Context:
对于一个实数 $x$, 令 $\{x\}=x-[x]$. 记 $S=\int_{0}^{2024} \min \left(\left\{\frac{x}{8}\right\},\left\{\frac{x}{4}\right\}\right) \mathrm{d} x$, 则 $[S]=$ $\qquad$.
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Context:
Given the sequence \(a_k\) where \(a_k a_{k-1} + a_{k-1} a_{k-2} = 2a_k a_{k-2}\) for \(k \geq 3\) and \(a_1 = 1\), let \(S_q = \sum_{k=1}^{q} \frac{1}{a_k}\). If \(\frac{S_{2q}}{S_q}\) is independent of \(q\), find \(\frac{1}{a_{2016}}\).
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Context:
Let $X$ and $Y$ be independent random variables, each following a standard normal distribution, $N(0,1)$. Calculate the probability $P(|Y| \ge \frac{1}{\sqrt{3}}|X|)$. Provide your answer as a single probability value.
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Context:
Find the asymptotic equivalent of the sequence \((u_n)_{n \geq 1}\) defined by \(u_1 > 0\) and \(u_{n+1} = \frac{u_n}{n} + \frac{1}{n^2}\) as \(n \to +\infty\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence for the power series representation of the function \( f(x) = x^{0.6} \) centered at \( a = 4 \).
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Context:
已知下列结论成立：在复平面上的多项式 $f(z), g(z)$ 和实数 $r>0$ ，若对 $|z|=r$ ，都有 $|g(z)|<|f(z)|$ ，则在 $\{z||z|<r\}$ 中， $f(z)$ 与 $f(z)+g(z)$ 的零点数相等 (计算重数)。现已知多项式 $z^{9}+2 z^{5}-8 z^{3}+3 z+1$, 求其在 $\{z|1<|z|<2\}$ 上的零点个数（计算重数）.
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Context:
Find the number of different possible rational roots of the polynomial:
\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\]
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Context:
Calculate the curvature of a surface with isometric coordinates given by \(\frac{1}{x^2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) using the Laplace–Beltrami operator.
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Context:
20. [10] Find the volume of the set of points $(x, y, z)$ satisfying
$$
\begin{array}{l}
x, y, z \geq 0 \\
x+y \leq 1 \\
y+z \leq 1 \\
z+x \leq 1
\end{array}
$$
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Context:
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
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Context:
Determine the dimension of the vector space $V_r$ over the complex numbers $\mathbb{C}$, where $V_r$ consists of all holomorphic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $|f(z)| < \infty$ for all $z \in \mathbb{C}$. Provide your answer as a single number.
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Context:
$\underline{\text { Gordrin P.K. }}$.

Between two parallel lines, a circle of radius 1 is placed, touching both lines, and an isosceles triangle, the base of which lies on one of the lines, and the vertex on the other. It is known that the triangle and the circle have exactly one common point and that this point lies on the inscribed circle of the triangle. Find the radius of the inscribed circle of the triangle.
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Context:
In the following figure, the square \(ABCD\) is divided into three rectangles of equal area.

If the length of the segment \(BM\) is equal to 4, calculate the area of the square \(ABCD\).
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Context:
A jar was filled with jelly beans so that  $54\%$  of the beans were red,  $30\%$  of the beans were green, and  $16\%$  of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now  $20\%$  of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?
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Context:
Compute the sum
 \( \mathop{\sum }\limits_{{k = 0}}^{{1273}}\frac{1}{1 + {\tan }^{2548}\left( \frac{k\pi }{2548}\right) } \)
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Context:
When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man?
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Context:
A unit square is divided into rectangles, each colored either yellow or blue. For a blue rectangle, its associated number is the width divided by the height. For a yellow rectangle, the number is the height divided by the width. Let $x$ be the sum of all these numbers. If the total area of the blue rectangles equals the total area of the yellow rectangles, find the smallest possible value of $x$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximal value of the sum \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) is the distance from a real number \( a \) to the nearest number of the form \( \frac{m}{2^n} \) with \( m \in \mathbb{Z} \).
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Context:
Calculate the expectation \( \mathbb{E}\left[|\langle a, Cx \rangle|^2\right] \) where \( a \sim \mathcal{N}(0,I_m) \) is independent and \( C \in \mathbb{R}^{m \times n} \) has iid \( \mathcal{N}(0,1) \) entries. Fix \( x \in \mathbb{R}^n \).
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Context:
在一个圆周上给定十二个红点. 求 $n$的最小值，使得存在以红点为顶点的 $n$ 个三角形，满足以红点为端点的每条弦，都是其中某个三角形的一条边。
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Context:
Find the value of $x$ in terms of $a$ that maximizes the function $$y=\left(1-e^{-\frac{x}{a}}\right)^{\frac{x}{a}}.$$
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Context:
Find the minimum value of \( M_{a,b} \) for the function \( f_{a,b}(x) = \{x+a\} + 2\{x+b\} \), where \( \{x\} \) denotes the fractional part of \( x \), and the range of \( f_{a,b}(x) \) is \([m_{a,b}, M_{a,b})\). Consider all real numbers \( a \) and \( b \).
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Context:
Let \( U, V, W, Y, Z \) be independent random variables with the following distributions:

- \( U \sim \text{Exp}(1/5) \)
- \( V \sim \text{Exp}(2/3) \)
- \( W \sim \mathcal{U}(1, 5) \) (uniform distribution on \([1, 5]\))
- \( Y \sim \mathcal{N}(-13, 1) \)
- \( Z \sim \mathcal{N}(0, 5) \)

Determine the expected value:
\[ \mathbb{E}[(Z + \min(U+W, V+W)) \cdot (Y+Z)] \]

Hint: Use the properties of independence and linearity of expectation to simplify the computation. You do not need to derive any density or distribution functions by hand; all necessary means can be computed directly via the parameters of the distributions.
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Context:
Player A has a utility function given by \( u(x) = \sqrt{x} \). Determine the maximum amount that player A would be willing to pay to play first in the game.
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Context:
Let  $k>1$  be a given positive integer. A set  $S$  of positive integers is called *good* if we can colour the set of positive integers in  $k$  colours such that each integer of  $S$  cannot be represented as sum of two positive integers of the same colour. Find the greatest  $t$  such that the set  $S=\{a+1,a+2,\ldots ,a+t\}$  is *good* for all positive integers  $a$ .
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Context:
The feet of the altitudes of an acute-angled triangle $ABC$ form the vertices of another triangle, whose perimeter is $2p$. Find the area of triangle $ABC$ if the radius of its circumscribed circle is $\boldsymbol{R}$.
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Context:
Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$.
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Context:
I have chosen five of the numbers $\{1,2,3,4,5,6,7\}$. If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product?
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Context:
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?
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Context:
Find all positive integers $n > 1$ such that the polynomial $x^4 + 3x^3 + x^2 + 6x + 10$ belong to the ideal generated by the polynomial $x^2 + x + 1$ in $\Bbb Z_n[x]$.
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Context:
For which values of \( x \) does the series \( \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2} \) converge?
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Context:
Find the greatest value of \( n \) such that the equation \( 2\tan^2x - 5\sec x = 1 \) has exactly 7 distinct solutions for \( x \in [0, \frac{n\pi}{2}] \), where \( n \) is a natural number.
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Context:
Find the derivative with respect to the time-varying variable $x_t$ of the integral $\int_0^T y(x_s) \, ds$, where $T > t$ and $y(x_t)$ is a function of $x_t$, and $x_t$ is a function of $t$. Express your answer in terms of $x_t$ and $y(x_t)$.
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Context:
Shivani has a single square with vertices labeled $ABCD$. She is able to perform the following transformations:
$\bullet$ She does nothing to the square.
$\bullet$ She rotates the square by $90$, $180$, or $270$ degrees.
$\bullet$ She reflects the square over one of its four lines of symmetry.
For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
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Context:
Determine the number of possible ordered pairs \((a,b)\) of integers such that the function \(f(x)=\sqrt{\frac{x^2+ax+4}{x^2+bx+16}}\) is defined for all real \(x\).
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Context:
In the quadrilateral  $ABCD$ ,  $AB = BC = CD$  and  $\angle BMC = 90^\circ$ , where  $M$  is the midpoint of  $AD$ . Determine the acute angle between the lines  $AC$  and  $BD$ .
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Context:
Find all ordered pairs  $(a,b)$  of positive integers that satisfy  $a>b$   and the equation  $(a-b)^{ab}=a^bb^a$ .
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Context:
Find the equation of the hyperplane that contains the skew lines given by \( \mathbf{p}(s) = (s, 2s, 0, 1) \) and \( \mathbf{r}(t) = (1, 0, t, t+1) \).
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Context:
9. A positive integer that can be written in the form $20a + 8b + 27c$ (where $a, b, c$ are non-negative integers) is called a "good number." Then the number of good numbers in the set $\{1, 2, \cdots, 200\}$ is $\qquad$.
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Context:
Given that the focus of a parabola is on the y-axis, and the distance from point $M(m,-3)$ on the parabola to the focus is $5$, then $m=$ ______.
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Context:
Evaluate the limit for a positive integer $p \geq 2$:

$$ \lim_{n\to\infty}\left(\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{pn}\right) $$

