Context:
Navi and Ozna are playing a game where Ozna starts first and the two take turns making moves. A positive integer is written on the board. A move consists of either:

1. Subtracting any positive integer at most $2015$ from it, or
2. If the integer on the board is divisible by $2014$, dividing by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \geq 2015$, ensuring that all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T: P_k \rightarrow P_k$ be a linear operator defined as $T(p(x)) = p(x+a)$, where $a$ is a constant. If $L$ is a differential operator, find the dimension of the kernel of the operator $L - T$. \( \text{(Express your answer as a single integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jan's three-eyed lizard has 3 times more wrinkles than eyes, and a number of spots equal to seven times the square of the number of wrinkles. The lizard's cousin has 2 times more wrinkles than eyes, and a number of spots equal to 5 times the square of the number of wrinkles. How many fewer eyes do both lizards have combined, compared to the combined total of spots and wrinkles on both lizards?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third? Provide your answer in the form of a fraction \frac{k}{m}, and then calculate the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Benny plans to visit one of 8 different zoos each week for the next 13 weeks. Determine the number of ways Benny can schedule his visits such that he visits each zoo at least once and at most three times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Convert the following triple integral from Cartesian coordinates to cylindrical or spherical coordinates, choosing the system that simplifies the integral the most:
\[ \int_0^1 \int_0^x \int_0^{ \sqrt{x^2+y^2}} \frac{(x^2+y^2)^{3/2}}{x^2+y^2+z^2}\,dz\,dy\,dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for \( \alpha \) such that the improper integral \( \int_{0}^{\infty}\frac{\sin x}{x^{\alpha}} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A_1, A_2, A_3, \ldots, A_n \) be subsets of a set \( S \) with \(|S| = 2019\). The union of any three of the \( A_k \) is equal to \( S \), but the union of any two is not equal to \( S \). Determine the maximum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the magnitude of the vertex angle \( \alpha \) of an isosceles triangle with a given area \( A \) such that the radius \( r \) of the inscribed circle is maximized.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A waiter had 44 customers in his section. If 12 of them left, his remaining customers occupied tables with 8 people at each table and some tables with just 4 people due to social distancing. If 25% of the remaining tables had only 4 people, how many total tables did he have in his section?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of digits a natural number can have such that all the digits are different and the number is divisible by each of its digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
24. $[\mathbf{1 4}]$ In how many ways may thirteen beads be placed on a circular necklace if each bead is either blue or yellow and no two yellow beads may be placed in adjacent positions? (Beads of the same color are considered to be identical, and two arrangements are considered to be the same if and only if each can be obtained from the other by rotation).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum height of an isosceles right triangle that can contain all non-overlapping circles with radii $1/n$ for $n \in \mathbb{N}$, given that the total area of these circles is $\frac{\pi^3}{6}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point A chases point B at a unit velocity. Point B moves to the right at a unit velocity. Initially, at time $t = 0$, point A is at $(0,1)$ and point B is at $(0,0)$. Determine the limiting distance $||B(t) - A(t)||$ as $t$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence \(\{a_n\}\) satisfying
\[
a_{n+1} + (-1)^n a_n = 2n - 1,
\]
and the sum of the first 2019 terms of the sequence \(\{a_n - n\}\) is 2019, find the value of \(a_{2020}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jamie accidentally misinterprets the rules of the order of operations, and adds or subtracts before multiplying or dividing. What would be her result for the equation $4 + 3 \times 1 - 2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can the sequence $1,2,3,4,5$ be arranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of growth of the entire function defined by the infinite product \( \prod_{n=0}^{\infty} \left(1 - \frac{z}{(n+1)!}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the Riemann-Stieltjes integral \( \int_{0}^{2}{\alpha\left(\dfrac{x}{\sqrt{2}}+1\right) \mathrm{d}\alpha(x)} \), where \( \alpha(x) = 3[x] \) and \([x]\) is the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A soft drink can is $h$ centimeters tall and has a radius of $r$ cm. The cost of material in the can is $.0005$ cents per cm$^2$ and the soda itself costs $.001$ cents per cm$^3$. The cans are currently $9$ cm tall and have a radius of $4$ cm. Use calculus to estimate the effect on costs of increasing the radius by $0.4$ cm and decreasing the height by $0.7$ cm. Round your answer to $3$ decimals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$p, q, r$ are real numbers satisfying the following equations:
\[ \frac{(p+q)(q+r)(r+p)}{pqr} = 24 \]
\[ \frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10. \]
Given that \( \frac{p}{q} + \frac{q}{r} + \frac{r}{p} \) can be expressed in the form \( \frac{m}{n} \), where \( m, n \) are relatively prime positive integers, compute \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1 + x^2 + x^4 + \cdots + x^{2k}} \, \text{dx} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the index of $K^{*n}$ in $K^*$ for an infinite extension field $K/\mathbb{Q}_p$ where $p\nmid n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Today's date is written as: 22.11.2015. Name the last past date that is written with the same set of digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the curve $y=(x-a)\ln x$ has two tangent lines passing through the origin, then the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum likelihood estimate (MLE) of $m$ in the linear model $y = mx + e$, where $m$, $x$, and $e$ are independent and identically distributed as $N(0,1)$, given a single observation $(x, y)$. Assume $x = 1$ and $y = 3$. Provide the analytical expression for the MLE of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f\left(x\right)=\cos 2x+\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\left(x\right)$, then $\cos (x_{1}+x_{2})=$____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the infinite sum: $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\sum_{n=0}^\infty \frac{1}{k2^n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
We say a non-negative integer $n$“contains” another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ contains $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not contain a multiple of $7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function \( \mu(x,y) \) such that the differential equation \( \mu(x,y)(3x^2y+2xy+y^3)dx + \mu(x,y)(x^2+y^2)dy = 0 \) is exact.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle at the apex of a regular quadrilateral pyramid if it is equal to the angle between a lateral edge and the base plane of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\arccos \left( \frac{\sqrt{5}-1}{2} \right)}


Context:
Evaluate the following double integral, where $D$ is a domain in $\mathbb{R}^2$ that is bounded, closed, and symmetric with respect to the line $y=-x$:

$$
I:=\iint_{D} \frac{x^2+y^2}{1+x^4+y^4} \mathrm{d}x \mathrm{d}y
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the vertex of $\mathcal{P}$ to all four intersection points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rancher has 500 head of cattle. He was about to sell them all for $300,000 when 350 of them fell sick and died. Because of the sickness, his customers lost confidence in his cattle, forcing him to lower his price by $250 per head. How much money would the devastated farmer lose if he sold the remaining cattle at the lowered price compared to the amount he would have made by selling them at the original price?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers $n$, let the numbers $c(n)$ be determined by the rules:
- $c(1) = 1$
- $c(2n) = c(n)$
- $c(2n+1) = (-1)^n c(n)$

Find the value of \( \sum_{n=1}^{2013} c(n) c(n+2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \in \mathcal{C}[0, \infty) \) be such that \( \lim\limits_{x \to \infty} f(x) \in \mathbb{R} \) and let \( a > 0 \). Compute \[ \lim_{n \to \infty} \int_0^a f(nx)\,dx. \]
Please reason step by step, and put your final answer within \boxed{}.

