Context:
In a $5$-regular graph $G$ with $m$ edges and a girth of at least $6$, determine the number of paths of length $5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the operator norm of the linear operator $B: \ell^2 \to \mathbb{R}$ defined by \[ Bx = \sum\limits_{k=0}^\infty ((1+\sin(\pi(1/2)^{k+1}))\eta_{k+1}). \]
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Context:
Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$: $$x_{n+1} = \sum_{k=1}^{n} \sqrt[n]{x_k}.$$ Evaluate $\lim_{n \to \infty} \frac{x_n - n}{\ln n}$. Provide your answer as a single value.
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Context:
Evaluate the triple integral: $$\int_{1}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty} \exp\left(-\frac{x^{2}}{y^{2}}-\frac{y^{2}}{z^{2}}-\left[z-1-\log(z)\right]\right) \,{\rm d}x\,{\rm d}y\,{\rm d}z$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the Lambert W function at the point where the function \(f(x) = e^x\) intersects the line \(y = x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set \( T \) contains natural numbers \( a, b, c, d, e, \) and \( f \) such that the sum of their squares equals \( f^2 \). Another set \( S \) contains all possible values of the product \( a \times b \times c \times d \times e \times f \). Determine the greatest common divisor (GCD) of all elements of set \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the infinite series \( S = \sum_{k=1}^\infty \frac{k^{k-1} e^{-k}}{k!} \) and determine its value.
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Context:
Determine the number of symmetric, positive definite $10 \times 10$ matrices with a trace of 10 and a determinant of 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $g:\mathbb{D} \to \mathbb{D}$ is an analytic function with $g(0)=0$ and $|g'(0)|=\frac{1}{2}$. Determine the maximum possible value of $|g(\frac{1}{4})|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a convex quadrilateral \(ABCD\), points \(E, F, G, H\) are located on sides \(AB, BC, CD, DA\) respectively, such that:
\[ \frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1 \]

Another convex quadrilateral \(E_1F_1G_1H_1\) has points \(A, B, C, D\) on its sides \(H_1E_1, E_1F_1, F_1G_1, G_1H_1\) respectively, with \(E_1F_1 \parallel EF, F_1G_1 \parallel FG, G_1H_1 \parallel GH, H_1E_1 \parallel HE\). Given that \(\frac{E_1A}{AH_1} = \lambda\), determine the value of \(\frac{F_1C}{CG_1}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a+\cos \theta)^2}$.
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Context:
Determine the radius of convergence for the power series: $$ \sum_{n = 1}^{\infty} \left(1 + \frac{1}{n^{1/n}} \right)^{n} \sin^{2n}\left(\frac{n \pi}{3}\right) z^{n} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a Markov chain on the integers, the stochastic process \(X_n\) starts at \(X_0 = 3\). If \(|X_n|\) is even, the transition probabilities are \(P(X_{n+1} = i + 1|X_n = i) = \frac{3}{4}\) and \(P(X_{n+1} = i - 1|X_n = i) = \frac{1}{4}\). If \(|X_n|\) is odd, the transition probabilities are \(P(X_{n+1} = i + 1|X_n = i) = \frac{1}{4}\) and \(P(X_{n+1} = i - 1|X_n = i) = \frac{3}{4}\). Calculate the probability of reaching state 5 before reaching state 0, starting from state 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For lines $l_1: x + ay + 3 = 0$ and $l_2: (a-2)x + 3y + a = 0$ to be parallel, determine the values of $a$.
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Context:
Given the polynomial \((1-z)^{b_{1}}(1-z^{2})^{b_{2}}(1-z^{3})^{b_{3}} \cdots(1-z^{32})^{b_{32}}\), where \( b_{i} \) are positive integers for \( i = 1, 2, \ldots, 32 \), and this polynomial has the property that after expanding it and removing the terms with \( z \) raised to powers higher than 32, what remains is \( 1 - 2z \). Determine \( b_{32} \), and express your answer as the difference of two powers of 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S (x^2 + y^2 + z^2) \, dS \), where \( S \) is the part of the cone \( z = \sqrt{x^2+y^2} \) above the circle \( x^2 + y^2 \leq 2x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\int_{S}\frac{dxdy}{z}+\frac{dydz}{x}+\frac{dxdz}{y}\) over the surface of the sphere \(S\) with radius \(R\) and outward orientation, without using the Gauss-Ostrogradsky formula.
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Context:
Determine the minimum number of terms needed in the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \) to approximate \( \ln 2 \) within an error of 0.001, using the inequalities:

\[ 1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+1}}{2k} \approx \ln 2 \]

