Context:
Determine the number of monic primitive quadratic polynomials in \( \mathbb{Z}_7[x] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $M$  be a finite subset of the plane such that for any two different points  $A,B\in M$  there is a point  $C\in M$  such that  $ABC$  is equilateral. What is the maximal number of points in  $M?$
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Context:
Carol spends eight hours filling up her pool. During the first hour, the pool fills at a rate of 8 gallons of water per hour. For the next two hours, the pool fills at 10 gallons of water per hour. From the fourth hour to the fifth hour, the pool fills at a rate of 14 gallons of water per hour. During the sixth hour, the pool starts filling at a rate of 12 gallons of water per hour. However, during the seventh hour, the pool develops a leak and loses 8 gallons of water. In the eighth hour, the leak slows down and the pool loses another 5 gallons of water while continuing to fill at 12 gallons of water per hour. At the end of the eight hours, how many gallons of water are still left in the pool?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the conditional expectation \( E[N(6) \mid N(4) = 4, N(10) = 12] \) for a Poisson process \( \{N(t), t \ge 0\} \) with rate \( \lambda = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest prime number \( p \) such that 7 is the least quadratic nonresidue modulo \( p \).
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Context:
Find the joint density $g(u, v)$ for the transformed random variables $U = \exp(X) + 2\exp(Y)$ and $V = 2\exp(X^2) + \exp(Y^2)$, where $X$ and $Y$ are independent Exponential random variables with mean 1. Evaluate $g(3e, 3e)$.
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Context:
Find the smallest \( \lambda  \in  \mathbb{R} \) such that for all \( n \in  {\mathbb{N}}_{ + } \) ,there exists \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) satisfying \( n = {x}_{1}{x}_{2}\ldots {x}_{2023} \) ,where \( {x}_{i} \) is
 either a prime or a positive integer not exceeding \( {n}^{\lambda } \) for all \( i \in  \{ 1,2,\ldots ,{2023}\} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hexagon is inscribed in a circle of radius \( r \). Four of the edges of the hexagon are 10 feet long, and two of the edges are 20 feet long, but the exact arrangement of the edges is unknown. Using either Ptolemy's Theorem or the Law of Cosines, determine the value of \( r \) to three decimal places.
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Context:
On a standard die, one of the dots is removed at random, with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Express your answer as a fraction in simplest form, \frac{k}{m}, and provide the value of k + m.
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Context:
In Philosopher's Chess, the chessboard is an infinite grid of identical regular hexagons. A piece called the Donkey moves as follows: starting from one hexagon, it moves $m$ cells in one of the 6 directions, then turns 60 degrees clockwise and moves $n$ cells in this new direction to reach its final cell. Given $m$ and $n$ are nonnegative integers, determine the maximum number of cells on the chessboard such that one cannot reach any other cell from them using a finite number of Donkey movements.
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Context:
How many distinct roots does the equation x^{5}+3 x^{4}-4044118 x^{3}-12132362 x^{2}-12132363 x-2011^{2}=0 have, given that the roots of the equation are all integers?
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Context:
Evaluate the integral of \( f(z) = \frac{z^2}{z^3 - z - 1} \) over a circle of radius \( R \) centered at the origin, where \( R \) is greater than the maximum modulus of the roots of \( z^3 - z - 1 \).
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Context:
A bag of potatoes weighs some amount divided by half of its weight. The bag of potatoes weighs 12 lbs. What is the original weight of the bag of potatoes before dividing by half of its weight?
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Context:
In an $n \times n$ idempotent matrix, what is the minimum number of entries that need to be known to determine the rest of the matrix?
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Context:
Find the norm of the operator $T: l^q \rightarrow l^q$ defined by $T((b_{n})) = b_{n+1} - 2b_{n} + b_{n-1}$, where $l^q$ is the space of sequences with the $q$-norm.
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Context:
Calculate the expected value of the integral of a pure birth process from time 0 to the first time it reaches state 4, given that the process starts at state 1 and has birth rates \( \lambda_k = k \) for \( k=1,2, \dots \).
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Context:
The operation $\oslash$, called \u0022reciprocal sum, is useful in many areas of physics. If we say that $x=a\oslash b$, this means that $x$ is the solution to
$$\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$$
Compute$4\oslash2\oslash4\oslash3\oslash4\oslash4\oslash2\oslash3\oslash2\oslash4\oslash4\oslash3$.
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Context:
Determine the maximum number of solutions that the equation \(\max \{a_{1} x+b_{1}, \ldots, a_{10} x+b_{10}\} = 0\) can have, given that \(a_{1}, \ldots, a_{10}, b_{1}, \ldots, b_{10}\) are real numbers and all \(a_{i} \neq 0\).
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Context:
Find the residue field for the local ring $\mathbb{Z}_{\langle 5\rangle}$ with maximal ideal $5\mathbb{Z}_{\langle 5\rangle}$.
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Context:
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
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Context:
Given the expression $\frac{\cos \alpha \cos \frac{\beta}{2}}{\cos \left(a-\frac{\beta}{2}\right)} + \frac{\cos \beta \cos \frac{\alpha}{2}}{\cos \left(\beta-\frac{\alpha}{2}\right)} - 1$, find the value of $\cos \alpha + \cos \beta$.
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Context:
Borya and Vasha play a game with a positive integer \( n \geq 2 \). Borya selects \( 2n \) non-negative numbers \( x_1, x_2, \ldots, x_{2n} \) such that their sum is 1. Vasha arranges these numbers in a circle and calculates the product of each pair of adjacent numbers, writing the largest product on the blackboard. What is the number written on the blackboard, assuming both players use optimal strategies?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设数列$\left\{ {{a}_{n}} \right\}$的前$\textit{n}$项之积为${{An}}$，且${{A}_{n}}{{An+1}}+n{{A}_{n}}=\left( n+1 \right){{An+1}}$，${{a}_{1}}=\frac{1}{2}$，${{bn}}={{a}_{n}}+\frac{1}{{{an}}}$，数列$\left\{ {{b}_{n}} \right\}$的前$\textit{n}$项和为${{Sn}}$，则${{S}_{n}}+\frac{1}{{{An}}}=$______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{y \to 0} \frac{3y}{5-\left(\displaystyle\sum_{n=0}^{\infty}\frac{(2y)^n+4y^n}{n!}\right)}$$ without using L'Hospital's rule or Taylor series.
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Context:
Determine the approximate value of $T(n)$ for large values of $n$ given the recursive equation $T_n = \sqrt{n}T_{\sqrt{n}} + 1$, excluding $n=1$. Provide your answer in terms of asymptotic notation.
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Context:
Let $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer function.
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Context:
Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{1-\left(\cos x\right)^{\ln\left(x+1\right)}}{x^{4}} \]
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Context:
A tailor trims 4 feet from opposite edges of a square piece of cloth, and 3 feet from the other two edges. If 120 square feet of cloth remain, what was the length of a side of the original piece of cloth?
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Context:
Given the function f(x) = e^x cos x and x in (0, π), find the interval where f(x) is increasing. Express your answer as an interval in standard notation.
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Context:
A quadrilateral connecting the midpoints of the sides of trapezoid $\mathrm{ABCD}$ is a rhombus. Find its area if the height of the trapezoid $\mathrm{BH}=5 \mathrm{c}$, the smaller base $\mathrm{BC}=6 \mathrm{~cm}$, and the angle $\mathrm{ABC}$ is $120^{\circ}$.
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Context:
The polynomial \( f(x) = x^{2007} + 17 x^{2006} + 1 \) has distinct zeroes \( r_1, \ldots, r_{2007} \). A polynomial \( P \) of degree 2007 has the property that \( P\left( r_j + \frac{1}{r_j} \right) = 0 \) for \( j = 1, \ldots, 2007 \). Determine the value of \( P(1) / P(-1) \).
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Context:
47. In how many ways can $n$ traffic lights be lit, of which $k$ can be in one of three states, and the remaining $n-k$ in one of two?
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Context:
Three players are playing table tennis. The player who loses a game gives up their spot to the player who did not participate in that game. In the end, it turns out that the first player played 10 games, and the second player played 21 games. How many games did the third player play?
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Context:
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division?
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Context:
Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$.
Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and
$A+B=\{0,1,2,\dots, 2019,2020\}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{\sin \left\{ \frac{2}{n} \right\}}{\left[ 2n \tan \frac{1}{n} \right] \left( \tan \frac{1}{n} \right)} + \frac{1}{n^2 + \cos n} \right)^{n^2}, \] where \([.]\) denotes the greatest integer function and \(\{.\}\) denotes the fractional part function.
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Context:
Define a *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that the answer is greater than $1000$.
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Context:
Find the number of 1's in the binary representation of the 1717th term in the sequence of positive integers whose binary representation is a palindrome.
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Context:
In a regular octagon, diagonals connect the second and third neighboring vertices to form two convex octagons. Determine the ratio of the areas of these two octagons.
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Context:
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is  $I$ . Now, she pulls her arms in and her moment of inertia is now  $\frac{7}{10}I$ . What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)?