\boxed{a \cdot \lim_{x \to \infty} f(x)}


Context:
In the store "Everything for School," three types of chalk packs are sold: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the number of packs of excellent chalk increased by 80%, and the number of regular chalk packs decreased by no more than 10 packs. How many total packs of chalk were in the store initially?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal value of \( \alpha_3 \) such that the function \( u(x,y,z) = x^\alpha(1-x)y(1-y)z(1-z) \) belongs to the Sobolev space \( H^3(\Omega) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) and \( g \) be continuous functions over the reals such that \( g(x) = \int_0^x f(y)(y-x) \, dy \) for all \( x \), and \( g \) is three times continuously differentiable. What is the greatest integer \( n \) such that \( f \) is \( n \) times continuously differentiable?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface area of the portion of the cone defined by \( z^2 = x^2 + y^2 \), where \( z \geq 0 \), that is contained within the cylinder \( y^2 + z^2 \leq 81 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any subset \( S \subseteq \{1, 2, \ldots, 15\} \), a number \( n \) is called an "anchor" for \( S \) if \( n \) and \( n+|S| \) are both members of \( S \), where \( |S| \) denotes the number of members of \( S \). Find the average number of anchors over all possible subsets \( S \subseteq \{1, 2, \ldots, 15\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
108 singers participate in an art festival. Arrange them into performances with 4 singers in each performance such that among any 8 singers, any two singers perform together the same number of times. What is the minimum number of performances needed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit:
$$\lim_{n\rightarrow\infty}\frac{\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)^2}{\prod_{k=1}^{2n}\left(1-\frac{1}{2k}\right)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{a_n\}$ be a geometric sequence and $\{b_n\}$ be an arithmetic sequence with $b_1 = 0$. Define $\{c_n\} = \{a_n + b_n\}$, and suppose that $\{c_n\}$ is the sequence 1, 1, 2, .... Find the sum of the first 10 terms of $\{c_n\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The tetrahedron $ABCDE$ has congruent side edges and a volume of one unit. Its base is the rectangle $ABCD$. The tetrahedron is intersected by a plane passing through the edge $BC$, creating a cross-section that intersects the side edges $EA$ and $ED$ at points $A_1$ and $D_1$, respectively. The lines $BA_1$ and $CD_1$ intersect at point $M$. What is the volume of the tetrahedron $EMA_1D_1$ if $EM = 2AB$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to 0^+} \int_0^\infty \frac{e^{-xt}}{\sqrt{t^2 +t}} \, dt \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi}\frac{\sin^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\sin^2(x/2)} \, dx \) for any \(N\in\mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of lines of symmetry in a regular octahedron such that a rotation of at most $180^{\circ}$ around these lines maps the octahedron onto itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two concentric circles have radii of 1 and 2 units, respectively. What is the minimum possible area of a cyclic quadrilateral inscribed in the larger circle that also contains the smaller circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer $k$, determine the condition on the positive integer $m$ such that there exists exactly one positive integer $n$ satisfying the divisibility condition \( n^m \mid 5^{n^k} + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $a>0$, $b>0$, $c>2$, and $a+b=2$, find the minimum value of $$\frac {ac}{b}+ \frac {c}{ab}- \frac {c}{2}+ \frac { \sqrt {5}}{c-2}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $\iiint_E z\,dV,$ where $E$ lies above the paraboloid $z=x^2+y^2$ and below the plane $z=2y.$ In cylindrical coordinates the paraboloid is given by $z=r^2$ and the plane by $z=2r\sin\theta$ and they intersect in the circle $r=2\sin\theta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Lebesgue integral \( \int_{[0,\pi/2]} f \, d\mu \) where the function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases}
\sin(x) & \text{if } \cos(x) \text{ is rational} \\
\sin^2(x) & \text{if } \cos(x) \text{ is irrational}
\end{cases}
\]

Evaluate the integral and provide the result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the length of the normal chord of the parabola \(y^2 = 8x\) that subtends a right angle at the vertex of the parabola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the invariant factors of the module \( N = \langle a,b \rangle / \langle 5a-2b \rangle \), where \( \langle a,b \rangle \) and \( \langle 5a-2b \rangle \) are \( \mathbb{Z} \)-modules.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \( a = \frac{{(1+t^2)}^3}{t^4} \) using basic inequalities, without employing calculus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ring homomorphisms from the ring \( \mathbb{Z}_{12} \) to the ring \( \mathbb{Z}_{24} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what values of $c \in [-1,1]$ does the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) given by \[ f(x) = \begin{cases} \sin \left( \frac{1}{x} \right), & \text{if } x \neq 0; \\ c, & \text{if } x = 0 \end{cases} \] have an antiderivative?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the condition on \(\alpha\) such that the function \(|x|^{-\alpha}\) is in the \(L^1\) space with respect to the measure \(\mu\) on the set \(\{x \in \mathbb{R}^n : |x| \ge 1\}\), given that \(\mu\) is a polynomially bounded measure satisfying \(\mu(|x| \le A) \le C(1+A^N)\) for constants \(C, N > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\pi e^{2\cos(\theta)}\cos(2\sin(\theta) + \theta)\,d\theta \) using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six lines are drawn in the plane. Determine the maximum number of points, through which at least $3$ lines pass.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit:
$$L=\lim_{x\to 0} \frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $x+4y=4$ ($y > 0$), $0 < t < z$, then the minimum value of $\dfrac {4z^{2}}{|x|}+ \dfrac {|xz^{2}|}{y}+ \dfrac {12}{t(z-t)}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the diameter of the base of a cone is 4, and the slant height (generatrix) is 6, find the central angle of the sector in the lateral surface development (net).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the series \( \sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)} \) converges, where \( k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)} \) and \( a \in \mathbb{R} \). Use the approximations \( \cos\frac{1}{n} \sim 1-\frac{1}{2n^2} \) and \( \sin^a\left(\frac{1}{n}\right) \sim \frac{1}{n^a} \) as \( n \to +\infty \) to analyze the convergence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample of size 66 with a frequency distribution as follows: $(11.5, 15.5]$: $2$, $(15.5, 19.5]$: $4$, $(19.5, 23.5]$: $9$, $(23.5, 27.5]$: $18$, $(27.5, 31.5]$: $11$, $(31.5, 35.5]$: $12$, $[35.5, 39.5)$: $7$, $[39.5, 43.5)$: $3$, estimate the probability that the data falls in [31.5, 43.5).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cyclic quadrilateral $ABCD$ inscribed in circle $O$, where $AC=\sqrt{3}BD$ and $\angle ADC=2\angle BAD$, and given that $AB\cdot CD+BC\cdot AD=4\sqrt{3}$, calculate the radius of circle $O$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \to \infty}\left(\frac {\int_{x^{2}}^{2x}t^{4}e^{t^{2}}dt}{e^{x}-1-x - \frac{x^2}{2}- \frac{x^3}{6}-\frac{x^4}{24}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the absolute maximum value of \( f(z) = \left| \frac{5z^4 + 4}{z^2 + 1} \right| \) on the closed disk defined by \( |z - 2i| \leq 0.5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of equally spaced interpolation nodes required to approximate the function \( f(x) = e^x \) on the interval \([-4, 0]\) using Lagrange interpolation, such that the interpolation error is less than 0.05.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set
$$
S=\left\{A=\left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i}=0 \text{ or } 1, i=1,2, \cdots, 8\right\},
$$
for any two elements \(A\) and \(B\) in \(S\), define the distance between \(A\) and \(B\) as
$$
d(A, B)=\sum_{i=1}^{8}\left|a_{i}-b_{i}\right|.
$$

Determine the maximum number of elements that can be chosen from \(S\) such that the distance between any two of them is at least 5.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the product of the elements $(1,1,1,0,1)$ and $(0,1,0,1,0)$ in the Galois Field $GF(2^5)$. What is the result?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A \) be the set of any 20 points on the circumference of a circle. Joining any two points in \( A \) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these chords.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to \infty} \left(\frac{(2n!)}{n!^2}\right)^{\frac{1}{4n}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the series: $$\sum_{n=1}^\infty \arccos\sqrt{\frac{n}{n+1}}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain kingdom, there were 32 knights. Some of them were vassals to others (a vassal can only have one liege, and the liege is always wealthier than the vassal). A knight with at least four vassals held the title of baron. What is the maximum number of barons that could exist under these conditions?

(In the kingdom, the law stated: "the vassal of my vassal is not my vassal.")
Please reason step by step, and put your final answer within \boxed{}.