and

\[ 1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+2}}{2k+1} \leq \ln 2. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*} \) such that for any positive integers \( s \) and \( t \), the equation \( f\left(t^{2} f(s)\right) = s(f(t))^{2} \) holds, determine the minimum possible value of \( f(1998) \).
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Context:
In an obtuse triangle $\triangle ABC$, let $I$ be the incenter. Among the triangles $\triangle IAB$, $\triangle IBC$, and $\triangle IAC$, one of them is similar to $\triangle ABC$. Determine the ratio of the angles of $\triangle ABC$.
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Context:
Determine all real values of \(\lambda\) such that the inequality \(x + y + z \leq \frac{3}{2}\) holds for all non-negative real numbers \(x, y, z\) satisfying \(x^2 + y^2 + z^2 + 2\lambda xyz = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the last two digits of the sum \(a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}\), where \(a_1, a_2, \ldots, a_{100}\) are 100 consecutive natural numbers.
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Context:
Evaluate the limit \( \lim_{n \to \infty} J_n \), where \( J_n = \int_{-\infty}^\infty \frac{|\cos \left( \frac{x}{n} \right) \cos(x)|}{\frac{x}{n} x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An inspector selects every \( n \)-th item in a production line for a detailed inspection. The time between item arrivals follows an exponential distribution with a mean of 1 minute. It takes 2 minutes to inspect an item. Determine the smallest value of \( n \) such that the probability of completing the inspection before the arrival of the next item that requires inspection is at least 90%.
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Context:
Compute the limit: $$ \lim_{N\rightarrow\infty}N^2\sum^{N-1}_{k=1}\left(\frac{k}{N}\right)^{N\ln N} $$
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Context:
Determine the dimension of the vector space of conics in the real projective plane with homogeneous coordinates \(\{x, y, z\}\) that are singular at the points \([1, i, 1]\) and \([i, -1, i]\).
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Context:
Let \( N = \min\{n \in \mathbb{N}_0: X_1 + \ldots + X_n > T\} \). Calculate \( E[N] \) for \( T > 1 \).
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Context:
Find the asymptotic upper and lower bounds for the recurrence relation \( T(n) = T(\sqrt{n}) + n \).
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Context:
Calculate \( \lim_{n \to \infty} \int f_n \, d\mu \) where \( S = N, A = P(N), \mu = h \, dv \) with \( v \) being the counting measure, \( h(k) = \frac{1}{4^k} \), and \( f_n(k) = \left(\frac{k}{n} - 2\right)^k 1_{(1,2,3,...,n)}(k) \).
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Context:
Determine the order of the two-step Adams-Bashforth method given by the formula:
\[
u_{n+1} = u_n + \frac{h}{2}(3f_n - f_{n-1})\] where \(t_n = t_0 + nh\) is the \(n\)-th node, \(y_n = y(t_n)\), \(f_n\) stands for \(f(t_n,y_n)\), and \(u_n\) is an approximation of \(y_n\).
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Context:
Find the minimal polynomial of \( \alpha = \frac{1-\sqrt{5}}{2} \) over the finite field \( \mathbb{F}_{7} \).
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Context:
Find the residue of the function \( f(z) = \frac{e^{-\alpha/z}e^{-\alpha z}}{z} \) at infinity.
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Context:
Let $(Z_i)_{i \ge 1}$ be a sequence of i.i.d. random variables uniformly distributed on $\{1, -1\}$, with $P(Z_i = 1) = P(Z_i = -1) = \frac{1}{2}$. Define the Markov chain $X_n$ for $n \ge 1$ with $X_0 = 0$ as follows:
\[ X_n = X_{n-1} + Z_n \]
Let $a \in \mathbb{Z}$ be a non-zero integer and define the stopping time $T = \inf\{n \ge 1: X_n = a\}$, which is the first hitting time of $a$. Find $E[X_T]$. \(\text{Express your answer as a number.}\)
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Context:
In a tetrahedron \(ABCD\) with a volume of 12, points \(E\), \(F\), and \(G\) are located on edges \(AB\), \(BC\), and \(AD\) respectively. It is given that \(AE = 2EB\), \(BF = FC\), and \(AG = 2GD\). A cross-section \(EFHG\) is formed through points \(E\), \(F\), and \(G\). If the perpendicular distance from point \(C\) to this cross-section is 1, find the area of the cross-section \(EFHG\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a rectangle PQRS in the first quadrant with adjacent sides PQ and QR having slopes 1 and -1, respectively, and the function \( u(x,t) \) satisfying the wave equation \( \frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0 \), where \( u(P)=1 \), \( u(Q)=1/2 \), and \( u(R)=1/2 \), find \( u(S) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $d$ for which the field $\mathbb{Q}(\sqrt{d})$ has an integral basis of the form $\alpha, \alpha'$ where $\alpha'$ is the conjugate of $\alpha$. Provide the condition on $d$.
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Context:
Compute the first Chern class, \( c_1(\mathcal{O}_C(1)) \), where \( C \) is the twisted cubic defined by the map \( \nu: \mathbb{P}^1_{\mathbb{C}} \to \mathbb{P}^3_{\mathbb{C}} \) given by \( \nu(s,t)=(s^3,s^2t,st^2,t^3) \). Assume \( c_1(\mathcal{O}_{\mathbb{P}^3}(1))=1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability \( P(T_1 < T_{-1} < T_2) \), where \( T_a = \inf\{t \geq 0 | B_t = a\} \), and \( B_t \) denotes a Brownian motion.
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Context:
Let $x, y, z$ be nonzero real numbers such that $ x + y + z = 0$ and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1 -xyz + \frac{1}{xyz}.$$ Determine the value of the expression $$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$
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Context:
Given the differential equation \( y^{\prime}=4\sqrt{y-x^2} \) and the initial condition \( y(1)=2 \), find the value of \( y(3) \).
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Context:
Determine the number of integer solutions for the equation $a^b b^c c^a = p$, where $p$ is a prime number, and exactly two of $a, b, c$ are negative.
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Context:
Evaluate the integral \( \int_{\Gamma} \mathrm{Log}z \, \mathrm{d}z \) where \( \Gamma(t) = e^{2i\pi t} \) for \( 0 \leq t \leq 1 \), considering that the principal argument of \( e^{2i\pi t} \) is \( 2\pi t \).
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Context:
Determine the isomorphism class of the line bundle \(L\) over \(\mathbb{CP}^1\) with the transition function \(\psi_{12}([z_1, z_2]) = \frac{z_1/z_2}{|z_1/z_2|}\) on \(U_1\cap U_2\).
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Context:
Find the product of the 5th roots of \((1+\sqrt{2}i)^3.\)
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Context:
In a checkered plane painted with 10 different colors, each cell is colored with one color, and all colors are present. Any two adjacent cells are colored in different colors. A pair of colors is considered "good" if there are two neighboring cells painted in these colors. What is the least number of good pairs possible?
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Context:
Define \(\{ a_{n}\} \) as an arithmetic sequence where all terms are positive integers. If \(a_{10}-a_{1}=225\), find \[\lim _{n\rightarrow \infty }\dfrac {na^{2}_{n}+1}{\sum\limits^{n}_{k=1}\left( 1+2+3+\ldots +k\right) }\]
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Context:
Compute the sum of all 4-digit numbers that can be formed using the digits from the multiset \(\{1^2, 2^2, 3^2, 4^2\}\), where each digit can appear at most twice.
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Context:
Find the number of horizontal tangents to the function \( f(x) = xe^{-x^2-x^{-2}} \) for \( x \neq 0 \) and \( f(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit of the integral: \[ \lim_{n \rightarrow \infty} \int_{0}^{\pi/4} \frac{\sin^2(\frac{nx}{2})}{n\,\sin^2(\frac{x}{2})} \,dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n_0 \) such that for all \( n > n_0 \), the inequality \( \sqrt{n^3+1} + \sqrt{n^3} > 10^3 \) holds.
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Context:
Find the smallest variance that can be achieved by an unbiased estimator of the parameter \( \varphi \) given the probability density function: \[ f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{cases} \]
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Context:
Find the residue of the function \( \frac{\sin z}{z^2} \) at \( z = \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( {\left\{  {a}_{n}\right\}  }_{n = 0}^{\infty } \) be the sequence defined by the recurrence relation \( {a}_{n + 3} = 2{a}_{n + 2} - {23}{a}_{n + 1} + 3{a}_{n} \) for all \( n \geq  0 \) ,with initial
 conditions \( {a}_{0} = {20},{a}_{1} = 0 \) ,and \( {a}_{2} = {23} \) . Let \( {b}_{n} = {a}_{n}^{3} \) for all \( n \geq  0 \) . There exists a unique positive integer \( k \) and const
 \( {c}_{0},\ldots ,{c}_{k - 1} \) with \( {c}_{0} \neq  0 \) and \( {c}_{k - 1} \neq  0 \) such that for all sufficiently large \( n \) ,we have the recurrence relation
 \( {b}_{n + k} = \mathop{\sum }\limits_{{t = 0}}^{{k - 1}}{c}_{t}{b}_{n + t} \) . Find \( k + \sqrt{\left| {c}_{k - 1}\right| } + \sqrt{\left| {c}_{0}\right| } \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to0}(x\Gamma(x))^\frac{1}{x} \]
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Context:
Two individuals, A and B, start at opposite poles of a sphere with radius \( r \). They move randomly along either a latitude or a longitude, but only one individual moves at a time, with a step length of 1 unit. What is the expected time for them to meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $\overrightarrow{a}=(1, \sqrt {3}), |\overrightarrow{b}|=1, |\overrightarrow{a}+ \overrightarrow{b}|= \sqrt {3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.
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Context:
Determine the smallest odd integer \( k \) such that for every cubic polynomial \( f \) with integer coefficients, if there exist \( k \) integers \( n \) such that \(|f(n)|\) is a prime number, then \( f \) is irreducible in \( \mathbb{Z}[n] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \n\n\\[\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a}=4\\] and $ac=bd$. Find the maximum possible value of \n\n\\[\\frac{a}{c}+\\frac{b}{d}+\\frac{c}{a}+\\frac{d}{b}.\\]
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Context:
Let \( f: [0,1] \to \mathbb{R} \) be defined by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 9^n \) if \( x \) is irrational, where \( n \) is the number of zeros immediately after the decimal point in the decimal representation of \( x \). Compute the Lebesgue integral \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the center $Z(G)$ of the group $G$ generated by the elements $\sigma$ and $\tau$, where $\sigma$ has order $5$, $\tau$ has order $4$, and $\tau\sigma\tau^{-1} = \sigma^2$. The group $G$ is of order $20$, and every element $g$ in $G$ can be uniquely expressed as $g = \sigma^k\tau^m$, where $0 \le k < 5$ and $0 \le m < 4$. Determine $Z(G)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the function \(f(x) = (x^2-1) \log\left(\frac{1+x}{1-x}\right)\), defined on \((-1,1)\), can be continuously extended at \(x = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( k \) be an integer greater than 1, and let \( a \) be a root of the equation \( x^{2} - kx + 1 = 0 \). For any natural number \( n \) greater than 10, the units digit of \( a^{2^{n}} + a^{-2^{n}} \) is always 7. Determine the units digit of \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( I = \int_0^\infty \frac{dx}{(x+1)^2(x+2)} \) given that there are two first-order poles at \( x = -1 \) and \( x = -2 \) on the real axis, and their residues cancel each other out.
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Context:
Find a polynomial $p(x)$ in $\mathbb{Z}_4[x]$ such that $p(x)$ is a unit and $\deg p(x) > 1$. Provide an example of such a polynomial.
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Context:
Determine the number of group homomorphisms from $A_5$ to $S_5$. Provide your answer as a single integer.
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Context:
Calculate the mass of the tile in the first quadrant defined by the inequalities \(1 < xy < 3\) and \(1 < x^2 - y^2 < 3\) using the density function \(p(x, y) = (x^4 + y^4)(x^2 + y^2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the degree of the field extension \([L:\mathbb{Q}]\), where \(L\) is the splitting field of the polynomial \(x^4 - 2\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the iteration formula \(x_{n+1}=x_n-(\cos x_n)(\sin x_n)+R\cos^2x_n\), where \(R\) is a positive constant, determine the derivative of the function \(f(x)\) that is being approximated using Newton's method.
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Context:
Determine the number of dimensions in which the statement "a projectivity that fixes any three points is the identity" is true.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the point on the surface defined by the equations \(x_1x_3-x_2x_4=1\) and \(x_2x_3+x_1x_4=0\) that is furthest from the origin.
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Context:
In a contest with n questions, each question is worth a specific positive number of points, and no partial credit is awarded. By redistributing the question scores, any desired contestant ranking can be achieved. What is the maximum possible number of contestants, m?
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Context:
Compute the intersection multiplicity of the curves $V(f)$ and $V(g)$ at the point $p = [1:0:-1]$, where $f = x^2 + y^2 - z^2$ and $g = (x-z)^2 - y^2$.
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Context:
Given a real \(2\times2\) matrix \(A\) such that \(A^6 = I\), where \(I\) is the identity matrix, determine the number of distinct possibilities for the characteristic polynomial of \(A\). The characteristic polynomial must divide the annihilating polynomial \(x^6-1 = (x-1)(x^2+x+1)(x+1)(x^2-x+1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest integer $k$ such that $\Bbb{Z}^{k}$ can be embedded in $\mathcal{GL}(n,\Bbb{Z})$. What is this largest value of $k$?
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Context:
Find the minimum value of \(|A \cap B \cap C|\) given that \(n(A) + n(B) + n(C) = n(A \cup B \cup C)\), \(|A| = |B| = 100\), and \(n(S) = 2^{|S|}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all values of $a$ such that there exist positive integers $a$ and $b$ satisfying the equation $(a - b)\sqrt{ab} = 2016$. Express your answer as a single integer.
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Context:
Given the matrix \( A = \begin{bmatrix} 1 & -1 & -5 & 1 & 4\\ -1 & 2 & 8 & -3 & -4\\ 3 & -1 & -9 & 0 & 4 \\ 2 & 2 & 2 & -5 & -10\\ 0 & -3 & -9 & 5 & 13 \end{bmatrix} \), define the subspace \( W_2 \) as \( W_2 = \{ Y \in M_{5 \times 5} \mid YA = 0 \} \). If the rank of matrix \( A \) is 3, determine the dimension of \( W_2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a country, there are 28 cities and between some cities there are two-way flights. In
 every city there is exactly one airport and this airport is either small or medium or big. F
 every route which contains more than two cities, doesn't contain a city twice and ends
 where it begins; has all types of airports. What is the maximum number of flights in this
 country?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x,y) \) be defined as follows:
\[
f(x,y) =
\begin{cases}
\frac{xy(x^2-y^2)}{x^2+y^2}, & \text{if } (x,y) \ne (0,0) \\
0, & \text{if } (x,y) = (0,0)
\end{cases}
\]
Find the value of \( \frac{\partial^2 f}{\partial x \partial y} \) at the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle that provides $T$ with a $(m,n)$-polarization.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, compute the the largest possible value of $m+k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the upper boundary of the function \( f_n(x) = \frac{\sin nx}{\sin x} \) for \( n \in \mathbb{Z} \), where \( x \) is not a rational multiple of \( \pi \). Express your answer in terms of a function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the Cauchy problem for the partial differential equation:

$$\frac{\partial u}{\partial t}+ \bigg(\frac{\partial u}{\partial x}\bigg)^2=1 \quad x\in \mathbb{R},t>0$$

with the initial condition:

$$u(x,0)=-x^2 \quad \quad x \in \mathbb{R}$$

Determine the value of $t^*$ such that $|u(x,t)| \to \infty$ as $t \to t^*$. \(t^*\) is the time at which the solution becomes unbounded.
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Context:
Calculate the limit: $$\lim \limits_{n \to \infty}\left(1-e^{\left(\frac{t+ \log^2(n)}{t-\log(n)}\right)}\right)^n$$
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Context:
Determine the minimum number of times two fair six-sided dice must be rolled to ensure a 95% probability that the proportion of times they show the same face is between \( \frac{5}{36} \) and \( \frac{7}{36} \) using the central limit theorem.
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Context:
David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work?
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Context:
Given the probability density function \( \frac{1}{\pi a_0^3}e^{-2r/a_0} \), find the most probable value of \( r \) for the wavefunction of the lowest energy state of hydrogen, \( \psi(r) = \sqrt{\frac{1}{\pi a_0^3}}e^{-r/a_0} \), where \( r \) is the radial distance between the electron and nucleus and \( a_0 \) is the Bohr radius.
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Context:
Determine the value of \( n > 1000 \) that maximizes the probability \( P(x=10) = \frac{\binom{1000}{10}\binom{n-1000}{140}}{\binom{n}{150}} \), where \( n \) is the total number of identical balls in a box. Initially, 1000 balls are marked and returned to the box, and then 150 balls are randomly selected. Find the value of \( n \) that makes this probability largest.
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Context:
Find the probability that a random variable \( \xi \sim \text{Beta}(1,2) \) is less than another independent random variable \( \eta \sim \text{Exp}(1) \). Calculate \( P(\xi < \eta) \).
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Context:
On a 3D chess board of size \(N \times N \times N\), determine the minimum number of rooks required to ensure that every empty cube on the board can be reached by a rook in a single move.
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Context:
Let $f: \mathbb{D} \to \mathbb{R}$ be defined as \( f(x) = \frac{x^2 + 2x + a}{x^2 + 4x + 3a} \), where \( \mathbb{D} \) is the domain of \( f \). Determine the range of the parameter \( a \) such that \( f \) is a surjective mapping.
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Context:
Determine the number of natural numbers \( n \) such that there is no natural number \( M \) for which \( n^2 \) can be expressed as a sum of distinct squares of natural numbers \( a_i \).
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Context:
Consider the set $\mathcal F$ of all holomorphic functions $f$ in the unit disc $\Bbb D$ such that $f(\Bbb D)\subset\Bbb D$, and satisfying $f\left(\frac{1}{2}\right)=f'\left(\frac{1}{2}\right)=0$. Determine the supremum of $|f(0)|$ over all functions $f$ in $\mathcal F$. What is this supremum?
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Context:
Determine the center of the algebra $PB(H)P$, where $P$ is a projection on a Hilbert space $H$. Express your answer in terms of $P$.
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Context:
Evaluate the integral \( \int_{-\infty}^\infty |f\ast f'|^2(x)\,dx \) given that \( \mathcal{F}[f](\omega)=\frac{1}{1+|\omega|^3} \) using Plancherel’s theorem.
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Context:
Determine the largest exponent \(d\) such that the volume of the set \(X(r)\), consisting of matrices \(A \in M(n \times m)\) with \(n \leq m\), where the largest singular value of \(A\) is at most 1 and the smallest singular value is less than \(r\), divided by \(r^d\) remains uniformly bounded as \(r\) approaches 0.
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Context:
Determine the minimum value of $n$ for which the probability $P(0.39 \leq \overline{Y} \leq 0.41)$ is at least $0.95$, where $\overline{Y}$ is the sample mean of $n$ independent and identically distributed Bernoulli random variables with success probability $p = 0.4$. Assume the mean and variance of a Bernoulli variable are $np$ and $np(1-p)$, respectively.
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Context:
Find the direction of a light beam that should be projected onto the plane \(\pi: x-2y+4z=0\) such that its reflection has the direction \((-2,1,2)\).
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \frac{(2 n)! e^n (n)^n}{n! (2 n)^{2 n}}$$
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Context:
Determine a closed formula for the nth value of x such that \( 15^x \equiv x \pmod{1000} \) for \( x \geq 3 \).
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Context:
Find the argument of the complex number $z$ that satisfies the equation $|z| = |z - 2|z||$. Express your answer in radians.
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Context:
Given the system of equations \(a x + b y = 1\) and \(x^{2} + y^{2} = 50\), determine the number of real pairs \((a, b)\) such that there is at least one solution, and all solutions \((x, y)\) are integer solutions.
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Context:
Simplify the ordinal expression \(((\omega + 1) + (\omega + 1))^\omega\) and express it in Cantor Normal Form.
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Context:
Let a board game has 10 cards: 3 skull cards, 5 coin cards and 2 blank cards. We put these 10 cards downward and shuffle
 them and take cards one by one from the top. Once \( 3\mathrm{{skull}} \) cards or coin cards appears we stop. What is the possibility of it
 stops because there appears 3 skull cards?
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Context:
Determine the highest degree of the polynomial for which the quadrature formula \( \int_{-1}^1 |x| f(x) \, dx \approx \frac{1}{2}[f(x_0) + f(x_1)] \) is exact, given that \( x_0 \) and \( x_1 \) are the quadrature points.
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Context:
Find an entire function that is bounded on the imaginary axis.
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Context:
Given the function \( H(s) = \frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi) = \sum_{n=1}^{\infty} \frac{h(n)}{n^s} \), determine the smallest integer \( n \) as a function of \( q \) such that \( h(n) \neq 1 \).
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Context:
What is the least positive integer \( m \) such that the following is true?
 Given \( m \) integers between 1 and 2023,inclusive,there must exist two of them \( a,b \) such that \( 1 < \frac{a}{b} \leq  2 \) .
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Context:
Determine the value of $d$ for which the generalized Cantor set, constructed by retaining intervals of size $d<1$ at both ends and removing an interval of size $1-2d$ at each stage, has Lebesgue measure zero. Specifically, find the value of $d$ that satisfies the equation:

$$ \sum_{n=1}^{\infty} 2^{n-1} (1-2d^n) = 1. $$
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Context:
Evaluate the double integral of the floor function over the rectangular region: \[ \int_{1}^{3} \int_{2}^{5} \lfloor x+y \rfloor \, dy \, dx \]
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Context:
Let \( E \) be the set of all continuous functions \( f: [0,1] \to \mathbb{R} \) such that \( f(x) + f(y) \ge |x-y| \) for all \( x, y \in [0,1] \). Find the minimum value of \( \int_0^1 f(x) \, dx \) for \( f \in E \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( k \) such that for any coloring of the squares of a \( 2n \times k \) chessboard with \( n \) colors, there exist 2 columns and 2 rows where the 4 squares at their intersections all have the same color.
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Context:
In triangle $ABC$, the vertices are located at the centroids of three houses, and the points are not collinear. Let $N = 2017$. Define the $A$-ntipodes as the points $A_1, A_2, \ldots, A_N$ on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$. Similarly, define the $B$-ntipodes and $C$-ntipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-ntipode, and similarly for qevians through $B$ and $C$. Determine the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, and $C$, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of moves required to achieve a reasonably uniform distribution of configurations for a 2x2x2 Rubik's cube.
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Context:
Find the gradient of the function \( G(Y) = Y^{-3} \), where \( Y \) is an \( m \times m \) matrix.
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Context:
The Great Kingdom is a rectangular island divided into smaller rectangular landscapes. Each landscape has a road running along one of its diagonals. These roads form a closed loop that does not intersect itself. If the Great Kingdom has an odd number of landscapes, what is the minimum number of landscapes it can have?
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Context:
Let $A$ be the number of 2019-digit numbers composed of exactly two different digits. Determine the highest power of 3 that divides $A$. For example, the number $10\underbrace{1...1}_{2016}0$ is such a number.
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Context:
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the surface composed of the paraboloid \( y = x^2 + z^2 \) for \( y \in [0,1] \) and the disk \( x^2 + z^2 \leq 1 \) at \( y = 1 \), oriented with outward normal vectors. The vector field is given by \( \mathbf{F} = (0, y, -z) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $v$ for which the rank of the matrix \( \begin{bmatrix} -\sin u + \sin (u+v) & \sin(u+v) \\ \cos u - \cos(u+v) & -\cos(u+v) \\ 1 & 0 \end{bmatrix} \) is equal to 2.
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Context:
Evaluate \( A^2 \) if \( A = \lim_{n\to\infty} \left(\prod_{r=0}^n \binom{n}{r}\right)^\frac{1}{n(n+1)} \).
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Context:
In \( \bigtriangleup {ABC} \) with \( {AB} = {10},{AC} = {13} \) ,and \( \measuredangle {ABC} = {30}^{ \circ  },M \) is the midpoint of \( \overline{BC} \) and the circle with diameter \( \overline{AM} \) meets
\( \overline{CB} \) and \( \overline{CA} \) again at \( D \) and \( E \) ,respectively. The area of \( \bigtriangleup {DEM} \) can be expressed as \( \frac{m}{n} \) for relatively prime positive
 integers \( m,n \) . Compute \( {100m} + n \) .
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Context:
Given \(z = e^{\frac{2\pi i}{101}}\) and \(w = e^{\frac{2\pi i}{10}}\), evaluate the following product modulo 101:
\[ A = \prod_{a=0}^{9}\prod_{b=0}^{100}\prod_{c=0}^{100}(w^a + z^b + z^c) \]
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Context:
Evaluate the limit \( \lim_{(x, y) \to (0, 0)} (1 + x^2 + y^2)^{\frac{1}{x^2 + y^2 + xy^2}} \) using polar coordinates.
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Context:
Given the companion matrix \(A\) of a monic polynomial \(f(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + x^n\), where \(A\) is defined as:

\[A=\begin{bmatrix} 0&\cdots&0&0&-a_{0}\\1&0&\cdots&0&-a_{1}\\ 0&1&\ddots&\vdots&-a_{2}\\ \vdots & \ddots& \ddots&0&\vdots \\ 0& \cdots&0&1&-a_{n-1}\end{bmatrix}\]

and the characteristic polynomial of \(A\) is \(\chi(\lambda)=(-1)^{n}f(\lambda)\). Determine the minimal polynomial of \(A\).
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Context:
Calculate the integral of the function \( \operatorname{sgn}(\sin(\pi/x)) \) over the interval \((0, 1)\).
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Context:
Calculate the expected value of the integral of the inverse squared norm of a 3-dimensional Brownian motion from 0 to \( t \). Specifically, find \[ \int_0^t \mathbb{E}\left[||(B^1_s,B^2_s,B^3_s)||^{-2}\right]ds. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct cards can be constructed by punching holes in two of the small triangles of an equilateral triangle divided into 9 congruent sub-triangles?
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Context:
Find the coefficient of \(x^{18}\) in the expansion of \((1+x^{3}+x^{5}+x^{7})^{100}\).
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Context:
An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths.