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Context:
Find the exhaustive range of $k$ such that the function $f(x)=\frac{x-1}{k-x^2}$ never takes values within the interval $[-1, -\frac{1}{3}]$.
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Context:
King Middle School has1200students. Each student takes5classes a day. Each teacher teaches4classes. Each class has3(
 students and 1 teacher. How many teachers are there at King Middle School?
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Context:
Evaluate the infinite series: \[ S = \sum_{n=1}^{\infty} \left( \frac{2}{n} - \frac{4}{2n+1} \right) \]
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Context:
In the positive geometric sequence {a_n}, if a_1 + a_2 + a_3 = 18 and 1/a_1 + 1/a_2 + 1/a_3 = 2, find the value of a_2. Express your answer as a single numerical value.
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Context:
Evaluate the set \( \bigcup_{s \in \mathbb{R}} \bigcap_{t \geq s} A_t \), where \( A_t = (\cos t - 2, \cos t + 2) \) for \( t \in \mathbb{R} \).
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Context:
Consider a vector $\mathbf{X} \in \mathbb{R}^n$ where each component is subGaussian with parameter $s$. Determine if the vector $\mathbf{X}$ is subGaussian, and if so, find its subGaussian parameter.
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Context:
A list of five positive integers has all of the following properties:

$\bullet$ The only integer in the list that occurs more than once is $7,$ and it occurs three times.
$\bullet$ Its median is $10,$
$\bullet$ Its average (mean) is $11.$

What is the largest possible integer that could appear in the list?
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Context:
Determine the number of ways to rearrange the letters in the word INDIVISIBILITY such that no two 'I's are adjacent.
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Context:
Find the minimal extension field of \(\mathbb{F}_2\) that contains an element of order 21.
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Context:
Given that \( \lim_{x\rightarrow \infty} f(x)=0 \), where \( f'(x)=\dfrac{-f(x)}{\sqrt{f^2(x)+g^2(x)}} \) and \( g'(x)=1-\dfrac{g(x)}{\sqrt{f^2(x)+g^2(x)}} \) for all \( x \in \mathbb{R} \), with initial conditions \( g(0)=0 \) and \( f(0)=10 \), find the value of \( \lim_{x\rightarrow \infty} g(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( C \) be the smooth projective curve in \( \Bbb{P}^2 \) defined by the equation \( y^2z = x^3 - xz^2 \). Consider the rational map \( \phi: \Bbb{P}^2 \dashrightarrow \Bbb{P}^1 \) given by \( (x:y:z) \mapsto (x:y) \), which is undefined at the point \( P = (0:0:1) \). The restriction \( \psi = \phi|_C \) extends to a morphism \( \psi: C \to \Bbb{P}^1 \). Determine the value of \( \psi(P) \).
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Context:
Let \( g: \mathbb{R}^m \rightarrow \mathbb{R} \) be a differentiable function. Define \( v: \mathbb{R}^n \rightarrow \mathbb{R} \) by \( v(x) = g(s) \), where \( s = \sqrt{\sum_{i=1}^m x_i^2} \), for all \( x \in \mathbb{R}^n \). Calculate \( \|\nabla v\|^2 \) in terms of \( g \) and \( s \) only.
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \int_{1}^{\infty} n e^{-x^{\delta}}(1 - e^{-x^{\delta}})^n \, dx \] where \(0< \delta <1\).
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Context:
Assume that $z$ and $w$ are differentiable functions of $x$ and $y$, satisfying the equations $xw^3+yz^2+z^3=-1$ and $zw^3-xz^3+y^2w=1.$ Find the value of $\frac{\partial z}{\partial x}$ at the point $(x,y,z,w)=(1,-1,-1,1)$. \( \boxed{?} \)
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Context:
In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\overline{YE}$ and $\overline{XD}$, respectively. Find $PQ$.
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} n \int_1^e x^a (\log (x))^n \,dx. \]
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Context:
A chord of a circle is at a distance $h$ from the center. A square is inscribed in each of the segments formed by the chord such that two adjacent vertices of the square lie on the arc and the other two vertices lie on the chord. Find the difference in the side lengths of the squares.
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Context:
Equilateral triangle $ABC$ has a side length of $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and the minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the "lune" that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p} - \frac{q\pi}{r}$, where $p, q$, and $r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
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Context:
A group of 100 readers meet monthly to discuss their book findings in pairs. Each pair must have read at least one book that the other hasn't. What is the minimum number of distinct books needed to ensure any two random members can start a discussion?
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Context:
In the game of coins, there are two bowls. The first player places 100 coins in the first bowl. The second player must decide how many coins to place in the second bowl. Each player, starting with the first, can make one of the following moves: take any number of coins from the first bowl, take any number of coins from the second bowl, or take the same number of coins from both bowls. The winner is the player who leaves no coins in either bowl after their move. How many coins should the second player place in the second bowl to guarantee a win with optimal play?
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Context:
Find the limit without using L'Hopital's rule or Taylor series: $\lim_{x \to 0} \left ( \frac{x}{e^x-1} -\frac{1}{\sin x}\right )$
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Context:
A student's final score on a 150-point test is directly proportional to the time spent studying multiplied by a difficulty factor for the test. The student scored 90 points on a test with a difficulty factor of 1.5 after studying for 2 hours. What score would the student receive on a second test of the same format if they studied for 5 hours and the test has a difficulty factor of 2?
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Context:
Find the asymptotic upper and lower bounds for the recurrence relation \( T(n) = T(\sqrt{n}) + n \). Use any appropriate method, noting that the master method does not apply.
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Context:
Let  $ABCD$  be a quadrilateral with  $AD = 20$  and  $BC = 13$ .  The area of  $\triangle ABC$  is  $338$  and the area of  $\triangle DBC$  is  $212$ .  Compute the smallest possible perimeter of  $ABCD$ .
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Context:
Let $f$ be a Lebesgue integrable function on $[0,1]$ with $\lim_{x\rightarrow 1}f(x) = c$, where $c$ is a constant. Determine the value of $\int_{0}^{1} nx^n f(x) \,dx$.
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Context:
There are 350 items that are members of set U. Of these items, 49 are members of set B, 59 are not members of either set A or set B, and 23 are members of both sets A and B. Additionally, 36 items are members of set C, 14 items are members of set A and set C, and 19 items are members of set B and set C. Moreover, 8 items are members of all three sets A, B, and C. How many of the members of set U are members of set A?
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Context:
If the negation of the proposition "There exists a real number $x_0$, such that $x + ax_0 + 1 < 0$" is true, then the range of the real number $a$ is ______.
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Context:
令
$$
S=\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{m^{2} n}{2^{m}\left(n 2^{m}+m 2^{n}\right)}
$$
则 $[100 S]=$ $\qquad$
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Context:
Evaluate the integral: \[ \int_{0}^b x\frac{2b(b^2-x^2)}{(b^2+x^2)^2} \, dx \]
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Context:
Find the remainder when the polynomial
\[p(x) = 1 + x^2 + x^4 + x^6 + \dots + x^{22}\]is divided by the polynomial $q(x) = 1 + x + x^2 + x^3 + \dots + x^{11}.$
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Context:
Determine all values of $b$ for which the infinite series $$\sum_{k=1}^\infty \frac{k!}{(b+1)(b+2)\cdots(b+k)}$$ converges.
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Context:
Determine the number of geodesics between two distinct points on a unit cylinder, given that the points do not lie on the same circular arc.
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Context:
Given that \( P \) is a moving point on the parabola \( y^2 = 2x \), and points \( B \) and \( C \) are on the \( y \)-axis, the circle \((x-1)^2 + y^2 = 1\) is the incircle of \( \triangle PBC \). Then, the minimum value of \( S_{\triangle PBC} \) is ______.
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Context:
Find the smallest positive integer consisting only of the digit '1' repeated multiple times (e.g., 111, 1111) that is divisible by 2019.
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Context:
Compute the integral \( \int_{F(S^1)} \omega \) given that the pullback of the 1-form \( \omega = dt_1 + dt_2 + dt_3 \) on \( S^1 \times S^1 \times S^1 \) by the function \( F: S^1 \rightarrow S^1 \times S^1 \times S^1 \) defined by \( F(p) = (p,p,p) \) is \( F^*\omega = 3\,dt \).
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Context:
Evaluate the contour integral \( \int_C \frac{1}{z} \, dz \) for any contour \( C \) in the right half-plane from \( z = -3i \) to \( z = 3i \).
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Context:
Find the area enclosed by the equation: \[ \left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3 \] where \(\lfloor\cdot\rfloor\) represents the greatest-integer function.
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Context:
A regular triangle is constructed on the diameter of a semicircle such that its sides are equal to the diameter. What is the ratio of the areas of the parts of the triangle lying outside and inside the semicircle?
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Context:
Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$.
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Context:
Let \(ABC\) be an equilateral triangle. \(D\) and \(E\) are midpoints of \([AB]\) and \([AC]\). The ray \([DE\) cuts the circumcircle of \(\triangle ABC\) at \(F\). The original answer is in the form \(\frac{\sqrt{k} - m}{n}\). Please find the value of \(k + m + n\).