Context:
A class went on a camping trip and walked along the road from the school to the base, then immediately returned to the school by the same route. If the distance traveled each day increased by 1 kilometer, and it took 4 days to go and 3 days to return, calculate the distance from the school to the base. Express your answer as a whole number in kilometers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\alpha$ and $\beta$ are both acute angles, $sin\alpha=\frac{3}{5}$, $cos(\alpha+\beta)=\frac{5}{13}$, then the value of $\cos \beta$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All the prime numbers are written in order,  $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers  $a$  and  $b$  with  $a - b \geq 2$ , such that  $p_a - p_b$  divides  $2(a-b)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum dimension of a subspace of endomorphisms of a 4-dimensional vector space over the complex numbers, where all elements of the subspace commute and at least one element has three distinct eigenvalues.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1$. On what day does his unit of blood expire? Provide the answer by converting the month to its corresponding number (e.g., January is 1, February is 2, etc.) and the day to a number, then give the sum of these two numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rook has traveled across an \( n \times n \) board, visiting each cell exactly once, where each of its moves was exactly one cell. The cells are numbered from 1 to \( n^2 \) in the order of the rook's path. Let \( M \) be the maximum difference between the numbers of adjacent (by side) cells. What is the smallest possible value of \( M \)?
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Context:
A circular coin with a circumference of about 5 cm starts rolling from a point on the boundary of a quadrilateral with a circumference of 20 cm. After making one complete rotation and returning to the starting point, the center of the coin will trace a closed curve. The length of this curve is ____ cm.
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Context:
Determine the minimal degree of a smooth curve in $\mathbb{CP}^3$ that is not contained in any cubic surface.
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Context:
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
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Context:
Evaluate the contour integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^2(e^{z}-1)}dz \) where the contour \(\gamma\) is defined by \(z=\frac{1}{2}e^{2\pi i t}\) for \(0\leq t\leq1\), and \(z=0\) is a pole of order 3. Determine the value of the integral without directly applying the Cauchy Integral Formula.
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Context:
Using the random walk approach, calculate the expected time for a mouse to reach one of the three exits in a maze, where each exit is an absorbing state.
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Context:
Example 8. Find the integral $\int \frac{d x}{\sqrt{3 x^{2}-6 x+9}}$.
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Context:
In the tetrahedron \(EFGH\), the side lengths are given as \(EF = GH = 7\), \(EG = FH = 10\), and \(EH = FG = 11\). The points \(K, L, M, N\) are the incenters of the triangles \(EFG\), \(EFH\), \(EGH\), and \(FGH\) respectively. Calculate the volume of the tetrahedron \(KLMN\). Round your answer to the nearest hundredth if necessary.
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Context:
A regular tetrahedron has all edges of equal length. There are two spheres that each touch every edge of the tetrahedron. What is the ratio of their volumes?
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Context:
How many distinct ways can the letters of the word COMBINATION be arranged such that none of the patterns CAN, BIN, or NIB appear?
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Context:
Find the limit: $$\lim_{x\to \infty} \frac{(x+3)^x}{\sum_{n=3}^{x+2}n^x}.$$
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Context:
The 6th problem of the 3rd China Northern Hope Star Mathematics Summer Camp:
Given non-negative real numbers $a, b, c, x, y, z$ satisfying $a+b+c=x+y+z=1$.
Find the maximum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$.
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Context:
Let $x_{1}$, $x_{2}$, $\\ldots$, $x_{n}$ be $n$ real numbers in $\\left(\\frac{1}{4},\\frac{2}{3}\\right)$. Find the minimal value of the expression: \\[ \\log_{\\frac 32x_{1}}\\left(\\frac{1}{2}-\\frac{1}{36x_{2}^{2}}\\right)+\\log_{\\frac 32x_{2}}\\left(\\frac{1}{2}-\\frac{1}{36x_{3}^{2}}\\right)+\\cdots+ \\log_{\\frac 32x_{n}}\\left(\\frac{1}{2}-\\frac{1}{36x_{1}^{2}}\\right). \\]
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\boxed{
The minimal value of the expression is \( 2n \).}


Context:
Consider an ergodic Markov chain \( \{Y_t\} \) with state space \( \{0, \dots, m\} \). Given that \( E(Y_{t+1}-Y_t)=\alpha \) for \( Y_t\in [1,m-2] \), \( E(Y_{t+1}-Y_t)>\alpha \) for \( Y_t=m-1 \), and \( E(Y_{t+1}-Y_t)=\gamma \) for \( Y_t=0 \), where \( \alpha \) is a small negative number, \( \gamma>\alpha \), and \( m \) is a large positive constant, determine the behavior of \( \lim_{t\to\infty}E(Y_t)/m \) as \( m\rightarrow\infty \).
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Context:
Set
\[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\]
Find \(\lim_{n\to\infty} A_n.\)
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Context:
Simplify the following limit:

