounds for \( f_3(n) \) provided.
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Context:
Suppose that for a certain Poisson process, the average arrival rate is once every $4$ seconds. If there are no arrivals during the first $10$ seconds, what is the probability of observing exactly $6$ arrivals in the subsequent $10$ seconds?
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Context:
Determine whether the limit \( \lim_{t \to 0^{+}} A_t \) exists for the surface area \( A_t \) of the surface \( P_t = \{(x,y,z) : (x^2 + y^2)z = 1, \, t^2 \le x^2 + y^2 \le 1\} \) in \( \mathbb{R}^3 \). If it exists, evaluate the limit.
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Context:
Find the coefficient of $x^{10}$ in the generating function $$\frac{1+x^3}{1-2x+x^3-x^4}.$$
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Context:
Find the value of the infinite series: \[ \frac{1}{2\cdot4} + \frac{1\cdot3}{2\cdot4\cdot6} + \frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8} + \cdots \]
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Context:
What is the greatest number of natural numbers, not exceeding 2016, that can be selected such that the product of any two selected numbers is a perfect square?
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Context:
for any integer k > 1 , the term “ length of an integer ” refers to the number of positive prime factors , not necessarily distinct , whose product is equal to k . for example , if k = 24 , the length of k is equal to 4 , since 24 = 2 × 2 × 2 × 3 . if x, y, and z are positive integers such that x > 1 , y > 1, z > 1, and x is a product of two distinct prime numbers, y is a product of three distinct prime numbers, and x + 3y + 5z < 5000 , what is the maximum possible sum of the length of x, the length of y, and the length of z?
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Context:
Determine the constant \( k \) such that for any sequence of positive numbers \( a_i \), the inequality \[ \sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 + \ldots + a_n} \leq k \sum_{n=1}^{\infty} \frac{1}{a_n} \] holds.
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Context:
设 a 是正整数，数列 \{b_n\} 定义为 b_n=\left(1+\df{1}{n}\right)^{2024n+a}. 若 \{b_n\} 单调递减，则 a 的最小值为__________.
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Context:
Given that the probability of drawing a first-class product is 0.65, the probability of drawing a second-class product is 0.2, and the probability of drawing a third-class product is 0.1, calculate the probability of drawing a non-first-class product. Express your answer as a decimal value between 0 and 1.
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Context:
Calculate the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 3 = 0,\] given that all roots are real and nonnegative.
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Context:
$ 500 is divided amongst a, b, c, d, and e so that a may get 5 / 7 as much as b, c, d, and e together, b may get 11 / 16 as much as a, c, and d together, c may get 3 / 8 as much as a, b, and e together, and d may get 7 / 12 as much as a, b, and c together. Find the share of a.
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Context:
Let $V$ be an infinite-dimensional vector space over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. If the dimension of $V$ is $\alpha$, a cardinal number, determine the dimension of $V^*$, the algebraic dual of $V$. Express your answer in terms of $\alpha$.
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Context:
A BWM tree is defined recursively:

- An empty tree is a BWM tree of height 0 and size 0.
- A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest.
- The height of a nonempty BWM tree is one more than the height of its tallest subtree, and the size of a nonempty BWM tree is one more than the sum of the sizes of the subtrees.

What is the minimum size of a height-10 BWM tree?
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Context:
Find the scalar value of \( a \) such that the vector set \( \{u_1+au_2, u_2+au_3, u_3+au_4, u_4+au_1\} \) is linearly independent, given that the vector set \( \{u_1, u_2, u_3, u_4\} \) is linearly independent.
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Context:
$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.
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Context:
Let \( X \sim \text{Poisson}(\lambda) \), where \( \lambda > 0 \) is unknown. If \( \delta(X) \) is the unbiased estimator of \( g(\lambda)=e^{-\lambda}(3\lambda^2+2\lambda+1) \), find the value of \( \sum_{k=0}^\infty \delta(k) \).
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Context:
There are 10 ping pong players participating in a round-robin tournament. The results show that there are no draws, and among any group of 5 players, there is one player who defeats the other 4 and one player who loses to the other 4. How many players have exactly two wins?
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Context:
Determine the number of solutions, counting multiplicity, in the domain \(\{z \in \mathbb{C} : 1 < |z| < 2\}\) for the equation:
\[z^9 + z^5 - 8z^3 + 2z + 1 = 0\]
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Context:
Given a triangle ΔABC with a fixed side length, determine the number of inscribed equilateral triangles that can be drawn with vertices on different sides of ΔABC.
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Context:
Let $k$ be a positive integer. A sequence $a_0, a_1, \ldots, a_n$, where $n > 0$, of positive integers satisfies the following conditions:

1. $a_0 = a_n = 1$;
2. $2 \leq a_i \leq k$ for each $i = 1, 2, \ldots, n-1$;
3. For each $j = 2, 3, \ldots, k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0, a_1, \ldots, a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
4. For any $i = 1, 2, \ldots, n-1$, $\gcd(a_i, a_{i-1}) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.

