Context:
Determine the condition on $p$ for which the series \( \sum_{n=1}^\infty \frac{|g(nx)|}{n^p} \) converges almost everywhere, where \( g:\mathbb{R}^m\rightarrow\bar{\mathbb{R}} \) is a Lebesgue integrable function with \( \int |g|>0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(a \in \mathbb{R}\) for which there exists a non-constant function \(f: (0, 1] \rightarrow \mathbb{R}\) such that \(a + f(x+y-xy) + f(x)f(y) \leq f(x) + f(y)\) holds for all \(x, y \in (0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:

$$ \lim_{x \to \infty} {\displaystyle{\large\left(1 + x\right)\left(1 + \frac{1}{x}\right)^{2x} - x\,{\rm e}^{2}} \over \displaystyle{\large{\rm e}^{2} - \left(1 + \frac{1}{x}\right)^{2x}}} $$
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Context:
A mouse is placed in one corner cube of a $3 \times 3 \times 3$ grid, and a piece of cheese is placed in the central cube. The mouse moves randomly, stepping into one of the neighboring cubes at each move. On average, how many steps will it take for the mouse to reach the cheese?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside regular hexagon $ZUMING$, construct square $FENG$. What fraction of the area of the hexagon is occupied by rectangle $FUME$?
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Context:
What is the probability that five randomly chosen points from a unit square will determine an ellipse when used to form a conic section?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rubber band is initially 1 meter long, with one end fixed to a wall. The other end is pulled away from the wall at a constant speed of 1 m/s. A bug starts at the wall and crawls along the rubber band toward the moving end at a speed of 0.001 cm/s. Determine if the bug will ever reach the end of the rubber band, and if so, calculate the time it takes for the bug to reach the end.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the bases $B_1 = \{1+x,1-x,1+x^2\}$ and $B_2 = \{1+2x,x,1+x+x^2\}$ for the vector space $P_2(\Bbb R)$, and the linear transformation $T$ represented by the matrix $M(T,B_1,B_2) = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 1 & 1 & -1 \\ \end{pmatrix}$, determine $T(1-x+3x^2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi} \cot(e^{it})ie^{it}dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic growth rate of the sequence defined by the recurrence relation: \[ a(n+1) = a(n) + \frac{1}{1+2^{a(n)}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \([0, \Omega) \setminus (\bigcup \{ U_\lambda : \lambda < \Omega \text{ is limit} \})\), where \(\Omega\) is the first uncountable ordinal with the order topology, and for each limit ordinal \(\lambda < \Omega\), \(U_\lambda\) is an open neighborhood of \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple summation \( \sum_{1 \le i < j \le k \le 20} a_i a_j a_k \) where \( a_i = 1 \) for \( 1 \le i \le 10 \) and \( a_i = 2 \) for \( 11 \le i \le 20 \). Find the value of the summation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \alpha \in \mathbb{R} \) such that the change of coordinates \( \varphi(p,q) \rightarrow (P,Q) \) is symplectic, where \( Q = q^2 + \alpha\sqrt{q^2+p} \) and \( P = q + \sqrt{q^2+p} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the line integral \( \int -4x^2y \, dx -(x^3+y^3) \, dy \) along the closed curve that starts at \((-2,0)\), follows the curve \(y=4-x^2\) to \((2,0)\), and returns to \((-2,0)\) along the curve \(y=x^2-4\), using Green's Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the rank of the elliptic curve given by the equation \( y^2 = x^3 + p^3x \) where \( p \) is a prime number such that \( p \equiv 7 \pmod{16} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to\infty}(x-\ln\cosh x)$$ where $$\cosh x=\frac{e^x+e^{-x}}{2}."
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Context:
Two equally strong tennis players play against each other until one player wins three games in a row. Each game is independent, and each player has a probability of \(\frac{1}{2}\) of winning any given game. What is the expected number of games they will play?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functional equation \(f(x)f(y) = f(x+y)\) and the condition \(f(0) = 1\), find the derivative \(f'(0)\) if \(f\) is differentiable and continuous at \(0\), and there exists a function \(g(x)\) such that \(f(x) = 1 + \sin(3x)g(x)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Bayes estimator for the parameter \( \theta \) given the loss distribution \( l(a,\theta)=\frac{(a-\theta)^2}{\theta(1-\theta)} \), a prior distribution \( \pi(\theta) \) that is Uniform from 0 to 1, and a likelihood function \( f(x|\theta) \) for a sequence of \( n \) independent Bernoulli trials, each with a success probability \( \theta \). The total number of ones in the sequence is \( X \sim \text{Bin}(n, \theta) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\int_M(x-y^2+z^3)dS\) over the portion of the cylinder \(x^2+y^2=a^2\) bounded by the planes \(x-z=0\) and \(x+z=0\), where \(a>0\), without using vector calculus theorems.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the random variable $X$, which represents the number of different digits that appear in two randomly and independently drawn 5-digit pins from a set where all digits must be unique. Each pin is selected from the set $A=\{(d_1, d_2, d_3, d_4, d_5): d_i \in \{0, \ldots, 9\}, i \in \{1, \ldots, 5\}, \forall (i \neq j, i, j \in \{1, 2, 3, 4, 5\}) d_i \neq d_j\}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval of convergence for the series \(\sum_{n=1}^\infty \frac{\ln(1+nx)}{n x^n}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A coin is flipped 100 times. Let $X$ be the number of heads in the first 70 flips and $Y$ be the number of heads in the last 50 flips. Compute the correlation of $X$ and $Y$. Provide your answer as a decimal rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x\to0^{-}}\frac { f(-x){ x }^{ 2 } }{ \left\{ \frac { 1-\cos(x) }{ [f(x)] } \right\} } \). Given that \( f(x) \) is a real-valued continuous function except at \( x=0 \), with \( f'(x)<0 \) for negative \( x \) and \( f'(x)>0 \) for positive \( x \). Also, \( \lim_{x\to0^{+}} f(x)=2 \), \( \lim_{x\to0^{-}} f(x)=3 \), and \( f(0)=4 \). Here, \([\cdot]\) denotes the greatest integer function, and \(\{\cdot\}\) denotes the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal polynomial of \(\alpha^{21}\) over \(\mathbb{Z}_2\), given that \(\alpha\) satisfies the equation \(\alpha^6 + \alpha + 1 = 0\) over \(\mathbb{Z}_2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the solution to the second-order ordinary differential equation \(2xx''-3(x')^2=4x^2\) with the initial conditions \(x(0) = 1\) and \(x'(0) = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with side length \( n \) (\( n \geq 3 \)) is divided into unit cubes by partitions. Determine the minimum number of partitions that must be removed so that it is possible to reach the boundary of the cube from any unit cube.
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Context:
Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5x^n} } \]
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Context:
Calculate \( \liminf_{n\to\infty} \left( \frac{p_{n}}{p_{n}+p_{n+1}} \right) \), where \( p_{n} \) is the \( n \)-th prime number and \( p_{n+1} \) is the next prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of the function \( f(x) = \sqrt{\frac{-x^2 + 2x + 3}{18x - 3x^3}} \) without using limits or derivatives.
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Context:
In how many ways can the letters of the word "INDIVISIBILITY" be rearranged such that no two 'I's are adjacent?
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Context:
Given an $n$-dimensional vector space $V$ with a basis $\{e_1, \ldots, e_n\}$, consider partitions of this basis into two subsets of $k$ and $n-k$ elements, which define subspaces $P$ and $Q$. Determine the maximum number of partitions for which a given $k$-dimensional subspace $W$ has a trivial intersection with $Q$. What is this maximum number?
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Context:
If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x \to 0} \left( x^2 \left( 1 + 2 + \cdots + \left[ \frac{1}{|x|} \right] \right) \right) \), where \([a]\) denotes the greatest integer less than or equal to \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit as \( n \) approaches infinity: \[ \lim_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^{n}{\left(\dfrac{n}{k}-\left\lfloor\dfrac{n}{k}\right\rfloor\right)}} \]
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Context:
Find the Bayes estimator of \(\theta\) given the following information:

- Loss function: \(l(a,\theta)=\frac{(a-\theta)^2}{\theta(1-\theta)}\)
- Prior distribution: \(\pi(\theta)\) is uniformly distributed from 0 to 1.
- Likelihood function: Bernoulli, where \(X\) is the total number of ones in a sequence of \(n\) independent Bernoulli trials, each with sample space \(\{0,1\}\), so \(f(x|\theta) = \theta^x(1-\theta)^{1-x}\), \(x=0,1\).