$$\lim _{ n\rightarrow \infty }{ \left( \sin ^{ 4 }{ x } +\frac { 1 }{ 4 } \sin ^{ 4 }{ 2x+\cdots+\frac { 1 }{ { 4 }^{ n } } \sin ^{ 4 }{ 2nx } } \right) } $$
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Context:
Find the closed-form expression for the limit of the series \( \sum_{m=M+1}^{\infty}\frac{(-1)^{m+1}}{2m+1}\sin((2m+1)\frac{\pi}{2}x) \) as \( M \to \infty \), given that \( \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}\sin((2k-1)x)=\frac{1}{2}\ln\tan(\frac{\pi}{4}+\frac{x}{2}) \).
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Context:
$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?
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Context:
Express \(\cos(x^2)\) in terms of \(\cos(x)\) and \(\sin(x)\), if possible.
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Context:
Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$.
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Context:
If 60% of (x - y) = 30% of (x + y + z), and 40% of (y - z) = 20% of (y + x - z), then what percent of x is (y - z)?
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Context:
14:There are N permutations $(a_{1}, a_{2}, ... , a_{30})$ of 1, 2, ... , 30 such that for $m \in \left\{{2, 3, 5}\right\}$, m divides $a_{n+m} - a_{n}$ for all integers n with $1 \leq n < n+m \leq 30$. Find the remainder when N is divided by 1000.
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Context:
Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets 4 spots, calculate the total number of different distribution schemes.
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Context:
Evaluate the limit \( \lim_{n \to \infty} \frac{\prod_{k=0}^{n-1} \binom{n-1}{k} \cdot \prod_{k=0}^{n+1} \binom{n+1}{k}}{\left(\prod_{k=0}^{n} \binom{n}{k}\right)^2} \).
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Context:
Alex, a self-described nerd and an avid comic book collector, has been attending Comic-Con annually since 1995. To celebrate his love for comics, each year at Comic-Con, Alex buys a set of comic books equal to the number of years he's been attending. In 2023, how many comic books did Alex purchase in total at Comic-Con?
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Context:
Since the implementation of quality education in high schools in a certain province, student clubs have developed rapidly. Five freshmen from a certain high school plan to join one of the four clubs: "Chunhui Literature Society," "Dancers Roller Skating Club," "Basketball Home," and "Go Garden." If each club has at least one student join, each student joins at least one club and only one club, and student A does not join "Go Garden," then the number of different participation methods is ______. (Fill in the number)
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Context:
Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.
What is the maximal possible number of points with this property?
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Context:
A number is *interesting*if it is a  $6$ -digit integer that contains no zeros, its first  $3$  digits are strictly increasing, and its last  $3$  digits are non-increasing. What is the average of all interesting numbers?
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Context:
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 54$. Express $x$ as a decimal.
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Context:
Let $S = \{1, 2, \dots, 9\}$. Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S$, $f(f(f(s))) = s$ and $f(s) - s$ is not divisible by $3$.
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Context:
Solve for \( x \) in the equation \( x\left(\frac{e^x+1}{e^x-1}\right)=4 \) without using numerical methods or series expansion.
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Context:
Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) positive random variables with expected value $E(X_i) = \mu$ and variance $Var(X_i) = \sigma^2$. Calculate the expectation of the ratio $\frac{X_1 + X_2 + \ldots + X_m}{X_1 + X_2 + \ldots + X_n}$, where $m < n$. \( \text{Express your answer in terms of } m, n, \mu, \text{ and } \sigma^2. \)
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Context:
If one takes 6 different letters from "CN Dream" and arranges them in a row, how many different permutations are there that include the letter combination "ea" (in the same order)? Provide your answer as a single integer.
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Context:
We define a number as "funny" if it is divisible by the sum of its digits plus one. For example, since $1 + 2 + 1 \mid 12$, the number $12$ is considered funny. What is the maximum number of consecutive funny numbers?
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Context:
There are $360$ permutations of the letters in $MMATHS.$ When ordered alphabetically, starting from $AHMMST,$ $MMATHS$ is in the $n$th permutation. What is $n$?
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Context:
Let $A B C D E F G H$ be an equilateral octagon with $\angle A \cong \angle C \cong \angle E \cong \angle G$ and $\angle B \cong \angle D \cong \angle F \cong$ $\angle H$. If the area of $A B C D E F G H$ is three times the area of $A C E G$, then $\sin B$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.
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Context:
Given the sets \(A\) and \(B\) defined as follows:
\[ A=\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\} \]
\[ B=\{(x, y) \mid x=n, y=a(a^{2}-n+1), n \in \mathbf{Z}_{+}\} \]
Determine the total number of integers \(a\) such that the intersection \(A \cap B\) is non-empty.
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Context:
A community plans to invest 500 yuan to purchase three types of books, A, B, and C. Type A costs 30 yuan per book, type B costs 25 yuan per book, and type C costs 20 yuan per book. The community must buy at least 5 books of type A and at most 6 books of type A, and all three types of books must be purchased. Calculate the total number of purchasing schemes that satisfy these conditions.
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Context:
What is the cardinality of the set of all sequences of real numbers that have a finite number of terms?
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Context:
Given a matrix $A$ with entries of either 0 or 1, where each row contains at most $r$ ones, and each row of the matrix product $AA^T$ contains at most $z$ non-zero entries, find an upper bound on the number of ones in each column of $A$ as a function of $r$ and $z$. The bound need not be tight but should be expressible as a simple function of $r$ and $z$. Provide your answer as a function of $r$ and $z$.
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Context:
Let $A \in \mathbb{R}^{m \times m}$ be an invertible matrix such that $A^k = I$ for some positive integer $k$. Define the linear map $T_A: \mathbb{R}^{m \times m} \to \mathbb{R}^{m \times m}$ by $T_A(X) = AXA^{-1}$. Determine the minimal polynomial of $T_A$ when restricted to the space of skew-symmetric $m \times m$ matrices.
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Context:
In a regular quadrilateral prism \( ABCDA_1B_1C_1D_1 \) \( (A \| A_1 \| B \| B_1 \| C \| C_1 \| D \| D_1 ) \), a plane passes through the midpoints of two adjacent sides of the base \( DC \) and \( AD \), and the vertex \( B_1 \) of the upper base. Find the angle between this plane and the base plane if the perimeter of the cross-section is three times the diagonal of the base.
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Context:
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs of \( y = f(x) \) and \( y = g(x) \) are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function \( y = (f(x))^2 \) touches the graph of the function \( y = 11g(x) \). Find all values of \( A \) such that the graph of the function \( y = (g(x))^2 \) touches the graph of the function \( y = Af(x) \).
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Context:
In the diagram, each of four identical circles touch three others. The circumference of each circle is 48. Calculate the perimeter of the shaded region formed within the central area where all four circles touch. Assume the circles are arranged symmetrically like petals of a flower.
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Context:
Determine the radius of convergence of the power series for \( \frac{1}{1+x^2} \) centered at \( x=1 \) without using complex analysis.
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Context:
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?
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Context:
Find a bijection \( g : \mathbb{N} \rightarrow \mathbb{N}/\sim \), where \( \mathbb{N}/\sim \) is the set of equivalence classes of the equivalence relation \( \sim \) defined by \( x \sim y \) if \( x = 2^{k}y \) or \( y = 2^{k}x \) for some \( k \in \mathbb{Z} \).
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Context:
Farmer Red has three milk cows: Bess, Brownie, and Daisy. Bess, the smallest cow, gives him two pails of milk every day. Brownie, the largest cow, produces three times that amount. Then Daisy makes some amount of milk compared to Bess. Farmer Red gets 77 pails of milk from them each week. How many more pails of milk does Daisy produce compared to Bess?
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Context:
Koshchey the Immortal has 11 large chests. In some of them, there are 8 medium chests. In some of the medium chests, there are 8 small chests. There is nothing else in the chests. Koshchey has a total of 102 empty chests. How many chests does Koshchey have in total?
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Context:
Karen’s work tote bag is twice the weight of her husband Kevin’s briefcase when the briefcase is empty. When Kevin puts his laptop and work papers in his briefcase, it is twice the weight of Karen’s tote. Kevin’s work papers are a sixth of the weight of the contents of his full briefcase. How much does Karen's tote weigh if Kevin's laptop weighs 2 pounds more than her tote?
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Context:
Find the number of idempotent elements in the ring \( \mathbb{Z}[\sqrt{-11}] / (1+\sqrt{-11}) \).
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Context:
Let \( \{s_n\} \) be a sequence in \([0,1]\) such that \( 2s_n \leq s_{n-1} + s_{n+1} \) for \( n \geq 1 \). Determine the supremum of the set \( \{s_{n+1} - s_n \mid n \geq 1\} \).
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given that $a=3$, $b=4$, and $\sin B= \frac{2}{3}$, then angle $A$ equals \_\_\_\_\_\_.
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Context:
At a distance of \( l \) meters from bridge \( A \) downstream, there is bridge \( B \). When an athlete was swimming past bridge \( A \) towards bridge \( B \), two balls were thrown to him. He caught the first ball and left the second one to float downstream. After swimming a certain distance with the first ball, the athlete left this ball and swam upstream to catch the second ball. After catching the second ball, he turned around and swam towards bridge \( B \), arriving there at the same time as the first ball which was floating freely. What distance did the athlete swim, given that his own speed was always \( k \) times greater than the speed of the current?
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Context:
In a regular pentagon \(ABCDE\), there is an equilateral triangle \(ABM\). Determine the measure of the angle \(BCM\).
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Context:
Let the continuous random variables \(X_1\), \(X_2\), \(X_3\), and \(X_4\) be independent such that \(X_1 \sim N(\mu, 1^2)\), \(X_2 \sim N(\mu, 2^2)\), \(X_3 \sim N(\mu, 3^2)\), and \(X_4 \sim N(\mu, 4^2)\), where \(\mu\) is an unknown parameter. Define the estimator \(\hat{\mu}\) as \(\hat{\mu} = (4X_1 + 3X_2 + 2X_3 + X_4)/10\). Calculate the 95% confidence interval for \(\mu\) using \(\hat{\mu}\) when \(x_1 = 77.47\), \(x_2 = 77.17\), \(x_3 = 79.38\), and \(x_4 = 81.87\).
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Context:
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$
 determine \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\]
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{1}{n}\log\left(\sum_{0<i<n}{\binom{2i}{i}}\right)$$
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Context:
Calculate the number of elements in the intersection $A_6 \cap N_{S_6}(\langle (1\ 2\ 3\ 4\ 5) \rangle)$ using the sign homomorphism and Lagrange's theorem.
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Context:
Find the distributional derivative of the indicator function $\Bbb1(Y \ge a)$. Provide your answer in terms of distributions.
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Context:
What is the maximum number of axes of symmetry that the union of \( k \) line segments on a plane can have?
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Context:
A semicircle has a radius of 11. What is the approximate perimeter of the semicircle?
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Context:
Determine the number of abelian subgroups of order 15 in the symmetric group $S_8$. Provide your answer as a single integer.
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Context:
In a group $G$, let $a$ and $b$ be non-identity elements satisfying the relation $ab = b^2a$. If the subgroup generated by $a$ has order 3, determine the order of the subgroup generated by $b$. \( \boxed{?} \)
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Context:
Evaluate the triple integral \( \iiint_S z \sqrt{x^2 + y^2} \, dx \, dy \, dz \) over the region \( S \) bounded by \( x^2 + y^2 = 1 \), \( z = 0 \), and \( z = x \) with \( z \ge 0 \).
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Context:
A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a factor of 3 rather than 4. If this process is repeated indefinitely, calculate the total area of the shaded triangles.
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Context:
Let $S$ be the ring $\mathbb{F}_2[x]/((x^2+x+1)(x^3+x+1))$, where $\mathbb{F}_2$ is the finite field of two elements. If $I$ is the ideal generated by $\overline{x}$, where $\overline{x}$ is the residue of $x$, what is the cardinality of the ring $S/I$?
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Context:
Determine the minimum number of vertices in a graph that can be expressed as the join of two graphs, each with at least 2 vertices.
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Context:
2. Let $n \geq 2$ be a positive integer, and let $a_{1}, a_{2}, \cdots, a_{n}$ be non-negative real numbers, satisfying $\sum_{i=1}^{n} a_{i}=1$. Find the maximum value of $\left(\sum_{i=1}^{n} i^{2} a_{i}\right) \cdot\left(\sum_{i=1}^{n} \frac{a_{i}}{i}\right)^{2}$.
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Context:
What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101?
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Context:
What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is a positive integer?
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Context:
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = x^2 + 2x$. If $f(2-a^2) > f(a)$, then the range of the real number $a$ is.
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Context:
Given that $X$ and $Y$ are independent and identically distributed standard normal random variables, find the conditional covariance $Cov(XY|X+Y>0)$. Express your answer in terms of known constants or functions.
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Context:
What is the value of $99^3 + 3(99^2) + 3(99) + 1$?
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Context:
Compute the limit of the function \( \frac{1}{A} \int_{1}^{A} A^{1/x}\, dx \) as \( A \) approaches infinity.
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Context:
Determine the values of $a$ for which the infinite series \( \sum_{n=1}^{\infty} \frac{a^n (\ln n)^{1/n}}{n^{0.5}} \) converges.
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Context:
An Englishman owns a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250,000 rubles per hectare. It is known that 1 acre = 4840 square yards, 1 yard = 0.9144 meters, and 1 hectare = 10,000 square meters. Calculate how much the Englishman will earn from the sale.
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Context:
Let \( M = \{ 1, 2, \cdots, 19 \} \) and \( A = \{ a_{1}, a_{2}, \cdots, a_{k} \} \subseteq M \). Determine the smallest integer \( k \) such that for every \( b \in M \), there exist elements \( a_{i}, a_{j} \in A \) satisfying \( b = a_{i} \) or \( b = a_{i} \pm a_{j} \).
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Context:
$\triangle A B C$ 中, $\angle C=90^{\circ}, \angle B=30^{\circ}, A C=2, M$ 是 $A B$ 的中点。将 $\triangle A C M$ 沿 $C M$ 折起,使 $A, B$ 两点间的距离为 $2 \sqrt{2}$ ，此时三棱雉 $A-B C M$ 的体积等于$\frac{m \sqrt{n}}{k}$.求$m+n+k$.
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Context:
Given triangle \( \triangle ABC \) with \( AB = 2 \) and \( \frac{2}{\sin A} + \frac{1}{\tan B} = 2\sqrt{3} \), find the minimum area of \( \triangle ABC \).
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Context:
# Problem 8. (4 points)