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Context:
Consider the function \(f:\mathbb{R}^2\to\{1,2,...,2012\}\) that satisfies the following rule: If \(a<b<c\), then \(f(a,c)=f(b,c)=f(a,b)\) and \(f(b,a)=f(c,a)=f(c,b)\). Let \(x_1,x_2,...,x_{2010}\) be a sequence of real numbers, which are all different. Find the number of possible ordered \(2010^2\)-tuples
\[\Bigl(f(x_1,x_1),f(x_1,x_2),\ldots,f(x_1,x_{2010}),f(x_2,x_1),f(x_2,x_2),\ldots,f(x_2,x_{2010}),\ldots,f(x_{2010},x_{2010})\Bigr)\]
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Context:
In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds.
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Context:
Find the minimum period of the periodic function $g(x)$ given that for all $x$, \( g(x+2023) = \frac{1-g(x)}{1+g(x)} \).
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Context:
Consider the circle defined by the equation \((x - r - 1)^2 + y^2 = r^2\). Let \((x_0, y_0)\) be a point on this circle such that \(4x_0 - y_0^2 \leq 0\). Find the minimum value of \(|r|\).
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Context:
Evaluate the contour integral \( \int_{|z|=1}\frac{\log z}{z}dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), \( O \) is the circumcenter of \( \triangle ABC \), and \( D \) is the midpoint of \( AC \). It is given that \( \overrightarrow{OB} \cdot \overrightarrow{OD} = \overrightarrow{OA} \cdot \overrightarrow{OC} \). Determine the minimum value of \( \frac{\sin B}{\tan A} + \frac{\sin B}{\tan C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral of the function \( f(x,y) = (x+y)^9(x-y)^9 \) over the region defined by \( 0 \leq x \leq \frac{1}{2} \) and \( x \leq y \leq 1-x \).
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Context:
Let \( n \) be a positive integer. Consider the points \((n, 0)\), \((n-1, n)\), \((n+1, n)\), and \((0, n)\). Connect the point \((n, 0)\) to \((n-1, n)\) and \((n+1, n)\), and connect the point \((0, n)\) to \((n-1, n)\) and \((n+1, n)\). What is the area of the quadrilateral enclosed by these four line segments?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For large $n$, approximate the probability that the expression \( Y = \frac{V_1 + V_2 + \cdots + V_n} {\sqrt{V_1^2 + V_2^2 + \cdots + V_n^2}} \) is greater than 0.5, where \( V_i \) are independent and identically distributed random variables with a uniform distribution over \([-2, 2]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The positive integer divisors of 252, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 14?
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Context:
Calculate the variance of the function \( f(X) = 100 \times \exp(x) + 100 \times \exp(2x) \) using the formula for variance:
\[
\operatorname{Var}(f(X))=[f'(EX)]^2\operatorname{Var}(X)+\frac{[f''(EX)]^2}{4}\operatorname{Var}^2(X)+\tilde{T}_3
\]
where the expected value of \( x \) is \( 0.05 \) and the standard deviation is \( 0.1 \). Assume \( x \) follows a normal distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The curve defined by the differential equation \( \frac{dy}{dx}=\frac{y^3}{e^x +y^2} \) passes through the point \((0,2)\). The line \(x=\ln 5\) intersects the curve at points where \(y=a\) and \(y=b\). Calculate the value of \(\frac{4(a^2+b^2)}{53}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
若复数 z 满足 \left|\df{z^2+1}{z+i}\right|+\left|\df{z^2+4i-3}{z-i+2}\right|=4，则 |z-1| 的最小值为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Evaluate the integral \( \int_{0}^{1} (1-x^{\frac{1}{a}})^{-\frac{1}{2}} \, dx \) as \( a \to 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the expression \(2(|AB - I_5| + |BA - I_4|) + 1\), where \(A\) is a \(5 \times 4\) matrix and \(B\) is a \(4 \times 5\) matrix. Here, \(I_n\) denotes the \(n \times n\) identity matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine how many zeros of the function \( f(z) = z^5 + 3z^4 + 6 \) lie within the circle \( |z| = 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the arithmetic sequence $\{a\_n\}$, $a_{66} < 0$, $a_{67} > 0$, and $a_{67} > |a_{66}|$. $S_{n}$ denotes the sum of the first $n$ terms of the sequence. Find the smallest value of $n$ such that $S_{n} > 0$.
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Context:
Determine the number of integer solutions for the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 41\), where \(0 \leq x_i \leq 20\) for all \(i\), and \(x_i\) is even when \(i\) is even, while \(x_i\) is odd when \(i\) is odd. Use a generating function to find the coefficient of \(x^{41}\) in the expanded form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 1}\frac{\Gamma(1+x)\Gamma(2+x)\Gamma(3+x) - 12}{x-1}$$ without using L'Hôpital's rule.
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Context:
The convex quadrilateral  $ABCD$  has area  $1$ , and  $AB$  is produced to  $E$ ,  $BC$  to  $F$ ,  $CD$  to  $G$  and  $DA$  to  $H$ , such that  $AB=BE$ ,  $BC=CF$ ,  $CD=DG$  and  $DA=AH$ . Find the area of the quadrilateral  $EFGH$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the constant \( \lambda > 0 \) such that \( 2\lfloor \lambda n \rfloor = 1 - n + \lfloor \lambda \lfloor \lambda n \rfloor \rfloor \) holds for all positive integers \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the squared integral of the function \( f(t) = (W_2 - W_1)1_{[2,3)}(t) + (W_3 - W_1)1_{[3,5)}(t) \) with respect to Brownian motion \((W_t)_{t\ge0}\), where \(1_{[a,b)}(t)\) is the indicator function for the interval \([a, b)\). Use the properties of Brownian motion to find the most efficient method.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many rearrangements of the letters of "HMMTHMMT" do not contain the substring "HMMT"? For instance, one such arrangement is $HMMHMTMT$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. Calculate $\frac{1}{1 \times 3 \times 5}+\frac{1}{3 \times 5 \times 7}+\frac{1}{5 \times 7 \times 9}+\cdots+\frac{1}{2001 \times 2003 \times 2005}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid in the first octant bounded by the surfaces \(z = x + y\), \((x^2 + y^2)^2 = 2xy\), and \(z = 0\) using polar coordinates.
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Context:
Masha wrote the numbers $4, 5, 6, \ldots, 16$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value that the sum of the remaining numbers on the board can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least possible value of \( \sum_{n=1}^{696} |nx - 1| \) for \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ columns and $4$ rows such that the $16$ squares of the intersections are all of the same color. Find the remainder when $n$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(a \in \mathbb{R}\) for which the function \(f_a(x,y) = (x^2 + y^2)^a\) is Lebesgue-integrable over the set \(\mathbb{R}^2 \setminus E\), where \(E = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le 1, y \le \sqrt{|x|} \}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi} e^{-\sin\theta} \cos(\theta-\cos\theta) \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chairs does Person B originally have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer  $n$ , let  $f(n)$  be the sum of the positive integers that divide at least one of the nonzero base  $10$  digits of  $n$ . For example,  $f(96)=1+2+3+6+9=21$ . Find the largest positive integer  $n$  such that for all positive integers  $k$ , there is some positive integer  $a$  such that  $f^k(a)=n$ , where  $f^k(a)$  denotes  $f$  applied  $k$  times to  $a$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coordinates of all points P from which two tangent lines to the ellipse \( \frac{x^2}{17} + \frac{y^2}{8} = 1 \) are perpendicular to each other.
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Context:
In the 3rd grade, the boys wear blue swim caps, and the girls wear red swim caps. The male sports commissioner says, "I see 1 more blue swim cap than 4 times the number of red swim caps." The female sports commissioner says, "I see 24 more blue swim caps than red swim caps." Based on the sports commissioners' statements, calculate the total number of students in the 3rd grade.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2 \pi i} \int_{|z|=2} \frac{4z^{50}}{15z^{51}-3z^{30}+z^2-9} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Pentagon ABCDE is such that all five diagonals AC, BD, CE, DA, and EB lie entirely within the pentagon. If the area of each of the triangles ABC, BCD, CDE, DEA, and EAB is equal to 1, calculate the area of the pentagon ABCDE. Express your answer as a decimal value rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( 5^{\log 30} \times \left(\frac{1}{3}\right)^{\log 0.5} = d \), find the value of \( d \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(X_n)_{n\geq1}$ be a sequence of independent random variables such that for every positive integer $k$, the following probabilities hold: $P(X_k=k^2)=\frac{1}{k^2}$, $P(X_k=2)=\frac{1}{2}$, and $P(X_k=0)=\frac{1}{2}-\frac{1}{k^2}$. Define $S_n=X_1+X_2+\dots+X_n$. Determine if there exists a real number $c$ such that $\frac{S_n}{n}\rightarrow c$ almost surely, and if so, find the value of $c$. \(\text{If } c \text{ exists, express it as } \boxed{c}.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a point  $p$  and a line segment  $l$ , let  $d(p,l)$  be the distance between them.  Let  $A$ ,  $B$ , and  $C$  be points in the plane such that  $AB=6$ ,  $BC=8$ ,  $AC=10$ .  What is the area of the region in the  $(x,y)$ -plane formed by the ordered pairs  $(x,y)$  such that there exists a point  $P$  inside triangle  $ABC$  with  $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Every day, you put either 1 or 2 dollars into a piggy bank, each with a probability of \( \frac{1}{2} \). What is the expected number of 2-dollar bills in the piggy bank when the total amount reaches at least 100 dollars for the first time?