Calculate the Bayes estimator of \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a plane, there are \( n+4 \) points, four of which are located at the vertices of a square, and the remaining \( n \) lie inside this square. You are allowed to connect any of these points with segments, provided that no segment contains any of the marked points other than its endpoints, and no two segments share any points other than their endpoints. What is the maximum number of segments that can be constructed in this way?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series $$\sum (n^2 \sin(2/n^2))^{n^2+2} x^{n^2}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $f(x)$ is defined on the reals such that

$$f\left(\frac{1-4x}{4-x}\right) = 4-xf(x)$$
for all $x \ne 4$. There exists two distinct real numbers $a, b \ne 4$ such that $f(a) = f(b) = \frac{5}{2}$. $a+b$ can be represented as $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $10p + q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of incongruent triangles that can be formed with integer side lengths and a perimeter of 1994.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the series:

$$\lim_{x\to\infty} \sum ^\infty_{n=3} \frac{n\cdot x^{n-2}\cdot(-1)^n\cdot(n-1)}{(2n)!}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three distinct points \( L, M, \) and \( N \) on a straight line, with \( M \) between \( L \) and \( N \) and \( LN \neq MN \), semicircles are constructed on diameters \( LM, MN, \) and \( LN \). The centers of these semicircles are points \( A, B, \) and \( C \) respectively, with \( C \) on one side of the line \( LN \) and \( A \) and \( B \) on the opposite side. Determine the ratio of the area of the region enclosed by these three semicircles to the area of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the positive time $T$, in years, such that the amount in Fund A, which accumulates at a rate of $3\%$ convertible monthly, equals the amount in Fund B, which accumulates with a force of interest given by $\delta_t = t^{20}$ for all $t$. At time $t = 0$, $\$1 is deposited in each fund. Calculate $T$. \( \left(1+\frac{0.03}{12}\right)^{12T} = e^{\int_0^T t^{20} \, dt} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
159 The area enclosed by the curve $(|x|-1)^{2}+(|y|-2)^{2}=5$ is $=$

The area enclosed by the curve $(|x|-1)^{2}+(|y|-2)^{2}=5$ is $=$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{B(0,r)} \frac{1}{|x|^{n-1}} \, dx \) and determine if it can be expressed as a constant times the radius \( r \).
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Context:
Determine the limit of the sequence \( \{x_n\} \) where \( x_n = \frac{M_n + a}{M_n(M_n + b)} \), given that \( M_n = o_p(\sqrt{n}) \), \( M_n > 0 \), and \( \mathbb{P}(\liminf\{M_n > K\}) = 1 \) for any integer \( K \). Here, \( a \) and \( b \) are positive constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\pi \int_0^\pi |\cos(x+y)| \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of growth for the error function, defined as \( \text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt \), when expanded into its Taylor series and considered as an entire function in the complex plane.
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Context:
In the binomial expansion of \((1+\sqrt{11})^{208}\), determine the value of \(k\) for which the term \(C_{208}^{k}(\sqrt{11})^{k}\) is maximized.
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Context:
Determine the number of surjective functions from the set $A = \{1,2,3,4,5,6\}$ to the set $B = \{1,2,3,4,5,6\}$, given that $f(1) = 2$ and $f(i) \neq i$ for all $i \in A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the topological group that is isomorphic to the quotient group $\mathbb{R}/\mathbb{Z}$.
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Context:
Given a $2\pi$ periodic function defined as $f(t) = t$ for $0 \leq t < \pi$ and $f(t) = 0$ for $\pi \leq t < 2\pi$, find the value of its complex Fourier series representation at $t = \pi$. The complex Fourier series is given by:
\[ f(t) = \frac{\pi}{4} + \sum_{k \neq 0} \frac{1}{2\pi} \left( \frac{(-1)^k - 1}{k^2} \right) e^{ikt}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Euler characteristic of the Grassmannian $G(3, 7)$ by counting the fixed points of a torus action.
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Context:
An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
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Context:
Given the sequence $a_n$ where \(\sum_{n=1}^\infty \left(na_n-\frac{n^2+1}{2n+1} \right)=3,\) determine the limit \(\lim_{n\rightarrow\infty}(a_n^2+2a_n+2).\)
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Context:
Let \( \sigma \left( n\right) \) be the sum of all the positive divisors of \( n \) . Let \( a \) be the smallest positive integer greater than or equal to 2015 for which there exists some positive integer \( n \) satisfying \( \sigma \left( n\right)  = a \) . Finally,let \( b \) be the largest such value of \( n \) . Compute \( a + b \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the sum of all ordinals up to \(\omega\), where \(\omega\) is the first transfinite ordinal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distance between the point \( p = (1,0,0,1) \) and the subspace \( L = (\operatorname{span}{(1,0,2,1),(2,1,2,3),(0,1,-2,1)})^\perp \) in \( \mathbb{R}^4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a box, there are eight real coins and two fake coins, with the fake coins having a different weight than the real ones. You randomly select a pair of coins without replacement from the ten coins, and then select another pair from the remaining eight coins. What is the probability that all four coins selected are real, given that the combined weight of the first pair is the same as the second pair?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \Delta z \) along the path \( y = x^2 \) from the point \((0,0)\) to \((1,1)\). Given the differential \( dz = xydx + xydy \), find the change in \( z \) as you move from \((0,0)\) to \((1,1)\) along the path \( y = x^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A motorist arrives at a toll bridge with a single toll booth in operation and finds 50 cars ahead of them. The toll collection time for each vehicle follows an exponential distribution with a probability density function given by \( f(x) = 2\exp(-2x) \), where \( x \) is in minutes. Using the central limit theorem, calculate the approximate probability that the motorist will have to wait more than 25 minutes to reach the toll booth.
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Context:
Evaluate the integral \( \frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos(\theta)}\cos(n\theta) \, d\theta \) for \( n \in \mathbb{N} \).
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Context:
Find the smallest positive integer \( n \) such that for any integer \( p \geq n \), a square can be partitioned into \( p \) non-congruent smaller squares.
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Context:
In an \( 8 \times  8 \) grid of squares, each square was colored black or white so that no \( 2 \times  2 \) square has all its squares in the same
 color. A sequence of distinct squares \( {x}_{1},\ldots ,{x}_{m} \) is called a snake of length \( m \) if for each \( 1 \leq  i < m \) the squares \( {x}_{i},{x}_{i} \) .
 are adjacent and are of different colors. What is the maximum \( m \) for which there must exist a snake of length \( m \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the region between the surfaces $z=0$ and $z=\frac{1}{(x^2+y^2)^9}$, and outside the cylinder $x^2+y^2=1$, using a triple integral.
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Context:
Given a metric on \(\mathbb{R}^2\) in polar coordinates of the form \(dr^2 + f(r,\theta)d\theta^2\), and a curve \(\gamma(t) = (t, \theta_0)\) where \(\theta_0\) is a constant, find the covariant derivative \(\nabla_{\gamma'(t)}\gamma'(t)\).
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Context:
Let $n > 1$ be an integer that is divisible by exactly two distinct primes, $p$ and $q$. If $G$ is a group with exactly $n$ elements of order $n$, determine the maximum possible number of cyclic subgroups of order $n$ in $G$. Provide your answer as a single integer.
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Context:
Evaluate the surface integral \( \iint_S F \cdot n \,ds \) for the vector field \( F(x,y,z)=(z^2-x,-xy,3z) \) over the surface \( S \) of the solid bounded by the equations \( z=4-y^2, x=0, x=3 \), and \( z=0 \), with the normal vector pointing outward. Do not use the Divergence Theorem.
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Context:
Let $U$ be the vector subspace of $\mathbb{R}^5$ generated by the set $S_U = \{(1,3,-3,-1,-4),(1,4,-1,-2,-2)\}$, and let $V$ be the vector subspace of $\mathbb{R}^5$ generated by the set $S_V = \{(1,6,2,-2,3),(2,8,-1,-6,-5),(1,3,-1,-5,-6)\}$. Determine the dimension of the intersection $U \cap V$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any positive integer \( a,\sigma \left( a\right) \) denotes the sum of the
 positive integer divisors of \( a \) . Let \( n \) be the least positive int
 such that \( \sigma \left( {a}^{n}\right)  - 1 \) is divisible by 2021 for all positive
 integers \( a \) . Find the sum of the prime factors in the prime
 factorization of \( n \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of permutations \(\alpha\) such that \(\alpha\sigma\alpha^{-1} = \tau\), where \(\sigma = (14735)(698)(2)\) and \(\tau=(16973)(485)(2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the real values of $p$ for which the series \( \sum_{n=2}^{\infty}\frac{1}{(n^2\log n)\cdot(n^{\frac{1}{n}}-1)^p} \) is convergent.
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Context:
Consider the polynomial \(f(x) = x^5 - 4x + 2\). Let \(L\) be the complex splitting field of \(f(x)\) over \(\mathbb{Q}\). Determine the number of subfields \(K\) of \(L\) such that \((K : \mathbb{Q}) = 2\).
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Context:
At a class reunion with 45 attendees, it is known that any two people with the same number of acquaintances among the attendees do not know each other. What is the maximum number of pairs of acquaintances that could exist among the attendees?
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Context:
How many people do you need in a room so that you expect there to be a shared birthday on a Monday, assuming there are 52 Mondays in a year and 365 days in a year?
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Context:
Identify the location and direction of the tangent vector $t_y$ to the surface $S$ defined by $z = \frac{1}{1+x^2+y^2}$, restricted to the plane $y=0$. The surface is given by $\mathbf{r}(x,y) = x\mathbf{i} + y\mathbf{j} + \frac{1}{1+x^2+y^2}\mathbf{k}$. Determine $t_y$ in this plane.
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Context:
In a country, 3% of the population has a certain disorder. A test for this disorder has a 97% true positive rate and a 2% false positive rate. If a person takes the test twice independently and receives one positive result and one negative result, what is the probability that they actually have the disorder?
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Context:
Determine the number of ordered pairs \((x, y)\) that satisfy the following system of equations:
\[\begin{align*}
\sin(x+y) &= \cos(x+y) \\
x^2 + y^2 &= \left(\frac{1995\pi}{4}\right)^2
\end{align*}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) be a function defined on \(\{0, 1, \cdots, 2005\}\) that takes non-negative integer values. The function satisfies the following conditions for any variable \( x \) within its domain:

\[ f(2x+1) = f(2x), \]
\[ f(3x+1) = f(3x), \]
\[ f(5x+1) = f(5x). \]