What is the maximum number of rooks that can be placed on an 8x8 board so that each one attacks no more than one of the remaining rooks?

#
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Context:
A phone fully charges in 1 hour 20 minutes on fast charge and in 4 hours on regular charge. Fedya initially put the completely discharged phone on regular charge, then switched to fast charge once he found the proper adapter, completing the charging process. Determine the total charging time of the phone, given that it spent one-third of the total charging time on the fast charge. Assume that both fast and regular charging happen uniformly.
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Context:
Alice, Bob, and Charlie are playing a game with 6 cards numbered 1 through 6. Each player is dealt 2 cards uniformly at random. On each player's turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob decide to tell their cards to each other, trying to prevent him from winning whenever possible. Compute the probability that Charlie wins regardless.
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Context:
Find the number of permutations \(a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}\) of the six integers from 1 to 6 such that for all \(i\) from 1 to 5, \(a_{i+1}\) does not exceed \(a_{i}\) by 1.
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Context:
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Ada has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.

Ada has written forty haikus so far and writes down seven haikus every hour. At which hour after she begins will she not have enough haikus done?
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Context:
Determine the value of the greatest integer function $[f(2)]$ for a differentiable function $f$ that satisfies the condition $$f(x+y) = f(x) + f(y) + 2xy(x+y) - \dfrac{1}{3}, \quad \forall x, y \in \mathbb{R},$$ and the limit $$\lim_{h \to 0} \dfrac{3f(h)-1}{6h} = \dfrac{2}{3}.$$
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Context:
Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
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Context:
已知$n\in N*$，集合${{M}_{n}}=\left\{ \frac{1}{2},\frac{3}{4},\frac{5}{8},\cdots ,\frac{2n-1}{{{2}^{n}}} \right\}$，集合${{Mn}}$所有非空子集的最小元素之和为${{T}_{n}}$，则使得${{Tn}}\ge 180$的最小正整数$n$的值为____________．
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Context:
Find natural numbers \( n \) such that for all positive numbers \( a, b, c \) satisfying the inequality

\[ n(ab + bc + ca) > 5(a^2 + b^2 + c^2) \]

there exists a triangle with sides \( a, b, c \).
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Context:
Given that \( \sum\limits_{n=2}^\infty \frac{1}{(n^2-n)^3} = 10 - \pi^2 \), find the closed form of \( \sum\limits_{n=1}^\infty \frac{1}{n^3} \).
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Context:
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.
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Context:
A dandelion blooms in the morning, remains yellow for three days, turns white on the fourth morning, and sheds its seeds by the evening of the fifth day. On Monday afternoon, there were 20 yellow and 14 white dandelions in the meadow. On Wednesday, there were 15 yellow and 11 white dandelions. How many white dandelions will be in the meadow on Saturday?
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Context:
If the total surface area of a cone is three times its base area, then the central angle of its lateral surface development is \_\_\_\_\_\_.
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Context:
Find the limit of the product as \( n \) approaches infinity: \[ \lim_{n\to\infty}\prod_{k=1}^{n}\frac{(k+1)(2k)^2}{k(2k+1)^2} \]
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Context:
.

With inspiration drawn from the rectilinear network of streets in New York, the Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be

$$
|a-c|+|b-d| \text {. }
$$

Suppose only two distinct Manhattan distances occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?
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Context:
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a negligible height above the equator, find the best estimate of the number of hours of flight.
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Context:
Two circles touch each other internally at point K. The chord \(A B\) of the larger circle is tangent to the smaller circle at point \(L\), and \(A L = 10\). Find \(B L\) if \(A K: B K = 2: 5\).
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Context:
In a tournament where each pair of teams played against each other twice, there were 4 teams participating. A win awarded two points, a draw one point, and a loss zero points. The team that finished in last place earned 5 points. How many points did the team that finished in first place earn?
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Context:
Determine the maximum number of points $y \in X(k)$ that can be removed from a smooth geometrically connected projective curve $X$ over a field $k$ of genus $g = 3$ such that $X \setminus y$ is an affine plane curve.
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Context:
Points \( D \) and \( E \) are marked on the sides \( AC \) and \( BC \) of triangle \( ABC \) respectively. It is known that \( AB = BD \), \(\angle ABD = 46^\circ\), and \(\angle DEC = 90^\circ\). Find \(\angle BDE\) given that \( 2 DE = AD \).
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Context:
Suppose there are 36 objects placed at equal distances along a circle. Determine the number of ways to choose 3 objects such that no two chosen objects are adjacent or diametrically opposite.
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Context:
Given the matrices \( A = \begin{bmatrix} 2 & 0 & 3 \\ -1 & -2 & 1 \\ 2 & 0 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & \frac{1}{3} \end{bmatrix} \), calculate the determinant of the matrix expression \( \det(\det(A)B[\det(B)A^{-1}]) \).
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Context:
18.4.11 * Find the integer solution of the equation $\left(1+\frac{1}{m}\right)^{m+1}=\left(1+\frac{1}{1988}\right)^{1988}$.
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Context:
Find the square root of the complex number \(5 - 12i\) using the square root definition. Express the square root in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
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Context:
Sabina is starting her first year of college that costs $30,000. Apart from the tuition, she also has to pay for her living expenses, which amount to $12,000 per year. She intends to complete a 4-year degree program.

She has saved $10,000 for her first year. She was awarded a grant that will cover 40% of the tuition for the first two years and 30% for the last two years. Additionally, in the second through fourth years, Sabina receives a partial scholarship amounting to a 20% reduction in her living expenses.