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Context:
Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can guarantee a profit of at least \$n in one year. Find the greatest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $p$  be a prime.  It is given that there exists a unique nonconstant function  $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$  such that  $\chi(1) = 1$  and  $\chi(mn) = \chi(m)\chi(n)$  for all  $m, n \not\equiv 0 \pmod p$  (here the product  $mn$  is taken mod  $p$ ). For how many positive primes  $p$  less than  $100$  is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual  $a^{-1}$  denotes multiplicative inverse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Kullback-Leibler divergence for $P \sim \mathrm{Bin}(n,p)$ and $Q \sim \mathrm{Bin}(n,p+h)$, where $h$ is a small positive number.
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Context:
Calculate the area inside the circle given by the polar equation \( r = 3a\cos(\theta) \) and outside the cardioid given by \( r = a(1+\cos(\theta)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the residue of the function \( \sin\left(\frac{1}{z}\right) \) at its poles.
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Context:
Construct an explicit isomorphism between the algebraic group $\mathbb{G}_m/\{\pm 1\}$ and $\mathbb{G}_m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of $2 \times 2$ involutory matrices over the ring $\mathbb{Z}_{26}$, where an involutory matrix satisfies $M^2 = I$, with $I$ being the identity matrix.
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Context:
Find the minimum value of \( M \) such that the inequality \( \cos \alpha_{1} \cos \alpha_{2} \cdots \cos \alpha_{n} + \sin \alpha_{1} \sin \alpha_{2} \cdots \sin \alpha_{n} \leqslant M \) holds for all angles \( \alpha_1, \alpha_2, \ldots, \alpha_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the eigenvalues of the matrix $A = (p_{ij})$ where $\sum_{i=1}^3 p_{ij} = 1$ for all $j = 1, 2, 3$. Assume $A$ is a $3 \times 3$ matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to 4-color the vertices of a square such that adjacent vertices have different colors, considering colorings equivalent under rotations and reflections.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{3} (x^2+1) \, d[x] \), where \([x]\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tim has a working analog 12-hour clock with two hands that run continuously. He also has a clock that runs at half the correct rate. At noon one day, both clocks show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of infinite arithmetic sequences of natural numbers such that the ratio of the sum of the first $2n$ terms to the sum of the first $n$ terms is constant for all $n$, and one of the terms in the sequence is 1971.
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Context:
Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e. there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of the players can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On an island, there are chameleons of five colors. When one chameleon bites another, the color of the bitten chameleon changes according to some rule, where the new color depends only on the color of the biting and the bitten chameleon. It is known that \$ 2023 \$ red chameleons can agree on a sequence of bites such that they all become blue. What is the smallest \$ \mathrm{k} \$ that guarantees that \$ \mathrm{k} \$ red chameleons can agree on a sequence of bites to become blue?

For example, the rules could be: if a red chameleon bites a green one, the bitten changes color to blue; if a green bites a red, the bitten remains red; if a red chameleon bites a red one, the bitten changes to yellow, and so on. (The specific color change rules may vary.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of squares that must be initially planted with grass in an n x n square garden, divided into a grid by footpaths, such that if at least two neighboring squares have full grass coverage, the grass will eventually spread to the entire garden.
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Context:
Let \( S = \left\{ p_1 p_2 \cdots p_n \mid p_1, p_2, \ldots, p_n \text{ are distinct primes and } p_1, \ldots, p_n < 30 \right\} \). Assume 1 is in \( S \). Let \( a_1 \) be an element of \( S \). We define, for all positive integers \( n \):
\[ a_{n+1} = \frac{a_n}{n+1} \quad \text{if } a_n \text{ is divisible by } n+1; \]
\[ a_{n+1} = (n+2) a_n \quad \text{if } a_n \text{ is not divisible by } n+1. \]
How many distinct possible values of \( a_1 \) are there such that \( a_j = a_1 \) for infinitely many \( j \)'s?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \mathrm{KLM} \) with angle \( \mathrm{L} = 120^{\circ} \), the angle bisectors \( \mathrm{LA} \) and \( \mathrm{KB} \) of angles \( \mathrm{KLM} \) and \( \mathrm{LKM} \) respectively are drawn. Find the measure of angle \( \mathrm{KBA} \).
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Context:
Given that $a_n(X_n-\theta)-a_n(Y_n-\theta)\xrightarrow[d]{}Z$, $a_n\xrightarrow[]{}\infty$, and $X_n, Y_n\xrightarrow[p]{}\theta$, with $o(X_n-\theta)= (X_n-\theta)\cdot o(1)$, determine if $a_n(X_n-\theta)\cdot \phi_n-a_n(Y_n-\theta)\cdot \phi_n\xrightarrow[d]{}0$ when $\phi_n$ is a sequence that converges to zero arbitrarily slowly.
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Context:
A train of a certain length is running with a speed of 60 km/hr. In 14.998800095992323 seconds, it will pass a man who is running at 6 km/hr in the direction opposite to that in which the train is going. What is the length of the train?
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Context:
Determine the rank of the free group that is the fundamental group of the covering space \( \hat{X} \rightarrow X \), where \( X \) is a bouquet of 3 circles and the covering space is of degree 10.
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Context:
Given the hyperbola \( \frac{(3x-4y-1)^2}{100} - \frac{(4x+3y-1)^2}{225} = 1 \), find the length of the latus rectum.
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Context:
Given the circle $O: x^2 + y^2 = 1$ and the functions $f(x) = x^3$, $f(x) = \tan(x)$, and $f(x) = x\sin(x)$, how many of these functions have graphs that can bisect the area of circle $O$ due to their symmetry properties? Express your answer as a single integer.
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Context:
Calculate the degrees of freedom (DOF) for a natural cubic spline with knots at \{10, 20, 30, 50\}, given that the curve is constrained to be linear before the first knot and after the last knot.
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Context:
A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for 8 hours a day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand and the hour hand to coincide once. Therefore, the factory underpays each worker by ____ yuan every day.
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Context:
Janet, a third grade teacher, is picking up the sack lunch orders from a local deli for the field trip she is taking her class on. There are 35 children in her class, 5 volunteer chaperones, and herself. She she also ordered three additional sack lunches, just in case there was a problem. Among the lunch orders, 10 are vegetarian, 5 are gluten-free, and 2 are both vegetarian and gluten-free. The regular sack lunches cost $7, while the vegetarian and gluten-free ones cost $8 each, and the lunches meeting both dietary restrictions cost $9 each. Calculate the total cost of all the lunches considering the various dietary restrictions and preferences.
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Context:
20. N6 (POL) ${ }^{1 \mathrm{MO}}$ Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2, \ldots, 2 p\}$ such that the sum of all elements of $A$ is divisible by $p$.
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Context:
In a school, a certain percentage of students study sociology, 44% study mathematics, and 40% study biology. If 30% of students study both mathematics and sociology, the largest possible number of students who study biology but do not study either mathematics or sociology is 180. How many students are there in the school?
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Context:
How many 4-permutations can be formed from the letters in the word "aaabbccdef"?
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Context:
Given the linear regression model with the design matrix \(X = [x_1 \mid x_2 \mid \cdots\mid x_p] \in\mathbb R^{n\times p}\), where the columns of \(X\) are centered (i.e., \(x^T_j 1_n = 0\) for \(j = 1, \ldots , p\)), and the model does not include an intercept, assume the following covariance matrix:
\[
\frac{1}{n}X^TX=\begin{bmatrix}I_{p-1} & z\\z & a^2\end{bmatrix}, z\in \mathbb{R}^{p-1}, a^2\in \mathbb{R}
\]
where \(a^2=10\|z\|^2\). Let \(R^2_j\) be the \(R^2\) obtained when regressing \(x_j\) onto the rest of the covariates \(\{x_1, \ldots , x_{j−1}, x_{j+1}, \ldots , x_p\}\). Find \(R^2_p\), the coefficient of determination resulting from regressing \(x_p\) onto \(\{x_1, \ldots , x_{p−1}\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The "Wuyang" building, if constructed jointly by team A and team B, would need more than 6 months. If constructed by team A and team C, it would need more than 5 months. If constructed by team B and team C, it would need more than 4 months. Calculate the maximum number of months it would take for teams A, B, and C to build it together. Express your answer as a whole number of months.