Determine the minimum good number.
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Context:
Consider the system\[x+y=z+u,\]\[2xy=zu.\]Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.
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Context:
Evaluate the integral \( \oint_{\gamma} \frac{z}{\cos z} \, dz \), where \( \gamma \) is the circle \( |z| = 5 \) oriented in the counterclockwise direction.
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Context:
Determine the maximum number of "almost centers of symmetry" that a finite set of points on the plane can have.
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Context:
Determine all values of $a$ for which the integral \( I = \int_{0}^{+\infty}\frac{dx}{x^2+x^a} \) converges.
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Context:
How many solutions does the equation system\[\dfrac{x-1}{xy-3}=\dfrac{3-x-y}{7-x^2-y^2} = \dfrac{y-2}{xy-4}\]have?
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Context:
Given the rotation matrix \(K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix},\) determine the axis of rotation by inspection.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}}


Context:
Consider a probability game where a target number \( x \) is chosen from the interval \((0, 1)\). A sequence of numbers \( c_1, c_2, \ldots \) is generated such that each \( c_i \) is picked uniformly at random from the interval \((0, c_{i-1})\), starting with \( c_1 \) from \((0, 1)\). The game stops when a number \( c_i \) is picked such that \( c_i < x \). What is the expected number of picks required to stop the game? Express your answer in terms of \( x \).
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Context:
Determine the value of $y_0$ such that the solution to the initial value problem $y'+\frac{2x^2-4xy-y^2}{3x^2}=0, \ x>0, \ y(1)=y_0$ has a limit of $y'(x)$ as $x$ approaches 0 that is not equal to 2.
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Context:
p14. Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened?
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Context:
For which values of \(a \in (0,2)\) must there exist \(x, y \in [0,2]\) such that \(|y - x| = a\) and \(f(x) = f(y)\), given that \(f\) is continuous on \([0,2]\) and \(f(0) = f(2)\)?
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Context:
Calculate the integral \( \int_{F}^{} \! a \cdot dn \) for the vector field \( a = [-z^2 - 2z, -2xz + 2y^2, -2xz - 2z^2]^T \) over the area \( F \) on the cylinder \( x^2 + z^2 = 4 \) that lies above the ground plane \( z = 0 \), in front of the plane \( x = 0 \), and between the planes \( y = 0 \) and \( y = 2 \).
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Context:
Find the number of ordered pairs of integers (p,q) satisfying the equation $p^2-q^2+p+q=2014$.
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Context:
In the ring of $2 \times 2$ matrices over the finite field $\Bbb{Z}_p$, consider the set of matrices of the form
$$
\begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}
$$
where $a, b \in \Bbb{Z}_p$. Determine the order of the centralizer of this set.
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Context:
Determine the maximum possible value of $k$ for which there exist distinct real numbers $x_1, x_2, \ldots, x_k$ greater than 1 such that for all $1 \leq i, j \leq k$, the equation \( x_i^{\lfloor x_j \rfloor} = x_j^{\lfloor x_i \rfloor} \) holds.
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Context:
Calculate the approximate proportion of positive integers less than 10,000 that are congruent to 11, 13, 17, or 19 modulo 30 and are prime.
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Context:
Find the values of $x$ such that the recurrence relation defined by \( f_n(x) = f_1(f_{(n-1)}(x)) \) with \( f_1(x) = \frac{2}{3} - \frac{3}{3x+1} \) satisfies \( f_{1001}(x) = x - 3 \) for \( x \ge 2 \).
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Context:
Determine the supremum of the expression \( \frac{\cos(2 \pi d t)-\cos(2 \pi (d+1) t)-\cos(4 \pi  t)+\cos(2\pi t)}{4\sin^2( \pi t)} \) as a function of \(d\) and \(t\).
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Context:
Determine the number of trees with vertex set \( \{v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8\} \) such that at least two vertices have a degree of 3.
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Context:
Determine the values of $p$ for which the function $d(x, y) = |x - y|^p$ defines a metric on the real line $\mathbb{R}$.
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Context:
A particle starts at the origin $(0,0)$ on the Cartesian coordinate plane. Every second, it moves with equal probability to one of the eight nearest lattice points. A square is drawn with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, and $(2, -2)$. The particle will eventually hit the square for the first time at either a corner or an interior point on a side. If the probability of hitting a corner is $\frac{m}{n}$, where $m$ and $n$ are coprime, find $m+n$. \( \text{Express your answer as a single integer.} \)
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Context:
Let $X$ and $Y$ be jointly continuous random variables with the joint density function given by: $$f_{X,Y}(x,y) = \begin{cases} \frac{1}{3}(x+y), & \text{for } x \in (0,1) \text{ and } y \in (0,2) \\ 0, & \text{otherwise}. \end{cases}$$ Compute the expected value $\mathbb{E}[\sin(\pi XY)]$. Express your answer as an integral if necessary.
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Context:
Solve the equation \( a^2 \cdot \arcsin \left(\frac{4}{a}\right) + 4 \cdot \sqrt{a^2 - 16} = 40 \).
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Context:
Let \( x_1 = 1 \) and define the sequence \( x_{n+1} = x_n + \left\lfloor \frac{x_n}{n} \right\rfloor + 2 \) for \( n = 1, 2, 3, \ldots \). Determine the value of \( x_{1997} \).
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} n^2\int_0^1 \frac{dx}{(1+x^2)^n }. \]
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Context:
Let $S$ be a set with infinite cardinality $\kappa_1$. Determine the cardinality $k$ of the set of subsets of $S$ with cardinality $\kappa \le \kappa_0 < \kappa_1$. Assume $\kappa_0$ can be either finite or infinite. What is $k$?
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Context:
Determine the order of the algebraic numbers $\sqrt[3]{3}(1+\zeta_4)$ and $\sqrt[3]{3}(1-\zeta_4)$ over the field $\mathbb{Q}$.
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Context:
 (b) If we place all of these seven-digit numbers in increasing order, find the seven-digit
 number which appears in the \( {2022}^{\mathrm{{th}}} \) position.
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Context:
Calculate the expected time to failure for a 3-out-of-5 system where each component has an exponentially distributed survival time with a mean of 5 hours.
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Context:
The graph of $r=2+\cos2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.
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Context:
Determine the degree of the field extension $[F(x):F(x^n)]$, where $F(x)$ is the field of fractions of polynomials over a field $F$.
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Context:
Use the divergence theorem to evaluate the surface integral \( \iint_S (2x+2y+z^2) \, dS \), where \( S \) is the sphere defined by \( x^2+y^2+z^2 = 1 \).
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Context:
Find \( \lim_{n \rightarrow \infty} \left( \frac{1}{\sqrt{n}} \sum_{k=1}^n \left| \frac{a_k}{\sqrt{k}} \right| \right) \), where \(a_n\) converges to \(a \in \mathbb{R}\).
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Context:
Let \( z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) \). Evaluate the sum \( \sum_{k = 0}^{13} \frac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} \) and express it in the form \( a - bi \). Find \( a + b \).
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Context:
Given a graph $T$ that is a spanning tree of a graph $G$ with a diameter of $x$, determine the diameter of the graph $$(T \cup T^c) + (T \cup T^c)$$ where $T^c$ is the complement of $T$, and $T$ and $T^c$ are different copies of $T$. What is the diameter of this graph?
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Context:
Let \( f(y) = \sum_{n=0}^{\infty} y^{b_n} \) where \( b_n \) is a strictly increasing sequence of positive integers. Given that \( f(1/2) = 5 \), find \( \lim_{y \rightarrow 1^-} f(y) \).
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Context:
Evaluate the integral \( \int_{0}^{1} C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \, dy \), where \( C(\alpha) \) is the coefficient of \( x^{1992} \) in the power series expansion of \( (1 + x)^{\alpha} \).
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Context:
What is the value of\[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]
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Context:
Determine the number of zeroes of the polynomial \( p(z) = z^8 - 20z^4 + 7z^3 + 1 \) that lie within the disk \( D(0,2) \), given that the polynomial has 4 real roots.
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Context:
Let \( {ABCD} \) be a convex quadrilateral such that the triangle \( {ABD} \) is equilateral and the triangle \( {BCD} \) is isosceles, with
