Context:
Given the equations:
\[ 2\sin^2 A - 2\tan A \sin A + 2 - \sin^2 A - \cos^2 B = 0 \]
\[ 2\cos^2 B - 2\tan A \cos B + 2 - \sin^2 A - \cos^2 B = 0 \]
If \( \sin A \neq \cos B \), find the value of \( \sin^2 A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) \) be a continuous function on \( \mathbf{R} \) such that \( f(0)=1 \) and \( f(x+y) \geqslant f(x) f(y) \) for all real numbers \( x \) and \( y \). Determine the function \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Jacobson radical of the ring \( R = \left\{ \frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0 \text{ and } q \nmid b \right\} \), where \( q \) is a prime different from \( p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a frequency response $H(f)$, find the inverse Fourier transform of $|H(f)|^2$ in terms of a convolution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}}$$ where $$H_k = \sum_{i=1}^{k} \frac{1}{i}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the region in the complex plane where the series \( \sum_{n=0}^{\infty} \frac{z^2}{(1+z^2)^n} \) is uniformly convergent.
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Context:
You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the splitting field of an irreducible Eisenstein polynomial $g$ with $p = 5$, given that one of its roots is $\sqrt{\frac{5}{2}+\frac{\sqrt{-5}}{2}}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_D \arctan e^{xy} \, dy \, dx \), where the region \( D \) is defined by \( \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 4x\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq  \alpha (x^2y + y^2z + z^2x).
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the series \( \sum_{n=2}^\infty \Bigg|\frac{x^n}{n(\log n)^2}\Bigg| \) converges.
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Context:
Determine the asymptotic behavior of the function \( f(x) = \sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1 \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a convex quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(P\). Given that \(PA = 1\), \(PB = 2\), \(PC = 6\), and \(PD = 3\), and that \(O\) is the circumcenter of \(\triangle PBC\). If \(OA\) is perpendicular to \(AD\), determine the circumradius of \(\triangle PBC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, X_2, \ldots, X_{10}$ be independent and identically distributed random variables, each uniformly distributed on $(-5,5)$. Determine the distribution of the random variable \( Y = -2\sum\limits_{i=1}^{10}\log\left(\frac{|X_i|}{5}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{n \to \infty} \frac{2^n + \left(1 + \dfrac{1}{n^2}\right)^{n^3} + \dfrac{4^n}{n^4}}{\dfrac{4^n}{n^4} + n^3 \cdot 3^n} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a set with cardinality $a$. Determine the cardinality of the set of countably infinite subsets of $A$. Express your answer in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ f(x)=\left(1+\frac{1}{x}\right)^{x}\ (x>0)$.

