Context:
Evaluate the integral \( \int^{\pi}_0 \log{(\sin{x})} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the third term in the Taylor series expansion of \( f(x) = \frac{(1+x)^4}{(1+2x)^3(1-2x)^2} \) around \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From $31$ students sitting around a circular table, how many different selections of $3$ students can be made such that there are at least $4$ students between any two selected students?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p(x)$ and $q(x)$ be two polynomials such that $p(2)=5$, $p(3)=12$, and $p(q(x))=p(x)q(x)-p(x)$. Find the value of $q(10)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{x\to 0}\frac{\frac{1}{1+x^2}-\int_{0}^{x}\arctan u\,du}{3\tan^{-1}x-3x-\frac{x^3}{3}} \) using L'Hopital's Rule.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \(\lim_{n \to \infty} \frac{f^{n}(0)}{2^{n}n!}\) given that \(\Omega = D(0,2) \setminus \left\{\frac{1}{2}\right\}\), \(f\) is a holomorphic function in \(\Omega\), and \(\frac{1}{2}\) is a simple pole with residue 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through the median $BE$ of the base $ABC$ of the pyramid $ABCD$ and the midpoint $F$ of the edge $DC$, a plane is drawn. Find the volume of the figure $ADBF E$ if the volume of the pyramid $ABCD$ is $40 \text{ cm}^3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle of intersection between the logarithmic spiral given by the equation \( r = Ce^{k\theta} \) and any line passing through the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of different ways to completely cover a regular hexagonal chessboard, consisting of 24 small equilateral triangles, using 12 rhombus-shaped paper pieces, each composed of 2 small equilateral triangles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $f(x)$ is a twice differentiable function with $f(a)=0$, $f(b)=2$, $f(c)=-1$, $f(d)=2$, and $f(e)=0$, where $a<b<c<d<e$, determine the minimum number of zeros of the function $g(x) = (f'(x))^2 + f(x)f''(x)$ in the interval $[a, e]$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{{n \to \infty}} n\int_1^2 \frac{dx}{x^2(1+x^n)} \) without using Taylor expansion.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
A piece is placed in the lower-left corner cell of a $15 \times 15$ board. It can move to cells that are adjacent either by sides or corners of its current cell. The piece must alternate between diagonal and horizontal moves, starting with a diagonal move. What is the maximum number of moves the piece can make without stepping on the same cell twice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Food, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: dancer, dressmaker, engineer, nurse
- Hobby: baking, filmmaking, hiking, puzzles
- Food: carrot, cauliflower, lettuce, mango
- Music-Genre: ambient, classical, jazz, rock
- Pet: fish, frog, lizard, turtle
- Transport: boat, car, helicopter, trike

Given the following premises about the line of people:
- Music-Genre:ambient == Hobby:puzzles
- Job:nurse is somewhere between Job:dancer and Job:engineer
- Hobby:filmmaking is on the left of Music-Genre:rock
- Pet:lizard is on the right of Music-Genre:jazz
- Job:dancer == Music-Genre:classical
- Pet:fish is on the right of Food:cauliflower
- Pet:frog is somewhere to the left of Transport:trike
- Transport:helicopter is somewhere to the right of Pet:turtle
- Job:dressmaker != Food:lettuce
- Transport:boat is on the left or right of Pet:turtle
- Transport:helicopter is on the far left or far right
- Music-Genre:rock is somewhere to the left of Hobby:hiking
- Food:carrot is on the left of Food:cauliflower
- Hobby:hiking == Food:cauliflower

Answer the following question:

Question: Question: What is the Music-Genre of the person who has/is filmmaking??
 Please provide your answer in the format: \boxed{X}, where X is the answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that each time you play a game, you win a prize, and the probability of winning a new type of prize is \((N-a)/N\), where \(N\) is the total number of unique prizes and \(a\) is the number of unique prizes already won, determine the formula to find the most likely number of unique prizes \(N\) based on the sequence of wins: (1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Napoleon has 56 jelly beans and Sedrich has 9 more jelly beans than 3 times what Napoleon has. Daphne has twice as many jelly beans as the difference between Sedrich's and Napoleon's jelly beans. Alondra has 8 fewer jelly beans than the average of Napoleon, Sedrich, and Daphne's jelly beans. If the sum of all four people's jelly beans is equal to 5 times the number of jelly beans that Mikey has, how many jelly beans does Mikey have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $b$ for which the series \( \sum_{k=1}^\infty \frac{k!}{(b+1)(b+2)...(b+k)} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\ln x- \frac {1}{2}ax^{2}-2x$ $(a\neq 0)$ has a decreasing interval, the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the polynomial congruence $\sum_{j=(p-1)/2}^{p-1}\left(\frac{j}{p}\right)j(j-1)...(j-(p-1)/2+1) \ (\textrm{mod} \ p)$, where $p$ is an odd prime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the product of all odd positive integers less than 10000?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the series \( \sum a_n x^{n^2} \), given that the series \( \sum a_n x^n \) has a radius of convergence \( R \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to choose $k$ numbers from the first $n$ natural numbers such that the longest string of consecutive numbers is exactly $m$ numbers long. For instance, if $k = 7$, $n = 14$, and $m = 3$, find the number of combinations of numbers that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $21^{-1} \equiv 15 \pmod{61}$, find $40^{-1} \pmod{61}$, as a residue modulo 61.  (Provide a number between 0 and 60, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory has more than 4 people but less than 8 people. The total number of students in the class participating in the study tour is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two points are chosen independently at random on the sides of a square with side length 1. The probability that the straight-line distance between the points is at least \( \frac{1}{2} \) is \( \frac{a-b\pi}{c} \), where \( a \), \( b \), and \( c \) are positive integers with \( \gcd(a,b,c)=1 \). Find \( a+b+c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence $(x_0 x_1 \cdots x_n)\sqrt{n}$ as $n$ approaches infinity, where $x_0 = \frac{1}{2}$ and $x_{n+1} = x_n^3 - x_n^2 + 1$ for all $n \in \mathbb{N} \cup \{0\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to \infty } \int_0^n \left(1-\frac{x}{n}\right)^n \cos\left(\frac{x}{n}\right)dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A random variable $Y$ has a moment generating function given by $e^{10t+2t^{2}}$. Calculate the probability that $25 < (Y - 10)^2 < 49$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $3:1$ and the ratio of the rectangle's length to its width is $3:2$. Calculate the percentage of the rectangle’s area that is inside the square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the leading order term of the asymptotic expansion for \( \int_{1/x}^1 e^{-t} \, dt \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine for which real values of \(\alpha\) the improper integral \(\int_0^\infty x^\alpha \cos x\, dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_0^1 \frac{\sin(\ln(|z|))}{\ln(|z|)} \, d|x| \), where \( z = x + iy \) and \( |z| \) denotes the modulus of \( z \). Express the result in terms of \( \pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of identical regular pentagons, each with natural numbers from 1 to 5 written at its vertices. The pentagons can be rotated and flipped. They are stacked vertex to vertex such that the sums of numbers at each of the five vertices are equal. Determine the number of pentagons in this stack.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the joint density function \( f_{X,Y}(x,y)=6e^{-x}e^{-2y} \) for \( 0<x<y< \infty \), where \( X \) and \( Y \) are the fail times for two components of a machine, find the expected time of failure of the machine. The component \( Y \) starts working if and only if component \( X \) fails, and the machine fails when component \( Y \) fails.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a hyperbola $x^{2}- \frac {y^{2}}{3}=1$ and two points $M$, $N$ on it are symmetric about the line $y=x+m$, and the midpoint of $MN$ lies on the parabola $y^{2}=18x$. Find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
[ Law of Sines ]