Determine the maximum number of distinct values this function can take.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( F = (x^3, 4y^3, z^3) \) through the surface defined by \( x^2 + 4y^2 + z^2 = 1 \).
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Context:
Find the norm of the differential operator \( \frac{d}{dx} \) on the space of polynomials of degree \( \leq n \) on the interval \([-a,a]\), where \( a > 0 \).
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Context:
For a bounded set \(\Omega \subset \mathbb{R}^d\), determine the maximum value of \(\epsilon\) such that there exists an infinite subset \(\mathcal{A} \subset \mathcal{B}(\Omega)\) with the property that \(|A \triangle B| \geq \epsilon\) for all distinct sets \(A, B \in \mathcal{A}\).
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Context:
Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$, we have $|a + b - c| > 10$. Determine the largest possible number of elements of $M$.
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Context:
Asif has cubes of three colors: red, green, and blue. He builds a tower by stacking the cubes, with the restriction that no more than 4 cubes of each color can be used. Asif stops building the tower as soon as it contains 4 cubes of any two colors. How many different towers can Asif build?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the series \( \sum_{k=1}^{\infty} e^{-\pi k^2} \left( \pi k^2 - \frac{1}{4} \right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Monty Hall problem with an infinite number of doors, if the contestant chooses a door with an infinitesimally small probability, what is the probability of winning by switching?
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Context:
Find a bound for the probability that the sum of independent and identically distributed random variables is negative, given that for each variable $a_i$, $Pr(a_i < 0) < e^{-d}$, where $d$ is a positive integer. Specifically, determine a bound for $Pr(\sum_{i=1}^{n}{a_i} < 0)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the image of the set \( \mathcal{F} \) under the map that takes a function \( f(x) \) in \( \mathcal{F} \) to \( \int_{-1}^{1} f(x) \, dx \). Here, \( \mathcal{F} \) is the subset of real-valued continuous functions on \([-1, 1]\) such that \( f(x) \geq 0 \) for all \( x \) in \([-1, 1]\) and \( \limsup_{n\rightarrow \infty}\int_{-1}^{1}f(x)^{n}dx <\infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $i^2 = -1$, then the sum
\[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \]
\[ + \cdots + i^{40}\cos{3645^\circ} \]
equals
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Context:
Find a sufficient statistic \(T(X)\) that serves as an unbiased estimator for \(1/p\) in a geometric distribution with probability mass function \(f(x;p) = p(1-p)^x\), where \(x = 0, 1, 2, \ldots\) and \(0 < p < 1\).
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Context:
In a land of knights (who always tell the truth) and liars (who always lie), 10 people are seated at the vertices of a regular decagon at a round table. Among them, there are liars. A traveler can stand at any point outside the table and ask the seated individuals: "What is the distance from me to the nearest liar among you?" Each person will answer the question. What is the minimum number of questions the traveler must ask to guarantee identifying all the liars?
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Context:
Find the lower bound for the limit below, given that the functions $u_\epsilon$ converge uniformly to $x \mapsto \xi \cdot x$:
$$\lim_{\epsilon \to 0} \epsilon \sum_{(x,y) \in (\epsilon \mathbb{Z}^3 \cap Q_1)^2, \ |x-y|=\epsilon} |u_\epsilon(x)-u_\epsilon(y)|^2$$
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Context:
Determine the kernel of the homomorphism \( f: K[X,Y] \rightarrow K[T] \) defined by \( f(X) = T^2 \) and \( f(Y) = T^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two matrices \( A \) and \( C \) of order \( n \times n \) and \( m \times n \) respectively, consider the block matrix:

\[ D=\left( \begin{matrix} C\\ CA\\ :\\ :\\ CA^{q-1} \end{matrix}\right) \]

where \( q \in \mathbb{N}^* \). Determine the smallest value of \( q \) such that \( \text{rank}(D) = n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of growth of the function $f(z) = \sin(z^2)$ using the definition of order of growth.
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Context:
Calculate

$$
\operatorname{tg} \frac{\pi}{47} \cdot \operatorname{tg} \frac{2 \pi}{47} + \operatorname{tg} \frac{2 \pi}{47} \cdot \operatorname{tg} \frac{3 \pi}{47} + \ldots + \operatorname{tg} \frac{k \pi}{47} \cdot \operatorname{tg} \frac{(k+1) \pi}{47} + \ldots + \operatorname{tg} \frac{2021 \pi}{47} \cdot \operatorname{tg} \frac{2022 \pi}{47}
$$
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Context:
Find an initial condition \( y(0) \neq (0,0,0) \) for the system of differential equations given by \( y_1'=12y_3 \), \( y_2'=y_1+13y_3 \), and \( y_3'=y_2 \) such that the solution \( y(x) \) approaches \( (0,0,0) \) as \( x \to \infty \).
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Context:
Determine the number of ways the sum \(1 + 2 + \cdots + 2007\) can be expressed as a sum of consecutive positive integers.
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Context:
Let  $ABC$  be an equilateral triangle, and let  $D$  and  $F$  be points on sides  $BC$  and  $AB$ , respectively, with  $FA=5$  and  $CD=2$ .  Point  $E$  lies on side  $CA$  such that  $\angle DEF = 60^\circ$ .  The area of triangle  $DEF$  is  $14\sqrt{3}$ .  The two possible values of the length of side  $AB$  are  $p \pm q\sqrt{r}$ , where  $p$  and  $q$  are rational, and  $r$  is an integer not divisible by the square of a prime.  Find  $r$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the double integral \( I = \int_{-\infty}^{\infty}\int_{0}^{\infty} \cos k\xi \cdot u(\xi ) \, dkd\xi \), where \( u(\xi) = a_{0} \operatorname{sech}^{2} \left( \sqrt{\frac{a_{0}}{6+4a_{0}}} \, \xi \right) \) and \( a_0 \) is a constant.
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Context:
Compute the area between the curves \( y = \cos(\sin(x)) \) and \( y = \sin(\cos(x)) \) on the interval \( 0 \leq x \leq 2\pi \).
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Context:
Evaluate the limit:
$$ \lim_{x\to0}\frac{\sqrt[n]{a+x} - \sqrt[n]{a-x}}{x},\ a>0,\ n\in\Bbb N $$
without using L'Hospital's rule or generalized binomial expansion.
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Context:
Evaluate the integral: $$\int_{0}^{\infty} \frac{\sin^4x}{x^4}dx$$
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Context:
How many different triangular pyramids can be formed using the vertices of a triangular prism?
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Context:
What is the probability that a subset $X$ of $\mathbb{R}$, formed by independently including each real number with a coin toss, is a Lebesgue measurable subset of $\mathbb{R}$?
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Context:
Find the limit \( \lim_{n \rightarrow \infty} \frac{\text{ex}(n; P)}{\binom{n}{2}} \), where \( \text{ex}(n; P) \) is the extremal number for the Petersen graph \( P \).
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Context:
存在 x_0<0，使得 x^2+|x-a|-2<0 恒成立，其中 a\in\bZ，则所有满足条件的 a 的和为__________.
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Context:
Find the function \( f(x) \) such that for all \( a > 1 \) and \( b > 1 \), the following equation holds:
\[ \int_{3}^{ab+2} f(x) \, dx = \int_{3}^{a+2} f(x) \, dx + \int_{3}^{b+2} f(x) \, dx. \]
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Context:
给定以正方形内的 2022 个点，且这 2022 个点以及正方形的顶点中任三点不共线。在这 2022 以及正方形的顶点之间连一些线段，将正方形划分为一些三角形，并要求这些线段之间只可以在端点处有公共点，则所连线段（不包含原正方形的边）的总数为 $\qquad$.
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Context:
Determine the interval in which the constant \( k \) must lie if the curve \( f(x) = k^3 x + k^3 - 2 \) intersects the curve \( g(x) = \frac{1}{2} \ln x^2 \) at exactly one point.
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Context:
Which positive integers satisfy that the sum of the number's last three digits added to the number itself yields $2029$?
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Context:
Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum \limits_{i=1}^{24}p_{i}^{99!}  \equiv a\, \, \left(mod\, 100\right)$
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Context:
Determine the largest integer $n$ such that $99^{(n+1)} > 100^n$ without using logarithms.
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Context:
Evaluate the sum \(\sum_{k=2}^{n}{(k-1)(n-k)!{n \choose k}}\) using properties of binomial coefficients.
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Context:
Evaluate the integral: $$\int_{-\infty}^{\infty} \frac{1}{x^6+1}dx$$
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Context:
Given that the probability of \(\gcd(a,b) \in P\) is \(0.2749\ldots\), the probability that \(\gcd(a,b) = 1\) is \(0.6079\ldots\), and the probability that \(\gcd(a,b) \in \{1\} \cup P\) is \(0.8822 \ldots\), find the probability that \(\gcd(a,b) \in 2P\), where \(2P = \{2p : p \in P\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \frac{\sqrt[n+1]{(n+1)! (2n+1)!!}}{n+1} - \frac{\sqrt[n]{n!(2n-1)!!}}{n} \right) \]
where \( n! \) is the factorial and \((2n-1)!!\) is the double factorial. Use Stirling's formula, \( n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \), to simplify the expression and determine the limit.
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Context:
If the graph of the function $f(x)=a+b\cos x+c\sin x$ passes through the points $(0,1)$ and $(-\frac{π}{4},a)$, and when $x\in[0,\frac{π}{2}]$, $|f(x)|≤\sqrt{2}$ always holds, then the range of real number $a$ is ______.
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Context:
A point light source is positioned at a distance from a sphere equal to three times the radius of the sphere. Determine the ratio of the illuminated area of the sphere to the lateral surface area of the cone of light formed by the light source and the sphere.
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Context:
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
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Context:
In a right triangle $ABC$ with a right angle at $C$, the leg $BC$ has a length of 26. A circle is constructed with $BC$ as its diameter. From point $A$, a tangent $AP$ is drawn to this circle, distinct from $AC$. A perpendicular $PH$ is dropped to segment $BC$, intersecting segment $AB$ at point $Q$. If $BH : CH = 4 : 9$, find the area of triangle $BPQ$. Express your answer as a single number.
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Context:
Suppose there are 3 bank tellers, all currently occupied with a customer. You are waiting in line (and there are no other customers that will come in after you). All service times are iid exponential random variables with parameter $\lambda$. What is the probability that you are the last to leave?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( \mathcal{R}[0,1] \) of all Riemann integrable real functions on the interval [0,1].
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Context:
Determine the number of triples of natural numbers \((m, n, k)\) that satisfy the equation \( m + \sqrt{n+\sqrt{k}} = 2023 \).
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Context:
Given the pressure function \( P(x, y, z) = \frac{x^2 + 2y^2}{1 + z^2} \) and the temperature function \( T(x, y, z) = 5 + xy - z^2 \), and the airplane's position described by \( (x(t), y(t), z(t)) = (2t, t^2 - 1, \cos t) \), find the time derivative of \((PT)^2\) at \(t = 0\) as observed from the airplane.
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Context:
Compute \( \lim_{|h| \to \infty} \int_{\mathbb{R}^n} |f(y+h) + f(y)|^p \, dy \) for a function \( f \in L^p(\mathbb{R}^n) \) where \( 0 < p < \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a constant \( C > 0 \) such that for Hardy's integral operator \( T \) defined on \( L^1(0, +\infty) \) by \( Tf(x) = \frac{1}{x}\int_0^x f(t) \, dt \), the inequality \( \lambda m(|Tf| > \lambda) \leq C{\Vert f \Vert}_1 \) holds for all \( \lambda > 0 \) and \( f \in L^1(0, +\infty) \), where \( m \) denotes the Lebesgue measure.
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Context:
In triangle \(ABC\), a bisector \(BD\) is drawn. In triangles \(ABD\) and \(CBD\), bisectors \(DE\) and \(DF\) are drawn, respectively. Given that \(EF \parallel AC\), determine the measure of angle \(DEF\).
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Context:
Given two independent exponential random variables $X_1$ and $X_2$ with mean 1, find the probability that the larger of the two, $X_{(2)}$, is greater than three times the smaller, $X_{(1)}$. That is, calculate $P(X_{(2)} > 3X_{(1)})$. Assume $X_1$ and $X_2$ are identically distributed with rate parameter $\lambda = 1$. Provide your answer as a probability value.
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Context:
Determine the minimum number of steps required for a random walker, who moves with a 50% probability in either direction, to have at least a 99% probability of being at least 10 meters away from the starting point.
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Context:
For the interval $\left[a,b\right]$, we define $b-a$ as the "length" of this interval. Given that $A$ and $B$ are both subsets of the set $\{x\left|\right.-1\leqslant x\leqslant 1\}$, $A=\{x\left|\right.a\leqslant x\leqslant a+1\}$, $B=\left\{{x|b-\frac{3}{2}≤x≤b}\right\}$, then the range of the "length" of the set $A\cap B$ is ______.
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Context:
Given a Markov chain \( \{X_n, n \geq 0\} \) with states 1, 2, 3 and the transition probability matrix \( P = \begin{bmatrix}0 & 0.4 & 0.6 \\ 1 & 0 & 0 \\ 0.3 & 0.3 & 0.4\end{bmatrix} \), and an initial distribution \( (0.5, 0, 0.5) \), calculate \( P(X_1=3|X_2=1) \).
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Context:
The drag force on a boat varies jointly as the wetted surface area and the square of the velocity of a boat. If a boat going $6.5 \mathrm{mph}$ experiences a drag force of $86 \mathrm{~N}$ when the wetted surface area is $41.2 \mathrm{ft}^{2}$, how fast must a boat with $28.5 \mathrm{ft}^{2}$ of wetted surface area go in order to experience a drag force of $94 \mathrm{~N} ?$
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Context:
Let $O$ be a point with three other points $A,B,C$ and $\angle AOB=\angle BOC=\angle AOC=2\pi/3$.  Consider the average area of the set of triangles $ABC$ where $OA,OB,OC\in\{3,4,5\}$.  The average area can be written in the form $m\sqrt n$ where $m,n$ are integers and $n$ is not divisible by a perfect square greater than $1$.  Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \(a, b, c, d \in \mathbf{R}\) such that \(a^{2} + b^{2} + c^{2} + d^{2} = 1\), find the maximum value of \(ab + bc + cd + da + ac + 4bd\).
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Context:
7.2. Find the volume of a regular triangular pyramid, the midpoint of the height of which is at a distance of 2 and $\sqrt{11}$ from the lateral face and the lateral edge, respectively. Round your answer to two decimal places if necessary.
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Context:
Let $$P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)\left(x^{24-j}+x^{24+j}\right).$$Let $z_1, z_2, \ldots, z_r$ be the distinct zeros of $P(x)$, and let $z_k^2=a_k+b_{k}i$ for $k=1, 2, \ldots, r$, where $i=\sqrt{-1}$, and $a_k$ and $b_k$ are real numbers. Find
\[\sum_{k=1}^{r}|b_k|.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a natural number $n$, you are allowed two operations:
1. Multiply $n$ by $2$, or
2. Subtract $3$ from $n$.