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Context:
Find the least number that must be subtracted from 427398 so that the remaining number is divisible by a certain divisor. The least number is 3. What is the divisor?
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Context:
Determine the number of zeros in the right half-plane for the polynomial \(z^4 + 2z^3 - 2z + 10\).
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Context:
Find the smallest  $\lambda \in \mathbb{R}$  such that for all  $n \in \mathbb{N}_+$ , there exists  $x_1, x_2, \ldots, x_n$  satisfying  $n = x_1 x_2 \ldots x_{2023}$ , where  $x_i$  is either a prime or a positive integer not exceeding  $n^\lambda$  for all  $i \in \left\{ 1,2, \ldots, 2023 \right\}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set $X$ with at most $\kappa$ operations, each having arity less than $\kappa$, and a subset $S$ of $X$ with cardinality less than or equal to $\kappa$, determine the maximum cardinality of the substructure of $X$ generated by $S$.
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Context:
Calculate the genus of a three-component unlink using the Seifert surface method.
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Context:
Given that \( \alpha, \beta, \gamma, \delta \) are the roots of the polynomial \( x^4 + 4x^3 - 6x^2 + 7x - 9 = 0 \), find the value of the product \( \prod(1+\alpha^2) \).
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Context:
Given four functions: $①y=-x$, $②y=- \frac{1}{x}$, $③y=x^{3}$, $④y=x^{ \frac{1}{2}}$, determine the probability of the event "the graphs of the two selected functions have exactly one common point" when two functions are selected from these four.
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Context:
Given the power function $y=(m^2-9m+19)x^{2m^2-7m-9}$, the graph does not pass through the origin. Then the value of $m$ is \_\_\_\_\_.
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Context:
Let \( S = \{1, 2, 3, \ldots, 20\} \) be the set of all positive integers from 1 to 20. Suppose that \( N \) is the smallest positive integer such that exactly eighteen numbers from \( S \) are factors of \( N \), and the only two numbers from \( S \) that are not factors of \( N \) are consecutive integers. Find the sum of the digits of \( N \).
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Context:
若${{x}_{1}}$满足$3x+{{3}^{x-1}}=7$,${{x2}}$满足$3x+3{{\log }_{3}}\left( x-2 \right)=7$,则${{x1}}+{{x}_{2}}=$__________．
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Context:
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 7$.  If $YZ = 100$, then what is $XY$?
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Context:
Find the determinant of a matrix that is similar to matrix \( A_n \), but with non-zero elements located on the left below the main diagonal instead of the bottom left corner.
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Context:
整数 $\left[\frac{10^{93}}{10^{31}+3}\right]$ 的末两位数是 $\qquad$.
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Context:
8. (for 9th grade. 2 points). In triangle $A B C$, angle $A$ is equal to $40^{\circ}$. The triangle is randomly thrown onto a table. Find the probability that vertex $A$ will be to the east of the other two vertices.
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Context:
The linear equation $Ax + By = 0$: if we choose any two different numbers from the set $\{1, 2, 3, 6, 7, 8\}$ for the values of $A$ and $B$, then the number of distinct lines that can be represented is __________.
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Context:
Evaluate the convolution integral \( \int_{-\infty}^{+\infty} \delta'(a-t) g(t) \, dt \) given that \( \int_{-\infty}^{+\infty} \delta'(t) g(t) \, dt = -g'(0) \).
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Context:
To investigate the number of students from various classes participating in an extracurricular calligraphy group, 5 classes were randomly selected from the school, and the number of students from each class participating in the group was taken as sample data. It is known that the sample mean is 7, and the sample variance is 4, and the sample data are all different. The maximum value among the sample data is ______.
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Context:
Suppose that $f(x)$ and $g(x)$ are functions which satisfy the equations $f(g(x)) = 2x^2$ and $g(f(x)) = x^4$ for all $x \ge 1$. If $g(4) = 16$, compute $[g(2)]^4$.
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Context:
Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points.
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Context:
What is the term for a lattice that is isomorphic to its dual lattice?
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Context:
Three equal circles are placed in triangle \(ABC\), each touching two sides of the triangle. All three circles have one common point. Find the radii of these circles if the radii of the inscribed and circumscribed circles of triangle \(ABC\) are \(r\) and \(R\), respectively.
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Context:
Arrange 6 volunteers $A$, $B$, $C$, $D$, $E$, $F$ to take care of 3 elderly people $X$, $Y$, $Z$. Each pair of volunteers takes care of one elderly person. Considering the distance between the volunteers' and the elderly people's residences, volunteer $A$ cannot be assigned to take care of elderly person $X$, and volunteer $B$ cannot be assigned to take care of elderly person $Y$. How many different arrangements are there? Express your answer as a single integer.
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Context:
Given that D is 1.3 meters, C is 0.6 meters, and the angle \( a \) is 30 degrees with a rate of change \( a' = 150 \) rad/s, calculate the rate of change of M. Use the following relationships:

\[ M = \cos(30^\circ) \cdot 0.6 + \cos(13.3424) \cdot 1.2 \]

\[ M' = -\left(\sin(30^\circ) \cdot 0.6 \cdot a' + \sin(13.3424) \cdot 1.3 \cdot B'\right) \]

where \( B' \) is calculated using:

\[ B' = \frac{\cos(30) \cdot 0.6 \cdot 150}{\cos(13.3424) \cdot 1.3} \]

Determine \( M' \) in rad/s or m/s.
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Context:
Evaluate the limit \( \lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1+\cos^2(nx)} \, dx \).
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Context:
Ned was helping the cafeteria workers pick up lunch trays, but he could only carry 5 trays at a time. He needs to pick up trays from 4 tables with different numbers of trays on each table. The first table has 7 trays, the second table has 10 trays, the third table has 12 trays, and the fourth table has 3 trays. How many trips will he make?
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Context:
Let \(I\) be the center of the inscribed circle in triangle \(ABC\). We assume that \(AB = AC + CI\). Determine the value of the ratio \(\frac{\widehat{ACB}}{\widehat{ABC}}\).
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Context:
Let $ABCDEF$ be a convex hexagon, whose opposite angles are parallel, satisfying $AF = 3$, $BC = 4$, and $DE = 5$. Suppose that $AD$, $BE$, $CF$ intersect in a point. Find $CD$.
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Context:
The probability of snow for each of the next four days is $\frac{3}{4}$. However, if it snows any day before the last day, the probability of snow on the last day increases to $\frac{4}{5}$. What is the probability that it will snow at least once during these four days? Express your answer as a common fraction.
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Context:
Find the singular integral of the differential equation \((xy^{\prime}-y)^{2}=x^{2}(x^{2}-y^{2})\).
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Context:
 Let $C$ be the answer to problem $18$. Suppose that $x$ and $y$ are real numbers with $y > 0$ and

$$x + y = C$$
$$x +\frac{1}{y} = -2.$$
Compute $y +\frac{1}{y}$.
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Context:
Is there a closed-form solution for this combinatorial limit of a sum?
$$\lim_{n\to\infty}\sum_{k=0}^n\binom nk\frac{3k}{2^n(n+3k)}$$
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Context:
Given real numbers $a$ and $b$ belong to $(0,1)$, and satisfy $(1-a)b > \frac{1}{4}$, then the relationship between $a$ and $b$ is ______.
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Context:
A set $S$ of natural numbers is called good, if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a good set which is a subset of the set $A=\{1,2,3, \ldots, 63\}$.
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Context:
A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?
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Context:
Determine the region of convergence for the power series \( \sum_{n=1}^{\infty} a_{n}(x-1)^{n} \), given that the sequence \( \{a_{n}\} \) decreases monotonically, \( \lim_{n \rightarrow \infty} a_{n}=0 \), and the series \( S_{n}=\sum_{k=1}^{n} a_{k} \) diverges.
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Context:
Alice claims, "The probability that I'm lying is greater than \( p \)." Determine the probability that Alice is lying.
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Context:
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at 0. Every minute, he has a $\frac{1}{3}$ chance of moving 1 unit left, a $\frac{1}{3}$ chance of moving 1 unit right and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to 0 (not including the start) before getting eaten.
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\left(e^n \left(n-ne\sqrt[n]{\ln{(1+e^n)}-n}\right) \right)$$
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Context:
Calculate the sum \(\sum^{7}_{n=2} (n-1)P (X = n)\) for the random experiment of tossing a coin until the same number of heads and tails have been observed.
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Context:
On the radius \( A O \) of a circle with center \( O \), point \( M \) is chosen. On the same side of \( A O \) on the circumference, points \( B \) and \( C \) are chosen such that \( \angle A M B = \angle O M C = \alpha \). Find the length \( B C \), given that the radius of the circle is 15 and \( \cos \alpha = \frac{3}{5} \).
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Context:
In a convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(DB\) are perpendicular to sides \(DC\) and \(AB\) respectively. From point \(B\), a perpendicular is dropped to side \(AD\), intersecting \(AC\) at point \(O\). Find \(AO\) if \(AB = 4\) and \(OC = 6\).