In a convex quadrilateral $A B C D$, the segment $C M$, connecting vertex $C$ with point $M$ located on side $A D$, intersects diagonal $B D$ at point $K$. It is known that $C K: K M = 2: 1, C D: D K = 5: 3$ and $\angle A B D + \angle A C D = 180^{\circ}$. Find the ratio of side $A B$ to diagonal $A C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be an $n$-dimensional vector space over a finite field $F$ of order $p^m$. Let $V_1$ be a $k$-dimensional subspace of $V$. Determine the number of $(n-k)$-dimensional subspaces $V_2$ of $V$ such that $V_1 + V_2 = V$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the area of the region in the first quadrant bounded by the line $y = \\frac{x}{2}$, the x-axis, and the ellipse $\\frac{x^2}{9} + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx,$ the y-axis, and the ellipse $\\frac{x^2}{9} + y^2 = 1.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum possible dimension of the annihilator $W^\circ$ of the subspace $W = \{A^m; m \geq 0\}$ within the vector space $V = M_n(R)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an equivalent expression for the remainder of the converging series \( R_n = \sum_{k=n+1}^{\infty} \frac{1}{k^3} \) given the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\iint_{S}(z-x^2-y^2)\,dS\) over the surface \(S\) of the cylinder \(x^2+y^2=4\) for \(-2\le z\le2\) using symmetry and geometry.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the total length of intervals on the number line where the minimum of the numbers \( 8 - x^{2} \) and \( \operatorname{ctg} x \) is not less than -1. Round your answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You roll a fair six-sided die repeatedly until you roll a 5. What is the expected value of the maximum number rolled before the first 5 appears? Exclude the roll of 5 from consideration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Frobenius norm \(||AA^+ - I||_F\), where \(A^+\) is the Moore–Penrose inverse of matrix \(A\). The matrix \(A\) has dimensions \(n \times m\) and rank \(r\). The Frobenius norm is defined as \(||A||_F = \sqrt{\text{Tr}(AA^T)}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function sequence defined by \( f_{1}(x)=|1-2x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), find the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the upper bound for the relative error in computing \((a+b)^2\) using floating point arithmetic, where \(a\) and \(b\) are arbitrary real numbers and \(\epsilon\) is the machine error.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of all power series with real coefficients.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability that a randomly selected integer is greater than a given integer $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A doctor gives a scientist a pack of 10 tablets for his absent-mindedness. Each time the scientist experiences absent-mindedness, he takes a tablet. If he sees only one tablet left, he orders a new pack. If the pack is empty, he discards it. What is the probability that at 10 a.m. on December 31st, the scientist will have exactly two packs of tablets in his cabinet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the inequality for $x > 0$: $$ \Gamma(x) \leq \frac{1}{x} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to partition the set \([8]\) into 3 disjoint pairs? For example, one such partition is \((12)(45)(67)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangle \(\Delta ABC\) inscribed in a circle of radius one unit, the internal angle bisectors of angles \(\angle A, \angle B, \angle C\) meet the circle at the points \(A_1, B_1, C_1\) respectively. Find the value of the expression:
\[ S = \frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the standard wave equation for small amplitudes, determine the position of a string $y(x,t)$ with the initial conditions $y(x,0) = \sin(x)$ and $\frac{\partial y}{\partial t}(x,0) = \cos(x)$. Express your solution in terms of $x$ and $t$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral of $z^{-1}$ over the contour defined by the square with vertices at $(1+i), (1-i), (-1-i), (-1+i)$ using Cauchy's Integral Formula and the calculus of residues.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence for the series \( \sum_{n\geq 1}\left(\frac{x}{\sin n}\right)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrix $A = I - uv^T$, determine the eigenvalues of $A^2$ in terms of the eigenvalues of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two particles start at the $x$-intercepts of a circle with a circumference of $12$ units, centered at the origin. Each second, both particles move independently one unit either clockwise or counterclockwise with equal probability. What is the expected number of seconds until the two particles simultaneously occupy the same point on the circle for the first time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal number of parties, $z$, that must be thrown to find out whether each pair of people from different groups like each other, given $n$ groups of $k$ people each, with no person in more than one group. Express $z$ in terms of $k$ and $n$. For example, if $n=k=2$, then $z=4$. Note that $k^2 \leq z \leq k^n$. Provide your answer as a function of $k$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest integer \( m \) such that \( n^{12} \equiv 1 \pmod{m} \) for all integers \( n \) that are relatively prime to \( m \), given that the Carmichael function \( \lambda(m) = 12 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two lines pass through point \(N\) and are tangent to a circle with center \(O\). Point \(A\) is on one line, and point \(B\) is on the other such that \(OA = OB\) and \(OA > ON\). Given \(NA = a\), \(NB = b\), and \(OA = c\) where \(a \neq b\), find the length of \(ON\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \[ g(g(y) - x) + 2x = g(g(x) + y) \] for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point is randomly selected from the interval [6, 11]. Let $k$ be the selected value. Determine the probability that the roots of the quadratic equation $(k^2 - 2k - 24)x^2 + (3k - 8)x + 2 = 0$ satisfy the condition $x_1 \leq 2x_2$. Express your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \frac{bc}{a^{5}} + \frac{ac}{b^{5}} + \frac{ab}{c^{5}} \) in terms of \( p \), where \( a, b, c \) are the roots of the polynomial \( x^3 - px + 1 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S (x^2 y^2 + y^2 z^2 + x^2 z^2)\, dS \) using the divergence theorem, where \( S \) is the entire surface of the sphere of unit radius centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the larger region cut from the solid sphere defined by \(x^2 + y^2 + z^2 = 4\) by the plane \(z = 1\). Use double integrals in spherical coordinates to find the solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the line integral \( \oint_C (y^2 \, dx - xy \, dy) \) where \( C \) is the boundary of the triangle with vertices \((1,0)\), \((4,0)\), and \((1,10)\), oriented counterclockwise. Use Green's Theorem to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=3} \frac{1}{2-z} e^{\frac{1}{z}} \,dz \) by finding the residue at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) \) be a polynomial with integer coefficients such that \( f(15) f(21) f(35) - 10 \) is divisible by 105. Given \( f(-34) = 2014 \) and \( f(0) \ge 0 \), find the smallest possible value of \( f(0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of permutations \(\sigma\) in \(S_6\) such that \(\gamma ' = \sigma \gamma \sigma ^{-1}\), where \(\gamma = (123)(45)(6)\) and \(\gamma ' = (1)(23)(456)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using ropes that burn irregularly, determine the least expensive way to measure exactly 1 minute. A rope that burns in 16 minutes costs 32 rupees, and a rope that burns in 7 minutes costs 14 rupees. You can light the ropes at one end, extinguish them, and relight them as needed. What is the minimum cost to measure exactly 1 minute?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Is there a base in which all prime numbers end with 5 different symbols? If such a base exists, identify it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given quadrilateral $ACDB$ and parallelogram $AEDF$, the following vector equations are satisfied: \[ 6 \overrightarrow{A E}+3 \overrightarrow{D E}=2 \overrightarrow{A C}, \] \[ 4 \overrightarrow{F B}+5 \overrightarrow{F D}=2 \overrightarrow{A D}. \] Determine the ratio of the area of quadrilateral $ACDB$ to the area of parallelogram $AEDF$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eight identical dice are rolled simultaneously. How many possible outcomes are there in which each of the six numbers appears at least once?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 1-dimensional characters for a nonabelian group of order 57.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 11 chess players and 5 distinct tables, in how many ways can the players be paired to play, considering that the order and position (color) of the players matter?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$ with circumcircle $\omega$, the bisector of $\angle ABC$ meets segment $AC$ at $D$ and circle $\omega$ at $M \neq B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E \neq B$, and $CE$ meets $\omega$ at $P \neq C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q \neq C$. Given that $PQ = MC$, find the degree measure of $\angle ABC$. \n\n\textit{(Ray Li)}
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values of $p > 0$ for which the improper integral \( \int_0^{\infty} x^p e^{-x^8 \sin^2 x} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1$ and $X_2$ be independent and identically distributed random variables with an exponential distribution, EXP(1). The joint PDF of $X_1$ and $X_2$ is given by $f(x_1, x_2) = e^{-x_1-x_2}$ for $x_1, x_2 \geq 0$. Find the joint PDF of $T_1 = \min(X_1, X_2)$ and $T_2 = X_1 + X_2$. Express your answer in terms of $t_1$ and $t_2$. Assume $t_1 \leq t_2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\int_{-\infty}^{+\infty} \frac{b}{1+2ax+(cx)^2} \cdot \frac{dx}{1-2ax+(cx)^2},$$ given that $a^2+b^2=c^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 110x110 array of guinea pigs, with 110 guinea pigs for each of the 110 species, determine the maximum integer n such that there is always a row or column containing at least n different species.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all values of \(a\) that satisfy the equation:
\[ \int_{0}^{\pi/2} (\sin x + a \cos x)^3 \, dx - \frac{4a}{\pi-2} \int_{0}^{\pi/2} x \cos x \, dx = 2. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int \pmb{f} \cdot d\pmb{\alpha} \) for the path \( \pmb{\alpha}(\theta) = R(\sin{\theta}\hat{i} + \cos{\theta}\hat{k}) \) where \( \theta \in [0, \pi] \), and the vector field \( \pmb{f}(r, \theta, \phi) = r^2(\cos{\theta}\hat{r} + \cos{\phi}\hat{\theta} - \cos{\theta}\sin{\phi}\hat{\phi}) \). Express \( \pmb{f}[\pmb{\alpha}(\theta)] \) when \( \pmb{\alpha} \) is in Cartesian coordinates and \( \pmb{f} \) is in spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \((AA^t)^r = I\) for an \(m \times n\) real matrix \(A\) and a natural number \(r\), determine the possible values of \(\det(AA^t)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of zeros of the polynomial \( h(z) = z^4 - z^3 + 1 \) in the region \( \{ z \in \mathbb{C} : \text{Re}(z) < 0 < \text{Im}(z) \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the lambda expression: $[[\lambda f.\lambda m. f(m + m^2))]([\lambda n.2n])](3)$. Perform the function evaluation correctly.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \to 0^+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of the term $x^3y^4z^5$ in the expansion of the polynomial $(x + y + z)^8(x + y + z + 1)^8$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the vector equation:
\[ (p^2 + q^2 |\mathbf{P}|^2 + (\mathbf{P} \bullet \mathbf{Q})^2)\mathbf{X} + (\mathbf{Q} \bullet \mathbf{X})(|\mathbf{P}|^2 \mathbf{Q} - (\mathbf{P} \bullet \mathbf{Q})\mathbf{P}) + (\mathbf{P} \bullet \mathbf{X})(|\mathbf{Q}|^2 \mathbf{P} - (\mathbf{P} \bullet \mathbf{Q})\mathbf{Q}) + (\mathbf{X} \bullet (\mathbf{P} \times \mathbf{Q}))(\mathbf{P} \times \mathbf{Q}) \]
where \(p\) and \(q\) are scalars, \(\mathbf{P}, \mathbf{Q}, \mathbf{X}\) are 3D vectors, and \(|\mathbf{P}|^2 \equiv (\mathbf{P} \bullet \mathbf{P})\). Given that \(p^2 + |\mathbf{P}|^2 = q^2 + |\mathbf{Q}|^2 = 1\), express the equation in terms of \(\mathbf{X}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the moment of inertia around the x-axis for the solid bounded by the planes $x=0$, $y=0$, $z=0$, and $x+y+z=1$, assuming a constant density.
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Context:
Determine the number of real numbers \(x\) that satisfy the equation:
\[
\left(|x^2-12x+20|^{\log{x^2}}\right)^{-1+\log x}=|x^2-12x+20|^{1+\log{\frac{1}{x}}}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the length of the longest simple circuit in a wheel graph $W_n$ for $n > 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of invertible $2\times2$ matrices over the field $\mathbb{Z}/p\mathbb{Z}$ that are unimodular, meaning their determinant is either $1$ or $-1$. Provide your answer as a single integer.
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Context:
Determine the ideal generated by \((2+3i, 4+5i)\) in the ring of Gaussian integers \(\mathbb{Z}[i]\).
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Context:
Evaluate the line integral \(\oint_{\partial T} \vec{F} \cdot \hat{a} \, ds\) for the vector field \(\vec{F}(x,y) = (2x^2 + 3x -2 \cos^4(y) \sin^3(y) , 4 e^{2x} \sinh(x) - 3y)\) and the triangle \(T\) with vertices \((-1,0)\), \((1,1)\), and \((0,2)\), traversed counterclockwise. The outward normal to \(T\) in the \(xy\) plane is \(\hat{a}\).
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Context:
Find the area of the region that lies between the polar curves \( r = 1 + \sin\theta \) and \( r = 1 + \cos\theta \).
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Context:
Given the function $f(x)=\frac{1}{2}x^{2}+(2a^{3}-a^{2})\ln x-(a^{2}+2a-1)x$, and $x=1$ is its extreme point, find the real number $a=$ \_\_\_\_\_\_.
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Context:
Determine the field to which the quotient \( \frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle} \) is isomorphic.
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Context:
Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monogrammed on their towels. Different letters may cost different amounts to monogram. If it costs \$$21$to monogram BRYAN, \$$25$ to monogram BARRY and \$$18$ to monogram SARAH, how much does it cost to monogram SHANA?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies within the government. Friendship and enmity are symmetric relations. A triplet of ministers is called **uniform** if all three are either mutual friends or mutual enemies. How many uniform triplets are there?
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Context:
In a one-dimensional lattice $\mathbb{Z}^1$, a thief starts at the origin (0) and $N$ policemen are initially positioned at 2. All participants perform independent random walks, where each step to the left or right occurs with equal probability (1/2). Let $\tau_N$ denote the first encounter time between the thief and a policeman. Given that $E\tau_1=\infty$, find the smallest $N$ for which $E\tau_N<\infty$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the mass of the tetrahedron bounded by the planes \(x=0\), \(y=0\), \(z=0\), and \(\frac{x}{10} + \frac{y}{8} + \frac{z}{3} = 1\), given that the mass density at each point is \(\rho = \left(1 + \frac{x}{10} + \frac{y}{8} + \frac{z}{3}\right)^{-6}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular triangular prism \(ABC A_1 B_1 C_1\) is inscribed in a sphere, with \(ABC\) as its base and lateral edges \(A A_1, B B_1, C C_1\). The segment \(CD\) is the diameter of this sphere. Point \(K\) is the midpoint of edge \(A A_1\), and point \(L\) is the midpoint of edge \(A B\). Given that \(D L = \sqrt{6}\) and \(D K = 3\), find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the supremum of \( \left| z + e^{i\theta} w \right| \) for any complex numbers \( z \) and \( w \), where \( \theta \in [0, 2\pi] \).
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Context:
Find the second derivative $F''(\pi)$ of the function $$ F(x) = \int_x^{\sin x} \left( \int_0^{\sin t} \sqrt{1+u^4} \, du \right) dt $$ using the fundamental theorem of calculus.
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Context:
An ant is sitting at the center of a circle with a radius of 3 meters. Every minute, the ant randomly chooses a direction and moves 1 meter in that direction. On average, how many minutes does it take for the ant to leave the circle? Assume that if the ant reaches the edge of the circle within a minute, it is considered to have left the circle at the end of that minute.
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Context:
If the complex number \( z \) satisfies \( |z| = 2 \), find the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \).
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Context:
Determine an upper bound for \(|\cot z|\) on the boundary of the square with vertices at \(k\pi(1+i)\), \(k\pi(-1+i)\), \(k\pi(-1-i)\), and \(k\pi(1-i)\), where \(k\) is a natural number.
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Context:
Let X be a random variable uniformly distributed over the interval [0,2]. Find the expected value of \( \max(X, X^3) \).
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Context:
Determine the index \([B : A]\) of the subgroup \(A\) in the subgroup \(B\) of the group \(\mathbb{Q}[x]\) of one variable rational polynomials under addition, where:

- \(A = \{p(x) \in \mathbb{Z}[x]\mid p\) has degree at most \(2\}\), and
- \(B = \{p(x) \in \mathbb{Q}[x] \mid p\) has degree at most \(2\), and \(p(\mathbb{Z}) \subseteq \mathbb{Z}\}\).
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Context:
In preparation for the World Cup, Coach F plans to evaluate seven players, labeled as \( A_{1}, A_{2}, \cdots, A_{7} \), over three training matches, each lasting 90 minutes. At any given time during a match, only one player is on the field. The total playing time for players \( A_{1}, A_{2}, A_{3}, A_{4} \) must be divisible by 7, and the total playing time for players \( A_{5}, A_{6}, A_{7} \) must be divisible by 13. With unlimited substitutions allowed, determine the number of different scenarios for the total playing time of each player.
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Context:
Given that $a^x = b^y = c^z$ and $a^3 = b^2c$, find the value of $\dfrac{3}{x} - \dfrac{2}{y}$.
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Context:
Find the number of distinct elements in the smallest subgroup of \(GL(2,\mathbb{R})\) that contains the matrices \(A = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the complex number $z$ that satisfies the equation $z^{1+i} = 4$.
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Context:
Determine the pointwise limit of the sequence of functions \( f_n(x) = x^2 - \frac{x^n}{n} \).
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Context:
Evaluate the triple integral \( \int_A |z| \,dx \, dy \, dz \) where \( A = \{(x,y,z): x^2+y^2+z^2\le4, x^2+y^2-2y\le0\} \).
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Context:
Gary purchased a large beverage, but drank only $m/n$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m+n$.
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Context:
Let $z_1$ and $z_2$ be two nth roots of unity that form a right angle at the origin. If the roots are represented as $z_k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right)$ for $k = 1, 2$, determine the condition on $n$ such that the angle between $z_1$ and $z_2$ is $\frac{\pi}{2}$ radians.
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Context:
Determine the values of $p$ for which a function in the Sobolev space $W^{1,p}(\Omega;\mathbb{R}^n)$ has a continuous representative, given that $\Omega$ is a bounded and smooth domain in $\mathbb{R}^n$.
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Context:
A crate contains blocks of four different dimensions: $13 \times 17 \times 21$, $13 \times 17 \times 37$, $13 \times 21 \times 37$, and $17 \times 21 \times 37$, with $33$ blocks of each kind. A tower is constructed by stacking $100$ of these blocks, one from each cross-section, to form a tower of height $h$. Determine the number of unique possible values for $h$. \(\boxed{\text{?}}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
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Context:
Evaluate the integral: $$\int_0^\infty \frac{\cos{kt}}{\pi}\,\mathrm{d}k.$$
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Context:
Calculate the integral \( \int_{0}^{1}\int_{0}^{2-2x}\frac{\left(2x-y\right)^{2}}{2x+y}dydx \) using the substitution \( u=2x-y \) and \( v=2x+y \).
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Context:
In an equilateral triangle $ABC$, a point $P$ is such that $|AP| = 2$ and $|BP| = 3$. Determine the maximum possible value of $|CP|$.
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Context:
The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by \[V=\left\{   \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.\]For each $j$, $1\leq j\leq 12$, an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected.