For example, starting with $8$, you can reach $13$ as follows: $8 \longrightarrow 16 \longrightarrow 13$. You need two steps, and you cannot do it in less than two steps.

Starting from $11$, what is the least number of steps required to reach $121$?
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Context:
Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.
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Context:
Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions.
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Context:
Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$.  Let $D$ be the point such that $AB\parallel CD$ and $CA\parallel BD$.  Let $R$ be a point within triangle $BCD$.  Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively.  Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively.  If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.
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Context:
Given the differential equation: $$y^{2}dy = x\left ( x dy - y dx \right ) e^{\frac{x}{y}}$$, determine the integrating factor that makes the equation exact.
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Context:
Given 5 differently colored balls and 3 different boxes, with the requirement that no box is empty, calculate the total number of different ways to place 4 balls into the boxes.
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Context:
The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence.
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Context:
Given the function $f(x)=\sqrt{2}\sin(2\omega x-\frac{\pi}{12})+1$ ($\omega > 0$) has exactly $3$ zeros in the interval $\left[0,\pi \right]$, determine the minimum value of $\omega$.
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Context:
If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of a certain number, the least possible value of n is 7. What is the multiple?
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Context:
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 11.879049676025918 seconds. If the length of the second train is 160 m, what is the length of the first train?
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Context:
Let \( f(x) = ax^3 + bx + c \) (where \( a, b, c \) are real numbers). When \( 0 \leq x \leq 1 \), \( 0 \leq f(x) \leq 1 \). Determine the maximum possible value of \( b \).
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Context:
Find all prime numbers \( p, q, \) and \( r \) such that \( pqr = 5(p + q + r) \).
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Context:
19. Let the number $a$ have the following property: For any four real numbers $x_{1}, x_{2}, x_{3}, x_{4}$, there always exist integers $k_{1}, k_{2}, k_{3}, k_{4}$, such that $\sum_{1 \leqslant i<j \leqslant 4}\left(\left(x_{i}-k_{i}\right)-\left(x_{j}-k_{j}\right)\right)^{2} \leqslant a$, find the minimum value of such $a$. (1990
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Context:
What is the ratio between perimeters of two squares, one having 2 times the diagonal than the other, and the other square having doubled side lengths compared to the original squares?
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Context:
Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89.
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Context:
Given that a normal vector of line $l$ is $\overrightarrow{n}=(\sqrt{3}, -1)$, find the size of the slope angle of line $l$.
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Context:
For any $x \in \mathbb{R}$, the function $f(x)$ satisfies $f(x+1)= \sqrt {f(x)-[f(x)]^{2}}+ \frac {1}{2}$. Let $a_{n}=[f(n)]^{2}-f(n)$, and the sum of the first $15$ terms of the sequence $\{a_{n}\}$ is $- \frac {31}{16}$. Find $f(15)$.
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Context:
Arthur decided to treat himself to dinner at a nice restaurant to celebrate his recent promotion. He ordered a nice appetizer for $8, a delicious ribeye steak for his entrée at $30, had two glasses of nice red wine with dinner for $4 each, and a slice of caramel cheesecake for dessert for $7. He had a voucher that offered a 40% discount on his entrée, and another voucher for a 10% discount on his total bill before tax. The local tax rate is 8%.