How much will Sabina need to apply for to receive a loan that will cover her tuition and living expenses for her entire college education?
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Context:
Determine the average length of threads criss-crossing a hollow sphere of radius $R$. The sphere has a large, random number of holes with a constant surface density, and threads pass randomly from one hole to another. Express the average length as a multiple of $R$.
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Context:
Determine the minimum number of prime factors that a negative square-free even integer must have to ensure that the class number of $\mathbb{Q}(\sqrt{n})$ is even.
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Context:
Find the closed form of the summation \( \sum\limits _{i=0}^{\lfloor n/2 \rfloor} {n-i \choose i} \).
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Context:
A triangular wire frame with side lengths of $13, 14, 15$ is fitted over a sphere with a radius of 10. Find the distance between the plane containing the triangle and the center of the sphere.
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Context:
Find the integral of the Cantor function over the interval \((0, 2)\) with respect to the Lebesgue-Stieltjes measure \(u\).
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Context:
Let $n$ be a positive integer greater than or equal to 5. Determine the number of subgroups of order $n$ in the group $\mathbb{F}_n \times \mathbb{F}_n$. \( \mathbb{F}_n \) denotes the finite field with $n$ elements.
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Context:
Evaluate the surface integral \( \int_S \mathbf{A} \cdot \mathbf{n} \, dS \) over the entire surface of the region above the \(x\)-\(y\) plane bounded by the cone \(x^2+y^2=z^2\) and the plane \(z=4\), where \(\mathbf{A} = 4xz\mathbf{i} + xyz^2 \mathbf{j} + 3z \mathbf{k}\).
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Context:
A square with side length \(10 \text{ cm}\) is drawn on a piece of paper. How many points on the paper are exactly \(10 \text{ cm}\) away from two of the vertices of this square?
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Context:
Determine the positive integer $n$ for which any continuous map from $S^n$ to $S^1 \times S^1$ is nulhomotopic.
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Context:
Consider the symmetric group $G = S_5$. Let $L$ be the set of subgroups of $G$ that are isomorphic to the non-cyclic group of order 4. Two subgroups $H$ and $K$ in $L$ are considered equivalent if they are conjugate in $G$. Determine the number of equivalence classes in $L$.
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Context:
Determine the smallest value of \(k\) such that \(\ln(x) = O(x^k)\).
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Context:
Determine the minimum degree of the Lagrange polynomial required to approximate \( f(x) = \sin(x) \) on the interval \([0, \pi]\) such that the interpolation error \( | \sin(x) - L_n(x) | \) is less than or equal to 0.001.
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Context:
Let \(ABC\) be a triangle with \(AB = 13\), \(BC = 14\), and \(CA = 15\). Company XYZ wants to locate their base at the point \(P\) in the plane minimizing the total distance to their workers, who are located at vertices \(A\), \(B\), and \(C\). There are 1, 5, and 4 workers at \(A, B\), and \(C\), respectively. Find the minimum possible total distance Company XYZ's workers have to travel to get to \(P\).
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Context:
Find the primary decomposition of the ideal \((x, y^2)\) in the polynomial ring \(K[x, y]\), where \(K\) is a field.
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Context:
Evaluate the integral: $$\int_0^\infty\dfrac{x^2}{e^x-1}\mathrm{dx}$$
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Context:
Find the remainder when \(50!\) is divided by \(1904\), given that \(1904 = 4 \times 8 \times 59\). Use the Chinese Remainder Theorem.
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Context:
A line passing through the focus of the parabola $x = 4y$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_1, y_2)$, if $y_1 + y_2 = 6$, then the length of segment $AB$ is ______.
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Context:
Determine the number of minimal elements in the ordered set \((A\times A, \preceq)\), where \(A=\{n\in\mathbb{N} \mid n\geq 2\}\) and \((a,b)\preceq (c,d)\) if and only if \(a\mid c\) and \(b\leq d\).
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Context:
A circle with radius $R$ is tangent to a line $l$ at point $A$. $AB$ is a diameter of this circle, and $BC$ is an arbitrary chord. Let $D$ be the foot of the perpendicular dropped from $C$ to $AB$. Point $E$ lies on the extension of $CD$ beyond point $D$, such that $|ED| = |BC|$. Tangents to the circle passing through $E$ intersect the line $l$ at points $K$ and $N$. Find the length of the segment $|KN|$.
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Context:
Determine the limit of the following expression as $n$ approaches infinity: $$\lim_{n\to +\infty}\left(\frac{1}{n^3}\sum_{k=1}^n\left(3k^2-2k\right) \cos\left(\frac{k}{n}\right)\right)$$
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Context:
What is the remainder when $34!$ is divided by $71$?
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Context:
Suppose  $w,x,y,z$  satisfy \begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of  $w$  can be expressed in lowest terms as  $w_1/w_2$  for some integers  $w_1,w_2>0$ .  Find  $w_1+w_2$ .
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Context:
If the consecutive integers from $50$ to $1$ were written as $$5049484746...,$$ what would be the $67^{\text{th}}$ digit to be written?
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Context:
If $f(x)=3x+2$ for all real $x$, then the statement:
"$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$"
is true when
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Context:
What is the minimum number of times you need to roll a six-sided die so that the probability of rolling each number at least once is greater than 0.5?
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Context:
The electronic clock on the International Space Station displayed time in the format HH:MM. Due to an electromagnetic storm, the device malfunctioned, and each digit on the display either increased by 1 or decreased by 1. What was the actual time of the storm if the clock displayed 20:09 immediately after it?
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Context:
In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$? If the answer is $x^\circ$, what is the value of $x$?
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Context:
Find the number of ideals in the ring $R[X]/(X^3)$, where $R$ is a field.
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Context:
A time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle between the hour hand and minute hand on a clock is less than $\frac{360}{11}$ degrees is $\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$.
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Context:
Determine the Galois group of the polynomial $x^4 - 5x^2 + 6$ over the field of rational numbers $\mathbb{Q}$.
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Context:
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\( \mathop{\sum }\limits_{{i = 1}}^{{2012}}\left| {{a}_{i} - i}\right| \)
 then compute the sum of the prime factors of \( S \) .
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Context:
An acute-angled isosceles triangle and a trapezoid are inscribed in a circle. One base of the trapezoid is the diameter of the circle, and the lateral sides of the trapezoid are parallel to the lateral sides of the triangle. Find the ratio of the areas of the trapezoid and the triangle.
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Context:
Max likes to collect model trains.  He asks for one for every birthday of his, and asks for two each Christmas.  Max always gets the gifts he asks for, and asks for these same gifts every year for 5 years.  At the end of the 5 years, his parents give him double the number of trains he already has.  How many trains does Max have now?
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Context:
Evaluate the line integral \( \oint_C x\,dx \) where the curve \( C \) is composed of the segments \( x=0 \), \( y=0 \), and \( y=-x+1 \).
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Context:
Evaluate the improper integral $$ I=\int_{-\infty}^{\infty} \frac{x^2 e^x}{(e^x+1)^2} dx $$
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Context:
Compute the first Chern class \( c_1(L_D) \) of the line bundle \( L_D \) associated with the effective divisor \( D = [F] \) on \( \mathbb{C}\mathbb{P}^3 \), where \( F = z_0z_1^2 + z_2z_3(z_2 - z_3) \).
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Context:
Determine the modulo classes for which the product of two twin primes plus 2 is always congruent to 1.
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Context:
Find the local or global extrema of the function \( f(x_1, \ldots, x_n) = (1+x_1)\cdots(1+x_n) \) using the method of Lagrange multipliers, given the constraint \( x_1\cdots x_n = a^n \), where \( a > 0 \).
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Context:
Calculate the flux of the vector field \( \mathbb{v} = (x^2-2xz, -2xy, z^2-x) \) downwards through the paraboloid \( z = 1 - x^2 - y^2 \) for \( x \geq 0 \) and \( z \geq 0 \).
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Context:
Given that in triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, the radius of the circumcircle is $1$, and it satisfies $\frac{\tan A}{\tan B} = \frac{2c - b}{b}$, then the maximum area of $\triangle ABC$ is ______.
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Context:
One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters in the first minute and in each subsequent minute covered 0.5 meters more than in the previous minute. How many minutes will it take for the second particle to catch up with the first?
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Context:
Calculate the fundamental group \( \pi_1 \) of the space \( S^2 - \{N, S\} \), where \( N \) and \( S \) are the north and south poles of the 2-sphere \( S^2 \).
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Context:
Evaluate the contour integral \( \oint_{|z|=2} \frac{\sum_{k=0}^\infty (k^2/2^k)z^k}{z^3} \, dz \).
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Context:
How many diagonals in a regular 32-gon are not parallel to any of its sides?
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Context:
Weng earns $12 an hour for babysitting. Yesterday, she just did some babysitting and earned $10. How much time did she spend on babysitting?
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Context:
Evaluate the Legendre symbol \( \left( \frac{2017}{5003} \right) \).
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Context:
What is the value of
$$\log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27?$$
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Context:
棱长为 $\sqrt{2}$ 的正四面体 $A B C D$ 中, 已知 $\overrightarrow{A P}=\frac{1}{2} \overrightarrow{A B}, \overrightarrow{A Q}=\frac{1}{3} \overrightarrow{A C}, \overrightarrow{A R}=\frac{1}{4} \overrightarrow{A D}$, 若点 K 为 $\triangle B C D$ 的重心, 则四面体 $K P Q R$ 的体积为  ?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:
In the arithmetic sequence $\{a_n\}$, $a_1 > 0$, and $S_n$ is the sum of the first $n$ terms, and $S_9=S_{18}$, find the value of $n$ at which $S_n$ is maximized.
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Context:
Determine the number of non-constant polynomial factors with leading coefficient 1, where the other coefficients may be complex, for the polynomial \(x^{2015} + 18\).
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Context:
Suppose there exists a convex $n$-gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of $n$.
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Context:
Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt{5}$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt{5}$. The original answer for the length of $|AD|$ is in the form $k\sqrt{m}$. Please find the value of k + m.
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Context:
11. In the Cartesian coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and the circumcircle of $\triangle A O B$ intersects the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B$, find all possible values of $|P F|$.
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Context:
In triangle $ABC, AB=32, AC=35$, and $BC=x$. What is the smallest positive integer $x$ such that $1+\cos^{2}A, \cos^{2}B$, and $\cos^{2}C$ form the sides of a non-degenerate triangle?
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Context:
Determine the number of tournaments with five vertices that have at least one vertex with an in-degree of zero, up to isomorphism.
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Context:
The entries of a $7 \times 5$ table are filled with numbers such that in each $2 \times 3$ rectangle (either vertical or horizontal), the sum of the numbers is $0$. For $100$ dollars, Peter may choose any single entry and learn the number in it. What is the least amount of dollars he should spend to determine the total sum of numbers in the table for sure?
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Context:
In a right triangle, medians are drawn from point $A$ to segment $\overline{BC}$, which is the hypotenuse, and from point $B$ to segment $\overline{AC}$. The lengths of these medians are 5 and $3\sqrt{5}$ units, respectively. Calculate the length of segment $\overline{AB}$.
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Context:
Determine the values of \( \kappa \) for which the integral \( \int_{0}^{+\infty} z^{\kappa-1} \sin(z) \, dz \) converges, given that \( z > 0 \).
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Context:
Given the hyperbola equation x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), let the right focus be F, and the left and right vertices be A1 and A2, respectively. A perpendicular line to A1A2 passing through F intersects the hyperbola at points B and C. If A1B is perpendicular to A2C, calculate the slope of the asymptotes of the hyperbola. Express your answer as a slope (±value).
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Context:
What is the maximum area of the orthogonal projection of a rectangular prism with dimensions $3 \times 4 \times 12$?
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Context:
In trapezoid $ABCD$, $AD // BC$ and $\angle B=40^{\circ}$, $\angle C=50^{\circ}$. Points $E, M, F, N$ are the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $EF = a$ and $MN = b$, then express $BC$ in terms of $a$ and $b$.
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Context:
In a finite sequence of real numbers, the sum of any three consecutive terms is negative, and the sum of any four consecutive terms is positive. Find the maximum number of terms \( r \) in this sequence.
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Context:
In the acute-angled triangle $ABC$, the altitudes $AL$ and $CN$ are drawn. Find the radius of the circle passing through the points $B, L,$ and $N$, given that $AC = a$ and $\angle ABC = \alpha$.
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Context:
on a certain island , 5 % of the 10000 inhabitants are one - legged and half of the others go barefooted . what is the least number of shoes needed in the island ?
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Context:
Pipe A can fill a tank in 12 minutes and Pipe B can empty it in some minutes. If both the pipes are opened together, Pipe B should be closed after 36 minutes so that the tank is filled in 30 minutes. How many minutes does it take for Pipe B to empty the tank?
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Context:
Determine the number of subgroups of index \( p \), where \( p \) is a prime, in the additive group \( \mathbb{Z}^2 \).
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Context:
Multiples of 4 are written sequentially. Those multiples are greater than 0 and less than 100. How many numbers are there between the 5th from the left and the 8th from the right?
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Context:
Consider a  $2 \times n$  grid where each cell is either black or white, which we attempt to tile with  $2 \times 1$  black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer  $N$  where  $N$  takes the value  $n$  with probability  $\frac{1}{2^n}$ . We then take a  $2 \times N$  grid and randomly color each cell black or white independently with equal probability. Compute the probability the resulting grid has a valid tiling.
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Context:
Determine the convergence and value, if any, of the integral $$\int_{0}^{+\infty }\frac{x \log(x)}{(x^{2}+1)^{2}}dx.$$
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Context:
Let \( n \) be any positive integer. Consider a sequence containing only the letters \( \mathrm{A} \) and \( \mathrm{B} \), where at most three consecutive \( \mathrm{A} \)s and at most three consecutive \( \mathrm{B} \)s are allowed. Let \( S(n) \) represent the number of such sequences of length \( n \). Find the remainder of \( S(2015) \) when divided by 12. Express your answer as a single integer.
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Context:
Find the smallest real number \(\lambda\) such that there exists a sequence \(\{a_n\}\) with all terms greater than 1, and for any positive integer \(n\), the following inequality is satisfied:
\[
\prod_{i=1}^{n+1} a_i < a_n^\lambda.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $f(x)$ is an even function defined on $\mathbb{R}$ and satisfies $f(x+2)=f(x)$ for any $x\in[0,+\infty)$. If $f(x)=|x^{2}-x-1|$ when $x\in[0,2)$, find the number of zeros of the function $y=f(x)-1$ on $[-2,4]$.
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Context:
Let $T$ be the smallest positive real number such that the tangent to the helix \( \cos t \mathbf{i} + \sin t \mathbf{j} + \frac{t}{\sqrt{2}} \mathbf{k} \) at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Find the line integral of the vector field \( \mathbf{F} = x \mathbf{j} - y \mathbf{i} \) along the section of the helix from \( t = 0 \) to \( t = T \).
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Context:
In triangle $\triangle ABC$ with $|BC| > |BA|$, $D$ is a point inside the triangle such that $\angle ABD = \angle DBC$, $\angle BDC = 150^\circ$, and $\angle DAC = 60^\circ$. If the measure of $\angle BAD$ is $x^\circ$, what is the value of $x$?
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Context:
[ Properties and characteristics of a parallelogram ] [ Isosceles, inscribed, and circumscribed trapezoids ]