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Context:
Determine the number of compact Hausdorff spaces of cardinality \( \kappa \), up to homeomorphism, where \( \kappa \) is an uncountable cardinal.
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Context:
Alice has a deck of 36 cards, consisting of 4 suits with 9 cards each. She selects any 18 cards and gives the remaining 18 cards to Bob. In each turn, Alice picks one of her cards and places it face-up on the table, followed by Bob doing the same with one of his cards. Bob scores a point if the two cards have the same suit or the same value. What is the maximum number of points Bob can guarantee, regardless of Alice's actions?
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Context:
Evaluate the triple integral of \( f(x,y,z) = (x^8 + y^6 + z^4)\sin(z^3) + 2y - 3 \) over the unit ball.
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Context:
Given the joint moment generating function of random variables $X$ and $Y$:
\[ M(t_1,t_2) = \frac{1}{2}e^{t_1+t_2} + \frac{1}{4}e^{2t_1+t_2} + \frac{1}{12}e^{t_2} + \frac{1}{6}e^{4t_1+3t_2} \]
find the probability $P(X < Y)$. Provide your answer as a simplified expression or numerical value.
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Context:
Determine the Galois group of the polynomial \( f = x^5 + x^4 + x^3 + x \) over the field \( \mathbb{Z}_2 \).
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Context:
Use Green's theorem to calculate the line integral \( \oint\limits_{C2} \frac{-y}{(x^{2}+y^2)^2} \, dx +\frac{x}{(x^{2}+y^2)^2} \, dy \), where \( C1: x^{2}+y^2 = 1 \) and \( C2: r = \frac{2}{\sqrt{2-\cos(\theta)}} \). The region \( R \) is the area between \( C1 \) and \( C2 \), and both curves are oriented positively. Given that \( \oint\limits_{C1} \frac{-y}{(x^{2}+y^2)^2} \, dx +\frac{x}{(x^{2}+y^2)^2} \, dy = 2\pi \) and \( \iint\limits_R \frac{1}{(x^{2}+y^2)^2} \, dx \, dy = \frac{\pi}{2} \), find the value of the line integral over \( C2 \).
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Context:
Find a possible expression for the differentiable function $g(x)$ given the integral relationship: \[ \int g(x)e^{\frac{x}{4}}dx = 4g(x)e^{\frac{x}{4}} - \int 8x^2e^{\frac{x}{4}}dx, \] where integration by parts is applicable.
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Context:
Evaluate the sum \( \sum_{k=0}^{4}\csc^2\left(\frac{\pi}{6}+\frac{k\pi}{5}\right) \) given that \( \csc^2(5 \times \frac{\pi}{6}) = 5 \).
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Context:
Find the limit \( \lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \, dx \).
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Context:
Construct a field extension $E$ of the field $\mathbb{F}_{11}$ such that the degree of the extension, $[E:\mathbb{F}_{11}]$, is equal to $3$. Provide the irreducible polynomial used for this extension.
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Context:
Evaluate the expectation $E[AX^TBA]$ in terms of $E[X]$, given that $A \in \mathbb{R}^{m \times n}$, $X \in \mathbb{R}^{n \times p}$, $\|A\| \leq 1$, and $E[X] \geq 0$. Assume $B$ is a known matrix of appropriate dimensions.
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Context:
Evaluate the double integral \( \int \int_D xydA \) where \( D \) is the union of the regions defined by the following bounds: \( \int_{-1}^{1}\int _{0}^{\sqrt{1-x^2}}xydydx + \int_{-1}^{1}\int _{-x/3-1/\sqrt{3}}^{\sqrt{x^2-1}}xydydx \).
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Context:
Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.
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Context:
Find a polynomial in two variables with integer coefficients whose range is the positive real numbers.
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Context:
Given the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}_2[x] \) defined by \( T(1,0) = 1+x \) and \( T(1,1) = 1-x \), find \( T^{-1}(2x) \).
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Context:
Determine the set of real numbers \(\alpha\) for which the function \(f_\alpha(x) = \sum_{n=1}^\infty n^\alpha x e^{-nx}\) is continuous on \([0, \infty)\) and integrable with respect to the Lebesgue measure on \([0, \infty)\).
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Context:
Evaluate the integral \( \oint_C \frac{zf'(z)}{f(z)} \,ds \) using the argument principle, where \( f(z) = z^4 - 2z^3 + z^2 - 12z + 20 \) and \( C \) is the circle \( |z| = 5 \).
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Context:
Evaluate the integral \( \int_{\lvert z \rvert = \frac{1}{2}} \frac{dz}{z \sin\left( \frac{1}{z} \right)} \).
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Context:
Suppose that vectors $x_1, x_2, \dots, x_n$ have the following property: for each $i$, the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1, x_2, \dots, x_n$ are not parallel, what is the value of the sum $x_1 + x_2 + \dots + x_n$?
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Context:
A subset $S$ of $\{1, 2, 3, \ldots, 2025\}$ is called balanced if for all elements $a$ and $b$ both in $S,$ there exists an element $c$ in $S$ such that $2025$ divides $a+b-2c.$ Compute the number of nonempty balanced sets.
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Context:
Find the direct limit of the sequence of groups \( \frac{1}{n}\mathbb{Z} \).
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Context:
Calculate the limit \( \lim_{n \to \infty} \int f_n \, d\mu \) where \( S = \mathbb{N} \), \( A = P(\mathbb{N}) \), \( \mu = h \, dv \) with \( v \) being the counting measure and \( h(k) = \frac{1}{4^k} \), and \( f_n(k) = \left(\frac{k}{n} - 2\right)^k \cdot \mathbf{1}_{(1,2,3,...,n)}(k) \).
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Context:
Find the equation of a polynomial of degree 6 whose roots are the products of the roots of the polynomial \(g(x) = x^3 - 2x^2 + 3x - 4\) taken three at a time.
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Context:
Find the greatest integer $k$ such that among 2016 customers who visited a store, we can select $k$ customers where either all $k$ were in the store at the same time, or no two of them were in the store at the same time.
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Context:
Given a complex number ω such that ω^5 = 1 and ω ≠ 1, evaluate the expression:

\[ \frac{\omega}{1+\omega^2} + \frac{\omega^2}{1+\omega^4} + \frac{\omega^3}{1+\omega} + \frac{\omega^4}{1+\omega^3} \]
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Context:
Find the range of the function, $f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos x+t^2}dt$.
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Context:
Determine the least number of terms needed to approximate the integral \( \int_0^{1/2} \frac{x - \arctan(x)}{x^2} \, dx \) using the series expansion \( \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \) for all \( x \in [-1,1] \), such that the error is less than \( 10^{-5} \).
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Context:
The inclination angle of the line $x- \sqrt {3}y+3=0$ is \_\_\_\_\_\_.
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Context:
Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$.
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Context:
In measuring the sides of a rectangle, one side is taken 15% in excess, and the other 11% in deficit. Find the error percent in the area calculated from these measurements.
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Context:
Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix} \, dx \]
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Context:
The price of a math textbook in the school bookshop is $45. If those sold in bookshops outside the school cost 20% less, Peter is considering buying from other bookshops. Additionally, there is a science textbook sold at the school bookshop for $60, which costs 25% less in external bookshops, and a literature textbook priced at $35 in the school bookshop, which costs 15% less in outside bookshops. The school bookshop has a tax rate of 7%, while alternative bookshops have a tax rate of 6%. If Peter needs to pay a shipping cost of $10 for the books purchased from the alternative bookshop, how much can Peter save in total (including tax and shipping) by purchasing these three textbooks from alternative bookshops instead of the school's?
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Context:
Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. She continues rolling the dice and recording the product until one of the products appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
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Context:
Given that the three vertices of $\triangle ABC$ are all on the hyperbola $\frac{x^{2}}{2} - \frac{y^{2}}{4} = 1$, the slopes of the three sides $AB$, $BC$, $AC$ all exist and are non-zero, and their sum is $-1$; furthermore, the midpoints of $AB$, $BC$, $AC$ are $M$, $N$, $P$ respectively, and $O$ is the coordinate origin. The slopes of the lines $OM$, $ON$, $OP$ are $k_{1}$, $k_{2}$, $k_{3}$ respectively, and are all non-zero. Find $\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}} = \_\_\_\_\_\_$.
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Context:
Using arithmetic operation signs, write the largest natural number using two twos.
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Context:
Given the joint moment generating function of random variables X and Y:
\[ M_{X,Y}(a,b) = \frac{4}{5}\left[\frac{1}{(1-a)(1-b)}+\frac{1}{(2-a)(2-b)}\right] \]
find the expected value of the product E(XY).
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Context:
When Erick went to the market to sell his fruits, he noticed that the price of lemons had risen by 50% and the price of grapes had increased by 25%. In anticipation of these changes, he brought different quantities of each fruit to the market. He had 80 lemons, 140 grapes, 60 oranges, and 100 apples in his basket. He originally planned to sell the lemons at $8 each, grapes at $7 each, oranges at $5 each, and apples at $4 each. However, since the price of oranges increased by 10% and apples by 20%, he needed to update his selling prices.