The probability that $P=-1$ can be expressed in the form
\[\frac{a}{p^b},\]where $a,$ $b,$ $p$ are positive integers, $p$ is prime, and $a$ is not divisible by $p.$  Find $a + b + p.$
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Context:
Given a random sample \( X_1, X_2, \ldots, X_n \) from a population with density function \( f(x) = \begin{cases} e^{\theta-x}, & x \geq \theta \\ 0, & \text{otherwise} \end{cases} \), find the maximum likelihood estimator for \( \theta \).
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Context:
Determine the values of $z$ for which the function $\exp(-2 z\sin^2 t)$ is the characteristic function of a discrete random variable.
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Context:
A company had a total sales revenue of 10 million yuan last year, with annual costs consisting of 5 million yuan in production costs and 2 million yuan in advertising costs. If the annual profit is taxed at a rate of p% and any annual advertising expenses exceeding 2% of the annual sales revenue are also taxed at p%, with no other taxes applied, and the company paid a total tax of 1.2 million yuan last year, calculate the tax rate p%. Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all polynomials \( P(x) \) that satisfy
$$
(x-1)P(x+1) - (x+2)P(x) \equiv 0, \quad x \in \mathbb{R}.
$$
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Context:
Consider a finite field \( F \) with cardinality \( q \). Let \( F_k[x] \) be the set of polynomials in \( F[x] \) with degree less than or equal to \( k \). Given distinct elements \( x_1, \ldots, x_n \) in \( F \), define a code \( C \) as in the original question. Determine the minimum distance of this code.
Please reason step by step, and put your final answer within \boxed{}.


Context:
30 students participated in the mathematical Olympiad. Each of them was given 8
 problems to solve. Jury estimated their work with the following rule:
 1）Each problem was worth \( k \) points,if it wasn’t solved by exactly \( k \) students;
 2) Each student received the maximum possible points in each problem or got 0 in it
 Lasha got the least number of points. What's the maximal number of points he could
 have?
 Remark: 1) means that if the problem was solved by exactly \( k \) students,than each of
 them got \( {30} - k \) points in it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a power function $y=f(x)$ that passes through the point $(4, \frac{1}{2})$, find the value of $f\left( \frac{1}{4} \right)$.
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Context:
8-8. A chess tournament is held according to the following system: each of the 15 students from the "White Rook" school must play one game with each of the 20 students from the "Black Bishop" school, i.e., a total of 300 games should be played. At any given time, no more than one game is played.

After $n$ games, a spectator named Sasha, who has watched all the games and knows all the participants, exclaims: "I can definitely name one of the participants of the next game!" What is the smallest $n$ for which this could happen?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $ A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $ 7$ times as long as it is wide, inside rectangle $A$, so that all four vertices of $ B$ lie on sides of $ A$, and again none of the sides of $B$ are parallel to any side of $A$. Find the length of the shortest side of rectangle $ B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Cindy has a collection of identical rectangular prisms. She stacks them, end to end, to form $1$ longer rectangular prism. Suppose that joining $11$ of them will form a rectangular prism with $3$ times the surface area of an individual rectangular prism. How many will she need to join end to end to form a rectangular prism with $9$ times the surface area?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distribution of the random variable $Y = X^2$, where $X$ is a standard normal random variable.
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Context:
What is the value of $\left(\left((3+2)^{-1}-1\right)^{-1}-1\right)^{-1}-1$?
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Context:
Determine the big-O bound for the following expression:
\[ \sum^{ N}_{k=0}{2^k \log_2\left(\frac{2^N}{2^k}\right)} \]
where \( N = \log M \).
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Context:
Find the sum of the eccentric angles \(\alpha, \beta, \gamma,\) and \(\delta\) of four concyclic points on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
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Context:
For how many integers $1\leq n \leq 2010$, $2010$ divides $1^2-2^2+3^2-4^2+\dots+(2n-1)^2-(2n)^2$?
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Context:
Suppose you draw random variables $x_i$ from the interval $[0,1]$, where each $x_i$ is uniformly distributed. If $x_i > x_{i-1}$, you draw again; otherwise, you stop. What is the expected number of draws before the sequence stops?
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Context:
Find a closed-form expression that bounds the sum \( \sum_{i=1}^{\log^* N} \log^{(i)}N \), where \( \log^* \) denotes the iterated logarithm function and \( \log^{(i)}N \) represents the logarithm function applied iteratively \( i \) times.
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Context:
Solve the complex equation \((z+1)^4 = (\bar{z}+i)^4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. The base of the pyramid $P A B C D$ is a trapezoid $A B C D$, the base $A D$ of which is twice the length of the base $B C$. The segment $M N$ is the midline of the triangle $A B P$, parallel to the side $A B$. Find the ratio of the volumes of the solids into which the plane $D M N$ divides this pyramid.
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Context:
Find a space that is homeomorphic to the quotient space $\mathbb{R}^3 / \sim$, where the equivalence relation $\sim$ is defined by $(x, y, z) \sim (x, y', z')$ for any $y', z' \in \mathbb{R}$.
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Context:
Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}=\angle{PBC}=\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}+ S_{B_1CA}+ S_{C_1AB} \geq \lambda S_{ABC}$.
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Context:
Evaluate the limit of the series as \( x \to \infty \):
\[ \lim_{x \rightarrow \infty} \sum_{n=0}^\infty (-1)^{n} \frac{x \lvert x \rvert^{2n}}{2n+1} \]
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Context:
Given a parabola \( y^2 = 6x \) with two variable points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the x-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \).
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Context:
1. Evan's analog clock displays the time $12: 13$; the number of seconds is not shown. After 10 seconds elapse, it is still $12: 13$. What is the expected number of seconds until $12: 14$ ?
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Context:
Determine the smallest number of subsets, $n$, into which the set $S = \{1, 2, \dots, 2022\}$ can be partitioned such that for each subset $S_i$, either all pairs of distinct elements have a GCD greater than 1, or all pairs of distinct elements have a GCD equal to 1.
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Context:
Let $S$ be the set of all points $(x,y)$ in the coordinate plane such that $0\leq x\leq \frac{\pi}{2}$ and $0\leq y\leq
\frac{\pi}{2}$. What is the area of the subset of $S$ for which \[
\sin^2 x -\sin x \sin y +\sin^2 y \leq\frac{3}{4}\,?
\]
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Context:
The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased?
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Context:
Determine the number of distinct real roots of the equation:
\[ \sum\limits_{i = 1}^{40}|a_i - x| = \sum\limits_{i = 1}^{40}|b_i - x|. \]
Here, \(a_i\) and \(b_i\) are real numbers.
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Context:
In a right triangle ABC with \(\angle B=90^\circ\), two equilateral triangles ABD and BEC are drawn externally on the legs of triangle ABC. Let G, H, and F be the midpoints of BE, BC, and DC, respectively. If the area of triangle ABC is 32, find the area of triangle GHF.
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Context:
Let $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$. Then there are rational numbers $A$ and $B$ such that:

$$\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}.$$

Find $A + B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the iterated integral \( \iint f \, \mathrm{d}\nu \, \mathrm{d}\mu \) for the function \( f(m,n) \) defined as follows: \( f(m,n) = 1 \) if \( m = n \), \( f(m,n) = -1 \) if \( m = n+1 \), and \( f(m,n) = 0 \) otherwise, where \( \mu = \nu \) are counting measures on \( \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 4.1. Along the road connecting Masha's and Sasha's houses, there are 17 apple trees and 18 poplars. When Masha was going to visit Sasha, she took photos of all the trees. Right after the tenth apple tree, Masha's phone memory ran out, and she couldn't photograph the remaining 13 trees. The next day, when Sasha was going to visit Masha, starting from the eighth apple tree, he picked one leaf from each tree. How many leaves did Sasha pick?
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Context:
In a knockout tournament with 55 boxers, each match is held sequentially. In every match, the difference in the number of previous victories between the two participants is at most 1. What is the maximum number of matches the tournament winner could have fought?
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Context:
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, the right vertex is $A(a, 0)$. There exists a point $P$ on the ellipse such that $\angle APO = 90^{\circ}$. Determine the range of values for the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{\sqrt{2}}{2}, 1 \right)}