Arthur thoughtfully tipped his waiter a full 20% of what the cost of his meal would have been without any discounts, but including tax. How much, including the tip, did Arthur spend on dinner?
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Context:
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she starts working at 7:25 A.M. and takes her lunch break at noon, what time will her workday end? Express your answer in the format of time (e.g., 4:10 P.M.).
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Context:
Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a + b?$ (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)
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Context:
Compute \(\left\lfloor \frac{1000!}{1!+2!+\cdots+999!} \right\rfloor.\)
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Context:
Compute
\[\sum_{j = 0}^\infty \sum_{k = 0}^\infty 2^{-3k - j - (k + j)^2}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Evaluate the triple integral \( \iiint_D x^2yz \,dx\,dy\,dz \) over the region \( D = \{(x,y,z):0 \leq x \leq y+z \leq z \leq 1\} \).
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Context:
In each cell of a $5 \times 5$ table, an invisible ink natural number is written. It is known that the sum of all the numbers is 200, and the sum of the three numbers located inside any $1 \times 3$ rectangle is 23. What is the central number in the table?
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Context:
In triangle \(ABC\), a median \(AM\) is drawn (point \(M\) lies on side \(BC\)). It is known that the angle \(\angle CAM\) is \(30^\circ\) and side \(AC\) is 2. Find the distance from point \(B\) to line \(AC\).
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Context:
Cyclists Petya, Vlad, and Timur simultaneously started a warm-up race on a circular bike path. Their speeds are 27 km/h, 30 km/h, and 32 km/h, respectively. After what minimum time will they find themselves at the same point on the track again? (The length of the bike path is 400 meters.)
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Context:
Find the following limit $$\lim_{n \rightarrow \infty} (\frac{2^{1/n}}{n+1} +\frac{2^{2/n}}{n+1/2}+\dots+\frac{2^{n/n}}{n+{1/n}})$$
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Context:
A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:

$$
0,2,3,4,4,7,8,8,9
$$

What is this number?
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Context:
Find the limit of the sequence \( x_n = \left(\frac{e\sqrt[3]{e}\cdots\sqrt[n]{e}}{n}\right) \) as \( n \to \infty \).
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Context:
John has three part-time jobs. From job A, he initially makes $60 a week, and after a raise, he now makes $80 a week. Job B gives biweekly raises, so he initially makes $100 a week, after the first raise, he makes $110, and after the second raise, he makes $130 a week. Job C pays hourly, and John works 20 hours per week initially at a rate of $8 per hour, then his hourly rate is increased to $10. What is the overall average percentage increase in his weekly income after all the raises?
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Context:
Use a parameterisation to find the flux of $F = (3xy, 0, -z)$ outward (normal away from the z-axis) through the cone $z = 9\sqrt{x^2 + y^2}$, $0 \le z \le 9$.
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Context:
Factorize: $a+(a+b) x+(a+2 b) x^{2}+(a+3 b) x^{3}+3 b x^{4}+2 b x^{5}+b x^{6}$.
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Context:
Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. For integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Words cannot cross between lines but may be repeated. Compute the number of distinct haikus Ted can make, and find the remainder when this number is divided by $100$.
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Context:
Let $M=\{1,2, \cdots, 2005\}$. Subset $A$ of $M$ satisfies the condition: if $x \in A$, then $15x \notin A$. What is the maximum number of elements in $A$?
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Context:
Spencer is making burritos, each consisting of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos. In how many ways can he make exactly five burritos?
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Context:
On the game show $\text{\emph{Wheel of Fortunes}}$, you encounter a spinner divided equally into 6 regions labeled as follows: "Bankrupt," "$600", "$100", "$2000", "$150", "$700". What is the probability that you will accumulate exactly $1450$ in your first three spins without landing on "Bankrupt"? Assume each spin is independent and all outcomes have equal likelihood.
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Context:
Calculate the following expression (accurate to 8 decimal places):
$$
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right)
$$
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Context:
Simplify first, then evaluate $\left(\frac{{a^2-4}}{{a^2-4a+4}}-\frac{1}{{2-a}}\right) \div \frac{2}{{a^2-2a}}$, where $a$ satisfies $a^{2}+3a-2=0$.
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Context:
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[ x = \sqrt{29} + \frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{x}}}}}}}}}.\]
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Context:
Two trains 140 m and another train of certain length run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 10.799136069114471 seconds. What is the length of the second train?
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Context:
Express the following as a common fraction: $\sqrt[3]{5\div 15.75}$.
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Context:
已知 a 、 b 、 c 分别为 $\triangle A B C$ 的三个内角 A 、 B 、 C 的对边长， $a=2$ ，且 $(b+2) \sin \frac{C}{2} \cdot \sin \frac{A-B}{2}=c \cos \frac{A}{2} \cdot \cos \frac{B-C}{2}$. 若 D 是 B C 的中点,求 A D 长的最小值。原始答案为 $\frac{\sqrt{m}}{n}$ 的格式，请给出 $m + n$ 的值。
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Context:
In the diagram, P is on RS so that QP bisects ∠SQR. Also, PQ = PR, ∠RSQ = 2y°, and ∠RPQ = 3y°. Find the measure of ∠RPQ in degrees.
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Context:
Long ago, nine identical books cost 11 rubles and some kopecks, and thirteen such books cost 15 rubles and some kopecks. How much did one book cost?
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Context:
Four runners start at the same position on a circular track that is 1 km long, running in the same direction. Their speeds are \( \sqrt{5} \), \( e \), 3, and \( \pi \) meters per second. Determine the first time, in seconds, when the runner with speed \( e \) is at least 250 meters away from each of the other runners.
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Context:
Determine the interval in which the solution to the equation \(e^x = \frac{1}{x}\) lies for \(x > 0\).
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Context:
Given a trapezoid \(ABCD\) where angle \(BAD\) is a right angle. A circle is constructed with diameter \(AB\) and it intersects the diagonal \(BD\) at point \(M\). It is known that \(AB = 3\), \(AD = 4\), and \(BC = 1\). Find the angle \(CAM\).
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Context:
Calculate the definite integral:

$$
\int_{0}^{\frac{2\pi}{3}} \frac{\cos^2 x \, dx}{(1 + \cos x + \sin x)^2}
$$
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Context:
Given that the point $\left(\frac{1}{2}, 2\right)$ is on the graph of the power function $y=f(x)$, and the point $(-2, \frac{1}{4})$ is on the graph of the power function $y=g(x)$, find the value of $f(2)+g(-1)$.
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Context:
We define \( a @ b = a \times (a + 1) \times \ldots \times (a + b - 1) \). Given \( x @ y @ 2 = 420 \), then \( y @ x = \) ?
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Context:
Let  $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form  $\frac{b}{2^a}$ , where  $a$  is a non-negative integer and  $b$  is an odd integer (not necessarily positive). Find  $a+b$ .
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Context:
Let $P$ and $A$ be $3 × 3$ real matrices such that $PAP^t = -A^t$, where $P^t$ denotes the transpose of $P$. Find the possible values of $\det(P)$.
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Context:
Given the sequence \( \{ a_n \} \) of real numbers for \( n \ge 1 \) defined by the recurrence relation:
\[ a_{n+2} = a_{n+1} - a_n \]
and the conditions:
\[ \sum_{n=1}^{999} a_n = 1003 \]
\[ \sum_{n=1}^{1003} a_n = -999 \]
find the value of \( \sum_{n=1}^{2002} a_n \).
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Context:
Given an equilateral triangle \( \triangle ABC \) with side length 1 and \( PA \perp \) plane \( ABC \), and \( PA = \frac{\sqrt{6}}{4} \). Let \( A' \) be the reflection of point \( A \) over plane \( PBC \). Determine the angle formed between line \( A'C \) and \( AB \).
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Context:
I have found that the curve given by $x^3+x+1=y^2$ over $\mathbb{F_5}$ has 9 points. Now I am supposed to find the number of points of the same curve on $\mathbb{F}_{125}$. Using Hasse and the fact that 9 has to divide this order I have 5 possibilities: $108,117,126,135,144$ but here I don't know how to proceed.
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Context:
Find the norm of the linear operator \( T: C([0,1],\mathbb{R}) \rightarrow C([0,1],\mathbb{R}) \) defined by \( Tf = gf \), where \( g \in C([0,1],\mathbb{R}) \), with respect to the \( L^2 \) norm \( \|f\|_2 \).
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Context:
Compute the limit: \[ \lim_{n\to\infty}\sum_{i=1}^{2n^4+n^2}\frac{5n^2+1}{n^4+i} \]
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Context:
Evaluate the integral \( I = \int_{0}^{1} f(x) \, dx \) given that the function \( f : [0,1] \rightarrow \mathbb{R} \) satisfies \( f'(x) = \sqrt{1+f^2(x)} \) for all \( x \in [0,1] \) and \( f(0) + f(1) = 0 \).
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Context:
Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate.
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Context:
Given a permutation group \( G \) acting on a finite set \( D \), and a finite set \( R \) of colors, a permutation group \( \hat{G} \) is induced to act on \( R^D \) (the set of functions from \( D \) to \( R \)). For any \( \sigma \in G \), the function \( \hat{\sigma} \) is defined by \( \hat{\sigma}(F)(d) = F(\sigma(d)) \) for all \( F \in R^D \). Determine the relationship between \( |G| \) and \( |\hat{G}| \).
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Context:
In the quadrilateral PQRS, the length of PQ is 11 cm, the length of QR is 7 cm, the length of RS is 9 cm, and the length of SP is 3 cm. Both ∠QRS and ∠SPQ are 90°. What is the area of the quadrilateral PQRS?
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Context:
Determine the number of real roots of the polynomial \( P(x) = x^7 + x^6 - 7x^5 - 7x^4 + 8x^3 + 8x^2 + 16x + 16 \), given that its Sturm's sequence ends with \( P_5(x) = -x^2 + 4 \).
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Context:
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a}|=2$, $| \overrightarrow {b}|=1$, and $\overrightarrow {b} \perp ( \overrightarrow {a}+ \overrightarrow {b})$, determine the projection of vector $\overrightarrow {a}$ onto vector $\overrightarrow {b}$.
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Context:
Starting from the natural numbers beginning with 1, remove multiples of 4, and then group the remaining numbers into sets of three. Determine the group number and position within the group where the number 2022 falls. Express your answer in the format "group number, position within group" (e.g., 506th group, 2nd number).
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Context:
求最小的正整数$n$，满足：

(1)$n$恰有144个不同的正因数；
(2)在$n$的正因数中有10个连续整数．
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Context:
From vertex \( B \) of the triangle \( ABC \), a line segment antiparallel to \( BC \) is drawn and intersects side \( b \) at point \( B_1 \). From \( B_1 \), a similar segment parallel to \( BC \) is drawn, intersecting side \( c \) at point \( C_1 \). This procedure can be repeated infinitely.