Determine the value of this limit.
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Context:
Let  $S(M)$  denote the sum of the digits of a positive integer  $M$  written in base  $10$ . Let  $N$  be the smallest positive integer such that  $S(N) = 2013$ . What is the value of  $S(5N + 2013)$ ?
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Context:
On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the given grid, all circles share the same center. Express the area of the highlighted gray region as a fraction of the area of the largest circle.
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Context:
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square. Find the area of quadrilateral WXYZ. Express your answer in square units.
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Context:
Based on the data provided, what were the average daily high and low temperatures in Addington from September 15th, 2008 to September 21st, 2008, inclusive? The high temperatures in degrees Fahrenheit for these days are 49, 62, 58, 57, 46, 55, 60 and the corresponding low temperatures are 40, 47, 45, 41, 39, 42, 44. Express your answer as a decimal to the nearest tenth.
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Context:
Let  $S$  be a finite set of points in the plane. A linear partition of  $S$  is an unordered pair  $\{A,B\}$  of subsets  of  $S$  such that  $A\cup B=S,\ A\cap B=\emptyset,$  and  $A$  and  $B$  lie on opposite sides of some straight line disjoint from  $S$  ( $A$  or  $B$  may be empty). Let  $L_{S}$  be the number of linear partitions of  $S.$  For each positive integer  $n,$  find the maximum of  $L_{S}$  over all sets  $S$  of  $n$  points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the limit: $$ \lim _{n \rightarrow \infty} \frac{96}{n^4}\left[1\left(\sum_{k=1}^n k\right)+2\left(\sum_{k=1}^{n-1} k\right)+3\left(\sum_{k=1}^{n-2} k\right)+\cdots+n \cdot 1\right] $$
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Context:
Evaluate the contour integral \( \displaystyle\int_{|z|=4} \frac {z} {\sin z}\,\mathrm dz \), given that \( \sin z \) has three zeros within the circle \( |z|=4 \), making the integrand non-analytic at those points.
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Context:
Find the remainder when 25! is divided by 78125.
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \int_{\frac{1}{(n+1)^2}}^{\frac{1}{n^2}} \frac{e^x\sin^2(x)}{x^{7/2}}\, \mathrm{d}x$$
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Context:
Determine the number of zeroes of the function \( f(z) = z^4 + e^z \) within the unit disc \( |z| < 1 \).
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Context:
马休先按顺序写出了1到10000的全部数字，然后擦去了那些既不能被5整除，又不能被11整除的数，在剩下的数中，位于第2004位的数是多少？
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Context:
Use the divergence theorem to compute the flux integral \( \iint_S F \cdot dS \), where \( F(x,y,z) = yj - zk \) and \( S \) consists of the union of the paraboloid \( y = x^2 + z^2, 0 \leq y \leq 1 \), and the disk \( x^2 + z^2 \leq 1, y = 1 \), oriented outward. What is the flux through just the paraboloid?
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Context:
If the complex number $z$ and $(z + 2)^2 - 8i$ are both purely imaginary, then $z = \boxed{\text{\_\_\_\_\_\_\_\_}}$.
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Context:
How many lattice points $(v, w, x, y, z)$ does a $5$-sphere centered on the origin, with radius $3$, contain on its surface or in its interior?
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Context:
Evaluate the real convergent improper integral \(\int_0^\infty \frac{\sqrt{x}}{x^3+1}\ \mathrm{d}x\) using residue theory with a keyhole contour that excludes the branch cut along the positive real axis.
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Context:
Evaluate the integral \( \frac{1}{2i\pi} \oint_\gamma \frac{2i}{z^2 +1}dz \) where \( \gamma \) is a closed and continuously differentiable path in the upper half-plane \( \{z \in \mathbb{C} : z = x + iy, x, y \in \mathbb{R}, y > 0\} \) that does not pass through the point \( i \).
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Context:
Determine the radius of convergence of the series: $$1+ \frac{a.b}{1.c}z + \frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+.... $$
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Context:
Initially, one of the two boxes on the table is empty and the other contains 29 different colored marbles. Starting with the full box, in each move, one or more marbles are selected from that box and transferred to the other box. What is the maximum number of moves that can be made without selecting the same set of marbles more than once?
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Context:
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the kernel of the matrix \(A\) defined as \[A:=\begin{pmatrix} x_1^0 & x_1^1 & \ldots & x_1^n\\ x_2^0 & x_2^1 & \ldots & x_2^n\\ \vdots & \vdots & \ddots & \vdots\\ x_m^0 & x_m^1 & \ldots & x_m^n \end{pmatrix}\] where \(n < m - 1\) and the elements \(x_i\) are distinct.
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Context:
Given two independent random variables, where \(X\) follows a uniform distribution \(U(0,\pi)\) and \(Y\) follows a uniform distribution \(U(0,2\pi)\), calculate the probability \(\Pr(Y \in [\pi, X+\pi] \mid X)\).
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Context:
In the rectangular coordinate plane $xOy$, two circles are given:
$\odot O: x^{2}+y^{2}=1$,
$\odot O_{1}:(x-3)^{2}+y^{2}=4$.

From a point $M$ on the left half of the $x$-axis, a tangent is drawn to $\odot O$, touching $\odot O$ at point $A$ and intersecting $\odot O_{1}$ at points $B$ and $C$. Given that $AB = BC$, find the coordinates of point $M$.
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Context:
设 $x \neq y$, 且两数列 $x, a_{1}, a_{2}, a_{3}, y$ 和 $b_{1}, x, b_{2}, b_{3}, y, b_{4}$ 均为等差数列, 那么 $\frac{b_{4}-b_{3}}{a_{2}-a_{1}}=$ $\qquad$.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:
Determine the VC-dimension of the class $H_{N-parity}$ consisting of all parity functions $h_I: \{0,1\}^N \rightarrow \{0,1\}$, where $I \subseteq \{1,2,..., N\}$, defined by $h_I(x) = \bigg(\sum_{i\in I}x_i\bigg) \mod 2$. Provide your answer as a single integer.
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Context:
Calculate the derivative of the function \( F(x) = \int_0^2 \sin(x+l)^2\ dl \) with respect to \( x \) at \( x=0 \). Use the Leibniz rule to differentiate \( F(x) \).
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Context:
Let \( f(x, y) \) be a homogeneous function of degree 3. Given that \( f'_x(2, 1) = 1 \), \( f(1, 0.5) = 2 \), \( f(4, 2) = 128 \), and \( f(2, 1) = 16 \), compute \( f'_y(4, 2) \).
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Context:
在已知数列1,4,8,10,16,19,21,25,30,43中，相邻若干数之和能被11整除的数组共有＿组．
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Context:
A regular hexagon has side length \(6\). Congruent arcs with radius \(3\) are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. The original answer is in the form \(a\sqrt{b}-c\pi\). Please find the value of a + b + c.
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Context:
从0、2、4中取一个数字，从1、3、5中取两个数字，组成无重复数字的三位数，则所有不同的三位数的个数是______（用数字作答）
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Context:
A family consists of 5 brother-sister pairs. You have 10 identical Barbie dolls and 10 identical toy cars to distribute among the sisters and brothers, respectively, ensuring that each child gets a gift. Additionally, each brother-sister pair must receive at least 3 gifts in total. Calculate the number of ways to distribute the gifts.
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Context:
Select 2 elements from each of the sets {O, P, Q, R, S} and {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to form a sequence (both letters and numbers cannot be repeated). The number of different sequences where at most one of the letters O, Q, and the numbers 0 can appear is ___ . (Answer with a number).
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Context:
Let the set \( A = \{1, 2, \cdots, 2016\} \). A 1008-element subset \( X \) of \( A \) is called a "good set" if there exist \( x, y \in X \) such that \( x < y \) and \( x \mid y \). Determine the largest positive integer \( a \in A \) such that every 1008-element subset containing \( a \) is a "good set".
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Context:
Evaluate the integral:

$$\int_{-\infty}^\infty\frac{2021^{-|x|}}{1+3^{\arctan(\tan^3(x))}}dx$$
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Context:
4. A cylinder is intersected by a plane, and the shape of the intersection is an ellipse. The eccentricity of this ellipse is $\frac{\sqrt{5}}{3}$. The acute angle formed between this intersection and the base of the cylinder is . $\qquad$
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Context:
Two triangular pyramids \( MABC \) and \( NABC \) have a common base and no other common points. All vertices of both pyramids lie on the same sphere. Find the lengths of the edges \( MA \) and \( MB \) if it is known that they are equal to each other, and the lengths of all other edges of these pyramids are \( \sqrt{3} \).
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Context:
A $7 \times 7$ grid is colored black and white. If the number of columns with fewer black squares than white squares is $m$, and the number of rows with more black squares than white squares is $n$, find the maximum value of $m + n$.
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Context:
Dr. Banks had some toothbrushes to give away to his patients. He gave away 53 toothbrushes in January, 67 toothbrushes in February, 46 toothbrushes in March, and the remaining toothbrushes were given away half each in April and May. The difference between the busiest and slowest month was 36 toothbrushes. How many toothbrushes did Dr. Banks have initially?
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Context:
In the diagram, the area of square \( QRST \) is 36. Also, the length of \( PQ \) is one-half of the length of \( QR \). What is the perimeter of rectangle \( PRSU \)?
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Context:
In the given equation, each letter represents a number between 1 and 9, with different letters representing different digits. Find the value of $A+B+C=$ $\qquad$ .
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Context:
Consider the number obtained by writing the numbers  $1,2,\ldots,1990$  one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?
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Context:
Find the Fourier transform of the product $f(x)\delta(x)$, where $f(x)$ is a continuous function and $\delta(x)$ is the Dirac delta function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Out of the 800 emails that Antonia received in her mail, 3/7 were spam emails, while 5/11 of the remaining emails were promotional messages. If the rest of the emails were important emails, calculate the total number of important emails in her inbox.
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Context:
Consider the function \( f(x) = x\sqrt{x\sqrt{x\sqrt{x...\infty}}} \). If \( f(x) \) is defined as the limit of the expression as the number of roots approaches infinity, determine if \( f'(5) \) exists. If it does, find its value.
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Context:
Find the number of solutions to the equation $\log_e |2\cos x| = - x^2 + x$ in the interval $[0, 2\pi]$. Do not use a graphing calculator.
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Context:
Determine the order of growth for the entire function represented by the infinite product \( f(z) = \prod_{n=1}^{\infty} \left(1-\frac{z}{n!}\right) \).
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Context:
Determine the condition on the $L^2$ norm of the Hilbert-Schmidt kernel $k$ that ensures the invertibility of the operator $I-T$, where $T$ is an integral operator with kernel $k$.
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Context:
Find the limit \( \lim_{n\rightarrow\infty}\int_{0}^{1}nx^{n}f(x)dx \) where \( f(x) \) is a Lebesgue integrable function on \([0,1]\) satisfying \( \lim_{x\rightarrow1^-}f(x) = \xi \) for some finite \( \xi \).
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Context:
Find the center of the group with the presentation \( \langle a, b \mid a^2 = b^3 \rangle \).
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Context:
What is the cardinality of the set of all infinite monotonically non-increasing sequences of natural numbers?
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Context:
Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.)
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Context:
In an isosceles trapezoid with an acute angle \(\alpha\) at the base, a circle constructed on one of the legs as its diameter is tangent to the other leg.

In what ratio does it divide the larger base of the trapezoid?
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Context:
Let  $f(x)=cx(x-1)$ , where  $c$  is a positive real number.  We use  $f^n(x)$  to denote the polynomial obtained by composing  $f$  with itself  $n$  times.  For every positive integer  $n$ , all the roots of  $f^n(x)$  are real.  What is the smallest possible value of  $c$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A swimming pool is 20 ft wide, 40 ft long, and has a variable depth: 3 ft at the shallow end and 9 ft at the deepest point. Water is being filled at a rate of 0.7 ft³/min. Determine how fast the water level is rising when the depth at the deepest point is 5 ft. Express your answer rounded to five decimal places.
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Context:
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ have radii of $20$ and $16$, respectively, find $\sqrt{1+BC/BD}$.
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Context:
Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of  $c$  such that for any positive integer  $n\ge 4$  and any set  $A\subseteq \{1,2,\cdots,n\}$ , if  $|A| >cn$ , there exists a function  $f:A\to\{1,-1\}$  satisfying  $$ \left| \sum_{a\in A}a\cdot f(a)\right| \le 1. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of weighings necessary to identify two groups of 4 boxes each, such that the total weight of the boxes in each group is equal, using a balance scale with two pans.
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Context:
Identify a function for which the Taylor series expansion does not represent the function within an interval containing the expansion point.
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Context:
Determine the value of the limit \( L = \lim \limits_{x \to 0}\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4} \) such that \( L \) is finite. What is the value of \( a \) that satisfies this condition?
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Context:
Determine the rank of the linear operator on the vector space of real 3x3 matrices that maps a matrix X to AX - XA, where \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \).
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Context:
Determine the maximum possible dimension of a residue field of the polynomial ring $k[x_1, x_2, \ldots]$ over an uncountable field $k$.
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Context:
Calculate the volume of the solid obtained by rotating the region bounded by the lines $x=2$, $y=3$, and the curves $xy=2$, $xy=4$ around the x-axis using the method of cylindrical shells.
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Context:
Find the limit of the sequence \(\{x_n\}\) as \(n\) approaches infinity, where \(x_n\) satisfies the equation \(x_n^3 - (n^2 + 3n)x_n - 3n^2 = 0\) for all \(n \in \mathbb{N}\).
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Context:
Consider the polynomial \( \prod_{n = 1}^{2023}\left( 1 + nx^{5^{n}}\right) = \sum_{n=0}^{m}a_{n}\,x^{k_{n}} \) where the \(k_{n}\) are in increasing order, and the \(a_{n}\) are nonzero. Determine the coefficient \(a_{2023}\).
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Context:
Evaluate the integral \( I = \int_{-\infty}^{\infty} \frac{e^{\cos(x)} \sin(\sin(x))}{x} \, dx \).
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Context:
In a semicircle with diameter $AB$ and center $S$, points $C$ and $D$ are located such that $C$ is on arc $AD$ and $\angle CSD = 120^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$, and $F$ be the intersection of lines $AD$ and $BC$. Determine the value of $\frac{EF}{AB}$.
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Context:
Evaluate the limit as \( n \to \infty \):
\[ \lim_{n \to \infty} \left[1- p(n) \right]^{n-1} \]
where \( p(n) = o \left( \frac{\log n}{n} \right) \).
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Context:
A plane passes through the vertex \( A \) of a right circular cone, creating a cross-section of maximum area. This area is twice the area of a cross-section that passes through the cone's axis. Determine the angle at the vertex of the cone's axial cross-section.
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Context:
Given the equation \(2\tan^2x - 5\sec x = 1\), which has exactly 7 distinct solutions for \(x\in[0,\frac{n\pi}{2}]\), where \(n\in\mathbb{N}\), find the greatest possible value of \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions of the polynomial equation \(z^4 - 7z^3 - 2z^2 + z - 3 = 0\) that lie inside the unit disc.
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Context:
Identify a non-commutative ring \( B \) such that its only left ideals are \((0)\) and \( B \) itself.
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Context:
Determine the values of \(a\) for which the function \(f(x,y) = -9x^2 + axy - y^2 + 4ay\) is convex everywhere and in all directions.
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Context:
From the $8$ vertices of a cube, choose any $4$ vertices. The probability that these $4$ points lie in the same plane is ______ (express the result as a simplified fraction).
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Context:
Evaluate or approximate the integral \( \int_{-\frac{b}{2}}^{\frac{b}{2}} \ln(f(x)) \, dx \) for the function \( f(x) = 350 + 5\left(1 - \frac{\cosh\left(\frac{x}{4 \cdot 10^{-6}}\right)}{\cosh\left(\frac{10^{-4}}{8 \cdot 10^{-6}}\right)}\right) \) over the interval \(-\frac{10^{-4}}{2} < x < \frac{10^{-4}}{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n \) such that for a random sample \( X_1, \ldots, X_n \) from a normal distribution with mean \( \mu = 0 \) and variance \( \sigma^2 = 2 \), the probability \( P(\min(X_1^2, \ldots, X_n^2) \leq 0.002) \geq 0.8 \).
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Context:
Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$.
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Context:
Find the harmonic conjugate of the function \( u(z) = \ln(|z|) \).
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Context:
On a $2020 \times 2020$ chessboard, what is the maximum number of warriors you can place such that no two warriors attack each other? A warrior can move either 3 steps forward and 1 step sideways in any direction, or 2 steps forward and 2 steps sideways in any direction. Determine the maximum number of warriors that can be placed on the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is a sufficient condition on the matrix $A$ using the $\ell^2$-norm that guarantees convergence of the sequence $\{x_k\}$ to $0$?
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Context:
Determine the residue of the function \( g(z) = \frac{\tan(z)\tanh(z)}{z^5} \) at \( z = 0 \) without using Cauchy's residue formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi} \frac{\sin^4(4x)}{\sin^2(x)} \, dx. \)
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Context:
Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.