Context:
Find the range of the expression $\frac{a \cos x - b \sin x + 2c}{\sqrt{a^2 + b^2 + c^2}}$ where $x, a, b,$ and $c$ are arbitrary numbers such that $a^2 + b^2 + c^2 \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Children are assembling robot toys. Initially, every 2 children cooperate to assemble 1 small robot toy, then every 3 children cooperate to assemble 1 large robot toy. A total of 30 robot toys of various sizes were assembled. How many children are there in total?
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Context:
Calculate the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, with no repeated digits.
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Context:
If real numbers \(x\) and \(y\) satisfy \(x^2 + y^2 = 20\), what is the maximum value of \(xy + 8x + y\)?
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Context:
Find the closed form of the following series: $$\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}.$$
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Context:
Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item.
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Context:
Seven old women are going to Rome. Each has seven mules, each mule carries seven bags, each bag contains seven loaves of bread, each loaf contains seven knives, each knife is in seven sheaths. How many total items are there?
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Context:
Tim and Allen are playing a match of *tenus*.  In a match of *tenus*, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability  $3/4$ , and in the even-numbered games, Allen wins with probability  $3/4$ .  What is the expected number of games in a match?
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Context:
An amount is invested in an account that earns 6% interest compounded semi-annually. The interest earned after one year is $6 greater than if the same amount had been invested at 8% simple yearly interest. What is the amount invested?
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Context:
A group of people are playing paintball. In a given situation, all distances between individuals are different. Everyone shoots at the person closest to them. What is the maximum number of people that can shoot at the same person?
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Context:
Find the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter is $A$, the last letter is $B$, and the sequence contains no three consecutive letters reading $ABA$. For example, count $AAABBABBAB$ and $ABBBBBBBAB$ but not $AABBAABABB$ or $AAAABBBBBA$.
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Context:
X is the number of 2 groups of 10 and 6 units, and Y is the number of 4 groups of 10 and 1 unit. Find the sum of X and Y.
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Context:
We drew 11 horizontal and 11 vertical intersecting straight lines on a sheet of paper. We call a segment of one of the straight line drawn that connects two neighboring intersections a “link”. What least number of links must we erase in order for each intersection to be a junction of not more than 3 links?
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Context:
设取整 $\left[\frac{10^{2022}}{10^{100}-9}\right]$ 被 $10^{n}$ 整除, 则非负整数 n的最大可能值为 $\qquad$.
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Context:
Determine the number of ring homomorphisms from $\mathbb{Z}_{12}$ into $\mathbb{Z}_{24}$ using the formula provided.
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Context:
What is the smallest number which when increased by a certain number is divisible by 27, 35, 25, & 21 and the result is 4728?
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Context:
24. Use red, pink, yellow, and black 4 colors to paint the 6 sides of a regular hexagon, so that the sides painted yellow and black each have one, and the sides painted red and pink each have 2. Mr. Li has some color blindness. For any two colored regular hexagons $A_{6}$ and $A_{6}^{\prime}$, if $A_{6}^{\prime}$ is obtained by changing the two sides painted red in $A_{6}$ to pink, and the two sides painted pink to red, Mr. Li, due to his color blindness, considers $A_{6}$ and $A_{6}^{\prime}$ to be the same. Find the number of distinct colorings of the hexagon under Mr. Li's identification method.
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Context:
10. In the Cartesian coordinate system $x O y$, it is known that the vertices of the equilateral $\triangle A B C$ (vertices in counterclockwise order) are $A(-2,2)$, vertex $B$ is on the $y$-axis, and $C B \parallel x$-axis; point $D(2,6)$, $E(2,2)$, and the radius of $\odot O$ is 2. Find the radius of the circle that is externally tangent to $\odot O$ and tangent to line segments $A B$ and $D E$ (accurate to $10^{-4}$).
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Context:
The racing magic takes some time to circle the racing track once. The charging bull makes 40 rounds of the track in an hour. If they left the starting point together, it took them 6 minutes to meet at the starting point for the second time. How many seconds does it take for the racing magic to circle the track once?
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Context:
A parallelogram contains two circles. One of them, with a radius of 3, is inscribed in the parallelogram, while the second circle touches two sides of the parallelogram and the first circle. The distance between the points of tangency on one side of the parallelogram is 3. Find the area of the parallelogram.
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Context:
Calculate the integral \( \int_{\gamma} \sqrt{z} \, dz \) where \( \gamma = C(2,1)^+ \), integrating over the principal value of \( \sqrt{z} \).
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Context:
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.

Hint: The equation of state for the gas is given by $\rho V=$ const, where $\rho$ is pressure and $V$ is volume.

Given:
$$
H=0.4 \text{ m}, \ h=0.2 \text{ m}, \ R=0.1 \text{ m}
$$
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Context:
A sphere is tangent to all the edges of the pyramid \( SABC \), specifically the lateral edges \( SA, SB, \) and \( SC \) at the points \( A', B' \), and \( C' \) respectively. Find the volume of the pyramid \( SA'B'C' \), given that \( AB = BC = SB = 5 \) and \( AC = 4 \).
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Context:
There were electronic clocks on the International Space Station, displaying time in the format HH:MM. Due to an electromagnetic storm, the device started malfunctioning, and each digit on the display either increased by 1 or decreased by 1. What was the actual time when the storm occurred if the clocks showed 00:59 immediately after it?
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Context:
设 $\overrightarrow{O A}$ 与 $\overrightarrow{O B}$ 的夹角为 $\theta,|\overrightarrow{O A}|=2,|\overrightarrow{O B}|=1, \overrightarrow{O P}=t \overrightarrow{O A}, \overrightarrow{O Q}=$ $(1-t) \overrightarrow{O B}$, 函数 $f(t)=|\overrightarrow{P Q}|$ 在 $t=t_{0}$ 时取最小值. 若 $0<t_{0}<\frac{1}{5}$, 求 $\theta$ 的取值范围。答案的标准形式为$\left(\frac{\pi}{a}, \frac{b \pi}{c}\right)$，请给出区间端点a+b+c之和。
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Context:
Remi prepared a tomato nursery and planted tomato seedlings. After 20 days, the seedlings were ready to be transferred. On the first day, he planted 200 seedlings on the farm. On the second day, while working alongside his father, he planted twice the number of seedlings he planted on the first day. On the third day, his father planted three times the amount Remi planted, and on the fourth day, his father planted four times the amount Remi planted. If the total number of seedlings transferred to the farm on these four days was 5000, how many seedlings did his father plant?
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Context:
[ $\left[\begin{array}{l}\text { Auxiliary circle } \\ {[\text { Law of Sines }}\end{array}\right]$

In quadrilateral $A B C D$, it is known that $\angle A B D=\angle A C D=45^{\circ}, \angle B A C=30^{\circ}, B C=1$. Find $A D$.
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Context:
Example 10 Let $x, y, z$ be positive real numbers, and satisfy $x y z + x + z = y$, find
$$
p=\frac{2}{x^{2}+1}-\frac{2}{y^{2}+1}+\frac{3}{z^{2}+1}
$$

the maximum value.
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Context:
In a physical education class, 25 students from class 5"B" stood in a line. Each student is either an honor student who always tells the truth or a troublemaker who always lies.