Suppose there is another sequence $b_0, b_1, \ldots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} > \frac{b_i}{a_i}$ for all $i = 0, 1, \ldots, n-1$. Find the minimum value of $b_n - b_0$.
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Context:
In an isosceles triangle \(ABC\), the angle at the base \(AC\) is \(\alpha\). The circle inscribed in this triangle touches the sides of the triangle at points \(A_1\), \(B_1\), and \(C_1\). Find the ratio of the area of triangle \(A_1B_1C_1\) to the area of triangle \(ABC\).
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Context:
9.4. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{3-\sqrt{2}} \sin x+1) \cdot(3+2 \sqrt{7-\sqrt{2}} \cos y-\cos 2 y)$. If the answer is not an integer, round it to the nearest integer.
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Context:
将时钟盘面上标有数字 $1,2, \cdots, 12$ 的十二个点分别染红、黄、蓝、绿四种颜色，每色三个点，以这些点为顶点作 $n$ 个凸四边形，使其满足：
$(1)$每个四边形的四个顶点四色都有；
$(2)$对于其中任何三个四边形，都存在某一色，染有该色的三个顶点所标数字互不相同。
求 $n$ 的最大值.
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Context:
In an Abelian category $\mathscr{A}$, determine the category in which the Hom functor $\mathrm{Hom}_\mathscr{A}(A,-)$ takes values.
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Context:
Shapovalov A.B.

Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them gets nine handfuls, after which the other takes all the remaining coins (the division may also end if the coins are divided before anyone gets nine handfuls). Hapok can grab any number of coins in a handful. What is the maximum number of coins he can guarantee himself regardless of Glazok's actions?
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Context:
Determine the cardinality of the set of all surjective functions from \( \mathbb{N} \) to \( \mathbb{R} \).
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Context:
Find the integral of \( f(x) = \frac{1}{1 + \tan(\sqrt{2} x)} \) over the interval from 0 to \(\pi/2\). In other words, find \(\int_0^{\pi/2} f(x) \, dx\).
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Context:
Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$.

Note: The circumcircle of a triangle is the circle that passes through its three vertices.
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Context:
When Jeffrey walks, for every three steps forward, he takes two steps backwards. However, after he covers the first 30 steps, he changes his pattern and starts taking only one step backwards for every four steps forward. If the distance between the house and the mailbox is 66 steps, what is the total number of steps Jeffrey takes when he goes from the house to the mailbox, considering the additional constraint and the change in his stepping pattern after the first 30 steps?
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Context:
For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x < 2010$ such that $x^2 - n$ is divisible by $2010$. Determine the remainder when $\sum_{n = 0}^{2009} n \cdot \theta(n)$ is divided by $2010$.
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Context:
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$.
Find $ \lim\limits_{n\to\infty}(a_0+a_1+\ldots+a_n)$.
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Context:
The function  $g\left(x\right)$  is defined as  $\sqrt{\dfrac{x}{2}}$  for all positive  $x$ .  $ $   $$ g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right) $$    $ $ can be expressed as  $\cos(b)$  using degrees, where  $0^\circ < b < 90^\circ$  and  $b = p/q$  for some relatively prime positive integers  $p, q$ . Find  $p+q$ .
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Context:
Let \( n = p_1^{k_1} + p_2^{k_2} + \ldots + p_m^{k_m} \), where \( p_1, p_2, \ldots, p_m \) are distinct prime numbers and \( k_1, k_2, \ldots, k_m \) are positive integers. Determine the number of ways \( n \) can be expressed as a product of two positive integers that have no common factors, considering that the order of the factors matters (i.e., \( a \cdot b \) and \( b \cdot a \) are regarded as different).
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Context:
A general population follows an exponential distribution with parameter \( \lambda \). A sample of six elements was drawn: \( 3.2, 2.5, 0.1, 5.1, 4.7, 4.1 \). Using the most-powerful test, verify the null hypothesis that \( \lambda = \frac{1}{2} \) against the alternative hypothesis that \( \lambda = \frac{1}{3} \). Assume a significance level of \( \alpha = 0.05 \). Determine the critical value \( c_2 \) for the test statistic \( \sum_{i=1}^{n} x_{i} \).
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Context:
In the three phrases "尽心尽力", "力可拔山", and "山穷水尽", each Chinese character represents a digit between 1 and 8. Identical characters represent the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each phrase is 19, and "尽" > "山" > "力", then the maximum value for "水" is $\qquad$
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Context:
Calculate the expectation of the falling factorial of a geometrically distributed random variable $X$ with success probability $p$, denoted by $E[(X)_3]$. The probability generating function of $X$ is $G(z) = \sum_{k=1}^{\infty} kp(1-p)^{k-1}z^k$, where $X$ represents the number of trials needed for the first success.
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Context:
A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \_\_\_\_\_\_ (answer in numbers).
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Context:
Compute\[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\](Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)
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Context:
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
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Context:
If $e^{i \theta} = \frac{3 + i \sqrt{8}}{4},$ then find $\sin 6 \theta.$
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Context:
Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Point $P$ is the unique point in the plane of $\triangle ABC$ such that $\triangle PBM \sim \triangle PNC$. What is $AP$?
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Context:
Find the following limit.
 \( \mathop{\lim }\limits_{{n \rightarrow  \infty }}\frac{1}{n}\log \)  \( \left( {\mathop{\sum }\limits_{{k = 2}}^{{2}^{n}}{k}^{1/{n}^{2}}}\right) \)
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Context:
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $|d - \sqrt{n}|$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$.
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Context:
## Problem Statement