How much money did he collect from selling the fruits at the new prices if he sold all the fruits in his basket?
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Context:
James decides to sell 80% of his toys. He bought them for $20 each and sells them for $30 each. He had a certain number of toys and made $800 more compared to before he bought them. How many toys did he have initially?
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Context:
Determine the minimal value of the maximum order (max(a, b, c)) among the subgroups $P_{23}$, $P_{13}$, and $P_{12}$, generated by permutations $p_1$, $p_2$, and $p_3$ in the symmetric group $S_n$ for $n > 7$. What is this minimal maximum order?
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Context:
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) of the vector field \( \mathbf{F}(x,y,z) = -x^2\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) along the boundary \( C \) of the surface \( S \) defined by \( f(x,y) = 4x - 8y + 30 \), within the rectangle \( R = \{(x,y): -2 < x < 3, 0 < y < 2\} \). The boundary \( C \) is oriented counterclockwise when viewed from above. Use Stokes' Theorem to find the value of the integral.
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Context:
Evaluate the limit: \[ \lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx \]
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Context:
Given that $x, y, z$ are positive real numbers satisfying $\frac{1}{3} \le xy + yz + zx \le 3$, determine the range of possible values for $x + y + z$. Provide your answer as an interval.
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Context:
A computer store offers employees different discount tiers off the retail price based on the length of their employment and additional conditions such as bulk purchases or store credit usage. The store has three computer models with different retail prices: Model A costs $500, Model B costs $650, and Model C costs $800. These retail prices are determined after applying varying markup percentages for each model (12% for Model A, 15% for Model B, and 18% for Model C). Employees who have worked for less than 1 year receive a 10% discount, employees who have worked for 1-2 years receive a 15% discount, and employees who have worked for 2 years or more receive a 20% discount. In addition, employees get an extra 5% discount if they purchase 5 or more computers, and employees can use their $100 store credit towards the purchase.

An employee who has worked for 1.5 years decides to purchase computers in bulk. If the employee purchases 5 Model A computers, 3 Model B computers, and 2 Model C computers, how much would they save on the total cost with their employee discount, bulk purchase discount, and store credit usage, compared to the final retail prices of all the computers?
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Context:
Let $f : \mathbb{N} \to \mathbb{R}$ be a function, satisfying the following condition: for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \left(\frac{n}{p}\right) - f(p)$. If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$.
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Context:
In the quarter circle defined by the equation $x^2 + y^2 = 16$, consider the quarter arc $AD$. Points $B$ and $C$ are arbitrary points on the arc $AD$, with $C$ located between $D$ and $B$. Determine the maximum possible area of the pentagon $OABCD$, where $O$ is the origin.
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Context:
Let $\odot$ be a binary operation on the natural numbers satisfying the properties that, for all $a, b,$ and $c$, $(a + b) \odot c = (a \odot c) + (b \odot c)$ and $a \odot (b + c) = (a \odot b) \odot c.$ Given that $5 \odot 5 = 160$, find the value of $7 \odot 7$.
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Context:
从 $1 \sim 9$ 这九个正整数中任选三个,则这三个数两两互素的概率是 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Determine the values of \(x\) for which the series \(\sum_{n=1}^\infty \frac{n! \> x^n}{(x+1)(2x+1) \cdots (nx+1)}\) converges.
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Context:
4. Among the natural numbers from 2021 to 9998, the numbers with the same tens and units digits are $\qquad$ in total.
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Context:
Provide an example of a group \( G \) that contains an element \( a \) of order 3 and an element \( b \) of order 4, such that the order of the product \( ab \) is less than 12.
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Context:
Deﬁne the determinant  $D_1$  =  $|1|$ , the determinant  $D_2$  =      $|1 1|$   $|1 3|$
, and the determinant  $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant  $D_n$  have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that  $D_n$   $\geq$  2015.
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Context:
Dave had some apps on his phone. After deleting 18 of them, he installed "x" number of new apps and then had 5 apps left. How many apps did he have on his phone at the start?
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Context:
Let  $\triangle ABC$  be a triangle with circumcenter  $O$  satisfying  $AB=13$ ,  $BC = 15$ , and  $AC = 14$ .  Suppose there is a point  $P$  such that  $PB \perp BC$  and  $PA \perp AB$ .  Let  $X$  be a point on  $AC$  such that  $BX \perp OP$ . What is the ratio  $AX/XC$ ?
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Context:
Given $2018$ ones in a row: $$\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}$$ in which plus symbols $(+)$ are allowed to be inserted in between the ones. What is the maximum number of plus symbols $(+)$ that need to be inserted so that the resulting sum is $8102$?
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Context:
Four copies of the triangle shown are joined together, without gaps or overlaps, to make a parallelogram. The triangle has edge lengths of 13 cm, 13 cm, and 12 cm. Find the largest possible perimeter of the parallelogram. Express your answer in centimeters as a whole number.
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Context:
In triangle \( A B C \) with side \( A C = 8 \), a bisector \( B L \) is drawn. It is known that the areas of triangles \( A B L \) and \( B L C \) are in the ratio \( 3: 1 \). Find the bisector \( B L \), for which the height dropped from vertex \( B \) to the base \( A C \) will be the greatest.
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Context:
Determine the signature of the matrix \( \begin{pmatrix} -3 & 0 & -1 \\ 0 & -3 & 0 \\ -1 & 0 & -1 \end{pmatrix} \). You may use methods such as Gaussian elimination to find the answer without computing eigenvalues.
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Context:
Let $T = \{9^k : k ~ {is ~an~ integer}, 0 \le k \le 4000\}$ . Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
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Context:
Calculate the expected present value of a continuous payment stream of £190 per annum for 20 years, beginning at time 0. For the first 12 years, the force of interest is 5%. After 12 years, the interest rate changes to 3% with probability 0.25, remains at 5% with probability 0.5, and increases to 7% with probability 0.25. What is the expected present value of this payment stream?
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Context:
Evaluate the second mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & \text{if }(x,y) \neq (0,0) \\ 0 & \text{otherwise} \end{cases} \).
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Context:
Calculate the Cauchy principal value of the integral: $$ \int_0^{\infty}\frac{\tan x}{x}\mathrm dx $$
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Context:
What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits?
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Context:
已知x、y是正实数，且满足$x y + x + y = 7 1 , x ^ { 2 } y + x y ^ { 2 } = 8 8 0$，则$x ^ { 2 } + y ^ { 2 } =$ ·
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Context:
How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$?
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Context:
Solve the recurrence relation \( T(n) = 2T(\sqrt{n}) + n \).
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Context:
Determine the values of \( p \in \mathbb{R} \) for which the integral \( \int_0^\infty \frac{\sin x}{x^p} \, dx \) converges.
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Context:
Find the minimum value of \( \int_{0}^{1} (f'(x))^2 \, dx \) given that \( f \in C^{(1)} [0,1] \), \( f(1) = b \), and \( \int_{0}^{1} f(x) \, dx = 0 \).
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Context:
若$\sin x+\sqrt{3}\cos x+1\le m$在$0\le x\le \frac{\pi }{2}$上有解，则实数$\textit{m}$的最小值为____________.
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Context:
Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum area of $\triangle PAB$.
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Context:
Evaluate the integer part of the sum \( \left[\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}}\right] \) using the concept of definite integrals.
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Context:
Each of the integers $1,2, \ldots, 729$ is written in its base-3 representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122 \ldots \ldots$ How many times in this string does the substring 012 appear?
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Context:
A cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth.
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Context:
Determine the dimension of the vector space $\mathfrak{m}/\mathfrak{m}^2$ for the curve $\mathcal{C}=\{ (x,y): x^3+y^3-1=0 \} \subset \mathbb{C}^2$, where $\mathfrak{m} = \langle x-a,y-b\rangle/\langle x^3 + y^3 -1\rangle$ and $p=(a,b) \in \mathcal{C}$. Provide your answer as a single integer.
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Context:
Find the determinant of the matrix \(A= \begin{bmatrix} 1 & x & x^2 & x^3\\ x & x^2 & x^3 & 1 \\ x^2 & x^3 & 1 & x \\ x^3 & 1 & x&x^2 \end{bmatrix}\).
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Context:
Determine the cardinality of the set \( S = \{ h: D \rightarrow D \mid h \text{ is analytic in } D \text{ and } h(z)^2 = \overline{h(z)} \text{ for all } z \in D \} \), where \( D = \{ z \in \mathbb{C} : |z| < 1 \} \).
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Context:
The base of a prism is a square with a side length of \( a \). One of the lateral faces is also a square, and another is a rhombus with an angle of \( 60^{\circ} \). Determine the total surface area of the prism.
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Context:
Evaluate the limit \( \lim_{Z \to Zi} (Z^2 - \bar{Z}) \).