At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."

At the end of the first round, Lele says: "The number of your gold coins is 6 times the number of my gold coins now."

At the end of the second round, Huanhuan says: "The number of my gold coins is 5 times the number of your gold coins now."

Find the minimum number of gold coins Huanhuan had at the beginning.
Please reason step by step, and put your final answer within \boxed{}.


Context:
14.Strike out a hundred digits from the number $12345678910111213 \ldots 5960 \quad$ so that the resulting number is the largest possible.
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Context:
Evaluate the double infinite summation: \[ \sum_{i=0}^\infty \sum_{j=0}^\infty (3 + i + j) \cdot \left( \frac{1}{3} \right)^{(1+i)} \cdot \left( \frac{2}{3} \right)^{(1+j)} \]
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Context:
A $4 \times 4$ square piece of paper is cut into two identical pieces along its diagonal. The resulting triangular pieces of paper are each cut into two identical pieces. Each of the four resulting pieces is cut into two identical pieces. Each of the eight new resulting pieces is finally cut into two identical pieces. What is the length of the longest edge of one of these final sixteen pieces of paper?
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Context:
Determine all real numbers $a$ for which the system of equations
$$3x^2 + 2y^2 + 2z^2 = a$$
$$4x^2 + 4y^2 + 5z^2 = 1 - a$$
has at least one solution $(x, y, z)$ in the real numbers.
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Context:
Let \( O \) be the circumcenter of a triangle \( {ABC} \) , \( P \) be the midpoint of \( {AO} \) , and \( Q \) be the midpoint of \( {BC} \) . If \( \angle {ABC} = 4\angle {OPQ} \)
 and \( \angle {ACB} = 6\angle {OPQ} \) , compute \( \angle {OPQ} \) .
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Context:
Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. The treasure was buried in a single cell of an $M \times N$ ($2 \le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure.

For each detector, you can set it up to scan a specific subgrid $[a,b] \times [c,d]$ with $1 \le a \le b \le 2$ and $1 \le c \le d \le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected.

You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. What is the minimum $Q$ required to guarantee to determine the location of the Blackboard’s legendary treasure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the only symmetric group that possesses a nontrivial outer automorphism group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bear is preparing to hibernate for the winter and needs to gain 1200 pounds. At the end of summer, the bear feasts on berries, small woodland animals, and insects. During autumn, it devours acorns, salmon, and honey. It gained a fifth of the weight it needed from berries during summer and one-tenth of the weight it needed from insects. During autumn, it gained twice the amount it gained from berries from acorns, and three times the amount it gained from insects from honey. Salmon made up a quarter of the remaining weight it had needed to gain. How many pounds did it gain eating small animals?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many positive integers  $n$ ,  $1\leq n\leq 2008$ , can the set \[\{1,2,3,\ldots,4n\}\] be divided into  $n$  disjoint  $4$ -element subsets such that every one of the  $n$  subsets contains the element which is the arithmetic mean of all the elements in that subset?
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Context:
Find the number of ways to choose two nonempty subsets $X$ and $Y$ of $\{1, 2, \ldots , 2001\}$, such that
$|Y| = 1001$ and the smallest element of $Y$ is equal to the largest element of $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the difference between the local values of a certain digit in the number 58408 if the difference is 7992?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X=\ell_1$ and $Y=\ell_2$. Define an operator $B$ as follows:

$$B(x_1,x_2,\ldots)=(x_2,x_3-x_2,x_4-x_3,\ldots)$$

Determine whether $B$ is a bounded linear operator from $X$ to $Y$ and find $\|B\|$ if it is bounded.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle relate to each other?
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Context:
求最大的正整数 n，使得存在 8 个整数 x_1, x_2, x_3, x_4 和 y_1, y_2, y_3, y_4，满足：\{0,1,\cdots,n\}\subseteq\{|x_i-x_j|\ |\ 1\leq i<j\leq 4\}\cup\{|y_i-y_j|\ |\ 1\leq i<j\leq 4\}.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Poisson process $\lbrace N(t)\rbrace_{t\geq 0}$ with intensity $\lambda = 3$, calculate the conditional probability:

$$P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right]$$
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Context:
George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among the other three. If a player has two hot potatoes on his turn, he only passes one. If George goes first, and play proceeds clockwise, what is the probability that Travis has a hot potato after each player takes one turn?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the bivariate probability density function for two independent random variables $X$ and $Y$: \( f(x,y) = 12x^2y^3 \) for \( 0 < x < 1 \) and \( 0 < y < 1 \), and \( f(x,y) = 0 \) otherwise, find the probability that the sample mean of a random sample of size \( n = 50 \) for the random variable $X$ is greater than 0.7.
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Context:
Syam invested Rs. 1800 in 9% stock A and Rs. 2300 in 7% stock B. His overall earnings from stock A and stock B were Rs. 120 and Rs. 160 respectively. If he later decided to invest Rs. 2500 in 6% stock C but lost track of his overall earnings, find out the rate at which the combined investment in stocks A, B, and C would be quoted.
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Context:
It takes Mary $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
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Context:
There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys?
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Context:
In isosceles triangle \( \triangle ABC \), \( AC = BC \) and \( \angle ACB = 40^\circ \). A circle with diameter \( BC \) intersects \( AC \) and \( AB \) at points \( D \) and \( E \) respectively. The diagonals of quadrilateral \( BCDE \) intersect at point \( F \). Find the measure of \( \angle BFC \) in degrees.
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Context:
Consider a symmetric random walk starting at 0. Let $T_a$ and $T_{-b}$ denote the first times the walk visits $a$ and $-b$, respectively, where $a, b > 0$. Find $\mathsf{E}(\min(T_a, T_{-b}))$. Use the fact that $P(T_a < T_{-b})$ can be computed by conditioning on the first step.
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Context:
The first term of a geometric sequence is 6, the sum of the first $n$ terms is $\frac{45}{4}$, and the sum of the reciprocals of these terms is $\frac{5}{2}$. What is this geometric sequence?
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Context:
Bob knows that Alice has 2021 secret positive integers $x_{1}, \ldots, x_{2021}$ that are pairwise relatively prime. Bob would like to figure out Alice's integers. He is allowed to choose a set $S \subseteq\{1,2, \ldots, 2021\}$ and ask her for the product of $x_{i}$ over $i \in S$. Alice must answer each of Bob's queries truthfully, and Bob may use Alice's previous answers to decide his next query. Compute the minimum number of queries Bob needs to guarantee that he can figure out each of Alice's integers.
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Context:
Find the mass percentage of Ca in Calcium carbonate
Please reason step by step, and put your final answer within \boxed{}.


Context:
The left and right foci of the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ are $F_1$ and $F_2$, respectively. Points $A$ and $B$ are on ellipse $C$, and $\overrightarrow{AF_1} = 3\overrightarrow{F_1B}$, $\angle BAF_2 = 90^{\circ}$. The eccentricity of ellipse $C$ is ______.
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Context:
Find the sum of all natural numbers $n$ such that the function $g(x) = 100|e^{x+1} - 1| - \sum_{k=1}^n |e^{x^k + 1} - 1|$ is differentiable over the entire real line $\mathbb{R}$. Here, $n \in \mathbb{N}$. Provide your answer as a single number.
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Context:
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
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Context:
Given the sequence \((a_{n})\) defined by the recurrence relation \(a_{n+1}=\frac{1}{2}(a_{n}+\frac{3}{a_{n}})\), which converges to \(\sqrt{3}\), determine the rate of convergence of the sequence.
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Context:
3. (4 points) In a parallelogram with sides 4 and 7, the bisectors of the four interior angles are drawn. Find the ratio of the area of the quadrilateral formed by the intersection of the bisectors to the area of the parallelogram.
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Context:
某校高一新生共有784人，每班分配56人．方法是：将每人的入学成绩从高分到低分依次编号（成绩相同的学生按姓氏笔画的顺序），然后按S形顺序编班．例如：若有8个班，按编号1至8号分别编在1至8班，9至16号分别编在8至1班，17至24号分别编在1至8班，···．该校高一新生中编号为300（每号只对应1人）的同学编在___班.
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Context:
Determine the smallest integer $k$ such that a $k$-vector, whose entries can be set to 0 or 1, can be used to communicate any integer between 1 and 1000 to a friend. The vector is subject to a transformation $f_i$, which sets the $i$-th component to 1 while leaving other components unchanged, and the value of $i$ is unknown. What is the smallest $k$ that ensures your friend can always understand the number?
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Context:
Misha made a homemade darts game during the summer at the dacha. A circular board is divided into several sectors by concentric circles, and darts can be thrown at it. Points are awarded based on the number written in the sector, as shown in the diagram.