 \( \angle C = {90}^{o} \) . If \( E \) is the midpoint of the side \( {AD} \) , determine the measure of the angle \( \angle {CED} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C F \cdot ds \) for the vector field \( F(x, y, z) = \left(4x\left(1+\frac{z^2}{8}\right), \frac{\pi}{3}\cos\left(\frac{\pi}{3}y\right), \frac{x^2z}{2}-e^{\frac{z}{3}}\right) \) along the curve \( C \) parameterized by \( \gamma(t) = (e^{t(t-3)}, t, 2t) \). Consider using a theorem from multivariable calculus to simplify the computation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum distance from the origin to the surface defined by the equations: \( xyza = 1 \) and \( x + y + z + a = 4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$ \int_{0}^{2 \pi} \sqrt{(-5\sin t-10\cos t\sin t)^2+(-5\sin^2t+5\cos^2t+5\cos t)^2} \, dt $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the region of convergence in \(\mathbb{R}^2\) for the power series \[ \lim\limits_{n \to +\infty} \sum_{m , n = 0}^n \frac{m^3 u^m v^n}{n!}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The fraction\[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\]where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $v \in \mathbb{C}^n$ be a complex unit norm vector, and let $G$ be an $m \times n$ matrix with i.i.d. $\mathcal{CN}(0,1)$ elements. Compute the expectation $\mathbb{E}\{G v v^H G^H\}$, where $v^H$ denotes the conjugate transpose of $v$. This expectation is a $m \times m$ matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( c \) that makes the Riemannian metric \( g = (1-x^2-y^2-z^2)^{2c}(dx \otimes dx + dy \otimes dy + dz \otimes dz) \) geodesically complete on the open unit ball \( B = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 <1 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_R (r^2) \, \mathrm{d}V \), where \( R \) is the region in cylindrical coordinates bounded by \( r = \theta \), \( z = 2 - r \), and \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer \( n \) such that in any two-coloring of the complete graph on \( n \) vertices \( K_n \), there always exist \( m \) lines of the same color, no two of which share a common endpoint.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Let \( {ABC} \) a scalene triangle and \( {AD},{BE},{CF} \) your angle
 bisectors, with \( D \) in the segment \( {BC},E \) in the segment \( {AC} \)
 and \( F \) in the segment \( {AB} \) . If \( \angle {AFE} = \angle {ADC} \) .
 Determine \( \angle {BCA} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive integers \( n \) such that \( 2n^3 + 5 \mid n^4 + n + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( K \) be a field, and consider the ideal \( I = (XY, (X-Y)Z) \subseteq K[X,Y,Z] \). Determine the radical of \( I \), denoted by \( \sqrt{I} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an infinite sequence of independent, fair coin tosses. Let $H_i$ denote the event that the $i$th coin toss results in heads. Determine the probability \( \Bbb{P}\left(\bigcap_{i=1}^{\log_2(n)} H_{n+i} \text{ i.o.}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the floating point number system \( F(10,10,-98,100) \), calculate the number of significant decimal digits lost when subtracting \( \sqrt{1001} \) from \( \sqrt{1002} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \[ \int_{(-\infty,\infty)^n} \frac{\prod_{k=1}^n \sin(a_k x_k)}{\prod_{k=1}^n x_k} \frac{\sin(\sum_{k=1}^n a_k x_k)}{\sum_{k=1}^n a_k x_k} \, dx_1 \cdots dx_n \] where \( a_1, \cdots, a_n > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{e^{ix}}{x} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest multiple of 2019 that cannot be expressed as the sum of a perfect square and the product of two consecutive natural numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the group of homomorphisms from the dihedral group $D_{2n}$ to the multiplicative group of complex numbers $\mathbb{C}^*$, where $n$ is an odd integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find\[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
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Context:
Given that $xf(x)=e^x-1$, calculate $\lim\limits_{n \to \infty}{nf^{(n)}}(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum rank of a matrix in $M(n \times n)$, where each entry is a distinct prime number, for $n > 3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312.
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Context:
Determine the number of non-unital ring homomorphisms from $\mathbb{Z}_{30}$ to $\mathbb{Z}_{42}$. Given that the number of group homomorphisms from $\mathbb{Z}_{30}$ to $\mathbb{Z}_{42}$ is $6$, find the specific count for non-unital ring homomorphisms.
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Context:
In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( \vec{F} = 2y\vec{j} + z\vec{k} \) over the surface \( S \), which is the portion of the surface defined by \( y = \ln(x) \) in the first octant. The projection of \( S \) onto the \( xz \)-plane is the rectangle defined by \( 1 \le x \le e \) and \( 0 \le z \le 1 \). The unit normal vector \( \vec{n} \) to \( S \) points outwards from the \( xz \)-plane. Compute the flux of \( \vec{F} \) over \( S \) in the direction of \( \vec{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of different types of trees that can be placed at the vertices of equilateral triangles, given that each type of tree must form a complete triangle and only one tree can grow in each hole.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a value of \( R \) such that a circle of radius \( R \) centered at the origin has exactly 32 integer points on its circumference.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^3 = 3xyz\) using a triple integral in spherical coordinates.
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Context:
Determine the minimum number of subspaces required to cover an entire vector space over a finite field.
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Context:
Find the number of non-isomorphic semi-direct products of two cyclic groups, one of order 4 and the other of order 9.
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Context:
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots   + a^{1!} + 1.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the moment generating function of a random variable \(X\) is \(M_X(t) = e^{2e^t - 2}\) and the moment generating function of a random variable \(Y\) is \(M_Y(t) = (0.2e^t + 0.8)^7\), and assuming \(X\) and \(Y\) are independent, compute \(\Pr(X + Y = 3)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a deck of $n$ cards numbered $1$ to $n$, calculate the probability of forming the maximum possible number of piles when drawing cards according to a specified rule.
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Context:
Determine the smallest integer $k$ such that the following conditions are satisfied for 14 students at the IMO training camp: (a) The students can be arranged in a circle such that every two adjacent students have different favorite numbers. (b) Among the students, 7 are boys and the rest are girls, and no boy and girl share the same favorite number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \bigtriangleup {ABC},D \) lies between \( A \) and \( C \) and \( {AC} = {3AD},E \) lies between \( B \) and
 \( C \) and \( {BC} = {4EC} \) .
 \( B,G,F,D \) in that order, are on a straight line and \( {BD} = {5GF} = {5FD} \) . Suppose the
 area of \( \bigtriangleup {ABC} \) is 900,find the area of the triangle \( \bigtriangleup {EFG} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid enclosed by the cone \( z = \sqrt{x^2 + y^2} \), the cylinder \( (x + \frac{3}{2})^2 + y^2 = \frac{9}{4} \), and the plane \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $t > 0$, how many times does the equation $S(t) = C(t)$ hold true, where $S(t) = -1200\cos\left(\frac{\pi}{6}t\right) + 1500$ and $C(t) = 1200(1.025)^{t}$? Here, $t$ represents the number of months since January 1, 2004.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint (x^2-2x+y^2+2)\,dA \) using the change of coordinates \((x-1)=r\cos{\theta}\) and \(y=r\sin{\theta}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of equivalence classes for statements formed with 5 variable letters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ be the product of all divisors of $2013^{2013}$. Determine the last two digits of $N$ in its decimal notation.
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Context:
Three circles are given by the equations: \( S_1: x^2+y^2+4y-1=0 \), \( S_2: x^2+y^2+6x+y+8=0 \), and \( S_3: x^2+y^2-4x-4y-37=0 \). These circles touch each other at points \( P_1, P_2, \) and \( P_3 \). Let \( C_1, C_2, \) and \( C_3 \) be the centers of these circles. Calculate the ratio \( \frac{\Delta P_1P_2P_3}{\Delta C_1C_2C_3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \frac{n+n^{\frac{1}{2}}+n^{\frac{1}{3}}+\ldots+n^{\frac{1}{n}}}{n} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $OAB$, where $\angle A = 90^\circ$, the altitude from $A$ meets the side of angle $O$ at point $M$. The distances from $M$ and $B$ to the second side of angle $O$ are 2 and 1, respectively. Determine the length of $OA$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U$ be a normally distributed random variable with mean $3$ and variance $16$ (i.e., $U \sim N(3, 16)$), and let $V$ be a chi-squared distributed random variable with $9$ degrees of freedom ($V \sim \chi_{9}^{2}$). Given that $U$ and $V$ are independent, calculate the probability $P(U - 3 < 4.33\sqrt{V})$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability of never losing an infinite game where you start with a single four-sided die numbered 1 to 4? In each round, rolling a 1 loses the die, a 2 keeps the die, and a 3 or 4 gains an additional die. You lose the game when you have no dice left.
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Context:
Let $X = \{(x,y) : x^2 + y^2 < 5\}$ and $K = \{(x,y) : 1 \leq x^2 + y^2 \leq 2 \text{ or } 3 \leq x^2 + y^2 \leq 4\}$. Determine the number of relatively compact connected components of $X \setminus K$ in $X$. Provide your answer as a single integer.
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Context:
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \frac{n \sin x}{x (1+n^2 x^2)} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A \) be a 2x2 square matrix with \( |A| \neq 0 \) such that \( |A + |A| \text{adj}(A)| = 0 \). Find the value of \( |A - |A| \text{adj}(A)| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T$ be the linear transformation represented by the matrix
$$ \left( \begin{array}{cccc}
1 & 1 & 0 & 3 \\
1 & 1 & 1 & 5 \\
2 & 2 & 1 & 8 \end{array} \right) $$
from $\mathbb{R}^4$ to $\mathbb{R}^3$. Consider the space $V$ of linear maps $B$ from $\mathbb{R}^2$ to $\mathbb{R}^4$ satisfying $AB=0$. Determine the dimension of the space $V$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to 2}\frac{\cos(\frac \pi x)}{x-2} $$
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Context:
Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}-1}{1+e^{x}}\ dx =\ln n$.