Find the indefinite integral:

$$
\int \frac{\sqrt[3]{1+\sqrt[5]{x}}}{x \sqrt[15]{x^{4}}} d x
$$
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Context:
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the angles $A$, $B$, and $C$ respectively. If $\dfrac {a\sin A + b\sin B - c\sin C}{a\sin B} = 2\sin C$, find the size of $\angle C$.
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Context:
For any positive integer n, let d(n) denote the number of positive divisors of n; and let φ(n) denote the number of elements from the set {1, 2, · · · , n} that are coprime to n. (For example, d(12) = 6 and φ(12) = 4.) Find the smallest positive integer n such that d(φ(n)) = 2017.
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Context:
Ben "One Hunna Dolla" Franklin is flying a kite KITE such that $I E$ is the perpendicular bisector of $K T$. Let $I E$ meet $K T$ at $R$. The midpoints of $K I, I T, T E, E K$ are $A, N, M, D$, respectively. Given that $[M A K E]=18, I T=10,[R A I N]=4$, find $[D I M E]$.
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Context:
Use Stokes' Theorem to compute the line integral \( \int_C (y \, \mathrm{d}x + z \, \mathrm{d}y + x \, \mathrm{d}z) \), where \( C \) is the curve of intersection of the cylinder \( x^2 + y^2 = 1 \) and the surface \( z = x \cdot y \), oriented counterclockwise around the cylinder.
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Context:
In a small pond there are eleven lily pads in a row labeled $0$ through $10.$ A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability \(\frac{N}{10}\) and to pad \(N+1\) with probability \(1-\frac{N}{10}\). Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Calculate the value of \(k+m\).
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Context:
Evaluate the line integral \( \int_C y^2 x^2 \, dy + 2xy^3 \, dx \) using Green's Theorem, where \( C \) is the negatively oriented triangle with vertices at \((0,0), (3,1), (3,0)\).
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Context:
In the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots\). Find \(a_{1999}\).
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Context:
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\frac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
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Context:
Determine the number of elements of order 10 in the group \( \mathbb{Z}_{50\mathbb{Z}} \times \mathbb{Z}_{20\mathbb{Z}} \).
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Context:
What is the value of the definite integral $\int_{0}^{1} \left(1 + x + x^2 + x^3 + \ldots + x^{n-1}\right) \left(1 + 3x + 5x^2 + \ldots + (2n-1)x^{n-1}\right) dx$?
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Context:
Determine the number of distinct ways to color the faces of a regular icosahedron using two colors.
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Context:
Find the last two digits of the sum of the eighth powers of the integers from 1 to 100, inclusive.
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Context:
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. Find the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$. The original answer is in the form of $k\sqrt{m}$, please give the value of k + m.
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Context:
There is a unique positive integer \( x \) such that \( {x}^{x} \) has 703 positive factors. What is \( x \) ?
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Context:
Let \( n \) be a positive integer. Determine the largest integer \( f(n) \) such that there exists a polynomial \( p \in \mathbb{Z}[x, y] \) satisfying \((x+2y)^{2n} - x^{2n} = 2^{f(n)} p\).
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Context:
Given a sequence of independent random variables \(\{X_i, i \geq 1\}\), where \(\mathbb{P}(X_i = i^2 - 1) = i^{-2}\) and \(\mathbb{P}(X_i = -1) = 1 - i^{-2}\), determine the constant \(c\) to which \(\frac{1}{n} \sum_{i=1}^n X_i\) converges almost surely.
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Context:
João wrote down all the powers of 2, 3, and 5 greater than 1 and less than 2017 on a sheet of paper. He then calculated all possible products of two distinct numbers from that sheet and recorded them on another sheet of paper. How many distinct integers did João write on the second sheet?
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Context:
Given the set {1, 4, 7, 10, 13, 16, 19}, how many different integers can be expressed as the sum of three distinct members of this set? Express your answer as a single integer.
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Context:
Given a sequence $\{a_{n}\}$, where ${a}_{n}=n•{a}^{n}$, $n$ is a positive integer, and the constant $a \gt 0$, $a\neq 1$. If $\{a_{n}\}$ is a strictly decreasing sequence, then the range of real number $a$ is ______.
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Context:
In triangle $XYZ$ with right angle at $Z$, $\angle XYZ < 45^\circ$ and $XY = 6$. A point $Q$ on $\overline{XY}$ is chosen such that $\angle YQZ = 3\angle XQZ$ and $QZ = 2$. Determine the ratio $\frac{XQ}{YQ}$ in simplest form.
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Context:
Given a sequence $\{a\_n\}$ that satisfies: for any $n∈N^{*}$, $a_{n+1}=ka_{n}+2k-2$, where $k$ is a constant not equal to $0$ or $1$. If $a_{i}∈\{-272,-32,-2,8,88,888\}$, where $i=2$, $3$, $4$, $5$, then the sum of all possible values of $a_{1}$ is $\_\_\_\_\_\_\_$.
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Context:
The convex pentagon ABCDE has all sides of equal length, and angles A and B are both 90 degrees. Calculate the degree measure of angle E. Express your answer in degrees.
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Context:
In a random tournament with $n$ nodes, where each pair of distinct nodes $a$ and $b$ has an edge connecting $a$ to $b$ or $b$ to $a$ with equal probability of 0.5, determine the expected length of the longest transitive chain. If a simple closed-form expression is not available, provide a tight upper bound for this expected length.
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Context:
Martha receives a daily allowance of $15. She decides to save varying portions of this amount every day for three weeks. In the first week, she saves half of this amount every day, except for one day when she saves only a third. During the second week, she saves 60% of her daily allowance for the first 5 days, and only 25% on the last 2 days. In the third week, Martha decides to save 75% of her allowance every day, but she has to pay for a $10 expense on the first day and a $20 expense on the fifth day of that week. How much has she saved at the end of these three weeks?
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Context:
A point $A = (a,b)$ lies on the graph $y = x^2 + 1$. A point $B = (c,d)$ lies within the area defined by $(x+2)^2 + (y+2)^2 = r^2$. Define a matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $M$ always has an inverse (i.e., the determinant of $M$ is not zero). What is the maximum possible value of $50r^2$?
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Context:
Determine the actual growth rate of \( \left|\frac{\zeta'}{\zeta}\left(-\frac{1}{2}+it\right)\right| \) as \( t \to \infty \), where \( \zeta \) is the Riemann zeta-function and \( t > 0 \).
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Context:
Find the sum of the digits of the smallest positive integer $k$ such that the sum of the coefficients $a_0 + a_1 + \ldots + a_{k-1}$ in the expansion of the polynomial \((1 + x)(1 + 2x)(1 + 3x) \cdots (1 + kx) = a_0 + a_1x + a_2x^2 + \cdots + a_kx^k\) is divisible by 2005.
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Context:
In acute triangle $ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and they satisfy $a\cos B = b(1 + \cos A)$. The area of triangle $ABC$ is $S=2$. Find the range of $(c + a - b)(c + b - a)$.
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Context:
Consider the group \( G = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} : a, b, c \in \mathbb{R} \right\} \) and its action \( \pi \) on \( \mathbb{R}^2 \) defined by \( \pi_g(x) = gx \) for all \( x \in \mathbb{R}^2 \). Determine the number of distinct orbits of this action on \( \mathbb{R}^2 \).
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Context:
Find the limit of the function \( f(x,y) = -\dfrac{y^4+x^3y-xy^3-x^4}{\left(y^2+x^2\right)^2} \) as \((x,y)\) approaches \((0,0)\) along the path \(x=y\).
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Context:
5. Given six cards, on which the digits $1,2,4,5,8$ and a comma are written. From them, all possible numbers are formed (each card must be used exactly once, the comma cannot stand at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?