A circle passing through the vertices $A, B$, and $C$ of parallelogram $A B C D$ intersects the lines $A D$ and $C D$ at points $M$ and $N$ respectively. Point $M$ is at distances 4, 3, and 2 from vertices $B, C$, and $D$ respectively. Find $M N$.
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Context:
Find the smallest positive value of \( a \) such that \( \left(1+\frac{1}{x}\right)^{x+a} > e \) for all \( x \geq 1 \).
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Context:
Determine the smallest dimension a non-commutative C*-algebra can have.
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Context:
An ant starts at the point $(0,0)$ in the coordinate plane. It can make moves from lattice point $(x_1,y_1)$ to lattice point $(x_2,y_2)$ whenever $x_2\geq x_1$, $y_2\geq y_1$, and $(x_1,y_1)\neq(x_2,y_2)$. For all nonnegative integers $m,n$, define $a_{m,n}$ to be the number of possible sequences of moves from $(0,0)$ to $(m,n)$ (e.g. $a_{0,0}=1$ and $a_{1,1}=3$). Compute
\[\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \frac{a_{m,n}}{10^{m+n}}.\]
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Context:
If  $x_1=5, x_2=401$ , and
\[
    x_n=x_{n-2}-\frac 1{x_{n-1}}
\]
for every  $3\leq n \leq m$ , what is the largest value of  $m$ ?
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Context:
Fresh grapes contain a certain percentage of water by weight, while dried grapes contain 20% water by weight. The weight of dry grapes available from 40 kg of fresh grapes is 5 kg. What is the percentage of water in fresh grapes by weight?
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Context:
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $\triangle XYZ$.
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Context:
Given that \(A^2=\begin{bmatrix} -2 & 2 & -4 \\ 2 & 1 & -2 \\ 4 & -6 & 6 \end{bmatrix}\) and the trace of matrix \(A\) is \(-1\), find the determinant of matrix \(A\).
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Context:
Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2015 \cdot 2016$.
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Context:
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?
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Context:
Evaluate \( \lim_{x \to \infty} \frac{\gamma(x+1, x)}{\Gamma(x+1)} \), where \( \gamma(x+1, x) \) is the lower incomplete gamma function and \( \Gamma(x+1) \) is the complete gamma function.
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Context:
In the 2016 art exam of a certain high school, there were 6 contestants, including 3 females and 3 males. Now, these six contestants are to perform their talents in sequence. If any two of the three males cannot perform consecutively, and the female contestant A cannot be the first to perform, then calculate the number of possible sequences for the contestants to perform.
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Context:
Let's call a number "remarkable" if it has exactly 4 different natural divisors, among which there are two such that neither is a multiple of the other. How many "remarkable" two-digit numbers exist?
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Context:
Three cones with vertex \(A\) touch each other externally, with the first two having an apex angle of \(\frac{\pi}{6}\). Each of the cones touches internally a fourth cone with a vertex at point \(A\) and an apex angle of \(\frac{\pi}{3}\). Determine the apex angle of the third cone.
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Context:
Find the large number from the problem below: The difference between two numbers is 20775. On dividing the larger number by the smaller, we get 23 as quotient and 143 as remainder.
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Context:
Determine the largest possible value of $k \in \mathbb{N}$ such that there exists an integer $n$ for which the digit sum of $k$ consecutive perfect squares, starting from $(n+1)^2$ to $(n+k)^2$, is a perfect square for each square in the sequence. Note that $k \geq 7$, and an example of $k=8$ has been found: $S(46045846^2) = 8^2, S(46045847^2) = 7^2, \ldots, S(46045853^2) = 8^2$. What is this largest $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose we flip 100 fair coins. If there are more heads than tails, what is the expected number of heads? Use the central limit theorem and the empirical rule to approximate this value.
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Context:
James goes to a pet store to adopt a puppy and a kitten. The adoption fee for the puppy is $200 and the adoption fee for the kitten is $150. The store offers a 10% discount on the total adoption fees for adopting multiple pets. Additionally, his friend agrees to cover 25% of the puppy's adoption fee and another friend contributes 15% towards the kitten's adoption fee. There is also a 7% sales tax applied to the total adoption fees after discounts. James is required to purchase pet supplies worth $95 alongside the adoptions. How much does James have to pay in total, including the sales tax and pet supplies?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A light ray is emitted from point $A(-3,3)$, reflected off a point $P$ on the $x$-axis, and then intersects with the circle $(x-1)^{2}+(y-1)^{2}=2$ at point $B$. Determine the minimum value of $|AP|+|PB|$.
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Context:
Given point $M(x_{0},2-x_{0})$, suppose there exists a point $N$ on circle $O$ : $x^{2}+y^{2}=1$ such that $\angle OMN=30^{\circ}$, then the range of real number $x_{0}$ is _____.
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Context:
The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square.
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Context:
Petya and Vasya came up with ten fifth-degree polynomials. Then, Vasya sequentially called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice, writing the resulting values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in this order). What is the maximum number of numbers Vasya could call out?
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Context:
If x = - 6 and y = - 3, the value of a certain number ( x - y ) ^ 2 - xy is 18. What is the value of that certain number?
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Context:
When \(0 < x < 1\), given \( f(x) = \frac{x}{\log x} \), what is the correct relationship between \( f(x) \), \( f(x^2) \), and \( f^2(x) \)?
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Context:
已知不全相等的三个实数 $a, b, c$, 满足 $a=a b+c, b=b c+a, c=c a+b$ ，则 $a+b+$ $c=$ $\qquad$.
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Context:
Consider a sequence of random variables $(X_n)$, where each $X_n$ has the distribution: $P(X_n = 1 - \frac{1}{n}) = P(X_n = 1 + \frac{1}{n}) = \frac{1}{2}$ for $n \in \mathbb{N}^*$. Determine the limiting distribution of this sequence as $n$ approaches infinity.
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Context:
Evaluate the limit: $$\lim_{n\to \infty} \sum_{i=1} ^ {n} \left(1- \frac{2i}{n} \right) ^ {\frac{n}{i}}.$$
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Context:
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = 2g(x) + g(g(y) - g(-x)) + y \] for all real numbers \( x \) and \( y \).