Find $ \lim\limits_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given elements $a, b, c$ in a unital non-commutative ring, where $c$ is an inverse of $1-ab$ such that $c(1-ab) = 1$, find an inverse for $1-ba$.
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Context:
Let \( p \) be a prime such that \( p^n \equiv 5 \pmod{6} \) for each odd positive integer \( n \). Determine the value of \( p^n \) modulo 36, i.e., find \( p^n \equiv ? \pmod{36}. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya has a triangle with vertices labeled 1, 2, and 3. Rotating the triangle 15 times clockwise around vertex 1 by the angle at this vertex returns it to its original position. Similarly, rotating the triangle 6 times clockwise around vertex 2 by the angle at this vertex also returns it to its original position. Determine the minimal number of times, n, that Vasya must rotate the triangle clockwise around vertex 3 by the angle at this vertex to return it to its original position.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of digits in the number \(2^{2^{22}}\) using number theory techniques, without employing logarithms.
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Context:
Let \( {ABCD} \) be a parallelogram. Point \( P \) is selected in its interior such that the distance from \( P \) to \( {BC} \) is exactly6times
 the distance from \( P \) to \( {AD} \) , and \( \angle {APB} = \angle {CPD} = {90}^{ \circ  } \) . Given that \( {AP} = 2 \) and \( {CP} = 9 \) , the area of \( {ABCD} \) can be
 expressed as \( m\sqrt{n} \) where \( m \) and \( n \) are positive integers and \( n \) is not divisible by the square of any prime. Find \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the five-digit number \(\overline{abcde}\) such that for the two-digit numbers \(\overline{ab}, \overline{bc}, \overline{cd}, \overline{de}\), the equation \((\overline{ab} + \overline{bc})(\overline{bc} + \overline{cd})(\overline{cd} + \overline{de}) = 157605\) holds. Note that multi-digit numbers cannot start with zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a probability density function (PDF) with a minimum value of $243.483K$ and a maximum value of $308.05K$, calculate the probability that the temperature lies between $281K$ and $291K$. Assume the PDF is defined over the range from $243.483K$ to $308.05K$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_{2-i\infty}^{2+i\infty} \frac{1}{t} \, \mathrm{d}t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $z \in \mathbb{C}$ be a root of the equation $z^2 - 2z + 3 = 0$. Find the modulus of the complex number $f(z) = z^{17} - z^{15} + 6z^{14} + 3z^2 - 5z + 9$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$ . (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action:
First, if your number is divisible by $3$  then turn around.
Then, if your number is divisible by $5$  then turn around.
Finally, if your number is divisible by $7$  then turn around.
How many students are face to coach now?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a, b, c$ are the roots of the cubic equation $f(x) = x^3 + 3x^2 - 7x + 1$, calculate the value of $f'(a)f'(b)f'(c)$. Provide your answer in a simplified form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence of polynomials \( \{f_n\} \) defined recursively by \( f_0(x) = 0 \) and \( f_{n+1}(x) = f_n(x) + \frac{x^2 - f_n^2(x)}{2} \) for \( n = 0, 1, 2, \ldots \), in the supremum norm on the interval \([-1, 1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of solutions for the equation \( g(x) = e^x - (px^3 + qx^2 + rx + s) \), where \( p, q, r, \) and \( s \) are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a row of n lamps, where n is a fixed positive integer and n ≥ 1, some lamps are on and some are off. In one move, you can choose a positive integer i (1 ≤ i ≤ n) and switch the state of the first i lamps from the left. Determine the smallest number k such that you can turn all the lamps on in at most k moves, regardless of the initial configuration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of points in the complex plane where the function \( f(z) = \sum_{n=0}^{\infty} \frac{z^{2n}}{(5n)^n} \) is holomorphic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the group defined by the presentation \( \langle x, y: xy=yx, x^5=y^3 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic relationship between the functions \( f(n) = \frac{\log n}{\log\log n} \) and \( g(n) = \log(n - \log n) \). Specifically, find whether \( f(n) = K(g(n)) \) for some asymptotic notation \( K \), such as \( \Omega, \Theta, \) or \( O \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{n=-\infty}^{\infty} \text{sinc}(\pi(2t-n)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the measure of the set \( F = \left\{x\in [-1, 1]: \text{there exist finitely many }p,q \in\mathbb{N} \text{ such that }\left|x -\frac{p}{q}\right|> \frac{1}{q^{2}} \right\} \). Denote this measure as \( m(F) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the product \( \prod_{k=0}^{2^{1999}}\left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the rate of convergence for the limit \( \lim_{k \rightarrow \infty} \left(1+\frac{2}{k}\right)^k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n \geq 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\ldots,e_{2n}$. The second exterior power $\Lambda^2(V)$ contains the element $$\omega=e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2n-1} \wedge e_{2n}.$$ Since $\Lambda^{2n}(V)$ is spanned by $x := e_1 \wedge \cdots \wedge e_{2n}$, the $n$-fold wedge product $\omega \wedge \cdots \wedge \omega$ is a scalar multiple of $x$. Compute this scalar.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the result of the convolution \( \delta_{im} \epsilon_{ijk} \) using Einstein summation notation, considering that two indices coincide.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sphere of radius $r$ centered at the origin. Two great circles of the sphere lie on the xy-plane and xz-plane. A part of the sphere is removed such that the cross-section of the remaining part, perpendicular to the x-axis, is a square with vertices on the great circles. Compute the volume of the remaining part of the sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the dimension of the space of cubic polynomials over $\mathbb{P}^5$ that vanish on $\mathbb{P}^2$, where $\mathbb{P}^2$ is embedded in $\mathbb{P}^5$ via the Veronese embedding.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct bracelets that can be made using six identical white beads, three identical blue beads, and five identical red beads, considering rotational and reflectional symmetries.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the smallest number system that fulfills the requirements of geometry and applied mathematics, excluding any numbers that are only used for mathematical convenience.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the product of the following integrals: \( \int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^4}} \) and \( \int_0^1 \frac{x^2\mathrm{d}x}{\sqrt{1-x^4}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the length of the longest increasing subsequence of a random permutation of \(n\) elements, divided by \(\sqrt{n}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a polynomial \( f \in \mathbb{Z}[x] \) that is irreducible over \( \mathbb{Q} \) but whose reduction modulo \( p \) is reducible for the primes \( p = 2, 3, \) and \( 5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game played within the unit disc $\mathcal{D}$, player $\mathcal{A}$ selects an initial point $p_0 \in \mathcal{D}$. At each step $n$, player $\mathcal{B}$ draws a line $\ell_n$ through $p_{n-1}$, and player $\mathcal{A}$ chooses a new point $p_n$ from the intersection of $\mathcal{D}$ and $\ell_n$. Player $\mathcal{B}$ wins if the sequence $(p_n)$ converges, while player $\mathcal{A}$ wins if it does not. Given that $\mathcal{B}$ has a winning strategy, determine the probability that $\mathcal{B}$ wins when both players make random choices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic behavior of the function \(a(n) = \sum_{i=1}^n d(i)\), where \(d(n)\) is the number of divisors of the positive integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of values needed to describe the orientation of a 3-dimensional rigid body in n-dimensional space.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit \(\lim_{r\to \infty}\frac{f(r)}{\pi r^2}\), where \(f(r)\) represents the number of integral points inside a circle of radius \(r\) centered at the origin. Given that the formula for the number of lattice points inside the boundary of such a circle is \(f(r)=1+4\lfloor r\rfloor+4\sum_{i=1}^{\lfloor r\rfloor}\lfloor \sqrt{r^2-i^2}\rfloor\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on the real numbers \(a\) and \(b\) such that the sequence \(X_n = \mathrm{e}^{a S_n - b n^2}\), where \(S_n = \sum_{k=1}^n \xi_k\) and \(\xi_1, \xi_2, \ldots\) are iid standard normal random variables, converges to 0 in \(\mathcal{L}^r\) for \(r \geq 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the covariance \( \text{cov}(5W_7 + 6W_9, W_7) \) where \( W_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( F=(x^3, y^3, z^3) \) through the surface of the sphere defined by \( x^2+y^2+z^2=x \) using the divergence theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set of distinct positive integers has a sum of 1987. Determine the maximum possible value of the expression \(3n + k\), where \(n\) is the total number of integers in the set and \(k\) is the number of odd integers in the set.
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Context:
Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. Points $A_1$, $B_1$, and $C_1$ are chosen on its incircle. Compute the maximum possible sum of the areas of triangles $A_1BC$, $AB_1C$, and $ABC_1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\infty \frac{t^2 - \sin^2 t}{t^4} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Markov chain with state space \( \{0, 1, 2\} \), where state 0 represents making no money, state 1 represents making a small amount of money, and state 2 represents making a large amount of money, the transition matrix is:
\[
\begin{bmatrix}
0.1 & 0.6 & 0.3 \\
0.4 & 0.4 & 0.2 \\
0.7 & 0.3 & 0
\end{bmatrix}
\]
Calculate \( P(X_5 = 2 \mid X_3 = 0, X_1 = 1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-isomorphic groups of order 42, given that there are 6 different homomorphisms from \( \mathbb{Z}_6 \) to \( \text{Aut}(\mathbb{Z}_7) \).
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Context:
In square \(ABCD\), lines \(EF\) and \(GH\) divide the square into four rectangles. Line \(EF\) is parallel to \(AB\), and \(GH\) is parallel to \(BC\). Given that \(\angle BAF = 18^\circ\) and the area of rectangle \(PFCH\) is twice that of rectangle \(AGPE\), find the nearest integer to the value of \(\angle FAH\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of individual equations needed to solve the geodesic equation in three dimensions on a spherical surface with coordinates $(r, \theta, \phi)$. The geodesic equation in Einstein notation is given by: $$\frac{d^2u^k}{d\lambda ^2}+\Gamma^k_{ij}\frac{du^i}{d\lambda}\frac{du^j}{d\lambda}=0$$ How many equations are there to solve for the geodesic paths on this surface?
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Context:
Given the function \( f(X)=\exp\left(\frac{-\gamma^2}{a^2X+b^2}\right) \) where \( X \sim \mathrm{Binomial}(n,p) \), use the central limit theorem to find an approximate expression for the value of \( \gamma^2 \) that maximizes the variance of \( f(X) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(X_1, X_2, \dots, X_n\) be a random sample from a distribution with probability density function (p.d.f.): \[ f(x, \theta) = e^{-(x-\theta)}; \quad \theta < x < \infty; \quad -\infty < \theta < \infty. \] Find the sufficient statistic for the parameter \(\theta\) using the factorization theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S = \{1, 2, 3, 4\} \). A sequence \( a_{1}, a_{2}, \ldots, a_{n} \) of \( n \) terms has the property that for any non-empty subset \( B \) of \( S \), there exists a sequence of \( |B| \) consecutive terms in the sequence that exactly forms the set \( B \). Determine the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a pyramid with a square base ABCD and apex S, the height of the pyramid is 8 units, and the angle between lines SA and SC is 48 degrees. Calculate the angle between the height SO (where O is the center of the square base) and the face SBC of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate
 \( {\oint }_{\left| z\right|  = 2}\frac{dz}{\sqrt{1 + {z}^{10}}} \)
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Context:
When tossing 3 coins 400 times, you score a point for 3 heads, lose a point for 3 tails, and get no points for any other outcome. The total score \( S \) is the difference between the number of 'triple heads' \( X \) and 'triple tails' \( Y \), both following a binomial distribution with parameters \( n=400 \) and \( p=1/8 \). Given that \( E(S) = 0 \), find the variance \( \text{Var}(S) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a closed form for the infinite series:
\[ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lambda = \lim\limits_{n \to \infty}{n\int_0^{\frac{\pi}{2}}(\sin x)^{2n} dx} \) without using Wallis' integral formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be a helicoid defined by the parametric equation $\sigma(u, v) = (u \cos v, u \sin v, v)$. Determine the normal curvature in the direction of a general unit tangent vector at the origin $p = (0, 0, 0)$. Provide your answer in terms of the components of the unit tangent vector.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given random variables $X_1, \dots, X_n$ and $Y_1, \dots, Y_n$ taking values in $\{0,1\}$, with the conditions $\sum_{i=1}^n X_i = \sum_{i=1}^n Y_i$, $\Pr[X_i=0] \geq \alpha$, and $\Pr[Y_i=0] \geq \beta$, determine the upper bound on $\alpha + \beta$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $q$ be the sum of the expressions $a_1^{-a_2^{a_3^{a_4}}}$ over all permutations $(a_1, a_2, a_3, a_4)$ of $(1,2,3,4).$ Determine $\lfloor q \rfloor.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $p$ such that the function \( f(x,y) = \frac{1}{(x^2+y^3-2)^p} \) is integrable over the region \( 1 \leq x \leq 2, 1 \leq y \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of edges in a directed graph with 7 vertices, 2 connected components, and 4 strongly-connected components.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0} \frac{\ln\left[\dfrac{(1-3x)(1+x)^3}{(1+3x)(1-x)^3}\right]}{x^3}$$ without using L'Hôpital's rule, Taylor series, or differentiation (or integration).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T$ be the linear transformation defined by $T=\begin{bmatrix}3&1\\1&3\end{bmatrix}$. Determine the area of the region enclosed by $\bigcap_{n\geq1}T^{n}D$, where $D$ is the unit disk in $\mathbb{R^{2}}$. \(\text{Provide your answer as a single numerical value.}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of positive integers \( n \leq 1988 \) for which the function \( f(n) = n \), given that \( f(1) = 1 \), \( f(3) = 3 \), \( f(2n) = f(n) \), \( f(4n + 1) = 2f(2n + 1) - f(n) \), and \( f(4n + 3) = 3f(2n + 1) - 2f(n) \) for all positive integers \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the function $f(X)$ that minimizes the conditional expected value $E[(Y-f(X))^2|X]$, given that $X$ and $Y$ are random variables with $E(Y) = \mu$ and $E(Y^2) < \infty$. Provide your answer in terms of $X$ and $Y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In parallelogram $ABCD$, $
abla ABC=105^{\circ}$. Triangle $BMC$ is equilateral, and $
abla CMD=135^{\circ}$. Point $K$ is the midpoint of $AB$. Determine the measure of $
abla BKC$. \( \text{(Express your answer in degrees.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest integer $k$ such that $PSU_3$ is a subgroup of $GL_k(\mathbb{C})$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the group of symmetries for the elements of the quaternion group, excluding 1 and -1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_\gamma \frac{-y^2 \, dx + 2xy \, dy}{x^2 + y^4} \) along the path \( \gamma: r(t) = (t, 2t^2 - 2) \) for \( -1 \leq t \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. What is the probability that exactly one of the stacks consists of two red boxes and two blue boxes? Express your answer as a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, and find \( m + n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the norm of the inverse operator \(A^{-1}\) for the operator \(A\) defined on \(L^2[0,1]\) by \((Ax)(t) = x(t) - \frac{1}{2}\int_{0}^{1}x(s)ds\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p,q \ge 2$ be such that $p+1$ is a divisor of $q+1$. Let $S$ be a star with $p$ edges. Calculate the Ramsey number $R(S,K_{1,q})$. Provide your answer as a single integer.
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Context:
Suppose \(X_1, \ldots, X_n\) are a random sample from a normal distribution with mean \(\theta\) and variance \(\sigma^2\), where \(\sigma^2\) is fixed and \(\theta > 0\) is a parameter. Find the maximum likelihood estimator of \(\sqrt{\theta}\).
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Context:
Given independent and identically distributed (i.i.d.) random variables $Z, \epsilon_1, \epsilon_2 \sim N(0,1)$, where $X = Z + \epsilon_1$ and $Y = Z + \epsilon_2$, find the expected value of $\exp(XY)$. \( \text{E}[\exp(XY)] \)
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Context:
Let \( X_1, X_2, \ldots, X_{48} \) be a random sample of size 48 from a distribution with probability density function \( f(x) = \frac{1}{x^2} \) for \( 1 < x < \infty \). Approximate the probability that at most 10 of these random variables have values greater than 4.
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Context:
A die is thrown repeatedly until the sequence "5 followed by 3" appears. What is the expected number of throws required for this sequence to occur?
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Context:
Evaluate the integral: $$\int_{-2}^2 (x^4+4x+\cos(x))\cdot \arctan\left(\frac{x}{2}\right)dx$$
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Context:
Calculate the mass of a half disc of radius $R$ centered at the origin $(0,0)$, with $y>0$, given that the density is \( \rho = \frac{\rho_oy}{\sqrt{x^2+y^2}} \) for \( y>0 \), where \( \rho_o \) is a constant with units of \( \text{kg.m}^{-3} \).
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Context:
Given the functions \( D(\rho) = g'(\rho)e^{ia\rho} - g'(-\rho)e^{-ia\rho} \) and \( S(\rho) = g'(\rho)e^{ia\rho} + g'(-\rho)e^{-ia\rho} \), determine if the ratio \( \frac{D(\rho)}{S(\rho)} \) is independent of the function \( g'(\rho) \) when \( g'(\rho) \neq g'(-\rho) \). If it is independent, find the ratio.
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Context:
Determine the condition under which the right cosets $Nu$ and $Nv$ are equal, given that $u, v \in G$ for a group $G$.
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Context:
Let $X_t, t \geqslant 0,$ be a Brownian motion. Define the stopping times $T_a := \inf \{t \mid X_t = a\}$. Calculate the probability $\mathbb{P}\{T_{2}< T_{-1} < T_{3}\}$.
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Context:
A stick of length 1 is randomly cut into 3 pieces. What is the expected length of the shortest piece?
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Context:
Evaluate the limit: $$\lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{\frac{1}{x^2}} $$
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Context:
Evaluate the improper integral: \[ I = \int_{0}^{\infty} \frac{x^3 \ln x}{(x^4+1)^3} \,dx \]
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Context:
In a box containing 91 balls, $m$ of them are red and the rest are blue. A sample of 19 balls is drawn, resulting in 7 red balls and 12 blue balls. Estimate the number of red balls, $m$, using the method of maximum likelihood.
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Context:
Consider a triangular tower with 11 rows, where each square on the bottom row contains either a 0 or a 1. Each square above the bottom row is the sum of the two squares directly below it. Determine the number of ways to fill the bottom row such that the number in the top square is a multiple of 3.
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Context:
Evaluate the expression \((\boldsymbol{x'}\cdot\nabla)^2\boldsymbol{x}\), where \(\boldsymbol{x'},\boldsymbol{x}\in \mathbb{R}^3\) and \(\boldsymbol{x'}\) is independent of \(\boldsymbol{x}\).
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Context:
Determine the indicial equation for the differential equation: $$ x^2y'' + x(1+x+x^2)y' + x(2-x)y = 0 $$
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Context:
Suppose $f(x)$ is a function such that $f^{(2)}(b), f^{(3)}(b), \ldots, f^{(n-1)}(b)$ are all zero. If $f^{(n)}(x)$ is continuous and non-zero at $x=b$, find the value of the limit:

$$\lim_{k \rightarrow 0} (\varphi_n-1)$$

where:

$$f (b+k)=f(b)+kf'(b)+\frac{k^2}{2!}f''(b)+\ldots+\frac{k^{n-1}}{(n-1)!}f^{n-1}(b+\varphi_{n-1} k)$$
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Context:
Find the area of the region enclosed by the ellipse given by the equation \(5x^2 + 6xy + 2y^2 + 7x + 6y + 6 = 0\).
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Context:
Find all \( n \in  \mathbb{N} \) so that 7 divides \( {5}^{n} + 1 \)
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Context:
Determine the maximum number of connected closed sets \(A_1, A_2, A_3, \ldots\) that can be drawn in \(\mathbb{R}^2\) such that for each pair \(i, j\), the interiors \(\operatorname{int}(A_i)\) and \(\operatorname{int}(A_j)\) are disjoint, but the intersection \(A_i \cap A_j\) is non-empty for all \(i \neq j\).
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Context:
In a classroom, there are 10 girls and 5 boys. Each of the 15 students plays against every other student exactly once, with each game having a single winner. Each participant has a 0.5 probability of winning any game. Let X be the total number of wins by the boys. What is the minimal value of Var(X), considering possible dependencies between the game results?
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Context:
Let \(X_1, X_2, \ldots\) be a sequence of independent random variables, each uniformly distributed on \([0, e]\). Define \(R_n = \prod_{k=1}^n X_k\). Determine the probability that the sequence \((R_n)\) converges.
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Context:
Calculate the limit as \( a \to \infty \) of \( a^3 \) times the area of the region \( S \) bounded by the parabola \( y = x^2 \), the line \( y = a^2 \), and the lines \( x = a \) and \( x = a + \frac{1}{2a} \).
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Context:
Provide an example of an irreducible polynomial \(g\) over \(\mathbb{Q}\) such that the extension \((\mathbb{Q}[x]/(g))/\mathbb{Q}\) is not Galois.
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Context:
How many natural numbers (ie. 1,2,3,...) equal three times the sum of the digits?
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Context:
Let $K$ be the field with exactly $7$ elements. Let $\mathscr{M}$ be the set of all $2 \times 2$ matrices with entries in $K$. How many elements of $\mathscr{M}$ are similar to the matrix \( \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \)?
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Context:
Let \( f:\mathbb{R}\to \mathbb{R} \) be a continuous and convex function such that \( f(0) \leq 0 \). Define the set \( \mathcal{D} = \{x \mid f(x) \leq 0\} \cap \{x \mid x \geq 0\} \). If \( \mathcal{D} \) is bounded, determine the set \( \mathcal{D} \).
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Context:
What is the fifth smallest positive integer with at least $5$ distinct prime divisors?
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Context:
Evaluate the limit: $$\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^{n} \left(\frac{k}{n}\right)^{1/k} - n}{\ln(n^2 + n)}$$
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Context:
Given a random sample $X_1, X_2, \cdots, X_n$ from a distribution with the probability density function $$f(x;\theta)=\frac{1}{\theta}x^{(1-\theta)/\theta}\, , 0<x<\infty\, , 0<\theta<\infty,$$ find the asymptotic variance of the maximum likelihood estimator $\hat\theta$ using the Rao–Cramér lower bound.
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Context:
Compute \(892^{(7918)(7909)} \mod (7919 \cdot 7907)\).
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Context:
Determine the number of distinct ways to color the 12 edges of a cube using the colors red, blue, and yellow, such that exactly 3 edges are red, 3 edges are blue, and 6 edges are yellow.
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Context:
Simplify the expression \((V' A V)^{-1}\) in terms of the matrices \(A^{-1}, V,\) and \(V'\), given that \(V \in \Re^{n \times k}\) with \(n > k\), \(V'V = I\), and \(\mathrm{rank}(V) = \text{rank}(V' A V) = k\). The matrix \(A\) is \(n \times n\) and its inverse \(A^{-1}\) is known.
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Context:
Determine whether the sequence \( \frac{2^n - 3n^3}{1-4^n} \) converges or diverges using the ratio test.
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Context:
Two identical sticks, each with the same length and mass density, are used in an experiment. One stick is broken at a uniformly random point, and the other stick is supported at a uniformly random point. The pieces of the broken stick are placed on the ends of the supported stick. What is the probability that the shorter piece of the broken stick will tip downwards?
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Context:
Given a circular list of coins, all initially showing Tails, determine the minimum number of moves required to make all coins show Heads. In each move, flipping the coin at position $i$ also flips the coins at positions $i-1$ and $i+1$. What is the minimum number of moves needed to achieve this for a list of $N$ coins?
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Context:
Determine the limit of the sequence defined by the recurrence relation: \[ a_{n + 1} = \int_{0}^{a_n} \left(1 + \frac{1}{4} \cos^{2n + 1} t\right) dt, \] where the initial term $a_0$ is in the interval $(0, 2\pi)$. Provide your answer as a single value.
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Context:
Find all solutions to the equation \(x^x = \frac{1}{256}\).
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Context:
Given the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ on the interval $(0, +\infty)$, determine the value of $m$ that makes it a decreasing function.
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Context:
Determine the number of zero divisors in the ring \( \frac{\mathbb{Z}_3[x]}{(x^4 + 2)} \).
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Context:
Evaluate the integral: \[ \int_{0}^{1} \frac{x^{a - 1}}{(1 - x)^{a + 1}} e^{\frac{x}{x - 1}} \, dx \] where \( a \in \mathbb{R} \) is a constant.
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Context:
$\text{What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?}$
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Context:
Let $f : \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ be a function satisfying $f(1)=1$ and \[f\Biggl(f(x+1) + f\Bigl( x + f(x)\Bigr)\Biggr) = x+2.\] Determine the value of $f(10)$. \(\boxed{\text{Your answer here}}\)
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Context:
Evaluate the integral of the function \( f(x, y, z) = x + y + z \) over the portion of the plane \( 2x + 2y + z = 8 \) that lies in the first octant.
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Context:
Find the infinite sum of the alternating series:
\[\sum_{n=0}^{\infty} \frac{(-1)^n \, (5 \pi)^{2n+1}}{6^{2n+1} \, (2n+1)!}\]
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Context:
Let $N$ be midpoint of the side $AB$ of a triangle $ABC$ with $\angle A$ greater than $\angle B$. Let $D$ be a point on the ray $AC$ such that $CD=BC$ and $P$ be a point on the ray $DN$ which lies on the same side of $BC$ as $A$ and satisfies the condition $\angle PBC =\angle A$. The lines $PC$ and $AB$ intersect at $E$, and the lines $BC$ and $DP$ intersect at $T$. Determine the value of $\frac{BC}{TC} - \frac{EA}{EB}$.
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Context:
In how many ways can 10 distinct toys be distributed among 4 children such that each child receives at least one toy and the oldest child receives the most toys?
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Context:
Find a cluster point of the sequence \( x_{n} = (\pi \cdot n) \mod 2 \).
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Context:
Compute the surface integral \( \int_S F \cdot n \, dS \) where \( F(x,y,z) = (x^2 + \sin y^2, y^2 + \sin z^2, z + \sin (xy)) \) and \( S \) is the surface of the bounded solid defined by the surfaces \( z = x^2 + y^2 + 1 \) and \( z = 2(x^2 + y^2) \), with the normal vector oriented outward from the solid.
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Context:
On one of the sides of triangle $ABC$, there is a point $P$. Draw a line through $P$ that cuts off a triangle similar to triangle $ABC$ from the original triangle $ABC$. Determine the number of possible solutions.
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Context:
Determine whether the poles of the function \( f(z) = \frac{z}{(1-e^z) \sin z} \) at \( z = 2\pi i k \), where \( k \in \mathbb{Z} \) and \( k \neq 0 \), are of order 1 or 2.
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Context:
Determine the asymptotic upper bound for the number of $m$-step walks in the given recursion.
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Context:
What is the name of a lattice where for every subset \( S \) of the lattice, both the infimum and supremum of \( S \) are elements of \( S \)?
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Context:
Find the value of the following limit: \[ \lim\limits_{n \rightarrow \infty}  \sum\limits_{k=1}^n \arctan\left(\frac{1}{\sqrt{k^2+1}}\right) \]
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Context:
Determine all real numbers $k$ such that the limit \( \lim_{n\to\infty}\frac{\tan n}{n^k}=0 \) exists.
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Context:
Compute the limit \( \lim_{n \to \infty}\int_{0}^{\pi} \frac{\sin(x)}{1 + 3\cos^{2}(nx)}\,\mathrm{d}x \).
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Context:
Determine the cardinality of the set of all countable subsets of the power set of the natural numbers.
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Context:
The angle of inclination of the line $x-y-1=0$ is ____.
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Context:
Given a convex quadrilateral \(ABCD\) with sides \(AB=70\) units, \(BC=100\) units, \(CD=35\) units, and \(DA=75\) units, and the projection of \(BC\) onto line \(AD\) is 96 units, find the distance \(MN\) where \(M\) and \(N\) are the intersection points of the angle bisectors at \(A, B\) and \(C, D\) respectively.
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Context:
What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$?
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Context:
Calculate the limit of the sum \(\sum_{k=1}^n \frac{k^3}{k^3+n^4}\) as \(n\) approaches infinity.
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Context:
The following list shows every number for which more than half of its digits are digits $2$, in increasing order:
$$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$If the $n$th term in the list is $2022$, what is $n$?
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Context:
Let $x$ be a uniformly distributed variable across the interval [0, 0.1], where $x$ represents the probability of a particular event occurring during a trial. Given that 528174 trials occur and the event does not occur in any of these trials, find the smallest real number $y$ such that $x < y$ with at least 95% certainty.
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Context:
Let \( A = K[x,y,z] \). Find the radical of the ideal \( I = (xyz, xy+xz+yz, x+y+z) \).
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Context:
Determine the cardinality of the set $A$, where a function $h$ belongs to $A$ if and only if there exists a natural number $k$ such that for all $i \geq k$, $f(i) = h(i)$, given a function $f: \mathbb{N} \rightarrow \{0,1\}$. Assume you cannot use cardinality arithmetic.
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Context:
Determine the natural number \( a \) such that the following equality holds for a given function \( f \) and a positive integer \( n \):
\[ \sum_{k=0}^n f(a+k) = 16(2^n - 1) \]
where \( f \) is a function satisfying the relation \( f(x + y) = f(x) \cdot f(y) \) for all natural numbers \( x \) and \( y \), and \( f(1) = 2 \).
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Context:
Given the transition matrix \(P\) for a Markov chain:
\[
P =
\begin{pmatrix}
0 & 1 & 0 & 0 \\
\frac{1}{3} & 0 & \frac{2}{3} & 0\\
0 & \frac{2}{3} & 0 & \frac{1}{3}\\
0 & 0 & 1 & 0\\ \end{pmatrix}
\]
and the limiting matrix \(\Pi = \lim_{n \to \infty} P^n\):
\[
\Pi =
\begin{pmatrix}
0.25 & 0 & 0.75 & 0 \\
0 & 0.75 & 0 & 0.25\\
0.25 & 0 & 0.75 & 0\\
0 & 0.75 & 0 & 0.25\\ \end{pmatrix}
\]
Calculate the long-run portion of time that the first urn is empty, denoted as \(\pi_0\).
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Context:
Given that \(a, b, c, d > 0\) and \(a + b = 3\), and \(\frac{a}{c} + \frac{b}{d} = 1\), find the minimum value of \(c + d\).
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Context:
Determine the largest positive integer \( N \) such that the expression \( \sqrt{64 + 32^{403} + 4^{N+3}} \) is an integer.
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Context:
Find the number of functions \( f : \{ 1,2,\ldots ,n\}  \rightarrow  \{ {1995},{1996}\} \) such that
 \( f\left( 1\right)  + f\left( 2\right)  + \ldots  + f\left( {1996}\right) \) is odd.
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Context:
Two cars, A and B, start at different positions on a road. Car A drops its first marker 5 feet from its starting position, and car B drops its first marker 9 feet from its starting position. Both cars then drop subsequent markers at increasing distances, with each subsequent marker being 2 feet further than the previous one. Determine the distance from the starting position where both cars drop a marker at the same location for the first time.
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Context:
Given that $X$ and $Y$ are independent random variables with a continuous uniform distribution on [0,1], and $Z = \max\{X, Y\}$, calculate $\mathbb{E}[X | Z]$.
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Context:
Find the dimension of the subspace \(W\) of \(\operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^3)\), the space of all linear transformations from \(\mathbb{R}^3\) to \(\mathbb{R}^3\), defined by:
\[
W = \{L \in \operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^3) : \operatorname{Im}(L) \subset S\}\]
where \(S = \{(x^1, x^2, x^3) \in \mathbb{R}^3 : x^1 + x^2 + x^3 - 1 \leq 0, \ x^1 + x^2 + x^3 + 1 \geq 0\}\).
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Context:
Calculate the probability that a random variable $X$ from a Gamma distribution with parameters $(3,3)$ is greater than a random variable $Y$ from an Exponential distribution with rate $1$. That is, find $P(X > Y)$. Assume $X \sim \text{Gamma}(3,3)$ and $Y \sim \text{Exp}(1)$. Express your answer as a probability.
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Context:
What is the limit, as \( n \to \infty \), of the expected distance between two points chosen uniformly at random within a unit-radius hypersphere in \( \mathbb{R}^n \)?
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Context:
Let $X, Y, Z \sim \mathcal{N}(0,1)$ be independent standard normal random variables. Calculate the probability $\mathbb{P}(X > Y \mid Y > Z)$. \( \boxed{?} \)
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Context:
What is the least common multiple of the orders of all possible move sequences in a Rubik's cube?
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Context:
Let \( X_1, X_2, \ldots, X_{40} \) be independent random variables, each uniformly distributed on \([-1/2, 1/2]\). What is the probability that the sum \( Y = X_1 + X_2 + \cdots + X_{40} \) is greater than 3?