(M. V. Karluкова)
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Context:
Kozhevnikov P.A

On each of two parallel lines $a$ and $b$, 50 points were marked.

What is the maximum possible number of acute-angled triangles with vertices at these points?
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Context:
In triangle \( ABC \) with sides \( AB = 8 \), \( AC = 4 \), \( BC = 6 \), the angle bisector \( AK \) is drawn, and on the side \( AC \) a point \( M \) is marked such that \( AM : CM = 3 : 1 \). Point \( N \) is the intersection point of \( AK \) and \( BM \). Find \( AN \).
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Context:
In how many ways can one fill a \(4 \times 4\) grid with a 0 or 1 in each square such that the sum of the entries in each row, column, and long diagonal is even?
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Context:
King Qi and Tian Ji are competing in a horse race. Tian Ji's top horse is better than King Qi's middle horse, worse than King Qi's top horse, Tian Ji's middle horse is better than King Qi's bottom horse, worse than King Qi's middle horse, and Tian Ji's bottom horse is worse than King Qi's bottom horse. Now, randomly select one horse from each side for a race, then the probability of Tian Ji's horse winning is ______.
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Context:
11. In triangle $ABC$, the altitude $BH$ is drawn, point $O$ is the center of the circumscribed circle around it, and the length of its radius is $R$. Find the measure in radians of the largest of the angles $ACB$ and $BAC$, given that $R=(2 / 3) \cdot B H=(4 / 3) \cdot O H$. If necessary, round the found value to two decimal places.
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Context:
Determine the values of $x$ for which the series \( \sum \frac{x^n}{x^{2n}-1} \) converges.
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Context:
Find the ratio of the area of $\triangle BCY$ to the area of $\triangle ABY$ in the diagram if $CY$ bisects $\angle BCA$. [asy]
import markers;

real t=34/(34+28);
pair A=(-17.18,0);
pair B=(13.82,0);
pair C=(0,30);
pair Y=t*B+(1-t)*A;

draw(C--A--B--C--Y);

label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$Y$",Y,NE);

label("$32$",.5*(B+A),S);
label("$34$",.5*(B+C),NE);
label("$28$",.5*(A+C),NW);
[/asy]
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Context:
Determine the set of real numbers \( s > 0 \) for which the double integral \(\iint_{(a,b)^2}\frac{1}{|x-y|^s}\,dx\,dy\) converges, where \((a,b)\) is an arbitrary finite open interval of \(\mathbb{R}\).
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Context:
How many natural numbers between 2 and 60 have no primitive roots and require at least four squares to be expressed as a sum of squares?
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Context:
In the repeating decimal 0.2017, if the sum of all digits from the $m$-th digit to the $n$-th digit after the decimal point is 2017, find the value of $n$ when $m$ takes the minimum value.
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Context:
Find the coefficient of \(x^{12}y^{24}\) in the expansion of \((x^3 + 2xy^2 + y + 3)^{18}\).
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Context:
Let $A_{1}, \ldots, A_{2022}$ be the vertices of a regular 2022-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)

In each round, Bob chooses a point in the plane (not necessarily among $A_{1}, \ldots$, $A_{2022}$ ) and Alice responds truthfully with the color of that point. What is the smallest number $Q$ for which Bob has a strategy to always determine the colors of points $A_{1}, \ldots, A_{2022}$ in $Q$ rounds?
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Context:
Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in  $6$  minutes, Mr. Rose in  $10$  minutes, and Mr. Schwartz in  $18$  minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?
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Context:
Consider a random walker starting at $S_0 = 2$ on the integers. The walker moves according to the following rules:

- If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or $p_{n+1}$ with equal probability.
- If the walker is on a composite number $x$, she moves to one of the prime factors of $x$, each with probability $1/\omega(x)$, where $\omega(n)$ denotes the number of distinct prime factors of $n$.

Determine whether the probability $\mathbb{P}(\sup_{n\ge 0} S_n = \infty)$ is 1 or less than 1.
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Context:
(15) The number of planes equidistant from the four vertices of a spatial quadrilateral is
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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. Find the probability that Amelia’s position when she stops will be greater than \(1\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
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Context:
Find the value of the third derivative \( y''' \) at \( x = 1 \) for the function defined by the equation \( x^2 + xy + y^3 = 1 \).
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Context:
Evaluate the integral in a closed form: \[ \int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx \]
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Context:
Let \( x \) be a non-zero real number such that

\[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \].