What is the sum of \( \overline{B C}+\overline{B B_{1}}+B_{1} C_{1}+\ldots \) to infinity, if \( b > c \)?
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Context:
Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \), what is \( m+n \)?
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Context:
Given two sets $A=\{1,x^{- \frac {1}{2}}\}$ and $B=\{0,1,2\}$, if $A \subseteq B$, then $x =$ __________.
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Context:
Marcy has three pets: a cat, a dog, and a parrot. She spends 12 minutes petting her cat, 1/3 of that time combing it, 1/4 of the combing time brushing its teeth, and 1/2 of the petting time playing with it. In addition to that, she feeds her cat for 5 minutes and spends 2/5 of the feeding time cleaning its food bowl.

She spends 18 minutes walking her dog, 2/3 of that time playing fetch, and 9 minutes grooming it, with 1/3 of the time trimming its nails. She spends 1/4 of the walking time training the dog for new tricks.

Her parrot needs 15 minutes of daily speech training, where she divides the time equally between repeating words and playing short tunes on a xylophone. Marcy also spends 8 minutes per day cleaning the parrot's cage, with half of that time spent setting up its toys in the cage.

How many minutes does Marcy spend with her pets in total?
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Context:
The integers whose absolute values are not greater than $3$ are ____.
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Context:
On a circle with a radius of 1, mark a point \( O \) and use a compass to make a mark to the right with a radius of \( l \). From the obtained point \( O_{1} \), make a second mark in the same direction with the same radius, and continue this process 1968 times. After this, the circle is cut at all 1968 marks, resulting in 1968 arcs. How many different lengths of arcs can be obtained in this way?
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Context:
Some camels cost as much as 24 horses, 16 horses cost as much as 4 oxen, and 6 oxen cost as much as 4 elephants. The cost of 10 elephants is Rs. 110000. If the cost of a camel is Rs. 4400, how many camels are we talking about?
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Context:
From point $A$, two rays emerge at an angle $\alpha$. Two points $B$ and $B_{1}$ are taken on one ray, and two points $C$ and $C_{1}$ are taken on the other ray. Find the length of the common chord of the circles circumscribed around triangles $ABC$ and $AB_{1}C_{1}$, if $|AB|-|AC|=\left|AB_{1}\right|-\left|AC_{1}\right|=a$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{a}{2 \sin \left( \frac{\alpha}{2} \right)}}


Context:
Calculate the spectral radius of the Volterra operator \( A \) defined on \( L^2[0,1] \) by \( A(f)(x) = \int_0^x f(t) \, dt \). Use the formula \( \rho(A) = \lim_{n\rightarrow \infty} \|A^n\|^{1/n} \) and the fact that \( \|(x-s)^n\|_\infty = 1 \) for \( s \in [0,x] \), \( x \leq 1 \).
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Context:
The height of a right triangle, dropped to the hypotenuse, is $h$.

What is the minimum length that the median, drawn from the vertex of the larger acute angle, can have?
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Context:
Determine whether the map \( \phi : L^3([0,1], \mathbb{C}) \to \mathbb{C} \) defined by \( \phi(f) = \int_{0}^{1} f(x^2) \, dx \) is a bounded linear functional on the Banach space \( L^3([0,1], \mathbb{C}) \). If it is bounded, find its norm.
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Context:
If ${e}^{x+1}-\frac{lnx+2k}{x}-k≥0$ always holds for any $x \gt 0$, then the range of real number $k$ is ____.
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Context:
Given two independent standard normal random variables, $X$ and $Z$, calculate the probability $P(X+Z<0, Z>0)$. Express your answer as a single probability value.
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Context:
From a pile of 20 100-yuan bills that includes 5 counterfeit bills, two bills are randomly selected. If one of the two bills is found to be counterfeit when checked with a counterfeit detector, the probability that both bills are counterfeit is _______.
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Context:
Given the vector space $P_3$ consisting of polynomials with a degree of 3 or less, and a basis $\{1, (1-x), (1-x)^2, (1-x)^3\}$. Consider the linear functional (evaluation function) $ev_3: P_3 \rightarrow \mathbb{R}$ that evaluates a polynomial at $x=3$. If $\{f_1, f_2, f_3, f_4\}$ is the dual basis of $P_3$, compute $f_3(p)$ for the polynomial $p = x^3 + 5x^2 + 3x + 4$.
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Context:
Given that \( n \) is a natural number and \( b = n^{3} - 4an^{2} - 12n + 144 \) is a prime number, find the value of \( b \).
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Context:
3 cloves of garlic have a 100% repulsion rate for vampires and wights but a 75% effectiveness rate for vampire bats, 60% for werewolves, and 50% for ghosts. How many cloves of garlic are needed to repel 30 vampires, 12 wights, 40 vampire bats, 20 werewolves, and 15 ghosts, considering their varying garlic compatibility percentages?
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Context:
Let  $\mathcal S$  be a set of  $16$  points in the plane, no three collinear. Let  $\chi(S)$  denote the number of ways to draw  $8$  lines with endpoints in  $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of  $\chi(\mathcal S)$  across all such  $\mathcal S$ .

*Ankan Bhattacharya*
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Context:
Determine the number of distinct ways to color a cube using four different colors, with each color appearing on at least one face of the cube. Consider two colorings to be the same if one can be obtained from the other by rotating the cube.
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Context:
Find the derivative of the quadratic form $z^TX^TXz$ with respect to the vector $z$.
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Context:
Find the number of positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect square.
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Context:
A 6-liter solution is 30% alcohol. How many liters of pure alcohol must be added to produce a solution that is 50% alcohol?
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Context:
The midpoints of the sides of a rectangle with a perimeter of 32 are connected by segments. Then, the same operation is performed on the resulting quadrilateral: the midpoints of its sides are connected by segments. How many times must this operation be repeated until the perimeter of the resulting quadrilateral is less than 1 for the first time?
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Context:
Find the sum of the values of $x$ such that $\cos^{3}3x+\cos^{3}5x= 8\cos^{3}4x\cos^{3}x$, where $x$ is measured in degrees and $100 ^\circ < x < 200^\circ$.
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Context:
The number of intersection points between the graphs of the functions \( y = \sin x \) and \( y = \log_{2021} |x| \) is:
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Context:
3. Given in the tetrahedron $D-ABC$, $\angle ACB=\angle ABD=90^{\circ}$, $CA=CB$, $\angle BAD=30^{\circ}$. If the projection of point $C$ on the plane $ABD$ is exactly on $AD$, then the sine value of the dihedral angle $C-AB-D$ is $\qquad$.
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Context:
Given an odd function \( f(x) \) with a domain of \([-2,2]\), which is decreasing on the interval \([-2,0]\), and satisfies \( f(1-m)+f\left(1-m^{2}\right)<0 \), determine the range of values for the real number \( m \).
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Context:
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \left( \frac{n}{1+n^2x^2} - \frac{ne^x\cos x}{1 + n^2x^2} \right) \, dx \).
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Context:
one bacterium splits into 8 bacteria of the next generation . but due to environment , only 50 % of one generation can produced the next generation . if the seventh generation number is 4096 million , what is the number in first generation ?
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Context:
Calculate the limit: $$\lim_{h\rightarrow0}\sup_{n\in\mathbb{N}}\left\|\sqrt{n}\,\mathbb{1}_{(0,\frac{1}{n})}(x+h)-\sqrt{n}\,\mathbb{1}_{(0,\frac{1}{n})}(x)\right\|_{L^2(\mathbb{R})}$$
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Context:
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is $\frac {π}{3}$, and the magnitudes of $\overrightarrow {a}$ and $\overrightarrow {b}$ are 2 and 1 respectively, find the dot product of $\overset{ -}{a}$ and ($ \overrightarrow {a}$$+2 \overrightarrow {b}$).
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Context:
Find the value of sum $\sum\limits_{n=1}^\infty A_n$, where
$$
A_n=\sum\limits_{k_1=1}^\infty\cdots\sum\limits_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}
$$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The answer is \( e^{\frac{\pi^2}{6}} - 1 \).