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Context:
Calculate the floor of the sum \(\left\lfloor{\sum_{A \subseteq S}^\ \frac {F(A)} {G(A)}}\right\rfloor\), where \(S = \{1, 2, 3, ..., 8\}\), \(F(A)\) is the sum of all elements in set \(A\), and \(G(A)\) is the product of all elements in set \(A\). Determine the floor value of this sum.
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Context:
Determine the interval of convergence for the power series:
\[ \sum_{n=2}^\infty(-1)^n\frac{(x-3)^n}{(\sqrt[n]{n}-1)n} \]
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Context:
Determine the minimum number of terms required in the series \( \sum_{n=1}^\infty \frac{1}{3+n^2} \) so that the error in approximating the sum of the series by the sum of the first \( N \) terms is less than \( 10^{-5} \).
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Context:
Calculate the surface integral of the vector field $\underline{F} = 3x\underline{i} -5z\underline{k}$ over the cylindrical region $x^2 + y^2\leq 1$, where $0\leq z \leq 1$, using parameterizations for all three surfaces.
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Context:
What is the maximum number of bottles that can be tested for poison using 50 rats within one hour, assuming that rats die exactly one hour after consuming poison and testing time is negligible?
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Context:
Determine the greatest integer \( k \) such that the inequality \( \left\lfloor \frac{n}{\sqrt{3}} \right\rfloor + 1 > \frac{n^2}{\sqrt{3n^2 - k}} \) holds for all positive integers \( n \geq 2 \).
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Context:
Given a quintic polynomial $g(x)$, it leaves a remainder of $-2$ when divided by $(x+2)^3$ and a remainder of $2$ when divided by $(x-2)^3$. Find the value of $g''(-1)$. \( \text{(The second derivative of } g \text{ at } x = -1) \)
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Context:
ABCD is a unit square. One vertex of a rhombus lies on side AB, another on side BC, and a third on side AD. Determine the area of the set of all possible locations for the fourth vertex of the rhombus.
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Context:
Find the exact value of \( C \) such that the sequence defined by the recursive relation \( x_{i} = \frac{x_{i-1} + x_{i-2}}{Cx_{i-1}} \), with initial conditions \( x_1 = 1 \) and \( x_2 = 2 \), converges to \( \frac{2}{C} \) as \( i \) approaches infinity.
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Context:
For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to
$
\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.
$
Proposed by Daniel Zhu.
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Context:
Find the dimension of the vector space \( \mathbb{Q}(t) \) over the field \( \mathbb{Q}(t^2) \).
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Context:
There are 100 towns labeled 1 through 100. The cost to build a bridge between town "i" and town "j" is given by max{|i − j|, 4}. However, a bridge between town "i" and town "2i" can be built at a cost of 2. Determine the minimum cost required to connect all the towns in a single connected graph.
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Context:
Given the cyclic group \( \mathbb{Z}_7 \) with generator 3, determine the smallest non-negative integer \( n \leq 5 \) such that \( 3^n \equiv 6 \pmod{7} \).
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Context:
Find the values of \(\beta\) for which the function \(g(x) = |x|^\beta\) is weakly differentiable on the unit ball in \(\mathbb{R}^m\).
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Context:
Let \( \phi: L \to L \) be a linear operator with a minimal polynomial \( m_\phi(x) = (x-1)x^8 \). Find the minimal polynomial of \( \phi^3 \).
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Context:
Evaluate the integral \( \int_{-1/2}^{1/\sqrt{2}} \frac{f(x)}{1-x^2} \, dx \) given that \( f:[0,1]\to [0,1] \) is a continuous function satisfying \( x^2 +(f(x))^2\le 1 \) for all \( x\in [0,1] \) and \( \int_0^1 f(x) \, dx=\frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of proper subgroups required to cover a noncyclic group \( G \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of a circle that passes through the intersection of the circle \( S: x^2 + y^2 - 12x - 4y - 10 = 0 \) and the line \( L: 3x + y = 10 \), and has the same radius as the circle \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the automorphism \( \sigma \) in the Galois group \( \text{Gal} \left( \frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}} \right) \), where \( \sigma(\sqrt[3]{3}) = \omega\sqrt[3]{3} \) and \( \sigma(\sqrt{-3}) = -\sqrt{-3} \).
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Context:
Determine the condition on the parameters $b$ and $k$ such that the lines $x-y=5$ and $bx^2+2kxy+by^2=0$ (where $b \neq 0$) form an isosceles triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the symmetric group $S_7$ to the alternating group $A_8$. Provide your answer as a single integer.
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Context:
Determine the non-trivial real numbers $x$ for which the $n \times n$ matrix $A$ with entries $A_{ij} = x^{|i-j|}$ is invertible.
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Context:
Evaluate the limit \( \lim_{r\rightarrow 0} \frac{1}{r} \int_{C_{r}} \frac{\sin(z)}{z}dz \), where \( C_r \) is the contour \( re^{i\theta} \) for \( 0 \leq \theta \leq \pi \).
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Context:
Compute the tensor product \( \mathbb{Z}[1/2] \otimes_{\mathbb{Z}} \mathbb{Z}/3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles with radii in the ratio 1:2 touch each other externally. Let the center of the smaller circle be $C$ and the center of the larger circle be $D$. The point of contact is $A$. A straight line $\overline{PAQ}$ is drawn such that $P$ is on the smaller circle and $Q$ is on the larger circle, and $\overline{PAQ}$ does not pass through $C$. Given that $\angle QAD = \alpha$, find the angle between the tangent at $Q$ to the larger circle and the diameter of the smaller circle that passes through $P$ (extended if necessary).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(\alpha \in \mathbb{R}\) for which the integral \(\int_{0}^{\infty} \frac{\sin u}{(u+1)^\alpha} \, du\) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^1 \frac{2x^3+x}{\tan x+x^{\frac{1}{3}}}\, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the polynomial $p(z)$ in the Hadamard product factorization of the function $e^z - 1$, given that the zeros are $\{2 \pi i n ,n\in \mathbb{Z}\}$, they are all simple, and the function is of order 1. The factorization is given by:
\[ e^z - 1 = e^{p(z)} z \prod_{n=1}^\infty \left(1 + \frac{z^2}{4n^2\pi^2}\right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\left(\sqrt[n]{n+2}\cdot\sqrt[n^2+2n]{\binom{n}{1}\binom{n}{2}\cdots\binom{n}{n}}\right) \]
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Context:
Let $G$ be a 5-regular graph with $m$ edges and a girth of at least 6. Find the number of distinct paths of length 5 in $G$. A regular graph is a graph in which each vertex has the same degree, and the girth of a graph is the length of the shortest cycle within the graph.
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Context:
In a school, there are \( n \) students, each assigned a unique student number. Each student number is a positive factor of \( 60^{60} \), and the highest common factor (H.C.F.) of any two student numbers is not a student number in the school. Determine the greatest possible value of \( n \).
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Context:
In a circle with center A and radius R, point C lies on the circle. Segment DE connects C to the center A, with CE having a fixed length H and AE = AD = R. Angle ECF intersects the circle at point F, and segment CF is formed. Given that EC × CD = GC × CF, where CF = x and GC = y, and the equation H(2R - H) = x(y) holds, find the length of CF.
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Context:
How many distinct triangles $\triangle ABC$ exist with $AB = 2011$ units such that both $\cos(2\angle A + 3\angle C)$ and $\sin(2\angle B + \angle C)$ are integers? Assume that $\angle A$, $\angle B$, and $\angle C$ are the interior angles of the triangle and satisfy the triangle inequality theorem.
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Context:
Evaluate the integral \( \int_0^1 x^{k - 1} e^{-x} \, dx \).
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Context:
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n + 2} = |a_{n + 1} - a_n|$ for $ n\ge 1$. If $ a_1 = 999, a_2 < 999,$ and $ a_{2006} = 1$, how many different values of $ a_2$ are possible?
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Context:
Given the polynomial equation $x^4 - 2x^3 + 3x^2 - 4x + 2 = 0$ with distinct roots $a$, $b$, $c$, and $d$, find the value of $a^4 + b^4 + c^4 + d^4$. Express your answer as a single number.
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Context:
Determine the smallest integer $n$ such that $e^{\sqrt{n}} > n^4$.
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Context:
Given integers \( x, y, z \) such that \( x + y + z = 3 \) and \( x^3 + y^3 + z^3 = 3 \), find the value of \( x^2 + y^2 + z^2 \).
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Context:
On a \( 4 \times  4 \times  3 \) rectangular parallelepiped,vertices \( A,B \) ,and \( C \) are adjacent to vertex
 \( D \) . The perpendicular distance from \( D \) to the plane containing
 \( A,B \) ,and \( C \) is closest to
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral of the vector field \( \vec{F} = (e^x \sin y + 24 y) \vec{i} + (e^x \cos y) \vec{j}\) along the curve \(C\), which is the upper half of the circle \(x^2 + y^2 = \frac{\pi^2}{4}\) oriented counterclockwise.
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Context:
Evaluate the limit: $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$
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Context:
In a diagram, two lines intersect with two circles at 9 distinct points. Determine the number of ways to select 4 points from these 9 such that no 3 points among the selected 4 are collinear or lie on the same circle.
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Context:
Let \( \textbf{Z}_n \) be the set \( \{0, 1, \ldots, n - 1\} \) with addition mod \( n \). Consider subsets \( S_n \) of \( \textbf{Z}_n \) such that \((S_n + k) \cap S_n\) is nonempty for every \( k \) in \( \textbf{Z}_n \). Let \( f(n) \) denote the minimal number of elements in such a subset. Determine \( \lim_{n \to \infty} {\ln f(n) \over \ln n} \).
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Context:
A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it?
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Context:
Determine the minimum number of lines required to touch every square of an \(n \times n\) grid, given that the lines must pass inside the squares and not on their edges or corners.
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Context:
In the plane, circle \(\omega\) is externally tangent to circle \(\Omega\) at point \(P\). The center \(O\) of \(\Omega\) is located inside \(\omega\). A diameter \(XY\) of \(\Omega\) is tangent to \(\omega\). Given that \(PY > PX\) and \(PY\) intersects \(\omega\) at \(Z\) such that \(PZ = 3YZ\), determine the measure of angle \(PYX\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{S}(x+y+z)\,dS \), where \( S \) is the region defined by \( 0\leq x,y,z\leq 1 \) and \( x+y+z\leq 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(\cos(97x) = x\).
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Context:
Determine the number of ways to arrange red balls in 6 places such that in any 3 consecutive places, there are at least 2 red balls. Use the minimum number of red balls possible.
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Context:
Find an integer \( n \) , where \( {100} \leq  n \leq  {1997} \) , such that
 \( \frac{{2}^{n} + 2}{n} \)
 is also an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given sequences of measurable sets $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ in a sigma-algebra $Q$, suppose $P(A_k \text{ infinitely often }) = 1$ and $P(B_k^c \text{ infinitely often }) = 0$. What is the probability that infinitely many of the joint events $A_k \cap B_k$ will occur?
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Context:
Given the recurrence relation with initial conditions $a_0 = a_1 = a_2 = a_3 = a_4 = 0$ and $a_5 = 1$, and the formula:
\[ a_{n+6} = \frac{a_{n+5} + a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_{n}}{6} \]
Find the limit of $a_n$ as $n$ approaches infinity.
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\left(\sum_{0<i,j<n}{\binom{2i}{i}\binom{2j}{j}}\right)^\frac{1}{n}$$
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Context:
Determine the values of \( \alpha > 0 \) for which the series \( \sum_{k=1}^{\infty} \frac{1}{(k+1)\ln(k+1)^\alpha} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the expansion \((3x^2 + 2x + c)^{12} = \sum_{r=0}^{24} A_r x^r\), if \(\frac{A_{19}}{A_5} = \frac{1}{2^7}\), find the value of \(c\).
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Context:
Determine the set of all numbers \( C \) such that the inequality \( |\alpha \sin x + \beta \cos 4x| \leq C \) is satisfied for all \( x \) and for any \( (\alpha, \beta) \) where \( |\alpha| \leq 1 \) and \( |\beta| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( {c}_{1},{c}_{2},\ldots ,{c}_{6030} \) be 6030 real numbers. Suppose that for any 6030 real numbers \( {a}_{1},{a}_{2},\ldots ,{a}_{6030} \) ,there exist 6030 real
 numbers \( \left\{  {{b}_{1},{b}_{2},\ldots ,{b}_{6030}}\right\} \) such that
 \( {a}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}{b}_{\gcd \left( {k,n}\right) } \)
 and
 \( {b}_{n} = \mathop{\sum }\limits_{{d \mid  n}}{c}_{d}{a}_{n/d} \)
 for \( n = 1,2,\ldots ,{6030} \) . Find \( {c}_{6030} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest natural number such that all its digits are different, and the sum of any two adjacent digits is a perfect square.
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Context:
A mixture of 30 liters of paint is $25\%$ red tint, $30\%$ yellow tint, and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint that is the mixture?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the capacity of the channel defined by the convolution operator \( C: \ell^1(G) \to \ell^1(G) \), where \( f \mapsto \mu * f \) and \( \mu \) is a probability measure on a finite group \( G \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \displaystyle \int\limits_{0}^{\infty} \dfrac{x^x}{x^{1+\frac{1}{x}}}\, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Gaussian random variable X with covariance matrix $C_{XX}$, determine a matrix A such that the random variable Y = AX has a covariance matrix $C_{YY} = I$. Provide the matrix A.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the maximum likelihood method, estimate the total number of fish in a pond. Initially, 20 fish are caught, marked, and released back into the pond. Later, 50 fish are caught, and 7 of them are found to be marked. Determine the total number of fish in the pond, \( N \).
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Context:
Let \(\left(a_{n}\right)\) be a sequence defined by \(a_{1}=3\), \(a_{2}=2\), and for \(n \geqslant 1\), \(a_{n+2}\) is the remainder of the Euclidean division of \(a_{n}+a_{n+1}\) by 100. Compute the remainder of the Euclidean division of:

\[ a_{1}^{2}+a_{2}^{2}+\cdots+a_{2007}^{2} \]

by 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of statistical cluster points of sub-sequences of the sequence \( \frac{n}{e}-\left[\frac{n}{e}\right] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an irreducible representation $V$ of a semisimple Lie algebra with highest weight $\lambda$, determine the highest weight of the dual representation $V^*$. Provide your answer in terms of $\lambda$.
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Context:
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \( f(x^3) - f(y^3) = (x^2 + xy + y^2)(f(x) - f(y)) \) for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle ADC, G is the centroid, and AE is perpendicular to FC. Given that BD = DC, AC = 12 units, and the midpoint of AC is H, with D, G, and H being collinear, find the length of AB.
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Context:
Estimate the upper bound of the number of solutions \(N_{\sqrt[3]{2}}(1)\) for the inequality:
\[ \left \lvert \sqrt[3]{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n \, dx \).
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Context:
Determine the number of (not necessarily abelian) extensions of $\mathbb{Z}$ by $\mathbb{Z}$ up to isomorphism.
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Context:
Find the value of $x$ that minimizes the integral \( \int_0^1 \left\lvert -x + e^\varphi \right\rvert d\varphi \).
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Context:
Find the minimum value of \((1+a^2)(1+b^2)(1+c^2)\) given that \(a, b, c \geq 0\) and \(ab + bc + ac = 27\).
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Context:
What is the maximum dimension for which nilpotent Lie groups have been classified?
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Context:
How many distinct garlands can be made using 5 blue, 3 yellow, and 2 pink flowers, such that no two flowers of the same color are adjacent?
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Context:
Determine all positive integers \( a,b,c,p \) ,where \( p \) and \( p + 2 \) are odd primes
 and
 \( {2}^{a}{p}^{b} = {\left( p + 2\right) }^{c} - 1. \)
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Context:
Triangle $ABC$ is given with angles $\angle ABC = 60^o$ and $\angle  BCA = 100^o$. On the sides AB and AC, the points $D$ and $E$ are chosen, respectively, in such a way that $\angle  EDC = 2\angle  BCD = 2\angle  CAB$. Find the angle $\angle  BED$.
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Context:
Evaluate the integral \( \int_{0}^{1}\int_{0}^{1} \frac{x-y}{(x+y)^3} \, dx \, dy \).
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Context:
Determine the number of distinct triangles with integer side lengths such that the sum of the lengths of any two sides exceeds the length of the third side by at least 5 units, and the area is numerically twice the perimeter. Consider two triangles to be the same if they are congruent.
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Context:
What is the probability that a subset $X$ of $\mathbb{R}$, formed by including each real number $x$ with a heads result from an independent coin toss, is a (Lebesgue) measurable subset of $\mathbb{R}$?
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Context:
Determine the order of the element $g$ in the group $G_2$. Provide your answer as a single integer.
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Context:
Determine the CW complex that is homotopy equivalent to the set defined by the equation $x^2 + y^2 - 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{n^n}{(n+k)!}, \ n,k\in\Bbb N$$
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Context:
Determine the number of connected components of the set \(\{(x,y) \in \mathbb{R}^2 : (xy)^2 = xy\}\) in \(\mathbb{R} \times \mathbb{R}\).
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Context:
Given a circle with the equation \(x^2 + y^2 = 1\), fix a point \(X = (-1,0)\). Draw a random angle \(\theta\) from a uniform distribution \(U(0,\pi)\) to create a second point \(Y = (\cos \theta, \sin \theta)\) on the circle. Choose a third point \(Z\) on the circumference such that the area of the triangle \(XYZ\) is maximized. What is the expected value of the area of this triangle?
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Context:
Find the sum of the rearranged alternating harmonic series given by:
\[ 1 + \frac{1}{3} + \frac{1}{5} - \frac{1}{2} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \frac{1}{4} + \frac{1}{13} + \frac{1}{15} + \frac{1}{17} - \frac{1}{6} + \cdots \]
Consider partial sums up to $4n$ terms to find the sum of the series.
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Context:
The periodic function \(g\) is defined on \(\mathbb{R}\) by \(g(x) = f(x)\) for \(0 \le x < a\) and \(g(x) = g(x+a)\) for all \(x\), where \(a > 1\). Given that \(g\) is a continuous function, determine the exact value of \(a\).
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Context:
Determine the isomorphism class of the abelian group $\mathbb{Z} \times \mathbb{Z}/ \langle (5,2), (10,4) \rangle$ using the fundamental theorem of finitely generated abelian groups.
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Context:
Consider the function $u(x,t)$ that satisfies the heat equation $$u_{t}=u_{xx}, \ x \in\mathbb{R}, \ t >0$$ with the initial condition $$u(x,0)= \begin{cases} 1, & x \in [0,1] \\ 0, & \text{otherwise} \end{cases}$$ Determine the value of $ \lim_{t \to 0^{+}} u(1,t)$.
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Context:
Given the permutations $A$, $B$, and $C$, where $AXB = C$, find the number of inversions in the permutation $X$. The permutations are defined as follows:

\[
A = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\
2 & 3 & 1 & 5 & 7 & 6 & 4
\end{pmatrix}, \\
B = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
7 & 6 & 5 & 4 & 3 & 2 & 1
\end{pmatrix}, \\
C = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
4 & 5 & 7 & 6 & 2 & 3 & 1
\end{pmatrix}
\]