Honor student Vlad took the 13th position. All the students, except Vlad, said: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are in the line in total?
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Context:
Two real numbers  $x$  and  $y$  are such that  $8y^4+4x^2y^2+4xy^2+2x^3+2y^2+2x=x^2+1$ .
Find all possible values of  $x+2y^2$
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Context:
If the sides of a square are multiplied by 10, how many times as large is the area of the original square compared to the area of the resultant square?
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Context:
For positive integer  $n$ , let  $s(n)$  denote the sum of the digits of  $n$ . Find the smallest positive integer  $n$  satisfying  $s(n)=s(n+864)=20$ .
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Context:
Mia is helping her mom clean up 30 toys scattered on the floor. Every 30 seconds, her mom puts three toys into the storage box, and at the same time, Mia takes out two of the toys that were put in during the previous 30 seconds. After how many minutes will Mia and her mom first put all the toys into the storage box?
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Context:
The bases of trapezoid \(ABCD\) are related by the equation \(AD = 4 \cdot BC\). The sum of the angles \(\angle A + \angle D\) is \(120^\circ\). Points \(M\) and \(N\) are chosen on the legs of the trapezoid such that \(CN:ND = BM:MA = 1:2\). Perpendiculars drawn at points \(M\) and \(N\) to the legs of the trapezoid intersect at point \(O\). Find \(AD\), given that \(AO = 1\).
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Context:
If a certain negative integer multiplied by a and -8b are negative integers, then the product of these integers plus ab is 89. What is the integer multiplied by a?
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Context:
 Let $ABCD$ be a square. Points $E$ and $F$ lie outside of $ABCD$ such that $ABE$ and $CBF$ are equilateral triangles. If $G$ is the centroid of triangle $DEF$, then find $\angle AGC$, in degrees
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Context:
What is $\sqrt[2015]{2^01^5}$?
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Context:
Given the sets $A=\left\{{x|{x=\frac{k}{6}+1,k∈Z}}\right\}$, $B=\left\{{x|{x=\frac{k}{3}+\frac{1}{2},k∈Z}}\right\}$, and $C=\left\{{x|{x=\frac{{2k}}{3}+\frac{1}{2},k∈Z}}\right\}$, determine the subset relationships among A, B, and C. Express your answer using ⊆ to denote subset relationships, and include all sets in your answer.
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Context:
What is the maximum number of non-attacking rooks that can be placed on an $n \times n \times n$ cube, where each rook can attack any square reachable vertically or horizontally, spanning a total of $4n$ squares in each direction?
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Context:
Given the proposition ∀x > 1, x² + x + 1 > 0, what is its negation? Express your answer as a logical statement.
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Context:
An elementary school teacher is making Halloween goodie bags for her class. She wants the bags to be personalized, so she surveys her students asking whether they'd like a vampire-themed bag or a pumpkin-themed bag. Of her 25 students, 11 indicate they want the vampire-themed bag and 14 indicate they want the pumpkin-themed bag. The store the teacher shops at sells packs of 5 of each theme at a price of $3 per package, as well as individual bags of each theme at a price of $1 each. The store has multiple special offers: "Buy 3 packages, Get 1 package Free", "Buy 5 individual bags, Get 1 Free", and a "10% discount on the entire purchase if you spend more than $10". What is the minimum amount of money the teacher can spend on the bags if she buys every student the theme they requested while maximizing the benefits of these special offers?
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Context:
For a set $A = \{1, 2, 3, \ldots, n\}$, determine the maximum size of a transitive relation $T$ on $A$ such that $T \neq A \times A$. What is the largest possible size for $T$?
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Context:
Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as \(\frac{m}{n}\) for relatively prime positive integers $m$ and $n$. Compute $m+n$.
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Context:
Find the minimal integer \( k \) such that the number of \( q \)-torsion points of a non-supersingular elliptic curve \( E \) over \( GF(q) \) is equal to \( q \).
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Context:
Given the set $S=\{(x,y)|1\leqslant x\leqslant 10,1\leqslant y\leqslant 10,x\in \mathbb{N},y\in \mathbb{N}\}$, find the maximum number of elements in a subset $A\subseteq S$ such that for any $(a,b)\in A$ and $(s,t)\in A$, the condition $(a-s)(b-t)\leqslant 0$ holds. Express your answer as a single integer.
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Context:
Determine the number of tuples \((k_1, k_2, \ldots, k_{26})\) such that each \(k_i\) is either 0, 1, or 3, and the sum \(k_1 + k_2 + \ldots + k_{26} = 15\).
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Context:
Let  $F$  be a finite field having an odd number  $m$  of elements. Let  $p(x)$  be an irreducible (i.e. nonfactorable) polynomial over  $F$  of the form   $$ x^2+bx+c, ~~~~~~ b,c \in F. $$   For how many elements  $k$  in  $F$  is  $p(x)+k$  irreducible over  $F$ ?
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Context:
Calculate the expectation of the Ito integral \( \int_0^3 B_t^3 \, dB_t \) using Ito's formula and the function \( f(t, B_t) = \frac{B_t^4}{4} \).
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Context:
Consider the linear system of differential equations $\vec{y}'(t) = A\vec{y}(t)$, where $A$ is a real $2 \times 2$ constant matrix with a repeated eigenvalue. Assume that solution trajectories have horizontal tangents on the line $y_2 = 2y_1$ and vertical tangents on the line $y_1 = 0$. Given that $a_{21} = -4$ and the repeated eigenvalue is nonzero, determine the matrix $A$.
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\boxed{\begin{bmatrix} 2 & 0 \\ -4 & 2 \end{bmatrix}}


Context:
A hall 36 m long and 15 m broad is to be paved with stones, each of a certain length and 5 dm broad. The number of stones required is 3600. What is the length of each stone?
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Context:
Evaluate the sum \( \displaystyle \sum_{n=2}^{\infty}\log\left(1-\frac{1}{n^2}\right) \).
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Context:
Evaluate the integral \(\int_0^{2\pi}\int_{\pi/3}^{\pi/2}\int_{2\sec(\varphi)}^4\cos(\varphi)\sin(\varphi)\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta\) using spherical coordinates.
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Context:
Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:

\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]

and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \).
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Context:
The decimal expansion of $37^9$ is $129A617B979C077$ for digits $A, B,$ and $C$. Find the three digit number $ABC$.
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Context:
Calculate the sum of the series \(\sum_{n=0}^{\infty} \frac{c_{n}d_{n}}{9^{n}}\), where \(c_{n} + id_{n} = (3 + i)^{n}\) and \(c_{n} - id_{n} = (3 - i)^{n}\).
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Context:
Henry walks $\tfrac{3}{4}$ of the way from his home to his gym, which is $2$ kilometers away from Henry's home, and then walks $\tfrac{3}{4}$ of the way from where he is back toward home. Determine the difference in distance between the points toward which Henry oscillates from home and the gym.
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Context:
Consider the function \( f(x,y,z) = 2^y3^za^xb^yc^z + 2^z3^xa^yb^zc^x + 2^x3^ya^zb^xc^y \) where \( a, b, c \in \mathbb{R} - \{0\} \) and \( x, y, z \in \mathbb{Z} \). Given that \( f(1,0,0) = 4 \) and \( f(2,0,0) = 6 \), find the maximum value of \( \lfloor a \rfloor^2 + \lfloor b \rfloor^2 + \lfloor c \rfloor^2 \).
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Context:
For which values of $c \in \mathbb{R}$ can the level set $M_c = \{(x, y) \mid x^2 + 2xy + 3y^2 = c\}$ be described locally as a $C^1$ graph at every point?
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Context:
Calculate the integral $\int_{-\infty}^\infty x^2e^{-\frac {(x-\mu)^2}{2\sigma^2}}dx$ using analytical methods.
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Context:
Evaluate \( \lim\limits_{n\to +\infty}\int\limits_{[0,n]}\left(1-\frac{x}{n}\right)^ne^{\frac{x}{2}}dx \) using the Dominated Convergence Theorem.
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Context:
Calculate the probability that a random permutation of the letters in the word POSSESSIONS contains exactly two occurrences of the substring "SS", with at least one letter separating the two occurrences.
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Context:
Determine the minimum number of points in a set \( S \) required to ensure at least \( m \) intersection points between two polynomials \( P(x) \) and \( Q(x) \), given that both polynomials pass through at least \( p \) points in \( S \).
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Context:
If x gets 25% more than y and y gets some percentage more than z, the share of z out of Rs. 925 is Rs. 250. What is the percentage by which y gets more than z?
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Context:
A bag contains four balls, each labeled with one of the characters "美", "丽", "惠", "州". Balls are drawn with replacement until both "惠" and "州" are drawn, at which point the drawing stops. Use a random simulation method to estimate the probability that the drawing stops exactly on the third draw. Use a computer to randomly generate integer values between 0 and 3, with 0, 1, 2, and 3 representing "惠", "州", "美", and "丽" respectively. Each group of three random numbers represents the result of three draws. The following 16 groups of random numbers were generated:
232 321 230 023 123 021 132 220
231 130 133 231 331 320 122 233
Estimate the probability that the drawing stops exactly on the third draw.
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Context:
Given nine cards with the numbers $5,5,6,6,6,7,8,8,9$ written on them, form three three-digit numbers $A, B, C$, each of whose digits are different. What is the minimum value of the expression $A + B - C$?
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Context:
Find $$\sum_{i=1}^{100} i \cdot \gcd(i, 100).$$
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Context:
Solve the inequality \( \sin x > \ln x \). Determine the solution set where the sine function is greater than the natural logarithm function.
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Context:
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(tx) \geq f(x)$ for all $t > 0$ and $x \in \mathbb{R}$. Given $a, b \in \mathbb{R}$ with $a < b$, determine the value of $\int_a^b f(x) \, dx$. \( \text{Provide your answer as a single value.} \)
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Context:
Evaluate the limit of the integral \( \int_{\pi}^0 \frac{e^{ire^{i\theta}}}{r^2e^{i2\theta}+1} d\theta \) as \( r \to 0 \), where the integral is taken along a small semicircular contour of radius \( r \) around the singularity at \( z=0 \).
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Context:
Provide an example of a nonabelian group in which all proper subgroups are normal.
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Context:
Calculate the gradient of the function \( f(X) = \text{tr}((AX)^t (AX)) \) with respect to the matrix \( X \).
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Context:
Martians love dancing dances that require holding hands. In the dance "Pyramid," no more than 7 Martians can participate, each with no more than three hands. What is the maximum number of hands that can be involved in the dance if each hand of one Martian holds exactly one hand of another Martian?
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Context:
Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not.
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Context:
A needle 2.5 cm long is dropped onto a piece of paper with parallel lines drawn 2.25 cm apart. What is the probability that the needle will lie entirely between two lines?
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Context:
Let $cov(null)$ be the minimum cardinality of a set of nullsets of $\mathbb{R}$ so that their union equals $\mathbb{R}$. In a model $M[G]$ obtained from a ctm $M$ with $M\models CH$ by adding $\aleph_2$ many Cohen reals with forcing, we have $M[G]\models 2^{\omega}=\aleph_2$. What is the value of $cov(null)^{M[G]}$?
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Context:
A can do a piece of work in 20 days, B can do it in some days, and C can do it in 10 days. They started the work together, and A leaves after 2 days and C leaves after 4 days from the beginning. The work lasts 15.000000000000002 days. How many days does it take for B to do the work alone?
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Context:
Given a positive integer S, determine the maximum value of N such that the sum of all elements in a 3D array A of dimensions N x N x N, where each element A[i][j][k] is defined as ik(i+j+k) for 1 <= i, j, k <= N, does not exceed S. Constraints: 3 <= S <= 10^{16}. For example, if S = 720, the maximum value of N is 3.
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Context:
Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]
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Context:
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ respectively. Let $R$ be the radius of the circumcircle of $\triangle ABC$, and given that $2R(\sin^2 A - \sin^2 B) = (\sqrt{2}a - c)\sin C$, find the measure of angle $B$.
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Context:
Evaluate the Cauchy principal value of the integral: $$\mrm{P.V.}\int_{0}^{\infty} \frac{\dd x}{x^3 - 1}.$$
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Context:
Given the joint probability density function of the lifetimes of two circuits $C1$ and $C2$ in an electronic device:
\[
f(x,y) = \begin{cases}
\frac{24}{(x+y)^4},  & \text{x,y > 1} \\
0, & \text{otherwise}
\end{cases}
\]
where $X$ and $Y$ are continuous random variables representing the lifetimes of $C1$ and $C2$, respectively. Calculate the expected value of the time until the device is completely corrupted.
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Context:
In the Cartesian coordinate system $xOy$, the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has its right focus at $F(c, 0)$. If there is a line $l$ passing through the point $F$ that intersects the ellipse at points $A$ and $B$, such that $OA \perp OB$, find the range of values for the eccentricity $e=\frac{c}{a}$ of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[ \frac{\sqrt{5} - 1}{2}, 1 \right)}