Misha threw 8 darts three times. In the second attempt, he scored twice as many points as he did in the first attempt, and in the third attempt, he scored 1.5 times more than in the second attempt. How many points did he score in the second attempt?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a^3 - a - 1 = 0 \). Find the exact value of the expression
\[
\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.
\]
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Context:
Jimmy finds that 8 bananas weigh the same as 6 oranges. If Jimmy has 40 oranges, and the weight of 4 oranges equals the weight of 5 apples, how many apples and bananas would Jimmy need to equal the weight of his 40 oranges?
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Context:
Find the coefficient of $x^{80}$ in the power series $$(1+x+x^{2}+x^{3}+x^{4}+	ext{...})(x^{2}+x^{4}+x^{6}+x^{8}+	ext{...})(1+x^{3}+x^{5}).$$
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Context:
Let  $a/b$  be the probability that a randomly chosen positive divisor of  $12^{2007}$  is also a divisor of  $12^{2000}$ , where  $a$  and  $b$  are relatively prime positive integers.  Find the remainder when  $a+b$  is divided by  $2007$ .
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Context:
There are 55 points marked on a plane: the vertices of a regular 54-gon and its center. Determine the number of ways to color a set of three marked points in red such that the colored points form the vertices of a regular triangle.
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Context:
Given a point $P$ and a circle $\mathcal{C}$ in the plane, the distance from $P$ to $\mathcal{C}$ is defined as the minimum length $PM$ between $P$ and a point $M$ on the circle $\mathcal{C}$. For example, if $P$ lies on the circle, then the distance from $P$ to $\mathcal{C}$ is zero, and if $P$ is the center of the circle, then the distance from $P$ to $\mathcal{C}$ is equal to the radius of $\mathcal{C}$.

Given four non-cyclic points $A, B, C, D$, what is the maximum number of circles that pass at equal distance from these four points?
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Context:
Let  $a,b,c$  be positive integers such that  $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$  are  $7$  distinct primes. The sum of two of  $a,b,c$  is  $800$ . If  $d$  be the difference of the largest prime and the least prime among those  $7$  primes, find the maximum value of  $d$ .
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Context:
A cylindrical container has a fixed inner volume \(V\) mm\(^3\), a 2 mm thick solid wall, and an open top. The bottom is a solid circular disc with the same thickness and radius as the outer radius of the container. The volume of the material used to construct the container is minimized when the inner radius of the container is 10 mm. What is the value of \(\frac{V}{250\pi}\)?
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Context:
Evaluate the limit: $$L = \lim_{x \to \infty}x^2\left(x^2 - x \cdot \sqrt{x^2 + 6} + 3\right)$$
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Context:
Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\left(0 \lt m \lt 1\right)$, point $A\left(0,2\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\frac{4}{3}$, then $m=$____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the genus of the curve \( y^{2}+y=\frac{\alpha}{x^{2}+x} \) over a finite field with even characteristic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Maria is a single mother who decided to renovate her kitchen. She hired a contractor who promised to complete the renovation for $8,000. Unfortunately, the contractor turned out to be fraudulent and disappeared after receiving a $2,500 deposit from Maria, leaving the work unfinished. Maria later found another reliable contractor who agreed to finish the renovation for $6,000. How much more money will Maria need to pay to finally complete her kitchen renovation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a closed-form expression for the series \( \sum_{n=0}^\infty\frac{(1/2)_n}{n!}(H_n-H_{n-1/2}) \), where \( H_z \) is the harmonic number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Please calculate the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$
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Context:
To which number can the expression

$$
\frac{(a+b)(b+c)(a+c)}{a b c}
$$

be equal if the numbers \( a \), \( b \), and \( c \) satisfy the condition of problem 84?
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Context:
Determine the value of \(a\) such that the quadrilateral \(ABCD\) is cyclic, where the line \(x + 2y + a = 0\) intersects the circle \(x^2 + y^2 = 4\) at points \(A\) and \(B\), and the line \(12x - 6y - 41 = 0\) intersects the circle \(x^2 + y^2 - 4x - 2y + 1 = 0\) at points \(C\) and \(D\).
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Context:
How many positive integers less than or equal to 100 have a prime factor that is greater than 4?
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Context:
Determine the number of terms required to approximate the integral \( \int_0^{0.5}\frac{1-e^{-x}}{x}dx \) with an error less than \( 10^{-5} \).
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Context:
Bob’s Rice ID number has six digits, each a number from $1$ to $9$, and any digit can be used any number of times. The ID number satifies the following property: the first two digits is a number divisible by $2$, the first three digits is a number divisible by $3$, etc. so that the ID number itself is divisible by $6$. One ID number that satisfies this condition is $123252$. How many different possibilities are there for Bob’s ID number?
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Context:
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
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Context:
Evaluate the expression: \[ \sum_{k=0}^{10} {30+k \choose k}{20-k \choose 10} \]
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Context:
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the answer is $x\pi$, what is the value of $x$?
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Context:
Given n balls of different colors arranged in a row, determine the expected number of swaps needed to rearrange them such that they satisfy a condition defined by a function f. Assume that at least one arrangement exists that satisfies the condition, and a function g provides all possible arrangements. What is the expected number of swaps required?
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Context:
Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$.
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Context:
Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).

Note: The factorials do not have to be distinct.  For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.
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Context:
Among all pairs of real numbers(x,y)such that
 \( \sin \sin x = \sin \sin y \) with \( - {10\pi } \leq  x,y \leq  {10\pi } \) , Oleg randomly
 selected a pair(X,Y). Compute the probability that \( X = Y \) .
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Context:
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
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Context:
A court astrologer calls a moment in time "good" if the hour, minute, and second hands of a clock are all on the same side of any diameter of the clock face (the hands rotate on a common axis and do not make jumps). Is there more "good" or "bad" time in a day?
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Context:
Consider an integer $n \ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \ldots, x_n$. An operation consists of eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3, x_7, x_{11}, \ldots$. For example, the sequence $4, 5, 9, 3, 6, 6, 1, 8$ produces the sequence $9, 1$.

Upon the sequence $1, 2, 3, \ldots, 1024$, the operation is performed successively $5$ times. Show that at the end only one number remains and find this number.
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Context:
Compute the expected value $E(N_1N_2)$ for a Poisson process $N_t$ with rate $\lambda = 2$.
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Context:
Let $S$ be a subset of the set $\{1,2,3, \ldots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a-b$ does not divide the sum $a+b$. Find the maximum possible size of $S$.
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Context:
In triangle  $ABC$ ,  $AB>AC$ . The bisectors of  $\angle{B},\angle{C}$  intersect the sides  $AC,AB$  at  $P,Q$ , respectively. Let  $I$  be the incenter of  $\Delta ABC$ . Suppose that  $IP=IQ$ . How much isthe value of  $\angle A$ ?
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Context:
Equilateral $\triangle A B C$ has side length 6. Let $\omega$ be the circle through $A$ and $B$ such that $C A$ and $C B$ are both tangent to $\omega$. A point $D$ on $\omega$ satisfies $C D=4$. Let $E$ be the intersection of line $C D$ with segment $A B$. What is the length of segment $D E$?
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Context:
Let a, b, c, and d represent the ages of four individuals. The total age of a and b is 10 years more than the total age of b and c, and the total age of c and d is 15 years less than the total age of a and d. If the ratio of a's age to d's age is 7:4, how many years younger is c compared to a?
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Context:
Determine the upper bound for numbers such that the sum of the fifth powers of their digits equals the number itself.
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Context:
Find the largest possible value in the real numbers of the term   $$ \frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} $$   with  $x^2 + y^2 \ne 0$ .
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Context:
Find the point \( w_0 \) that is excluded from the circle when the line \( x+y=1 \) is mapped to the circle minus a point under the mapping \( w=\frac{1}{z+2+3i} \), where \( z=x+iy \).
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Context:
Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\mathcal{R}$. If $t^{2}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.
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Context:
Given the joint probability density function of the bivariate random variable $(X,Y)$:

$$f_{XY}(x,y) = \frac{1}{\sqrt{3 \pi } } e^{ -\frac{2}{3} ( x^2 - xy + y^2 ) }$$

Calculate the variance of the random variable $Y$.
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Context:
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of half an hour at the same speed, calculate the total distance she would have ridden. Express your answer in miles.
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Context:
In 1978, a person introduced their family by saying: "I have one son and one daughter, and they are not twins. The cube of my son's age plus the square of my daughter's age equals the year of my birth. I was born after 1900. Both of my children are under 21 years old. I am 8 years older than my wife." Find the ages of each family member in 1978.
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Context:
Consider the sum
\[
  S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine  $\lfloor S_{4901} \rfloor$ .  Recall that if  $x$  is a real number, then  $\lfloor x \rfloor$  (the *floor* of  $x$ ) is the greatest integer that is less than or equal to  $x$ .
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Context:
Find the smallest normal subgroup of a group $G$ of order $p^{\alpha}m$, where $p$ is a prime and $p$ does not divide $m$, such that the quotient group is a $p$-group and the subgroup does not contain any Sylow $p$-subgroups of $G$.
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Context:
In an abelian group $G$, given that $(r + 1)|A| > |G|$ for some subset $A$, determine the smallest possible value of $k$ such that $kA = \langle A \rangle$.
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Context:
Solve for:

$$
\frac{(1+\operatorname{tg} \alpha)(1+\operatorname{tg} \beta)(1+\operatorname{tg} \gamma)}{1+\operatorname{tg} \alpha \operatorname{tg} \beta \operatorname{tg} \gamma}=2
$$

given that $\alpha+\beta+\gamma=45^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task B-3.4.

While preparing for the competition, Matko discovered a bookstore with good mathematical literature. The bookstore offers 7 different books with problems only in geometry, 4 only in number theory, and 5 only in combinatorics. Furthermore, the store also offers books with problems from exactly two areas. Thus, there are 6 different books with problems in number theory and combinatorics, and 7 with problems in geometry and combinatorics. In how many ways can Matko choose literature from two of all the mentioned areas if he can buy at most two books?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \sum _{n=-\infty}^\infty a_nz^n \) be the Laurent series expansion of \( f(z)=\dfrac{1}{2z^2-13z+15} \) in the annulus \( \frac{3}{2}<|z|<5 \). Find the value of \( \frac{a_1}{a_2} \).
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Context:
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break?
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Context:
The equation \(z(z+i)(z+3i)=2002i\) has a zero of the form \(a+bi\), where \(a\) and \(b\) are positive real numbers. The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).
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Context:
设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点, 已知 $B F=4, B D=2, B E=5, \angle B A D=\angle A C D, \angle B A F=\angle C A E$, 求 $B C$ 的长。
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Context:
Given that the function f(x) is differentiable on R with its derivative f'(x), and f(x) satisfies: (x-1)[f'(x)-f(x)] > 0, f(2-x) = f(x)e^(2-2x), determine the solution set of the inequality e^2f(ln x) < xf(2). Express your answer as an interval.
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Context:
The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? Provide your answer as an integer.
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Context:
$22$  football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.
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Context:
Determine the set of points where the function \( f(x,y) = \begin{cases} x^{4/3} \sin(y/x) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \) is differentiable.
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Context:
A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position?
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Context:
A 2022-digit natural number starts with the digit 4, and any two consecutive digits form a two-digit number divisible by either 19 or 23. Determine the last digit of this number.
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Context:
Find the periodic points of period 5 for the function \( f(x) = x^2 + 1 \) where \( f: \mathbb{R} \rightarrow \mathbb{R} \).
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Context:
Determine all the possible non-negative integer values that are able to satisfy the expression: $\frac{(m^2+mn+n^2)}{(mn-1)}$
if  $m$  and  $n$  are non-negative integers such that  $mn \neq 1$ .
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Context:
Find all natural numbers which are divisible by  $30$  and which have exactly  $30$  different divisors.
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Context:
Find the volume of the region bounded by the hemisphere \( y=\sqrt{4-x^2-z^2} \) and the planes \( y=x \) and \( y=\sqrt{3}x \) using spherical polar coordinates.
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Context:
A square is called proper if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.
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Context:
Given the general term formula of the sequence $\{a_{n}\}$ as $a_{n}=(-1)^{n-1}(n-1)$, let $S_{n}$ denote the sum of its first $n$ terms. Calculate the value of $S_{15}$.
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Context:
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A, B, C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.
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Context:
Determine the cardinality of the set of all partially ordered sets of natural numbers that have one least element and an infinite number of maximal elements.
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Context:
What is $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$?
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Context:
Let $f(x)$ be an odd function defined on $\mathbb{R}$ with $f(2)=0$. When $x > 0$, we have $\frac{x f''(x)-f(x)}{x^{2}} < 0$ holds true. Determine the solution set for $\frac{f(x)}{x} > 0$. Express your answer as an interval or union of intervals.
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Context:
Let $S$ be the region in the first quadrant bounded by the curves $xy=1$, $xy=3$, $x^2-y^2=1$, and $x^2-y^2=4$. Compute \[ \iint _S (y^2+x^2)\, dA. \]
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Context:
Big Head Son and Small Head Dad opened a McDonald's together. When they were calculating the day's revenue in the evening, they found that there was an extra $32.13 on the books. Later, they realized that the decimal point of a sum of money was misplaced by one digit. What was the original amount of this sum of money? Express your answer in dollars, rounded to two decimal places.
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Context:
The straight line $y=kx+1$ and the parabola $y^{2}=2x$ have at most one common point. Determine the range of values for $k$.
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Context:
The stadium seats 60,000 fans, but only a certain percentage of the seats were sold for the music show. Because of the threat of rain, 5,000 fans stayed home. 40,000 fans attended the show. What percentage of the seats were sold for the music show?
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Context:
There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.
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Context:
Evaluate the limit: $$\lim_{x \to 0}\lim_{n \to \infty} \frac{n}{\left(1^{\left(\cot^2x \right)}+2^{\left(\cot^2x \right)}+\ldots+n^{\left(\cot^2x \right)}\right)^{\left(\tan^2x \right)}}$$
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Context:
Evaluate the determinant:
\[\begin{vmatrix} \cos 2019^{\circ} & \cos 1131^{\circ}  & \cos 1869^{\circ}  & \cos  1101^{\circ}\\ -\cos 1131^{\circ} & \cos 2019^{\circ}  & -\cos 1101^{\circ}  & \cos  1869^{\circ}\\  -\cos 1869^{\circ} & \cos 1101^{\circ}  & \cos 2019^{\circ}  & -\cos  1131^{\circ} \\  -\cos 1101^{\circ} & -\cos 1869^{\circ}  & \cos 1131^{\circ}  & \cos  2019^{\circ}\end{vmatrix}.\]
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Context:
Let  $p = 9001$  be a prime number and let  $\mathbb{Z}/p\mathbb{Z}$  denote the additive group of integers modulo  $p$ .  Furthermore, if  $A, B \subset \mathbb{Z}/p\mathbb{Z}$ , then denote  $A+B = \{a+b \pmod{p} | a \in A, b \in B \}.$  Let  $s_1, s_2, \dots, s_8$  are positive integers that are at least  $2$ . Yang the Sheep notices that no matter how he chooses sets  $T_1, T_2, \dots, T_8\subset \mathbb{Z}/p\mathbb{Z}$  such that  $|T_i| = s_i$  for  $1 \le i \le 8,$   $T_1+T_2+\dots + T_7$  is never equal to  $\mathbb{Z}/p\mathbb{Z}$ , but  $T_1+T_2+\dots+T_8$  must always be exactly  $\mathbb{Z}/p\mathbb{Z}$ . What is the minimum possible value of  $s_8$ ?