Find the product of all possible values of \( x \).
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Context:
Given the function $y = x - 5$, let $x = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$, we can obtain 10 points on the graph of the function. Randomly select two points $P(a, b)$ and $Q(m, n)$ from these 10 points. What is the probability that $P$ and $Q$ lie on the same inverse proportion function graph?
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Context:
Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
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Context:
 Points $X$ and $Y$ lie outside regular pentagon $ABCDE$ such that $ABX$ and $DEY$ are equilateral triangles. Find the degree measure of $\angle XCY$ .
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Context:
Evaluate the integral \( \int g_\epsilon(x)f(x)\,\mathrm{d}^3x \) where \( g_\epsilon(x) = \epsilon^{-3}g(\epsilon^{-1}x) \), \( g \in L^1(\mathbb{R}^3) \cap L^2(\mathbb{R}^3) \) with \( \int g(x)\,\mathrm{d}^3x = 1 \), and \( f(x) \) is a smooth function with compact support.
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Context:
Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions:
(1)$a_1=1$
(2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$
what is the residue when we divide $f(2015)$ by $4$ ?
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Context:
Vanya thought of a seven-digit number and then subtracted the sum of all its digits except one. He got 9875352. What number would Vanya get if he subtracted the sum of all the digits except the second one from the left?
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Context:
Find the integral basis for the ring of algebraic integers in the quadratic field $\mathbb{Q}[\sqrt{D}]$ when $D \equiv 2 \pmod{4}$.
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Context:
How many arrangements of the letters in "DISAPPEARANCES" satisfy the following conditions: (i) there are at least two letters between each 'A', (ii) the arrangement ends with an 'A', and (iii) the consonants are not in alphabetical order?
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Context:
Find the mass percentage of C in CO
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Context:
Given that \( a_1, a_2, a_3 \) form an arithmetic sequence with \( a_1 + a_2 + a_3 = 15 \), and \( b_1, b_2, b_3 \) form a geometric sequence with \( b_1 b_2 b_3 = 27 \), if \( a_1 + b_1, a_2 + b_2, a_3 + b_3 \) are positive integers and form a geometric sequence, find the maximum value of \( a_3 \).
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Context:
A function \( f\left( x\right) \) satisfies \( \left\{  \begin{array}{l} f\left( x\right)  =  - {f}^{\prime \prime }\left( x\right)  - \left( {{4x} - 2}\right) {f}^{\prime }\left( x\right) \\  f\left( 0\right)  = a,f\left( 1\right)  = b \end{array}\right. \)
 Evaluate \( {\int }_{0}^{1}f\left( x\right) \left( {{x}^{2} - x}\right) {dx} \) .
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Context:
A convex polyhedron has more faces than vertices. What is the minimum number of triangular faces among its faces?
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Context:
Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?
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Context:
Calculate the value of \( \sqrt{M - 2} + 1 \), where \( M = \lim\limits_{x\to 0 } \left[\dfrac{1 - f(x)^3}{5 \tan^2x}\right] \), \( f(x) = \cos(2x) \cos(4x) \cos(6x) \cos(8x) \cos(10x) \), and \( M \) is finite.
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Context:
Find the kernel of the homomorphism $D : F[x] \to F[x]$ defined by $D(f(x)) = f'(x)$, where the characteristic of the field $F$ is a prime number $p$. What is the set of polynomials that form this kernel?
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Context:
Take the x-axis as horizontal and the y-axis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity \( v \). Assuming the only force on the pellet after firing is gravity (acceleration \( g \)), which points in the first quadrant can the gun hit?
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Context:
Given that \( P_n(0) = -\frac{5}{16} \), find the value of \( \int_{-1}^{1} P_n^2(x) \, dx \).
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Context:
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?
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Context:
The graph of the function y = sin(2x + π/3) is translated in such a way that the resulting graph is symmetric about the point (-π/12, 0). What is the translation type and amount? Express your answer in the format'move to the [left/right] by [amount]'.
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Context:
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. Given that India finished first, followed by Gertrude, Fiona, Janice, and lastly Hannah, calculate the minimum total number of points that could have been awarded.
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Context:
Point $P$ is a moving point on the parabola $y^{2}=4x$. The minimum value of the sum of the distances from point $P$ to point $A(0,-1)$ and from point $P$ to the line $x=-1$ is ______.
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Context:
Let $a > 1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n \geq 1}$ is defined such that $a_1 = a$ and for all $n \geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1, a_2, \ldots, a_7$ are all distinct.
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Context:
For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_{1}+r_{2}=\sqrt{2}$.
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Context:
What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped?
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Context:
Yesterday, Ben took his family fishing. They caught a total of 15 fish of different sizes and species:

- Ben caught 4 fish: two Bluefish that were 5 inches and 9 inches long, and two Yellowtail that were 9 inches long
- Judy caught 1 fish: a Red Snapper that was 11 inches long
- Billy caught 3 fish: a Bluefish that was 6 inches long, and two Yellowtail that were 6 inches and 10 inches long
- Jim caught 2 fish: a Red Snapper that was 4 inches long, and a Bluefish that was 8 inches long
- Susie caught 5 fish: a Red Snapper that was 3 inches long, two Yellowtail that were 7 inches and 12 inches long, and two Bluefish that were 12 inches long

Each fish species has its own unique minimum size limit to be kept:
- Bluefish: 7 inches
- Yellowtail: 6 inches
- Red Snapper: 8 inches

If each fish can give them 2 filets, how many fish filets will Ben and his family have from the fish that meet the size limits?
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Context:
Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$) and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares. What is the total area of these squares?
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Context:
How many elements of order $p$ are there in $GL_{2}(\Bbb Z_{p})$, where $p$ is a prime?
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Context:
In a grid of numbers and letters, starting from the letter A in the middle of the grid, calculate the total number of different paths that can spell out "AMC8". Each path must move to an adjacent (up, down, left, or right) letter or number only once (no diagonal moves allowed). Provide your answer as a single integer representing the total number of paths.
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Context:
用 $6$ 根等长的细铁棒焊接成一个正四面体形框架，铁棒的粗细和焊接误差不计。设此框架能容纳得下的最大球的半径为 $R_{1}$,能包容此框架的最小球的半径为 $R_{2}$ 。则 $\frac{R_{1}}{R_{2}}$等于。原始答案为 $\frac{\sqrt{m}}{n}$ 格式，请给出 $m + n$ 的值。
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Context:
For each pair of real numbers \((x, y)\) with \(0 \leq x \leq y \leq 1\), consider the set
\[ A = \{ x y, x y - x - y + 1, x + y - 2 x y \}. \]