}


Context:
Let  $F$  be the focus of parabola  $y^2=2px(p>0)$ , with directrix  $l$  and two points  $A,B$  on it. Knowing that  $\angle AFB=\frac{\pi}{3}$ , find the maximal value of  $\frac{|MN|}{|AB|}$ , where  $M$  is the midpoint of  $AB$  and  $N$  is the projection of  $M$  to  $l$ .
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Context:
A woman has $\$1.58$ in pennies, nickels, dimes, quarters, half-dollars and silver dollars. If she has a different number of coins of each denomination, how many coins does she have?
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Context:
Suppose in a right triangle $PQR$, $\cos Q = 0.5$. Point $Q$ is at the origin, and $PQ = 15$ units along the positive x-axis. What is the length of $QR$?
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Context:
Determine the number of involutory $3 \times 3$ matrices over the field $\mathbb{Z}_{13}$. An involutory matrix is one that is its own inverse.
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Context:
In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant must run to complete this. Express your answer to the nearest meter.
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Context:
8. $[7]$ Two circles with radius one are drawn in the coordinate plane, one with center $(0,1)$ and the other with center $(2, y)$, for some real number $y$ between 0 and 1 . A third circle is drawn so as to be tangent to both of the other two circles as well as the $x$ axis. What is the smallest possible radius for this third circle?
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Context:
There are 5 green candies, 3 blue candies, and 4 red candies in a bag. If Violet randomly picks a candy from the bag, how likely is it that it's blue?
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Context:
Students Alekseev, Vasiliev, Sergeev, and Dmitriev were cleaning the classrooms of the 7th, 8th, 9th, and 10th grades. During an inspection, it was found that the 10th grade was cleaned poorly. They started to find out who cleaned this classroom. Alekseev said, "I cleaned 7th grade, and Dmitriev cleaned 8th grade." Vasiliev: "I cleaned 9th grade, and Alekseev cleaned 8th grade." Sergeev: "I cleaned 8th grade, and Vasiliev cleaned 10th grade." Dmitriev left for home before the inspection. Who cleaned the 10th grade if exactly one part of each of these statements is true?
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Context:
Determine the winding number of the curve \( \alpha(t) = ((2 \cos t - 1) \cos t, (2 \cos t - 1) \sin t) \) with respect to the point \( p = (1/2, 0) \) using complex analysis.
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Context:
A water tank has the shape of the bottom half of a sphere with radius $r=1$ meter. If the tank is full, how much work is required to pump all the water out the top of the tank?
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Context:
A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \frac{1}{2},$ and
\[a_n = \frac{1 - a_{n - 1}}{2a_{n - 2}}\]for all $n \ge 3.$  Find $a_{120}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Given that \( f(0) = 0 \), \( f(1) = 2 \), and \( \int_0^1 f(x) \, dx = 3 \), find the value of \( \int_0^1 x f(x) \, dx \).
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Context:
Given \( f(x) \) is an odd function defined on \( \mathbf{R} \), \( f(1) = 1 \), and for any \( x < 0 \), it holds that
\[
f\left(\frac{1}{x}\right) = x f\left(\frac{1}{1-x}\right) .
\]
Then find the value of the sum
\[
\sum_{k=1}^{1009} f\left(\frac{1}{k}\right) f\left(\frac{1}{2019-k}\right).
\]
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Context:
The base of the pyramid \(TABC\) is a triangle \(ABC\), all sides of which are equal to 3, and the height of the pyramid, equal to \(\sqrt{3}\), coincides with the lateral edge \(TA\). Find the area of the cross-section of the pyramid by a plane that passes through the center of the circumscribed sphere of the pyramid, is parallel to the median \(AD\) of the base, and forms an angle of \(60^{\circ}\) with the base plane.
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Context:
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Let $E$ be the foot from $B$ to $AC$ and $F$ be the foot from $C$ to $AB.$ Denote $M$ the midpoint of $BC.$ The circumcircles of $\triangle BMF$ and $\triangle CME$ meet at another point $G.$ Compute the length of $GC.$
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Context:
Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?
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Context:
Calculate the difference $A - B$, where $A$ is the number of 6-digit tickets (from 000000 to 999999) that are considered lucky, and $B$ is the number of tickets that are considered medium. A ticket is lucky if the sum of its first three digits equals the sum of its last three digits. A ticket is medium if the sum of all its digits equals 27.
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Context:
In a plane, given a point Q and n distinct points P1, P2, ..., Pn. For each 1 ≤ k ≤ n, |QPk| is different. Furthermore, for each 1 ≤ k ≤ n, consider the circle with center Pk and radius PkQ. Suppose that none of the circles contain more than two points (these two points are Q and Pk), i.e., there is no other point inside or on the circle. Find the maximum value of n.
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Context:
Determine the cardinality of the set \( B \), where \( B \) is defined as: \[ B = \{g \in \Bbb N \to \Bbb N \mid \exists T\subseteq \Bbb N, \left.g\right|_T = \operatorname{Id}_T \land g(\Bbb N \setminus T)\subseteq \Bbb N \setminus T\} \]
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Context:
Convex quadrilateral \(EFGH\) has its vertices \(E\), \(F\), \(G\), and \(H\) situated on the sides \(AB\), \(BC\), \(CD\), and \(DA\) of convex quadrilateral \(ABCD\) respectively, and they satisfy the condition \(\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1\). Points \(A\), \(B\), \(C\), and \(D\) are located on the sides \(H_1E_1\), \(E_1F_1\), \(F_1G_1\), and \(G_1H_1\) of convex quadrilateral \(E_1F_1G_1H_1\) respectively, and they satisfy the conditions \(E_1F_1 \parallel EF\), \(F_1G_1 \parallel FG\), \(G_1H_1 \parallel GH\), and \(H_1E_1 \parallel HE\). Given that \(\frac{E_1A}{AH_1}=\lambda\), find the value of \(\frac{F_1C}{CG_1}\).
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Context:
14. How many polynomials $P$ with integer coefficients and degree at most 5 satisfy $0 \leq P(x)<120$ for all $x \in\{0,1,2,3,4,5\} ?$
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Context:
For which real numbers \(a\) does the series \( \sum_{n=0}^\infty \frac{1}{u_n} \) converge, where the sequence \( \{u_n\} \) is defined recursively by \( u_0 = 2 \) and \( \int_{u_n}^{u_{n+1}} (\ln u)^a \, du = 1 \)?
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Context:
Evaluate the double integral \( \iint \frac{(x^2+y^2)^2}{x^2y^2} \, dx \, dy \) over the region common to the circles \( x^2+y^2=7x \) and \( x^2+y^2=11y \).
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Context:
Let $k=\int_{0}^{\pi }{(\sin x-\cos x)}dx$. If ${{(1-kx)}^{8}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{8}}{{x}^{8}}$, find the sum of coefficients $a_1 + a_2 + \cdots + a_8$. Express your answer as a single integer.
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Context:
For any positive integer $n$, define
\[ E(n) = n(n+1)(2n+1)(3n+1)\cdots (10n+1). \]
Find the greatest common divisor of $E(1), E(2), E(3), \dots, E(2009)$.
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Context:
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$. If $S_7 > S_8 > S_6$, then the positive integer $n$ that satisfies $S_n \cdot S_{n+1} < 0$ is $\underline{\hspace{1cm}}$.
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Context:
It takes 10 minutes to wash a car, 15 minutes to change oil, 30 minutes to change a set of tires, 45 minutes to paint a car, and 60 minutes to service an engine. Mike works on sedans and SUVs. For sedans, he washes 9 cars, changes the oil on 6 cars, changes two sets of tires, paints 4 cars, and services 2 engines. For SUVs, he washes 7 cars, changes the oil on 4 cars, changes 3 sets of tires, paints 3 cars, and services 1 engine. However, since SUVs are larger, it takes 1.5 times longer to wash and paint them than it does for sedans. How many hours did Mike work?
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Context:
Determine the cardinality of all triangular subsets of the Sierpinski triangle.
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Context:
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?
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Context:
In a complete graph $G$ with 25 vertices, each edge is colored either red or blue. Given that there are no red triangles in $G$, determine the largest integer $m$ such that a blue complete subgraph $K_m$ is guaranteed to exist. Use Ramsey theory to find your answer.
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Context:
The coordinates of the point $A(-4,3)$ with respect to the $x$-axis are ____.
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Context:
In $\triangle Q R S$, point $T$ is on $Q S$ with $\angle Q R T=\angle S R T$. Suppose that $Q T=m$ and $T S=n$ for some integers $m$ and $n$ with $n>m$ and for which $n+m$ is a multiple of $n-m$. Suppose also that the perimeter of $\triangle Q R S$ is $p$ and that the number of possible integer values for $p$ is $m^{2}+2 m-1$. What is the value of $n-m$?
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Context:
What is the cardinality of the set of hyperreal numbers?
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Context:
Given two distinct lines OA and OB, take any point on OA and draw a perpendicular to OB. Then, from the foot of this perpendicular, draw another perpendicular to OA, and from the second foot, draw a perpendicular to OB, and so on, indefinitely. If the lengths of the initial two perpendicular segments are a and b (a!= b), what is the sum of these perpendicular segments as the process continues indefinitely? Express your answer in terms of a and b.
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Context:
A bicycle wheel has a certain diameter. It makes 522.0841599665866 complete revolutions in 1 km. What is the diameter of the wheel?
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Context:
Find the number of subgroups of the group $\mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_9$ that are isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$. Provide your answer as a single integer.
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Context:
Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\frac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
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Context:
Evaluate the integral \( \int (10x^2 + 3y^3 - 4z^2 + 8) \delta(2\vec{r}) \, d^3\vec{r} \), where \( \delta(2\vec{r}) \) is a three-dimensional delta function.
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Context:
The angle bisector of angle \( A \) of triangle \( ABC \) intersects side \( BC \) at point \( D \). A circle with radius 35, whose center lies on the line \( BC \), passes through points \( A \) and \( D \). It is known that \( AB^2 - AC^2 = 216 \), and the area of triangle \( ABC \) is \( 90 \sqrt{3} \). Find the radius of the circumcircle of triangle \( ABC \).
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Context:
Determine the Galois group of the splitting field of the polynomial $x^4 - 2x^2 - 2$ over the finite field $F_7$, where $F_7$ is the field with 7 elements.
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Context:
Cagney can frost a cupcake every 15 seconds and Lacey can frost a cupcake every 45 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes.
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Context:
For the integer numbers $i,j,k$ satisfying the condition $i^2+j^2+k^2=2011$, what is the largest possible value of $i+j+k$? Express your answer as a single integer.
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Context:
In the tetrahedron $ABCD$, it is given that $AB=BD=CD=3$, $AD=BC=4$, and $AC=5$. Determine the size of the dihedral angle $A-BD-C$.
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\boxed{\\pi - \\arccos \left( \frac{1}{10} \right)}


Context:
How many 10-digit numbers with distinct digits are divisible by 98765?
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Context:
Given a regular tetrahedron with edge length $a$, find the radius of a sphere centered at the center of the tetrahedron such that the total volume of the part of the tetrahedron outside the sphere and the part of the sphere outside the tetrahedron is minimized.
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Context:
B2. In a top hat, there are a hundred tickets numbered from 1 to 100. You want to have three tickets with the property that each of the three numbers is smaller than the sum of the other two numbers. For example, the three tickets with numbers 10, 15, and 20 would be suitable (since $10<15+20, 15<10+20$, and $20<10+15$), but the tickets with numbers 3, 4, and 7 would not (since 7 is not less than $3+4$). You may (without looking at the numbers) take a number of tickets from the hat.