Find $ \lim_{n\to\infty}(a_{n}-\ln n)$.
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Context:
You have 6000 red marbles, 6000 blue marbles, and 6000 green marbles. Determine the number of ways to select 3003 marbles such that either the number of red marbles is a multiple of 3 or the number of blue marbles is a multiple of 4.
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Context:
In a directed graph with 2013 vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. Given that from every vertex, you can reach exactly k vertices using at most two edges, determine the maximum value of k.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the curve of minimum length between the points (2, 5) and (6, 1) that passes through the point (4, 3) and has an integral from x = 2 to x = 6 equal to 12.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two bugs start at a distance of one foot away from each other on a beanstalk that is continuously growing at a constant rate, doubling its length every hour. The bugs crawl towards each other at a constant speed of x ft/hr. Determine the values of x for which the bugs will meet.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the highest order for which all groups have been classified, excluding groups of order 1024 and cube-free groups of order 50000.
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Context:
The three sides of the quadrilateral are equal, the angles between them are equal, respectively $90^o$ and $150^o$. Find the smallest angle of this quadrilateral in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a wrestling tournament, there are $100$ participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the alternating group $A_4$ to the cyclic group $\mathbb{Z}_{12}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \gcd(p, q) = 1 \) and consider the set \( Y = \{(t^p, t^q) \in \mathbb{C}^2 \} \). Determine the ideal \( I(Y) \) in \( \mathbb{C}[x,y] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of actions required to make a given triangle equilateral. The action $P(X)$ involves moving vertex $X$ to the intersection of the perpendicular bisector of the opposite side and a line parallel to that side passing through $X$. How many such actions are needed to transform the triangle into an equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the greatest common divisor of the polynomial \( p(x) = x^6 + x^5 + 2x^4 + x^3 + 2x^2 + x + 1 \) and its derivative \( p'(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability that the set of independent random variables $E_1, E_2, \cdots$, where $E_n \sim \text{Unif}\{1, 2, \cdots, 2n\}$, equals the set of all even numbers $\mathbb{E}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\text{Positive integers }a,b,c,\mathrm{and~}d\mathrm{~satisfy~}a>b>c>d,a+b+c+d=2010,\mathrm{and~}a^2-b^2+c^2-d^2=2010.\mathrm{Find~the}$
$\text{number of possible values of }a.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a base-3 number system represented by the symbols A, B, and C, each symbol corresponds to one of the digits 0, 1, or 2, but their exact values are unknown. You can ask questions involving A, B, and C, such as "What is A + B + C?", and receive answers in the same ABC base-3 system. What is the minimum number of questions required to determine the values of A, B, and C?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of group homomorphisms from the symmetric group \( S_3 \) to the quotient group \( \mathbb{Z}/6\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A particle of mass \(m\) starts from rest at time \(t=0\) and is moved along the \(x\)-axis with constant acceleration \(a\) from \(x=0\) to \(x=h\) against a variable force of magnitude \(F(t)=t^2\). Find the work done in moving the particle from \(x=0\) to \(x=h\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the directrix of a parabola with focus at $F(-1, -1)$ and a point $P(7, 13)$ on the parabola, where the equation of the tangent line at $P$ is $3x - y - 8 = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{46}\) in the expansion of \(\frac{1}{1 - x^3 - x^4 - x^{20}}\).
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Context:
Compute \( a(x)b(x) + c(x) \) in \( \mathrm{GF}(2^4) \) using the irreducible generator polynomial \( x^4 + x + 1 \). Given:

\[ a(x) = x^3 + x^2 + 1, \]
\[ b(x) = x^2 + x + 1, \]
\[ c(x) = x^3 + x^2 + x + 1 \]