Context:
例3 设S={1,,2,⋯,15}.从S中取出n个子集$A _ { 1 }$,$A _ { 2 } , \cdots , A _ { n }$，满足下列条件：

(1)$\vert A _ { i } \vert = 7$(i=1,2,⋯,n),

(2)$\vert A _ { i } \cap A _ { j } \vert \leq 3$(1≤i<j≤n)(,

（3）对S中任何3元子集M，存在某个$A _ { k }$，使$M \subset A _ { k } .$

求这样的子集个数n的最小值．（第40届IMO中国国家队选拔考试题）
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Context:
Given a cube of unit side. Let  $A$  and  $B$  be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point  $A$  and tangent to the three sides with common point  $B$ .
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Context:
Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2+xy+y^2=m$ intersects the hyperbola $xy=n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81\le n \le 81$.
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Context:
Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$.
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Context:
Let \( n \) be a positive integer. A sequence of \( n \) positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) is called a "full sequence" if it satisfies the following conditions: For any positive integer \( k \) (\( k \geq 2 \)), if \( k \) is an element of the sequence, then \( k-1 \) is also an element of the sequence, and the first appearance of \( k-1 \) in the sequence is before the last appearance of \( k \). For each \( n \), how many "full sequences" are there?
(43rd IMO Preliminary Problem)
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Context:
p runs at a certain percentage faster than q, so p gives q a 50-meter head start. If the race ends in a tie, p ran 300 meters in the race. What is the percentage by which p runs faster than q?
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Context:
Find a complex number $z=\_\_\_\_\_\_$ that satisfies the following two conditions:① $|z|=\sqrt{7}$; ② The point corresponding to $z$ in the complex plane is located in the second quadrant.
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Context:
A frog starts climbing from the bottom of a 12-meter deep well at 8:00 AM. For every 3 meters it climbs up, it slides down 1 meter due to the slippery walls. The time to slide down 1 meter is one-third the time taken to climb up 3 meters. At 8:17 AM, the frog reaches 3 meters from the well's top for the second time. Determine the total time the frog spends climbing from the bottom to the top of the well in minutes.
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Context:
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$.  Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$.  Find the smallest possible value of $m$.
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Context:
2. Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
(1 mark)
Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
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Context:
A freight train leaves the town of Jenkinsville at 1:00 PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at 10 miles per hour. At 1:12 PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at 3.5 miles per hour directly towards the road on which Julie is riding. Jim reaches the road at 2:12 PM, just as Julie is riding by. What is the speed of the train in miles per hour?
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Context:
Suppose that $(a_{1}, \ldots, a_{20})$ and $(b_{1}, \ldots, b_{20})$ are two sequences of integers such that the sequence $(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20})$ contains each of the numbers $1, \ldots, 40$ exactly once. What is the maximum possible value of the sum $\sum_{i=1}^{20} \sum_{j=1}^{20} \min (a_{i}, b_{j})$?
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Context:
Given an integer \( n \geq 2 \). There are \( n \) sets written on the blackboard. The following operation is performed: select two sets \( A \) and \( B \) on the blackboard that do not contain each other, erase them, and then write \( A \cap B \) and \( A \cup B \). This is referred to as one operation. Continue performing this operation until, in any two sets, one set contains the other. Find the maximum possible number of operations for all initial states and operation methods.
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Context:
Jackie can do 5 push-ups in 10 seconds. As she gets fatigued, her push-up rate decreases by 20% for each subsequent 10-second interval. How many push-ups can she do in one minute if she takes a 7-second break at the 22-second mark and a 12-second break at the 38-second mark, regaining 10% of her push-up rate during each break?
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Context:
In  $ \vartriangle ABC$ ,  $\angle B = 46^o$  and  $\angle C = 48^o$  . A circle is inscribed in   $ \vartriangle ABC$  and the points of tangency are connected to form  $PQR$ . What is the measure of the largest angle in  $\vartriangle P QR$ ?
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Context:
Using each of the eight digits $1, 3, 4, 5, 6, 7, 8,$ and $9$ exactly once, form a three-digit number $A$, two two-digit numbers $B$ and $C$ (where $B < C$), and a one-digit number $D$. The numbers are such that $A + D = B + C = 143$. In how many ways can this be done?
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Context:
Let $A$, $B$, and $C$ be independent random variables, each uniformly distributed between $-1$ and $1$. What is the probability that $A+B+C\geq1$?
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Context:
10.248. To two externally tangent circles with radii $R$ and $r$, a secant is constructed such that the circles intercept three equal segments on it. Find the lengths of these segments.
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Context:
Let \( a \) and \( b \) be real numbers such that \( a + b = 1 \). Then, the minimum value of
\[
f(a, b) = 3 \sqrt{1 + 2a^2} + 2 \sqrt{40 + 9b^2}
\]
is ______.
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Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+def$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Let \( a, b, c, d, e \) be nonnegative integers such that \( 625a + 250b + 100c + 40d + 16e = 15^3 \). What is the maximum possible value of \( a + b + c + d + e \)?
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Context:
Given the parabola $x^{2}=y$ with focus $F(0, \frac{1}{4})$, and points $A$ and $B$ on the parabola such that $|AF|+|BF|=3$, calculate the distance from the midpoint of line segment $AB$ to the x-axis. Express your answer as the absolute value of the y-coordinate of the midpoint, in fractional form.
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Context:
A queue has two people, Person 1 and Person 2. Person 1 enters the queue first, and their impatience is an exponential random variable with rate $θ$. Person 2 enters the queue $s$ minutes later, and their impatience is also an exponential random variable, independent of Person 1's impatience, with rate $μ$. What is the probability that Person 1 will abandon the queue before Person 2 does?
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\boxed{1-\frac{\mu  e^{-\theta  s}}{\theta +\mu }}


Context:
Given four non-collinear points on a plane, what is the maximum number of acute triangles that can be formed using these points? Provide your answer as a whole number.
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Context:
Given a polynomial equation \(z^{4} + a z^{3} + b z^{2} + c z + d = 0\) with real coefficients and all its roots lying on the unit circle in the complex plane centered at the origin, calculate the sum of the reciprocals of these roots. Express your answer in terms of the coefficients of the polynomial.
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Context:
${{\cos }^{2}}\theta +{{\cos 2}}\left( \theta +120{}^\circ  \right)+{{\cos }^{2}}\left( \theta +240{}^\circ  \right)$的值是______.
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Context:
7. (4 points) With the number written on the board, one of the following operations is allowed:
1) If there is a digit in the original number that is not equal to 9 and has two neighboring digits greater than 0, you can increase this digit by 1, and decrease the neighboring digits by 1.
2) Subtract 1 from any non-zero digit except the last one, and add 3 to the next digit.
3) Decrease any sufficiently large digit by 7.