How many tickets must you take at a minimum to be sure that there are three tickets among them that meet your requirement?
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Context:
Find the expected value of \( N = \min\{n: U_1 + U_2 + \cdots + U_n > 1\} \), where \( U_1, U_2, \ldots \) are independent uniform (0,1) random variables.
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Context:
Consider a continuous distribution on \((0,1)\) with probability distribution function \(f\) and cumulative distribution function \(F\). Define the function
\[ g(r) = \frac{\int_r^1 x f(x) \, dx}{2 - F(r)} \]
where \(r \in (0,1)\). Let \(r_M\) be the value of \(r\) that maximizes \(g(r)\). Determine if there exists a positive constant \(c\), independent of the distribution, such that
\[ \int_{r_M}^1 x f(x) \, dx \geq c \int_0^1 x f(x) \, dx. \]
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Context:
Alice and Bob take turns removing balls from a bag containing 10 black balls and 10 white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of black balls which Bob has can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\).
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Context:
A TV station is scheduled to broadcast two series for a company. Series A has a runtime of 80 minutes per episode, including 1 minute of commercials, attracting 600,000 viewers; Series B has a runtime of 40 minutes per episode, including 1 minute of commercials, attracting 200,000 viewers. If the company and the TV station agree that the station must broadcast at least 6 minutes of commercials per week and the station can only provide up to 320 minutes of program time for the company each week, what is the maximum number of viewers the TV station can attract per week by scheduling these two series accordingly? Express your answer in millions of viewers, rounded to the nearest tenth if necessary.
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Context:
7. (10 points) There is a sufficiently deep water tank with a rectangular base that is 16 cm long and 12 cm wide. Originally, the tank contains 6 cm of water and 6 cm of oil (the oil is above the water). If a block of iron with dimensions 8 cm in length, 8 cm in width, and 12 cm in height is placed in the tank, the height of the oil layer will be $\qquad$ cm.
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Context:
At the end of the month, a certain ocean desalination plant’s reservoir, which is irregularly shaped, contained 14 million gallons of water. This amount is twice the normal level due to an unusually high yield this month. It's known that this amount represents 70% of the reservoir’s total capacity, but it also should take into account a monthly 10% water loss due to evaporation. How many million gallons short of total capacity is the normal level, after factoring in the monthly evaporation loss?
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Context:
The number of non-empty proper subsets of the set $\{y \in \mathbb{N} | y = -x^2 + 6, x \in \mathbb{N}\}$ is ____.
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Context:
"That" and "this," and half of "that" and "this" - how many percent is this of three-quarters of "that" and "this"?
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Context:
What will be the difference between simple and compound interest at 8% per annum on a certain sum after 4 years if the difference is 40.48896000000036?
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Context:
Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.

[asy]
unitsize(0.2inch);
defaultpen(linewidth(0.7));
real f(real y)
{
  return (5*sqrt(3)-y)/sqrt(3);
}
draw((-5,0)--(5,0)--(0,5*sqrt(3))--cycle);
draw((-f(5*sqrt(3)/4),5*sqrt(3)/4)--(f(5*sqrt(3)/4),5*sqrt(3)/4));
draw((-f(5*sqrt(3)/2),5*sqrt(3)/2)--(f(5*sqrt(3)/2),5*sqrt(3)/2));
draw((-f(15*sqrt(3)/4),15*sqrt(3)/4)--(f(15*sqrt(3)/4),15*sqrt(3)/4));
label("$A$",(0,5*sqrt(3)),N);
label("$B$",(-f(15*sqrt(3)/4),15*sqrt(3)/4),WNW);
label("$C$",(f(15*sqrt(3)/4),15*sqrt(3)/4),ENE);
label("$D$",(-f(5*sqrt(3)/2),5*sqrt(3)/2),WNW);
label("$E$",(f(5*sqrt(3)/2),5*sqrt(3)/2),ENE);
label("$F$",(-f(5*sqrt(3)/4),5*sqrt(3)/4),WNW);
label("$G$",(f(5*sqrt(3)/4),5*sqrt(3)/4),ENE);
label("$H$",(-5,0),W);
label("$I$",(5,0),E);[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数 y=f(x) 的图像既关于点 (1,1) 中心对称，又关于直线 x+y=0 轴对称. 若 x\in(0,1) 时，f(x)=\log_2(x+1)，则 f(\log_210) 的值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
What is the absolute value of the difference between the squares of 101 and 99?
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Context:
In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ lie on segments $BH$ and $HF$, respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the infinite series representations:
$$a = 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + \cdots$$
$$b = \frac{x}{1!} + \frac{x^4}{4!} + \frac{x^7}{7!} + \cdots$$
$$c = \frac{x^2}{2!} + \frac{x^5}{5!} + \frac{x^8}{8!} + \cdots$$
Let $u = a^2 + b^2 + c^2$. Determine the value of $u(\ln 2)$.
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Context:
设U={1,2,3,⋯,n}(n≥3)，S是U的一个子集.如果不属于S的一个元素出现在U的一个排列P的某处，并且在该排列P中它处在S中的两个元素之间，那么我们称S是由排列P分裂出来的（例如排列13542可分裂出子集｛1,2,3｝，而不能分裂出子集｛3,4,5｝），试求最大正整数m，使得对U的任意m个子集（每个子集至少有2个元，至多有n-1个元），都存在U的一个排列分裂出所有这m个子集.
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Context:
Find the minimum possible value of $\sqrt{58-42 x}+\sqrt{149-140 \sqrt{1-x^{2}}}$ where $-1 \leq x \leq 1$
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Context:
In a certain month of a certain year, there are exactly 5 Fridays. Furthermore, the first and last day of this month are not Fridays. What day of the week is the 12th day of the month?
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Context:
Find the smallest positive integer  $n$  such that if we color in red  $n$  arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.
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Context:
Every cell of a $3 \times 3$ board is colored either red or blue. Find the number of all possible colorings such that there are no $2 \times 2$ squares where all cells are red.
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Context:
Let \( T \) be a right triangle with sides having lengths 3, 4, and 5. A point \( P \) is called awesome if \( P \) is the center of a parallelogram whose vertices all lie on the boundary of \( T \). What is the area of the set of awesome points?
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Context:
Add one pair of brackets to the expression:

\[1 + 2 \times 3 + 4 \times 5 + 6\]

so that the resulting expression has a valid mathematical value. For example, \(1 + 2 \times (3 + 4 \times 5) + 6 = 53\). What is the largest possible value that one can make?
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Context:
8. In a certain World Cup, the Brazilian team has a probability of $\frac{1}{2}$ to win against each opponent, a probability of $\frac{1}{3}$ to draw, and a probability of $\frac{1}{6}$ to lose. Winning a match earns 3 points, drawing earns 1 point, and losing earns 0 points. It is known that each team plays three matches in the group stage, and 4 points or more are required to advance from the group. In the knockout stage, if the teams draw, they go to a penalty shootout. It is known that Brazil's probability of winning a penalty shootout is $\frac{3}{5}$. What is the probability that Brazil will win the championship and have exactly one penalty shootout in the four knockout matches? $\qquad$
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Context:
Let  $S$  be a subset of  $\{0,1,2,\dots ,9\}$ . Suppose there is a positive integer  $N$  such that for any integer  $n>N$ , one can find positive integers  $a,b$  so that  $n=a+b$  and all the digits in the decimal representations of  $a,b$  (expressed without leading zeros) are in  $S$ . Find the smallest possible value of  $|S|$ .
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Context:
Given that the eccentricity of a hyperbola is $2$, what is the acute angle formed by its two asymptotes?
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Context:
Calculate the number of subgraphs of a graph with two disconnected components, one with 4 vertices and 4 edges, and the other with 2 vertices and 1 edge.
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Context:
Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2 + 4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$
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Context:
Let $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline
$$I_k=\int_{0}^{\infty}\frac{x^k}{p(x)}dx$$.
For which $k$ is $I_k$ smallest.
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Context:
Find the minimum possible value of
\[\sqrt{58 - 42x} + \sqrt{149 - 140 \sqrt{1 - x^2}}\]where $-1 \le x \le 1.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
The area of rhombus \(ABCD\) is 2. The triangle \(ABD\) has an inscribed circle that touches the side \(AB\) at point \(K\). A line \(KL\) is drawn through point \(K\) parallel to the diagonal \(AC\) of the rhombus (the point \(L\) lies on the side \(BC\)). Find the angle \(BAD\) given that the area of triangle \(KLB\) is \(a\).
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Context:
Mary just arrived at the beach. She has 4 times as many towels as Frances does. The total weight of their towels is 60 pounds. If Frances's towels weigh 128 ounces, how many towels does Mary have?
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