Permutations are applied from right to left, meaning $AB$ implies $B$ is applied first. Calculate the number of inversions in $X$. \( \text{Note: An inversion is a pair } (i, j) \text{ such that } i < j \text{ and } X(i) > X(j). \)
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Context:
Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$.
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Context:
Find the value of $a$ such that the solution to the differential equation \(9y''-12y'+4y=0\) with initial conditions \(y(0)=a>0\) and \(y'(0)=-1\) changes from positive to negative. The solution is given by \(y=ae^{\frac{2}{3}t}+(-1-\frac {2}{3}a)te^{\frac{2}{3}t}\). Determine the value of $a$ that separates solutions that become negative from those that remain positive.
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Context:
Consider a supermarket with three cash registers and a single queue for customers. Four people, A, B, C, and D, enter the supermarket in that order. A, B, and C immediately go to the available cash registers, while D waits for the first available one. The service time at each cash register is exponentially distributed with a mean of 5 minutes. What is the probability that D is the last person to leave the supermarket?
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Context:
Compute the double integral \( I = \iint_R \left|\cos(2x)-\cos(y)\right|\mathrm{d}x\mathrm{d}y \) over the region \( R = [0,\pi]\times [0,\pi] \).
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Context:
Given that $x + 2 = 18181818\ldots$ with $n$ digits, determine the number of trailing threes in the binomial coefficient $\binom{x}{x/3}$. Provide your answer as a single number.
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Context:
Suppose we have 2013 piles of coins,with the \( i \) th pile containing exactly \( i \) coins. We
 wish to remove the coins in a series of steps. In each step, we are allowed to take away  coins from as many piles as we wish, but we have to take the same number of coins
 from each pile. We cannot take away more coins than a pile actually has. What is the
 minimum number of steps we have to take?
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Context:
Given the equations \( x^2 + y^2 = t - \frac{1}{t} \) and \( x^4 + y^4 = t^2 + \frac{1}{t^2} \), find the value of \( \frac{dy}{dx} \) at the point \((1, 1)\).
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Context:
Find the discriminant of the number field \( \mathbb{Q}(\sqrt{3},\sqrt{5}) \).
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Context:
Determine the number of 10-digit numbers that contain the sequences "72" and "081" without any repetition of digits. Consider the constraints on placement due to the sequences and provide the total count of such numbers.
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Context:
Determine the number of closed subsets of \(\mathbb{R}\) up to homeomorphism.
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Context:
An urn contains the numbers 1, 2, 3, ..., 2019. Four numbers are drawn at random, without replacement, and in order from the urn, denoted as \(a, b, c, d\). What is the probability that the system of equations \((S)\): \(ax + by = ab\) and \(cx + dy = cd\) has a strictly internal solution (i.e., not on the axes) in the first quadrant?
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Context:
Evaluate the limit: \[ \lim_{x\to 0^-}\left(1+\frac{1}{x}\right)^x \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of convergence of the power series \( \sum_{n=0}^\infty (\frac{\pi}{2} - \arctan(n))(x)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic complexity of finding the value of $x$ using the extended version of the Chinese Remainder Theorem, given that $x$ is known modulo all primes less than $p$. Provide your answer in terms of big O notation.
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Context:
Evaluate the integral \( I = \int_{e^{-2\pi n}}^1 \left|\left(\cos\left(\ln{1\over x}\right)\right)'\right|dx \) for \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_C \frac{e^{1/z}}{(z-i)^3} \, dz \) where \( C \) is the circle \( |z|=5 \).
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Context:
Evaluate the surface integral \( \iint_S \bar{F} \cdot \bar{N} \, dS \) for the vector field \( \bar{F} = (4x^{3}+y^{2}-z , x^{2}+y^{3}+z , x^{3}+z+1) \) over the surface \( S \) defined by \( z = 4 - 4x^2 - y^2 \) with \( z \geq 0 \), where the normal vector \( \bar{N} \) is pointing upward. Consider using the divergence theorem to simplify the calculation.
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Context:
Determine the order of growth of the error function defined by \( \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt \).
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Context:
Evaluate the limit: \[ \lim_{x \to 0} \frac{(1+x)^{1/x} - e + \frac{1}{2}ex}{x^2} \] using a method other than Taylor series expansion.
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Context:
Find the trace of the matrix \( x(x^\top x)^{-1} x^\top \), where \( x_i^\top = (1, x_{1,i}, \ldots, x_{d,i}) \in \mathbb{R}^{1 \times (d+1)} \) for \( i \in \{1, \ldots, n\} \) are linearly independent, and \( x \in \mathbb{R}^{n \times (d+1)} \) with the \(i\)-th row of \(x\) equal to \(x_i^\top\).
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Context:
Find a field whose automorphism group is both infinite and countable.
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Context:
Evaluate the limit \( \lim\limits_{n\rightarrow\infty}\sum\limits_{k=2}^n\frac{1-e^{-(x_k-x_{k-1})}}{1+e^{-(x_k-x_{k-1})}} \) where \( \{x_1, \ldots, x_n\} \subset [0,1] \) is an ordered sequence satisfying \( x_1 < x_2 < \ldots < x_n \) and forms a partition of \([0,1]\) with the maximum difference \( x_j - x_{j-1} \) approaching zero.
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Context:
Determine the value of $s(1)$, where $s(d)$ is defined as follows: For $n \ge 1$ and $0 \le k < n$, define $B_{n,k} = [k/n, k/n + 1/n)$. Fix an integer $d \ge 0$ and suppose $(x_1, x_2, ..., x_{s+d})$ is a sequence with $x_i$ belonging to $[0,1)$ and with $s = s(d)$ chosen to be maximal such that for each $r \le s$ and each $k < r$, $B_{r,k}$ contains at least one point of the subsequence $(x_1, x_2, ..., x_{r+d})$. It is known that $s(0) = 17$. What is the value of $s(1)$?
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Context:
Determine the least integer \( n \) such that for any set of \( n \) lines in the 2D plane, there exists either a subset of \( 1001 \) lines that are all parallel, or a subset of \( 1001 \) lines that are pairwise nonparallel.
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Context:
In a group of 100 readers, each pair of individuals can start a discussion only if each has read at least one book that the other hasn't. What is the minimum number of distinct books required to ensure that any two members of the group can start a discussion?
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Context:
Determine the number of solutions to the equation \((\tan x)\sin^2(2x) = \cos x\) within the interval \(-2\pi \leq x \leq 2\pi\).
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Context:
该图显示了两个边长为 4 的等边三角形，它们安装在边长也为 4 的正方形的两个相邻边上。对于两个正整数 m 和 n，标记为 A 和 B 的两个顶点之间的距离可以写成$\sqrt{m}+\sqrt{n}$。找到 m+n。
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Context:
Determine the Banach-Mazur distance between the Banach spaces \(\ell_\infty = \ell_\infty(\mathbb{N})\) and \(L_\infty = L_\infty([0,1], \lambda)\), where \(\lambda\) is the Lebesgue measure on \([0,1]\). The Banach-Mazur distance is defined as \( d(X,Y) = \inf\{ \|\varphi\|\|\varphi^{-1}\| : \varphi\colon X\to Y \text{ isomorphism} \} \). What is \(d(\ell_\infty,L_\infty)\)?
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Context:
Find the limit of the expression \( \left(\cos\left(\frac{\pi}{x}\right)\right)^{x^2} \) as \( x \to 0 \) using a method other than L'Hopital's Rule.
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Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin x}{1+x^2} \, dx \) using complex analysis.
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Context:
In a race with 8 runners, each runner's finish time is independently and exponentially distributed with a rate of 1. Calculate the probability that the time difference between the first and second runners to finish is greater than 1.
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Context:
Let \( n = 2^7 \cdot 3^5 \cdot 11^3 \cdot 35 \). Determine the number of ways the cyclic group \( C_n \) can be expressed as a direct product of two or more nontrivial groups.
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Context:
When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?
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Context:
5. In an isosceles trapezoid $M N K L$ with bases $M L, N K$, the diagonals are perpendicular to the sides $M N$, $K L$ and intersect at an angle of $15^{\circ}$. Find the height of the trapezoid if the length $N Q=5$, where $Q-$ is the midpoint of the larger base.
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Context:
A cheetah takes strides of 2 meters each, while a fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. If the distance between the cheetah and the fox is 30 meters, calculate the distance the cheetah must run to catch up with the fox. Express your answer in meters as a whole number.
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Context:
The distance between the non-intersecting diagonals of two adjacent lateral faces of a cube is \( d \). Determine the total surface area of the cube.
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Context:
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} - d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
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Context:
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$?
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Context:
The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to  $1, 1$  and  $2$ , respectively. Determine the length of this hexagon's side.