Let the maximum value of the elements in set \(A\) be \(M(x, y)\). Find the minimum value of \(M(x, y)\).
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Context:
An ellipse satisfies the following optical property: a ray of light emitted from one focus of the ellipse, after being reflected by the ellipse, passes through the other focus. Now, consider a horizontally placed elliptical billiard table satisfying the equation: $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, where points $A$ and $B$ are its two foci. When a stationary ball is placed at point $A$ and starts moving in a straight line, after being reflected by the elliptical wall (not at the endpoints of the major axis) and returning to point $A$, what is the shortest distance the ball travels? Express your answer as a single number.
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Context:
Evaluate the integral $$\int_0^{\pi/2} \sin^{10}x\cdot \cos(12x)\,\text{d}x$$
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Context:
The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5.
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Context:
Let $c_1, c_2, \ldots, c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1, a_2, \ldots, a_{6030}$, there exist 6030 real numbers $\{b_1, b_2, \ldots, b_{6030}\}$ such that:
\[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\]
and
\[b_n = \sum_{d\mid n} c_d a_{n/d}\]
for $n=1, 2, \ldots, 6030$. Find $c_{6030}$.
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Context:
在 $\triangle A B C$ 中, $\angle B A C=30^{\circ}, \angle A B C=70^{\circ}, M$ 为形内一点, $\angle M A B=\angle M C A=20^{\circ}$, 求 $\angle M B A$ 的度数.
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Context:
Find all integer pairs \((x, y)\) such that \(2x^3 - 7y^3 = 3\).
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Context:
The cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$  Find
\[p(1) + p(2) + p(3) + \dots + p(21).\]
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Context:
Barry has three sisters. The average age of the three sisters is 27. The average age of Barry and his three sisters is 28. What is Barry's age?
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Context:
The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid.
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Context:
Differentiate the expression $\theta^TX^TX\theta$ with respect to the vector $\theta$. Provide the resulting expression.
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Context:
Given the sequence \( 1, (1+2), (1+2+2^{2}), \cdots, (1+2+2^{2}+\cdots+2^{n}) \), find the sum of the first \( n \) terms of this sequence in terms of \( n \). Express your answer as a formula in terms of \( n \).
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Context:
How many polynomials of degree 3 are contained in the ideal \( I = (x^5 + x^2 + x + 1, x^7 + x^5 + x^3 + x) \) in \( \mathbb{F}_2[x] \)?
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Context:
Solve the equation$$arc \sin (\sin x) + arc \cos (\cos x)=0$$
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Context:
A number of market women sold their various products at a certain price per pound (different in every case), and each received the same amount - 2s. 2 1/4d. What is the greatest number of women there could have been? The price per pound in every case must be such as could be paid in current money.
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Context:
An urn contains b blue balls and r red balls. You repeatedly and independently remove balls from the urn (without returning them) until the first blue ball is drawn. All balls currently in the urn have an equal probability of being selected each time a ball is removed. Define the random variable X as the number of balls that are drawn (number of red balls that are removed plus the first blue ball that is removed). Find E[X], the expected value of random variable X.
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Context:
Determine the values of $z \in \mathbb{C}$ for which the series \( \sum_{n=1}^{\infty} \frac{1}{(n+2)^4}(z-2i)^{n^3} \) converges using the root test.
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Context:
Determine the Lebesgue measure of Luzin's non-Borel set of reals, defined as follows: For $x \in [0,1]$, let \( x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} \) be the continued fraction expansion of $x$. Let $A$ be the set of numbers $x \in [0,1]$ whose corresponding sequence $a_0, a_1, a_2, \cdots$ admits an infinite subsequence $a_{k_1}, a_{k_2}, a_{k_3}, \cdots$ such that for each $i$, $a_{k_{i+1}}$ is divisible by $a_{k_i}$. What is the Lebesgue measure of the set $A$?
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Context:
Given the proposition $P$: "For all $x \in \mathbb{R}$, there exists an $m \in \mathbb{R}$ such that $4^{x}-2^{x+1}+m=0$". If the negation of the proposition $P$ is a false proposition, then the range of the real number $m$ is _____.
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Context:
Determine the limit of the sequence \( \frac{3^n+3.75^n+\frac{5^n}{n^5}}{\frac{5^n}{n^5}+n^44^n} \) as \( n \to \infty \).
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Context:
Let \( P \) be a regular polygon with 2006 sides. A diagonal of \( P \) is called good if its endpoints divide the perimeter of \( P \) into two parts, each having an odd number of sides of \( P \). The sides of \( P \) are also called good. Suppose \( P \) has been subdivided into triangles by 2003 diagonals that do not intersect at an interior point of \( P \). Find the maximum number of isosceles triangles that can appear in such a subdivision, where two sides of the isosceles triangles are good.
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Context:
Evaluate the limit: $$ \lim_{r\to 0}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 - \cos\left(2r\cos t\right)}{4r^{2}\cos t}\,\mathrm{d}t $$ without explicitly solving the integral.
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Context:
Find the residue of the function \( f(z) \log \frac{z-b}{z-a} \) at \( z = a \), where \( f \) is an entire function and \( a, b \in \mathbb{C} \).
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Context:
Calculate the minimum number of check bits required to create a correctable code with a minimum distance of 4 for a list of 10 key words.
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Context:
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.
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Context:
Given the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ where $a>b>0$, with left and right foci $F_{1}$ and $F_{2}$, and point $P$ in the first quadrant on the ellipse. Let $PF_{1}$ extend to meet the ellipse at another point $Q$, with $PF_{2}\perp F_{1}F_{2}$ and $|PF_{1}|=2|F_{1}Q|$. Calculate the eccentricity of the ellipse. Express your answer as a simplified fraction or decimal value.
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Context:
The ancient "Five Elements" theory posits that there are five types of properties in materials: metal, wood, earth, water, and fire. Metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal. If five different properties of materials are arranged in a row, but two materials with overcoming properties are not adjacent, then the number of such arrangements is $\boxed{\text{number}}$ (the result is expressed in numerical value).
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Context:
Let \( T: \mathbb{C}^{11} \rightarrow \mathbb{C}^{11} \) be a linear transformation such that \( \dim \ker T = 4 \), \( \dim \ker T^3 = 9 \), and \( \dim \ker T^4 = 11 \). Find \( \dim \ker T^2 \).
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Context:
Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?
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Context:
In a triangle with known sides \(a = 2\), \(b = 3\), and area \(S = \frac{3 \sqrt{15}}{4}\), the median drawn to its third side is less than half of that side.

Find the radius of the circumcircle of this triangle.
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Context:
At a certain school, there are 6 subjects offered, and a student can take any combination of them. It is noticed that for any two subjects, there are fewer than 5 students taking both of them and fewer than 5 students taking neither. Determine the maximum possible number of students at the school.
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Context:
Determine the Krull dimension of the ring $B = A[x,y,z]/\langle xy, xz \rangle$, where $A$ is a Noetherian commutative ring.
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