If, as a result of any of these operations, there are zeros in the number on one or more of the leading positions, they are automatically discarded.

Initially, the number consisting of one hundred nines was written on the board. In the end, a single-digit number remained. Which one exactly?
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Context:
An equilateral triangle is divided into smaller equilateral triangles with side length 1 by lines parallel to its sides, dividing each side into \( n \) parts (in the diagram \( n = 5 \)).

What is the maximum number of length 1 segments, with endpoints at the vertices of these smaller triangles, that can be marked so that no triangle is formed where all sides are marked segments?
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Context:
13. Let $n$ be the smallest positive integer of 4 digits greater than or equal to 2016 that has the following property: there exists a positive integer $S$ such that

$$
S=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d+S}}}}
$$

where $a, b, c, d$ are, in order, the thousands, hundreds, tens, and units digits of $n$. What is the value of $n$?
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Context:
If @ is a binary operation defined as the difference between an integer n and the product of n and a certain number, then the largest positive integer n such that the outcome of the binary operation of n is less than 10 is 2. What is the number that n is multiplied by in the product?
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Context:
How many diagonals in a regular 32-sided polygon are not parallel to any of its sides?
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Context:
The angle at the apex of the axial section of a cone is $60^{\circ}$. Inside the cone, there are three spheres with a radius of 1. Each sphere is tangent to the other two spheres, the base of the cone, and its lateral surface. Find the radius of the base of the cone.
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Context:
4. (10 points) A sequence of numbers: $8,3,1,4, . \cdots$, starting from the third number, each number is the unit digit of the sum of the two closest preceding numbers. What is the 2011th number? $\qquad$ .
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Context:
Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a, b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$.
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Context:
$a,b,c,x,y,z \in R$ 且满足$( x + y ) c - ( a + b ) z = \sqrt { 6 }$，求下式的最小值.

$$
a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + a x + b y + c z
$$
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Context:
Three students write on the blackboard three two-digit square numbers next to each other. In the end, they observe that the resulting 6-digit number is also a perfect square. Find this number!
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Context:
Through two vertices \( A \) and \( B \) of \( \triangle ABC \), perpendicular lines \( AD \) and \( BE \) are drawn on the same side of the plane \( ABC \), forming an equilateral triangle \( CDE \). Given \( AD = 1 \), \( BE = 2 \), and \( DE = 3 \), determine the dihedral angle between planes \( CDE \) and \( ABC \).
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Context:
Find the number of elements of order 12 in the automorphism group of the cyclic group \( \mathbb{Z}_{1080} \).
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Context:
Calculate the total mass of a thin plate with the shape defined by the region bounded by the line $y = x - 1$ and the parabola $y^{2} = 2x + 6$, given that the density of the plate is $
ho(x,y) = xy$.
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Context:
17 Let $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n}$ be real numbers, and $\sum_{i=1}^{n} x_{i}^{2}+\sum_{i=1}^{n-1} x_{i} x_{i+1}=1$, for each given positive integer $k, 1 \leqslant k \leqslant n$, find the maximum value of $\left|x_{k}\right|$.
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Context:
Find the area of the region intersected by two circles on the xy-plane given by the equations \(x^2 + y^2 = 1\) and \(x^2 + y^2 - 2x = 0\), using a double integral in polar coordinates.
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Context:
For reals  $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$ , let  $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$  for each  $k$ . If  $S_2 = 777$ , compute the least possible value of  $S_3$ .
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Context:
How many terms of the series $\sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!}$ are required to approximate $\int_{0}^{1}e^{-x^2}$ with an error less than $10^{-6}$?
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Context:
若 \triangle ABC 的内角 A,B,C 满足 \sin A=\cos B=\tan C，求 \cos^3A+\cos^2A-\cos A 的值.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:
Let $\triangle ABC$ be a right triangle with hypotenuse $AC$. A square is inscribed in the triangle such that points $D,E$ are on $AC$, $F$ is on $BC$, and $G$ is on $AB$. Given that $AG=2$ and $CF=5$, what is the area of $\triangle BFG$?
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Context:
Solve the equation $2^x\log(x) + \log(2) = 0$ for $x$.
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Context:
He wanted to make sure that he is protected from the cold evenings in the forest, so he decided to build a fireplace made of different types of cement. He bought 215 lbs of type-A cement, his son brought 137 lbs of type-B cement, and his neighbor contributed 150 lbs of type-C cement. They mixed these three types of cement together to create type-D cement. During the drying process, type-A cement lost 3% of its weight, type-B cement lost 2% of its weight, and type-C cement lost 5% of its weight. After all the weight losses, they had a certain amount of type-D cement. Find out what percentage of the total original cement they now have in the form of type-D cement.
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Context:
Suppose $ABCDEF$ is a regular hexagon with area $ 1$, and consider the diagonals $AC$, $BD$, $CE$, $DF$, $EA$ and $FB$. A star is formed by alternating between the vertices of the hexagon and the points of intersection of the diagonals. What is the area of the star?
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Context:
When a student multiplied the number $66$ by the repeating decimal,
$$\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$$
where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. Find the $2$-digit number $\underline{a} \ \underline{b}.$
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Context:
A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have?
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Context:
Let \( f(x) \) be a polynomial with integer coefficients. Suppose there are four distinct integers \( p, q, r, s \) such that \( f(p) = f(q) = f(r) = f(s) = 5 \). If \( t \) is an integer and \( f(t) > 5 \), what is the smallest possible value of \( f(t) \)?
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Context:
For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance of getting them right due to my prior knowledge. What is the minimum number of words I need to learn in order to guarantee at least a 90% score on this test?
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Context:
Find the closed form for the sum \( \sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k} \), given that \( k = 0, 1, \ldots, n \) and \( n \) is any natural number.
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Context:
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$.
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Context:
Find the area of the region defined by the set \( \{ (x, y) : |x + yt + t^2 | \leq 1 \text{ for all } t \in [0, 1] \} \).
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Context:
Given the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.
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Context:
Evaluate the triple integral \( \int \int \int _{E} \frac{1}{x^2+y^2+z^2 }dV \), where the region \( E \) is defined as \( E=\{(x,y,z) \in \mathbb{R}^3 : z\geq \sqrt{\frac{x^2+y^2}{3}}, 2z\leq x^2+y^2+z^2 \leq 9\} \).
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Context:
Find a function \( g: J \to \mathbb{R} \) (where \( J \) is an interval) that is Hölder continuous for some \( \alpha > 0 \), but not for any \( \alpha > 1 \).
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Context:
Find the degree of the field extension \( \mathbb{Q}(i, \sqrt[4]{3}, \sqrt[6]{3}) \) over \( \mathbb{Q} \).
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Context:
Find an imprimitive permutation group acting on a set of 12 elements such that the group has the maximum possible order. Provide the order of this group.
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Context:
The "Goldbach Conjecture" states that any even number greater than 2 can be written as the sum of two prime numbers. If we split 10 into the sum of two positive integers, calculate the probability that both addends are prime numbers under the condition that both addends are greater than 2. Express your answer as a simplified fraction.
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Context:
Determine the smallest size of a rectangular board on which a knight's tour, either open or closed, is possible.
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Context:
Embed a cube into a dodecahedron in such a way that all eight vertices of the cube are vertices of the dodecahedron. How many solutions does this problem have?
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Context:
If $S$ is a set which consists of $12$ elements, what is the maximum number of pairs $(a,b)$ such that $a, b\in S$ and $\frac{b}{a}$ is a prime number?
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Context:
Let \(I\) be the set of points \((x, y)\) in the Cartesian plane such that
\[ x > \left( \frac{y^4}{9} + 2015 \right)^{1/4} \]

Let \(f(r)\) denote the area of the intersection of \(I\) and the disk \(x^2 + y^2 \leq r^2\) of radius \(r > 0\) centered at the origin \((0,0)\). Determine the minimum possible real number \(L\) such that \(f(r) < L r^2\) for all \(r > 0\).
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Context:
What is the number that, when increased by 20%, equals twice the result of 20 less than the number decreased by 20%? Express your answer as a single number.
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Context:
Determine the value of $n$ for which the subnormal length of the curve $xy^n = a^{n + 1}$ is constant at any point on the curve.
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