[i]Proposed by Yang Liu
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Context:
A large playlist consists of songs with lengths that have a mean of 2 minutes and 10 seconds and a standard deviation of 15 seconds. What is the probability that more than 36 randomly chosen songs are required to fill a program that is 76 minutes long?
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Context:
Evaluate the integral by changing the order of integration:
\[ \int_{0}^{1} \int_{1-\sqrt{1-y^2}}^{1+\sqrt{1-y^2}} \frac{(x^2+y^2)x}{\sqrt{4x^2-(x^2+y^2)^2}} \, dx \, dy \]
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Context:
已知四面体 \(A B C D\) 内接于球 \(O\)，且 \(A D\) 是球 \(O\) 的直径。若 \(\triangle A B C\) 和 \(\triangle B C D\) 都是边长为 1 的等边三角形，则四面体 \(A B C D\) 的体积是多少？原始答案的形式为 \(\frac{\sqrt{c}}{b}\)，请给出a+b+c的值。
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Context:
For a set  $ P$  of five points in the plane, no three of them being collinear, let  $ s(P)$  be the numbers of acute triangles formed by vertices in  $ P$ .
Find the maximum value of  $ s(P)$  over all such sets  $ P$ .
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Context:
Let \( n=2017 \) and \( x_1,\dots,x_n \) be boolean variables. A \( 7 \)-CNF clause is an expression of the form \( \phi_1(x_{i_1})+\dots+\phi_7(x_{i_7}) \), where \( \phi_1,\dots,\phi_7 \) are each either the function \( f(x)=x \) or \( f(x)=1-x \), and \( i_1,i_2,\dots,i_7\in\{1,2,\dots,n\} \). For example, \( x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-x_3)+x_{12} \) is a \( 7 \)-CNF clause. Determine the smallest number \( k \) for which there exist \( 7 \)-CNF clauses \( f_1,\dots,f_k \) such that \[f(x_1,\dots,x_n):=f_1(x_1,\dots,x_n)\cdots f_k(x_1,\dots,x_n)\] is zero for all values of \((x_1,\dots,x_n)\in\{0,1\}^n\).
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Context:
John makes $30 a week from his job. He earns a 10% raise, followed by another 15% raise, and finally a 5% raise on his salary. What is the total percentage increase in John's weekly income after all three raises?
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Context:
Find the expectation of the random variable \( \frac{X_i}{\bar{X}} \), where \( X_i \) are IID random variables and \( \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \). Express \( E\left[\frac{X_i}{\bar{X}}\right] \) in terms of \( n \) and the distribution of \( X_i \).
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Context:
A convex pentagon $ABCDE$ is inscribed in a circle. The distances from point $A$ to the lines $BC$, $CD$, and $DE$ are $a$, $b$, and $c$, respectively. Determine the distance from point $A$ to the line $BE$. \( \text{Express your answer in terms of } a, b, \text{ and } c. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A_1, A_2, A_3, A_4$ be events such that for every $i, j, k = 1, 2, 3, 4$, $P(A_i) = \frac{1}{2}$, $P(A_i \cap A_j) = \frac{1}{3}$ for $i \ne j$, $P(A_i \cap A_j \cap A_k) = \frac{1}{4}$ for $i \ne j, j \ne k, k \ne i$, and $P(A_1 \cap A_2 \cap A_3 \cap A_4) = \frac{1}{5}$. Find $P(A^c_1 \cap A^c_2 \cap A^c_3)$. \( \text{Express your answer as a fraction.} \)
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Context:
已知x、y∈(0,+∞)，且$\frac { 1 9 } { x } + \frac { 9 8 } { y } = 1$，则x+y的最小值是多少？
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Context:
Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?
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Context:
Anakin's age and the ages of his three children are all prime numbers, and their combined ages sum to 50. After n years, the sum of the children's ages will be twice Anakin's age. Given that Anakin's age is a prime number, and the ages of his three children form a unique set of primes, determine the value of n.
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Context:
8、In tetrahedron $A B C D$, it is known that $D A \perp$ base $A B C$, side $A B D \perp$ side $B C D, B D=B C=2$, the sum of the squares of the areas of the three sides $D A B, D B C, D C A$ is 8, then $\angle A D B=$ $\qquad$
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Context:
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$ . Compute the area of the pentagon.
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Context:
Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
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Context:
A common chord of two circles subtends arcs of $60^{\circ}$ and $120^{\circ}$. Find the ratio of the areas of these circles.
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Context:
Evaluate the integral \( \int_\Gamma \frac{1}{z} \, dz \), where \( \Gamma \) is the semicircular contour in the right half-plane, traversed from \( -i \) to \( i \).
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Context:
Given 25 vessels with capacities $1, 2, \ldots, 25$ liters, determine the number of ways to select ten of them such that any two selected vessels can be used to measure exactly 1 liter of water. Assume an unlimited amount of water is available, and water can be transferred between vessels.
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Context:
一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率为_______。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:
Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor.
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Context:
Evaluate the integral \( I = \int_{C(0,2)^+} \frac{z^3}{z^3+z^2-z-1} \, dz \), where \( C(0,2)^+ \) denotes the positively oriented circle centered at the origin with radius 2.
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Context:
How many elements are in the conjugacy class of $(123)(456)$ in $A_7$?
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Context:
Find the interval(s) where the function $y = \sqrt{-x^2 + 2x + 3}$ is decreasing.
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Context:
Given a Markov chain with state space $E = \{1, 2, 3, 4\}$ and transition matrix $$P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0.4 & 0.6 & 0\\ 0.8 & 0 & 0.2 & 0\\ 0.2 & 0.3 & 0 & 0.5\\ \end{bmatrix},$$ compute the expected value of the product $E[f(X_5)f(X_6)|X_4 = 4]$, where $f$ assigns values $2, 4, 7,$ and $3$ to states $1, 2, 3,$ and $4$, respectively.
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Context:
Let \( f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} \) be a function such that for all \(n > 1\) there is a prime divisor \(p\) of \(n\) such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that \( f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 \). Determine \( f(2014^2)+f(2015^3)+f(2016^5) \).
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Context:
Determine the conditions under which the function \( \frac{f(z)}{g(z)} \), where \( f(z)=\sum_{k=m}^\infty a_kz^k \) and \( g(z)=\sum_{k=p}^\infty b_kz^k \), is holomorphic in a neighborhood of 0.
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Context:
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
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Context:
Find the number of $4 \times 4$ matrices $A$ such that $A^2$ equals a given matrix and $a_{11}, a_{33} > 0$.
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Context:
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$.
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Context:
Determine the radius of convergence of the series represented by the limit \( L_1 \) given below, for a positive integer \( k \):
\[ L_1=\lim_{n\rightarrow\infty} \left|\sqrt[n]{\frac{(n!)^k}{(kn)!}}\right| \]
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Context:
Petya cut three $2 \times 2$ squares from an $8 \times 8$ checkered square as shown in the figure. Olya wants to cut as many $1 \times 3$ rectangles as possible from the remaining part of the square (following the grid lines). What is the maximum number of such rectangles she can cut?
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Context:
Restore the digits. On the board, the product of three consecutive even numbers was written. During the break, Vasya erased some of the digits. As a result, the board shows $87*****8$. Help Petya find the missing digits in the product.
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Context:
Let \( f \) be a function that takes in a triple of integers and outputs a real number. Suppose that \( f \) satisfies the equations
\[
f(a, b, c) = \frac{f(a+1, b, c) + f(a-1, b, c)}{2}
\]
\[
f(a, b, c) = \frac{f(a, b+1, c) + f(a, b-1, c)}{2}
\]
\[
f(a, b, c) = \frac{f(a, b, c+1) + f(a, b, c-1)}{2}
\]
for all integers \( a, b, c \). What is the minimum number of triples at which we need to evaluate \( f \) in order to know its value everywhere?
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Context:
Find the limit of the sum of the areas of triangles formed by a line segment of length 1, with a ratio of 0.9, as the number of triangles approaches infinity.
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Context:
Determine the value of $b$ ($b \neq a$) such that the causal linear shift-invariant system described by the difference equation \( y(n) - ay(n-1) = x(n) - bx(n-1) \) is an allpass system, meaning the magnitude of its frequency response is constant, independent of frequency.
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Context:
Calculate the variance of the expression \((W_1W_2 - Z_1Z_2)^2\), where \(W_1, W_2, Z_1,\) and \(Z_2\) are independent random variables.
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Context:
An infinite sequence of positive integers $a_1, a_2, \ldots$ is defined such that for $k \geq 2$, the $k^\text{th}$ element is equal to $k$ plus the product of the first $k-1$ elements of the sequence. Given that $a_1 = 1$, determine the smallest prime number that does not divide $a_{2010}$. \( \text{Express your answer as a single prime number.} \)
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Context:
Let $p$ be a prime number and consider a group $G$ with order $p^4$ such that the center $Z(G)$ has order $p^2$. Determine the number of conjugacy classes of $G$.
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Context:
Calculate the double integral \( \int_{\mathbb{R}}\left(\int_{\mathbb{R}} f(x,y) \, dx\right)dy \) for the function \( f(x,y) \) defined as follows:

\[ f(x,y) = \begin{cases}
      1 & x\geq 0, \ x\leq y\leq 1+x \\
      -1 & x\geq 0, \ 1+x\leq y\leq 2+x \\
      0 & \text{otherwise}
   \end{cases} \]
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Context:
Calculate the expected value of the following limit:
\[ \lim_{N \to \infty} \left( \prod_{i = 1}^{N} X_{i} \right)^{\dfrac{1}{N}} \]
where each random variable $X_i$ for $i = 1, \ldots, N$ is an independently and identically distributed random integer from 1 to 9.
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Context:
Determine the conditions on the coefficients \(a\), \(b\), and \(c\) such that the integral \(\int \frac{ax^2+bx+c}{x^3(x-1)^2} \, dx\) represents a rational function.
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Context:
Evaluate the integral \( \int_C (x - y)(dx + dy) \) using Green's Theorem, where \( C \) is the semicircular part of \( x^2 + y^2 = 4 \) above the line \( y = x \), from \((-\sqrt{2}, -\sqrt{2})\) to \((\sqrt{2}, \sqrt{2})\).
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Context:
Suppose you have  $27$  identical unit cubes colored such that  $3$  faces adjacent to a vertex are red and the other  $3$  are colored blue. Suppose further that you assemble these  $27$  cubes randomly into a larger cube with  $3$  cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as  $\frac{1}{2^n}$  for some positive integer  $n$ . Find  $n$ .
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Context:
In how many ways can \((n+1)\) people be seated at two indistinguishable circular tables, ensuring that both tables have at least one person? Express your answer in terms of \(n\).
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Context:
Determine the number of distinct ways to insert $k$ vertices along the edges of a polygon with $n$ sides, considering that the order of insertion matters and vertices can be placed anywhere on the edges.
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Context:
Given a sequence of $n$ independent and identically distributed random variables $X_1, X_2, \ldots, X_n$, each following an exponential distribution with rate parameter $\lambda = 1$, let $M_n = \max(X_1, X_2, \ldots, X_n)$. The cumulative distribution function of $M_n$ is $F_{M_n}(x) = (1 - e^{-x})^n$ for $x \geq 0$. Calculate the expected value of $e^{M_n}$, i.e., $\mathbb{E}(e^{M_n})$. \( \text{Express your answer in terms of } n. \)
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Context:
In 1740, Knowall discovered a comet and calculated that it would be visible to humans in 1823, 1906, and so on, with a recurrence every 83 years. From now until the year 3000 AD, the number of times humans can see this comet is ______.
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Context:
Compute $$\sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)}$$
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Context:
Determine the range of values of \(\alpha\) for which the function \(f(x) = |x|^{\alpha}\) belongs to the Sobolev space \(H^1(B(0,1))\) in \(\mathbb{R}^2\).
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Context:
The tangent of the angle between the median and the altitude drawn to the side of an isosceles triangle is equal to $\frac{1}{2}$. Find the sine of the vertex angle.
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Context:
Find the limit of the sequence \( b_n = \sqrt[n]{\sum_{k=1}^{n}{\left(1 + \frac{1}{k}\right)^k}} \).
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Context:
Given the set $X=\{a,b,c,d\}$ and the topology $\tau=\{\emptyset, \{a,b\}, \{a,b,c\}, \{c,d\}, X\}$, find the connected component $C(a)=\bigcup \{A \mid a \in A , A \subset X, \text{$A$ connected}\}$.
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Context:
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college, one-third of the freshman class drops out of school, 40 students in the sophomore class drop out of school, and one-tenth of the junior class drops out of school. Given that the college only admits new freshman students and wants to begin each school year with 3400 students enrolled, how many students does it need to admit into the freshman class each year?
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Context:
Given is a chessboard 8x8. We have to place  $n$  black queens and  $n$  white queens, so that no two queens attack. Find the maximal possible  $n$ .
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)
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Context:
Given a non-constant sequence {a_i} satisfying a_{i+1}^2 - a_i a_{i+1} + a_i^2 = 0, with boundary conditions a_1 = a_{n+1} = 1, find the value of the sum ∑_{i=0}^{n-1} a_i. Express your answer as a single integer.
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Context:
Six different digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. Calculate the sum of the six digits used.
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Context:
Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying
$\forall x,y\in\mathbb{R}: \begin{cases} f(x+y)=f(x)f(y), \\ f(x)=xg(x)+1 & \end{cases}\quad\mathrm{and}\quad\lim_{x\to0}g(x)=1.$
Find the derivative of $f$ in an arbitrary point $x.$
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Context:
A and B play a game where each has a 50% chance of winning any given game. The game continues until one player wins n consecutive games. What is the expected total number of games played until this occurs?
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Context:
Let \( n \) be a natural number. A volleyball team consisting of \( n \) women and \( n \) men lines up for a game. Each team member occupies one of the positions \( 1, 2, \ldots, 2n \), whereby exactly positions \( 1 \) and \( n+1 \) are outside the court. During the game, all team members rotate, switching from position \( i \) to position \( i+1 \) (or from \( 2n \) to \( 1 \). How many possibilities are there for the starting line-up so that there are always at least \( n-1 \) women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.
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Context:
Evaluate the limit \( \lim_{x\to0}{\frac{x^2+2\ln(\cos x)}{x^4}} \) without using L'Hopital's rule or Taylor series.
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Context:
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex, the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Given that the sequence $\{a_n\}$ is an increasing sequence, and for any $n \in \mathbb{N}^*$, $a_n = 2n^2 + kn$ always holds, then the range of values for the real number $k$ is.
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Context:
Compute the residue of the function \( f(z) = e^{z + \frac{1}{z}} \) at \( z = 0 \) using the definition of a residue.
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Context:
Evaluate the double integral of the function \( f(x,y) = x^2 - y^2 \) over the region \( D \) bounded by the curves \( y = x \), \( y = x + 1 \), \( xy = 1 \), and \( xy = 2 \) in the first quadrant.
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Context:
Consider the quadratic function f(x) = ax^2 + bx + c, where a, b, and c are integers and x is a real number. Let p be a positive even integer with a positive units digit, such that 10 ≤ p ≤ 50. If the units digit of p ^ 3 minus the units digit of p ^ 2 is equal to 0, f(p^2) is equal to f(p^3), and f(p) = 0, what is the units digit of p + 5?
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Context:
Find the minimal polynomial of the linear map $T_P: M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ defined by $T_P(X) = P^{-1}XP$, where $P = \begin{pmatrix} i & 2 \\ -1 & -i \end{pmatrix}$ and $P^2 + 3I = 0$. Provide your answer as a polynomial.
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Context:
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts.  Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group.  Each of them has the same number of internet friends.  In how many different ways can this happen?
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Context:
Zsspasii A.A.

The number of the current Olympiad (70) is formed by the last digits of the year of its holding, written in reverse order.

How many more times will such a situation occur in this millennium?

#
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Context:
Convert the equation $2x(x-1)=3x$ into the general form of a quadratic equation, $ax^2 + bx + c = 0$, and provide the coefficients $a$, $b$, and $c$ in that order.
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Context:
Evaluate the limit: $$ \lim_{n \to \infty} \frac{\left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2}\right)^n }{ 2 \left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 + 1}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 (n^2 +1)}\right)^n } $$
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Context:
Evaluate the expression \( \sum\limits_{n=1}^{1009} \gcd(2n, 2020) - \sum\limits_{n=0}^{1008} \gcd(2n+1, 2019) \) without using calculus.
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Context:
Consider the set of complex numbers z satisfying $|1+z+z^{2}|=4$. The largest possible value for the imaginary part of z can be written as $\frac{\sqrt{m}}{n}$, where m and n are positive integers and m is not divisible by the square of any prime. What is m+n?
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Context:
For a Brownian motion $W_t\sim\mathcal{N}(0,t)$, determine the values of $c$ and $d$ such that $cW_2+W_4$ and $W_5-dW_7$ are independent.
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Context:
Calculate the integral \( \int_T db \wedge dt \) in \( \mathbb{R}^5 \) with coordinates \((p_1,q_1,p_2,q_2,t)\), where \( db = dp_1 \wedge dp_2 \) and \( T = \{ t \in [0,1], p_1^2 + q_1^2 = p_2^2 + q_2^2 \leq t^2 \} \).
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Context:
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers  $n < 100$  going through and "flipping" every  $n$ th locker. How many lockers are open after this process?
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Context:
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.

[i]
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Context:
Determine the Hausdorff dimension of the set $A$ of non-recurrent walks on $\mathbb{Z}$, where walks are identified with infinite binary sequences in the Cantor space $2^\mathbb{N}$. A non-recurrent walk is defined as a sequence $x$ for which $\frac{\sum_{k=1}^n x(k)}{n} = \frac{1}{2}$ for only finitely many $n$. What is the Hausdorff dimension of this set?
Please reason step by step, and put your final answer within \boxed{}.