(Mikhail Evdokimov)
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Context:
双曲线 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$ 的左、右焦点分别为 $A, B, P$ 是双曲线上一点, 若 $\triangle P A B$ 的内切圆圆心为 $(3,1)$, 则 $\triangle P A B$ 外接圆的半径为 ?原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is true: $A$ takes lessons from $B ; B$ takes lessons from $C ; C$ takes lessons from $A$. What is the largest number of players there can be?
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Context:
On a race track, a maximum of 4 horses can race together at a time. There are a total of 35 horses. There is no way of timing the races. What is the minimum number of races we need to conduct to get the top 3 fastest horses?
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Context:
Tycho wants to prepare a schedule for his jogging for the next few months. He wants to jog three times per week. Every week, he wants to jog on the same days of the week. He never wants to jog on two consecutive days. How many schedules can he choose from?
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Context:
The graph of $y = \frac{p(x)}{q(x)}$ where $p(x)$ is quadratic and $q(x)$ is quadratic is given conceptually (imagine a graph with necessary features). The function has vertical asymptotes at $x = -4$ and $x = 1$. The graph passes through the point $(0,0)$ and $(2,-1)$. Determine $\frac{p(-1)}{q(-1)}$ if $q(x) = (x+4)(x-1)$ and $p(x) = kx + m$.
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Context:
Let $k$ be a natural number. For which value of $k$ is $A_k = \frac{19^k + 66^k}{k!}$ maximized?
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Context:
Let  $p$  be an odd prime number. How many  $p$ -element subsets  $A$  of  $\{1,2,\ldots \ 2p\}$  are there, the sum of whose elements is divisible by  $p$ ?
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Context:
令 I=\displaystyle\sqrt{\df{2}{\pi}}\int_{-\infty}^{+\infty}e^{x-10\cosh(2x)}\dd x，则 [|\ln I|]=__________.
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Context:
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b) \in X \times X : x^2 + ax + b \text{ and } x^3 + bx + a \text{ have at least a common real zero.}\}$.

How many elements are there in $S$?
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Context:
The force exerted by the airflow on a sail can be calculated using the formula:

\[ F = \frac{C S \rho (v_0 - v)^2}{2}, \]

where \(C\) is the coefficient of aerodynamic force, \(S\) is the area of the sail (\(S = 5 \, \text{m}^2\)), \(\rho\) is the air density, \(v_0\) is the wind speed (\(v_0 = 6 \, \text{m/s}\)), and \(v\) is the speed of the sailboat. At some point in time, the instantaneous power of the wind reaches its maximum value. What is the speed of the sailboat at this moment?

Given:

\[ F = \frac{C S \rho (v_0 - v)^2}{2} \]

\[ N(t') = N_{\max} \]

\[ v_0 = 6 \, \text{m/s} \]

\[ \overline{v(t') = ?} \]
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Context:
集合A、B满足A∪B={1,2,3,⋯,10},A∩B=∅，若A中的元素个数不是A中的元素，B中的元素个数不是B中的元素，则满足条件的所有不同的集合A的个数为
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Context:
Three distinct vertices of a cube are chosen at random. What is the probability, in simplest form, that the plane determined by these three vertices contains points inside the cube? The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is the simplest form of the probability. Please give the value of \(k + m\).
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Context:
Given that our number system has a base of eight, determine the fifteenth number in the sequence.
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Context:
A number like 45132 is called a "five-digit wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. How many distinct "five-digit wave numbers" can be formed using the digits 1, 2, 3, 4, 5 without repetition?
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Context:
Let \(ABCD\) be a convex quadrilateral inscribed in a circle with shortest side \(AB\). The ratio \(\frac{[BCD]}{[ABD]}\) is an integer (where \([XYZ]\) denotes the area of triangle \(XYZ\)). If the lengths of \(AB, BC, CD\), and \(DA\) are distinct integers no greater than 10, find the largest possible value of \(AB\).
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Context:
A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square?
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Context:
Find the work done by the vector field $F = (x^2 + y^2)\mathbf{i} + (x^2 - y^2)\mathbf{j}$ along the closed curve formed by the line segments from (0, 0) to (2, 0), from (2, 0) to (2, 2), and from (2, 2) to (0, 0).
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Context:
Two objects start from points $A$ and $B$, and move towards each other. If the total distance between $A$ and $B$ is $m$ and the product of the individual distances traveled before meeting is $n^2$, determine the polynomial whose roots are the distances traveled by each object from $A$ and $B$ before meeting.
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Context:
A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed?
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Context:
In a table tennis match between players A and B, they follow the rules that: the winner of each game gets 1 point and the loser gets 0 points. The match stops when one player has 2 more points than the other or after a maximum of 6 games. Let the probability that A wins each game be $\frac{3}{4}$ and the probability that B wins each game be $\frac{1}{4}$, and assume the outcomes of each game are independent. Find the expected number of games $\xi$ played when the match stops, denoted as $\mathrm{E} \xi$.
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Context:
Given the inequality $e^{2x} - e^{-2x} - 4x - ae^{x} + ae^{-x} + 2ax \geq 0$ holds true for any positive real number $x$, determine the maximum value of the real number $a$. Express your answer as a single number.
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Context:
Consider all $1000$-element subsets of the set $\{1, 2, 3, ... , 2015\}$ . From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ .
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Context:
A collection of coins includes only pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). Using the coins in this collection, it is possible to create any amount of money less than one dollar (100 cents). What is the smallest possible number of coins in the collection?
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Context:
Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$.
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Context:
Inside a non-isosceles acute triangle \(ABC\) with \(\angle ABC = 60^\circ\), point \(T\) is marked such that \(\angle ATB = \angle BTC = \angle ATC = 120^\circ\). The medians of the triangle intersect at point \(M\). The line \(TM\) intersects the circumcircle of triangle \(ATC\) at point \(K\) for the second time. Find \( \frac{TM}{MK} \).
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Context:
Two circles with centers \( O_1 \) and \( O_2 \) intersect at points \( A \) and \( B \). The first circle passes through the center of the second circle and its chord \( BD \) intersects the second circle at point \( C \), dividing the arc \( ACB \) in the ratio \( AC:CB = n \). In what ratio does point \( D \) divide the arc \( ADB \)?
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Context:
A 100-litre mixture of milk and water contains 36 litres of milk. Some amount of this mixture is removed and replaced with an equal quantum of water. If the process is repeated once, then the concentration of the milk stands reduced at 9%. What is the volume of the mixture that was removed and replaced with water?
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Context:
A polynomial P satisfies the following criterion: Its coefficients are integers. For all real $(a, b, c, d)$ we have $(P(a) + P(b))(P(c) + P(d)) = P(ac - bd) + P(ad + bc)$. Determine all possible values of $P(2014)$.
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Context:
What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?
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Context:
Calculate the flow of the vector field $F(x,y,z)=(xz,yz,-z^2)$ out of the surface $S=\{(x,y,z) \in \mathbb{R}^3:z^2=1+x^2+y^2, 2 \leq z \leq 3\}.$
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Context:
In a box, there are 200 balls numbered consecutively from 101 to 300. Two people, A and B, each take balls from the box until they each have 100 balls. If person A takes ball number 102 and person B takes ball number 280, calculate the maximum difference between the sum of the numbers on the balls taken by A and the sum of the numbers on the balls taken by B. Provide your answer as a single integer.
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Context:
Given the function f(x) = x^5 / (5x^4 - 10x^3 + 10x^2 - 5x + 1), calculate the sum of f(i/2009) from i=1 to 2009. Express your answer as a single integer.
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Context:
Define
\[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \]
Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
In a pond, there are 6 ducks and they all eat different amounts of the total pieces of bread thrown into the water. The first duck eats 3/7 of all the pieces of bread, the second duck eats 1/5 of the remaining pieces after the first duck, the third duck eats 2/9 of the remaining pieces after the second duck, the fourth duck eats 1/7 of the remaining pieces after the third duck, the fifth duck eats 15 pieces of the bread, and the sixth duck eats 9 pieces of the bread. There are 50 pieces of bread left in the water after all the ducks have eaten. How many total pieces of bread were thrown in the pond?
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Context:
The numbers  $1, 2\ldots11$  are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
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Context:
In a positive geometric sequence {a_n}, it is known that a_1a_2a_3 = 4, a_4a_5a_6 = 12, and a_{n-1}a_na_{n+1} = 324. Find the value of n. Express your answer as a whole number.
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Context:
11.5. In a convex pentagon $P Q R S T$, angle $P R T$ is half the size of angle $Q R S$, and all sides are equal. Find angle $P R T$.
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Context:
If a number is selected from the whole numbers 1 to \( m \), and if each number has an equal chance of being selected, the probability that the number is a factor of \( m \) is \(\frac{p}{39}\). Find \( p \).
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Context:
A psychologist believes one in every four people has a disorder he calls ZYX syndrome. The clinic where he works has recently doubled its previous number of 26 patients. The distribution of the disorder is not even, as the clinic found that 20% of females and 30% of males have this syndrome. Out of the total patients, 60% are female. The psychologist's diagnostic accuracy is 80% for males and 70% for females to correctly identify patients with ZYX syndrome. On the other hand, the probability of false negatives (incorrectly classifying patients without the syndrome as having it) is 15% for males and 10% for females. Determine how many patients the psychologist is likely to diagnose with ZYX syndrome considering the varying diagnostic accuracies and probability of false negatives.
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Context:
In how many ways can the positive integers from 1 to 100 be arranged in a circle such that the sum of every two integers placed opposite each other is the same? (Arrangements that are rotations of each other count as the same.)
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Context:
To make the statement $(6\ ?\ 3) + 4 - (2 - 1) = 5$ true, replace the question mark with the correct arithmetic operation. Convert the correct arithmetic operation to its ASCII code value and provide this value as your answer.
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Context:
Evaluate the integral $$\int_0^{\pi/2}(\cos^{10}x )(\sin 12x )dx$$
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