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Context:
Evaluate the limit: $$\lim\limits_{x\to 0^{+}}\lim\limits_{n\to \infty}\frac{d^n}{dx^n}\left(\frac{e^{\sqrt{x}}+e^{-\sqrt{x}}}{x^n}\right)$$
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Context:
A man engaged a servant on the condition that he would pay him Rs. 900 and a uniform after one year of service. The servant served only for 9 months and received a uniform and a certain amount of money. The price of the uniform is Rs. 100. How much money did the servant receive?
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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 measure of angle \(ABC\) in degrees.
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Context:
Since the beginning of the school year, Andrey has been recording his math grades. Upon receiving a new grade (2, 3, 4, or 5), he called it unexpected if, up to that point, it had occurred less frequently than each of the other possible grades. (For example, if he received the grades 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 in order, the first 5 and the second 4 would be unexpected.) Over the entire school year, Andrey received 40 grades - exactly 10 fives, 10 fours, 10 threes, and 10 twos (the order is unknown). Can you definitively determine the number of grades that were unexpected for him?
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Context:
For all possible non-empty sets $A$ and $B$ such that $A \subseteq B$, how many possible subsets $K$ of $B\setminus A$ are there if $|A|+|B|=8$?
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Context:
A person draws cards one at a time from a standard deck of 52 cards without replacement, continuing until they draw the 3 of Hearts. What is the expected maximum value of the cards drawn? Provide your answer as a numerical value.
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Context:
Given that \( g(f(x)) = x \) and \( g'(x) = \frac{1}{1 + x^2} \), find \( f'(x) \).
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Context:
Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school, arriving 20 minutes late. By how many times does the speed of Uncle Vanya's car exceed Nikita's running speed?
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Context:
Calculate the expected number of days required for a stack of \( n \) books, initially arranged in order, to return to their original order. Each day, one book is picked uniformly at random and placed on top of the stack. Provide your answer as a function of \( n \).
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Context:
Let $V$ be an $n$-dimensional vector space over a field $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Suppose $\psi : V \to V$ is a linear transformation and $A$ is the matrix representing $\psi$ with respect to $\mathcal{B}$. Let $\psi^t : V^{*} \to V^{*}$ be the dual transformation and $B$ be the matrix representing $\psi^t$ with respect to $\mathcal{B}^{*}$. Determine the relationship between the matrices $A$ and $B$.
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Context:
Given a triangle $ABC$, an inscribed circle touches the sides $AB, AC, BC$ at points $C_1, B_1, A_1$ respectively. Find the radius of the excircle $w$, which touches the side $AB$ at point $D$, the extension of side $BC$ at point $E$, and the extension of side $AC$ at point $G$. It is known that $CE = 6$, the radius of the inscribed circle is $1$, and $CB_1 = 1$.
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Context:
Find the mass percentage of Ca in Calcium hydroxide
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Context:
The set $A=\{x|x\in\mathbb{N}, \text{ and } \frac{4}{2-x}\in\mathbb{Z}\}$ can be represented by listing method as $A= \_\_\_\_\_\_$.
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Context:
The sum of the positive numbers \(a, b, c, and d\) does not exceed 4. Find the maximum value of the expression

$$
\sqrt[4]{a^{2}+3 a b}+\sqrt[4]{b^{2}+3 b c}+\sqrt[4]{c^{2}+3 c d}+\sqrt[4]{d^{2}+3 d a}
$$
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Context:
Let \( x \) be a real number satisfying \( x^{2} - \sqrt{6} x + 1 = 0 \). Find the numerical value of \( \left| x^{4} - \frac{1}{x^{4}} \right|.
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Context:
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, with $a_1=1$ and $S_7=28$. Let $b_n=\left\lfloor \log a_n \right\rfloor$, where $\left\lfloor x \right\rfloor$ denotes the greatest integer less than or equal to $x$, such as $\left\lfloor 0.9 \right\rfloor = 0$ and $\left\lfloor \log 99 \right\rfloor = 1$. Find the sum of the first 1000 terms of the sequence $\{b_n\}$.
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Context:
Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, how many arrangements can Mereduth make?
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Context:
Let $X_1, X_2, \ldots$ be a sequence of independent uniform random variables on $(0,1)$. Define $N$ as the minimum index $n \geq 2$ such that $X_n < X_{n-1}$. Calculate the expected value $E(N)$. Provide your answer as a single number.
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Context:
Consider an m×n rectangular grid of points in the plane, where m and n are integers greater than 1. Some k of these points are colored red such that no three red points form the vertices of a right-angled triangle with two sides parallel to the sides of the grid. Determine the greatest possible value of k.
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Context:
Find the length of $BP$, in terms of the width $w$ of a sheet of paper, such that the area of triangle $B'PQ$ is minimized when the bottom right corner is folded over crease $\overline{PQ}$, assuming the sheet extends vertically indefinitely.
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Context:
Determine the number of distinct ways to arrange 9 black and 9 white rooks on a 6 × 6 chessboard such that no white rook can threaten any black rook. A rook threatens another piece if they share the same row or column, with no other pieces between them.
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Context:
Determine the smallest closed set containing $C^1[-1, 1]$ in the space $({\scr{B}}[-1, 1],||\cdot||_\infty)$. Provide your answer as a set description.
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Context:
Evaluate the integral \( \int_C \operatorname{Log}\left(1-\frac{1}{z} \right)\,dz \) where \( C \) is the circle \( |z|=2 \).
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Context:
Determine the degree of the splitting field of the polynomial \((x^{15}-1)(x^{12}-1)\) over the finite field \(\mathbb{F}_7\).
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Context:
Given the complete bipartite graph $G = K_{9,15}$ with $V(G) = 24$ vertices and $E(G) = 135$ edges, determine the minimum number of additional edges needed to make $G$ a Hamiltonian graph.
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Context:
For a given \( n \times n \) array, the first row is \( x_1, x_2, x_3, \ldots, x_{n-1}, x_n \). Each successive row is obtained by a cyclic shift of \( r \) places from the previous row. Determine the values of \( r \) for which this construction results in a Latin square.
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Context:
Two squares, \(ABCD\) and \(ABEF\), are positioned such that the angle between their planes is \(120^{\circ}\). Points \(M\) and \(N\) lie on the diagonals \(AC\) and \(BF\) respectively, with \(AM = FN\). Given that \(AB = 1\), determine the maximum possible value of the distance \(MN\).
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Context:
Jerry buys a bottle containing 150 pills. Using a standard 12-hour clock, he notices that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what time will the clock show when he takes the last pill from the bottle?
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Context:
Determine the number of zeros of the function \( f(z) = 2 + z^4 + e^{iz} \) within the set \( \{z : |z| \leq r, \Im z \geq 0\} \) for any \( r > 2 \).
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Context:
In a tetrahedron \(ABCD\), it is given that \(AB \perp BC\), \(BC \perp CD\), and \(AB = BC = CD = 2\sqrt{3}\). The volume of the tetrahedron is 6. Calculate the angle between the skew lines \(AD\) and \(BC\).
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Context:
Let $S$ be the set of all positive integer divisors of $100,000.$ Find how many numbers are the product of two distinct elements of $S.$
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Context:
With all angles measured in degrees, the product $\prod\limits_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$ , where $m$ and $n$ are integers greater than $1$. Find $m+n$ .
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Context:
Determine the number of inequivalent cusps of the modular group \(\Gamma_0(4)\).
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Context:
For $p > 0$, determine the range of values for $p$ that ensures the convergence of the integral \( \displaystyle\int_0^{\infty} x^pe^{-x^8\sin^2x}\,dx \).
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Context:
Let $G$ be a finite group with two generators $x$ and $y$ such that $|x|=2$, $|y|=2$, and $|xy|=4$. Determine the isomorphism class of the quotient group $G/Z(G)$. Provide your answer as the name of the group or its standard notation.
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Context:
Find the non-negative integer value(s) of \(a\) for which the following limit holds true:

\[\lim_{{x \to 1}} \left\{ \frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}\]
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Context:
Let $A_0=(0,0)$. Distinct points $A_1,A_2,\dots$ lie on the $x$-axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$. For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$?
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Context:
Let \( \phi(n) \) denote the number of positive integers less than or equal to \( n \) which are relatively prime to \( n \). Compute \( \displaystyle \sum_{i=1}^{\phi(2023)} \frac{\gcd(i,\phi(2023))}{\phi(2023)} \).
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Context:
On the bases \(AD\) and \(BC\) of trapezoid \(ABCD\), squares \(ADEF\) and \(BCGH\) are constructed outside the trapezoid. The diagonals of the trapezoid intersect at point \(O\). Find the length of segment \(AD\) if \(BC = 2\), \(GO = 7\), and \(GF = 18\).
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Context:
Determine the fractal dimension of a structure formed by repeatedly dividing a $2r \times 2r \times 2r$ cube into smaller cubes, removing the cubes along the axes, and continuing this process for every odd number raised to the power of three.
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Context:
Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the walls of the maze. After completing the maze, he noticed that it was possible to reach any unit square from any other unit square in exactly one way, excluding paths that pass through any unit square more than once. What is the side length of the initially drawn large square?
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