Let \( n \) be the number of possible values of \( g(2) \), and let \( s \) be the sum of all possible values of \( g(2) \). Find \( n \times s \).
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Context:
In a math competition, the possible scores for each problem for each participating team are 0 points, 3 points, or 5 points. By the end of the competition, the sum of the total scores of three teams is 32 points. If the total score of any single team can reach 32 points, how many different possible combinations are there for the total scores of these three teams?
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Context:
Find the maximum value of \( \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x} \) subject to the constraints \( \boldsymbol{q}^T \boldsymbol{x} = 1 \) and \( x_i \geq 0 \) for all \( i \), where \( \boldsymbol{x} \in \mathbb{R}^n \), \( \boldsymbol{A} \) is a positive definite matrix, and \( \boldsymbol{q} \) is a given vector. For example, if \( \boldsymbol{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \) and \( \boldsymbol{q} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \), determine the solution that maximizes the objective function.
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Context:
John and Ingrid have different income sources with varying tax rates. John makes $57,000 from his job, where he pays a tax rate of 30%, and he also earns $11,000 from rental income, which is taxed at a rate of 25%. On the other hand, Ingrid makes $72,000 from her employment, which attracts a 40% tax, and she earns $4,500 from her investments, which are subject to a 15% tax rate. Calculate their combined weighted average tax rate.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz \), where \( C \) consists of the contour \(|z|=2\) oriented counterclockwise and \(|z|=1\) oriented clockwise.
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Context:
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Given that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$, if \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, find $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the space of planes in \(\Bbb R^3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two random variables $X$ and $Y$, where $X$ follows a uniform distribution $U(1, 2)$, and the conditional distribution of $Y$ given $X$ is an exponential distribution $\operatorname{Exp}(X)$, find the joint distribution of $X$ and $Y$ conditioned on the event $X \leq 1.5$. Provide your answer in terms of the joint probability density function.
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Context:
Find a function $h$ such that $h$ is orthogonal to both $x$ and $x^2$ with respect to the inner product $\langle f,g \rangle = \int_{-1}^{1} g(x)f(x) \, dx$ defined on $\mathcal{C}[-1,1]$. Normalizing $h$ is not required.
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Context:
In the isosceles triangle \(ABC\), points \(K\) and \(L\) are marked on the equal sides \(AB\) and \(BC\) respectively so that \(AK = KL = LB\) and \(KB = AC\). Find the size of angle \(ABC\). Express your answer in degrees as a whole number.
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Context:
Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$24! = 620,448,401,733,239,439,360,000$ ends in four zeros, and $25!=15,511,210,043,330,985,984,000,000$ ends in six zeros. Thus, there is no integer $n$ such that  $n!$ ends in exactly five zeros. Let $S$ be the set of all $k$ such that for no integer n does $n!$ end in exactly $k$ zeros. If the numbers in $S$ are listed in increasing order, 5 will be the first number. Find the 100th number in that list.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle \(ABC\), determine the values of \(\alpha\) such that point \(A\) is equidistant from the circumcenter \(O\) and the orthocenter \(H\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the monic real polynomial $f(x)$ such that $f(1/x) = f''(x) = c$ as $x$ tends to infinity, and $f(x) \ge f(1)$ for all real $x$. Determine $f(x)$.
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Context:
Among six different quadratic trinomials, differing by permutation of coefficients, what is the maximum number that can have two distinct roots?
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Context:
Compute the unique positive integer \( n \) such that
\[
3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A robotic vacuum cleaner "Orthogonal" decided to clean the coordinate plane. It starts at point $O$, located at the origin, and travels a distance $X_{1}$ along a straight line. At the point where it stops, the vacuum turns and describes a circle with its center at the origin. After completing the circle, it continues straight without changing direction for a distance $X_{2}$. Then it turns to face the origin again to describe another circle centered at the origin, and after completing this circle, it travels another straight path for a distance of $X_{3}$. This pattern continues. It is known that the distances $X_{1}, X_{2}, X_{3}, \ldots$ follow a certain distribution with an expected value $a$ and variance $d$. Find the expected value of the area of the circle bounded by the $n$-th circle described by the vacuum cleaner.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice picks a number from the set \( \{1, 2, \ldots, 1000000\} \). Bob can ask yes/no questions to determine Alice's number, but Alice is allowed to lie once. What is the minimum number of questions Bob must ask to guarantee he correctly identifies the number?
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Context:
Determine the number of idempotent matrices in the matrix space $M_2(\mathbb{F}_p)$. An idempotent matrix $A$ satisfies $A^2 = A$. Calculate the total number of such matrices in this space.
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Context:
Evaluate the lambda calculus expression:
\[(\lambda x.\lambda y.x + ( \lambda x.x+1) (x+y)) ( \lambda z.z-4\ 5)\ 10\]
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n} \int_{0}^{n} \frac{x \ln(1 + \frac{x}{n})}{1 + x} \, dx.$$
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Context:
Find the coefficient of \(x^{24}\) in the expansion of \((1 + x + x^2 + x^3 + x^4 + x^5)^8\).
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Context:
A sequence $(a_n)$ is defined by means of the recursion\[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\]Find an explicit formula for $a_n.$
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Context:
What digit must be placed in place of the "$?" in the number $888\ldots 88?999\ldots 99$ (where the $8$ and $9$ are each written $50$ times) so that the resulting number is divisible by $7$?
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Context:
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches.
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Context:
Given that the line $l_{1}$: $3x-(k+2)y+3=0$ is parallel to the line $l_{2}$: $kx+(2k-3)y+1=0$, find $k=$____.
Please reason step by step, and put your final answer within \boxed{}.