Find the result of \( a(x)b(x) + c(x) \) modulo \( x^4 + x + 1 \) in \( \mathrm{GF}(2^4) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_T y^2 \, dx \, dy \, dz \), where \( T \) is the region bounded by \( 9y^2 + z^2 = 9 \), \( x + y = 3 \), and \( x = 0 \).
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Context:
A total area of \( 2500 \, \mathrm{m}^2 \) is available for building identical houses. The construction cost for a house with an area \( a \, \mathrm{m}^2 \) includes material cost \( 100 p_{1} a^{\frac{3}{2}} \) yuan, labor cost \( 100 p_{2} a \) yuan, and other costs \( 100 p_{3} a^{\frac{1}{2}} \) yuan, where \( p_{1} \), \( p_{2} \), and \( p_{3} \) are consecutive terms of a geometric sequence. The sum of these terms is 21, and their product is 64. If building 63 houses results in the material cost being less than the sum of the labor and other costs, determine the maximum number of houses that can be built to minimize the total construction cost.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following expression: \[ \sum_{m=0}^\infty \lim_{n\rightarrow\infty}\int_{2x^2+y^2<n}\left(1-\frac{2x^2+y^2}{n^2}\right)^{n^2}x^{2m}\,dx\,dy \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit of the characteristic function \( \varphi_{\bar{\xi}} \) as \( n \to \infty \), where \( \bar{\xi} \) is the sample mean of a sequence of independent and identically distributed random variables \( \xi_1, \xi_2, \ldots, \xi_n \) with finite second moments.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum distance between any two positions of a Rubik's cube, denoted as \( \max_{A,B \in P} d(A,B) \), where \( d(A,B) \) is the minimal number of moves required to reach position \( B \) from position \( A \)?
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Context:
Calculate the remainder when $666^{666}$ is divided by $1000$. Provide your answer as a single number.
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Context:
Determine the range of values for \( s \in \mathbb{R} \) such that the series \( \left(\sum_{k \ge 1} |k^s|^p \right)^{1/p} \) converges for \( 1 \le p < \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \to 1} \frac{x^x - x^{x^x}}{(1-x)^2} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \int\int (x^{4}-y^{4})xy \, dx \, dy \) over the region \( G = \{(x,y)\in\mathbb{R}^{2}: 0 \leq x, 1 \leq x^{2}-y^{2} \leq 2, 1\leq xy \leq 2\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a function $h$ is both $a$-strongly convex and $b$-strongly convex, determine the strong convexity constant of $h$.
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Context:
An investor buys a stock worth $100. Each month, the stock has a 50% chance of increasing by $1.10 and a 50% chance of decreasing by $0.90, with changes being mutually independent. What is the probability that, after 100 months, the stock's value exceeds $91?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x+y^{-99}=3$ and $x+y=-2$, find the sum of all possible values of $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest possible cardinality of a set of mutually disjoint perfect matchings in the complete graph $K_\omega$, where $K_\omega$ is the complete graph on an infinite set of vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider 2016 points arranged on a circle. You are allowed to jump ahead by either 2 or 3 points in a clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point?
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Context:
Determine the number of distinct subgroups of order 10 in a non-cyclic abelian group of order 20.
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Context:
Determine the range of values for \(\lambda\) such that \(\lambda < \sin(\sqrt{4n^2 + 1}\pi)\) for all natural numbers \(n\).
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Context:
Given a stack of 10 cards, where 3 are red, 5 are blue, and 2 are green, what is the expected number of draws, without replacement, before you draw both a red and a blue card? Assume that it does not matter whether the red or blue card is drawn first.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for the real number \( a \) such that the function \( f(x) = x^2 - 2x - |x - 1 - a| - |x - 2| + 4 \) is always non-negative for any real number \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral: $$ \int_{0}^{\infty}\frac{x^{2}\ln x}{(1+x^{3})^4}dx $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \frac{\partial z}{\partial x}(1, 2) \) where \( z \) is a function of \( (x, y) \) defined implicitly by the equation \( z^2f + zg + x - y = 0 \). Given that \( f(1, 2) = 2 \), \( g(1, 2) = 1 \), and at \( (1, 2) \), \( df = -2dx + 5dy \), \( dg = 4dx - 3dy \), and \( z(1, 2) = -1 \). What is \( \frac{\partial z}{\partial x}(1, 2)? \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos(\theta)}\cos(n\theta) \,d\theta,$$ where $n \in \mathbb{N}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to express 100 as a sum of positive even integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the principal value of the integral \( \int_{-\infty}^\infty \frac{\sin(2x)}{x^3} \, dx \) considering the third-order pole at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \(\sum_{d|n}\tau(d)\mu(d)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$ \int_{-1}^1 xde^{|x|} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of charts needed to parametrize the subset \( C \) of \( \mathbb{R}^{3} \) defined by \( C = \{(x,y,z) \in \mathbb{R}^{3} \mid 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z = x^{2} + y^{2} \} \). Consider the circular nature of the boundary where \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of distinct subgroups of order 8 in the symmetric group $S_5$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $f(d)$, the smallest possible size for an efficient substitute for $\Omega = \{0,1\}^d$, where $d$ is an integer $\geq 2$. An efficient substitute $\Omega'$ satisfies $p(\Omega',i,j,A)=p(\Omega,i,j,A)$ for all $i,j$ and $A$, with $p(\Omega,i,j,A)$ being the probability that $(x_i,x_j) \in A$ for a randomly selected $(x_1,x_2, \ldots, x_d) \in \Omega$. The trivial bound is $f(d) \leq 2^d$, and it is known that $f(3)=4$. What is $f(d)$ in general?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 25th power residues modulo 343, given that \(\gcd(25, 342) \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected residual service time for an arbitrary job in a queueing system with two types of arrivals. Type A arrivals occur at a rate of \(\lambda_A = 1/20\) per second and have a service time of exactly 8 seconds. Type B arrivals occur at a rate of \(\lambda_B = 1/30\) per second and have a service time of exactly 12 seconds. Use the formula for expected residual time: \[E(R) = \frac{E(B^2)}{2E(B)}\] where \(E(B)\) is the expected service time of an arbitrary job. Provide the expected residual time \(E(R)\) as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) \) be a function that is defined and has a continuous derivative on \((2, \infty)\). Assume that \( f(a) = b \), \( \vert f(x) \vert < x^9 + 9 \), and \( \int_{4}^{\infty} f(x) e^{-x/5} \, dx = -2 \). Determine the value of \( \int_{4}^{\infty} f'(x) e^{-x/5} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume the function $f : \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable on $\mathbb{R}$. Given that $f(0) = 0$ and $f(x)f'(x) \leq 2$ for all $x \in \mathbb{R}$, determine the largest possible value of $f(4)$. \( \text{Provide your answer in the form } \boxed{\text{value}}. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, $AL$ and $BM$ are the angle bisectors. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Determine the measure of angle $ACB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle with side lengths $AB=7$, $BC=8$, $AC=9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a,c)=gcd(b,d)=1$ and $b,d$ are not divisible by the square of any prime. Find $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \frac{u^{2n+1}\ln(u)}{2n+1} \bigg|^{u=1}_{u=0} \) at \( u=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{\infty}{xI(2x)e^{-x^2}\,dx} \), where \( I(x) = \frac{1}{\pi}\int_0^{\infty}{e^{x\cos\theta}\,d\theta} \) is a Bessel function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the polynomial congruence \( x^4 - 5x - 6 \equiv 0 \pmod{100^{100}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the generator of the kernel of the evaluation homomorphism \( \phi: \mathbb{R}[x] \to \mathbb{R} \) defined by \( f(x) \to f(1+\sqrt{2}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the numerical solution at \(x=1.2\) for the initial value problem \(y''=\frac{2}{x}y'-\frac{2}{x^2} y - \frac{1}{x^2}\), with \(y(1)=0\) and \(y'(1)=1\), using the Modified Euler's method. Use a step length of \(h=0.2\) and ensure the solution is accurate to 6 decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the ring to which the quotient ring \( R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z) \) is isomorphic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the quotient module \( \mathbb{Z}[1/6]/(2020)\mathbb{Z}[1/6] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be an even positive integer. Alice and Bob play a game where Alice chooses a set $S$ containing $m$ integers and announces it to Bob. Bob then chooses $i \in \{1,2,\dots, n\}$ that has not been chosen and sets the value of $v_i$ to either $0$ or $1$. After all $v_1, v_2, \dots, v_n$ have been set, the expression \( E = v_1 \cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1} \) is calculated. Determine the minimum $m$ such that Alice can always ensure that $E \in S$ regardless of Bob's choices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive integers \( n \) such that there exists a sequence \((x_k, y_k)\) defined by either \((P(x_k), Q(y_k)) = (x_k + 1, x_k^2 + 1)\) or \((Q(x_k), P(y_k)) = (x_k^2 + 1, x_k + 1)\), starting with \((x_1, y_1) = (1, 3)\), for which \( x_n = y_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the system of equations:
\[x_{1}+x^2_{2}+x^3_{3}+x^4_{4}+x^5_{5} =5\]
and
\[x_{1}+2x_{2}+3x_{3}+4x_{4}+5x_{5}=15\]
where \(x_{1},x_{2},x_{3},x_{4},x_{5}\) are positive real numbers. Determine the number of solutions this system has.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S = \{1, 2, 3, 4\} \). A sequence of \( n \) terms \( a_1, a_2, \ldots, a_n \) has the property that for any non-empty subset \( B \) of \( S \), there exist \( |B| \) consecutive elements in the sequence that exactly form the set \( B \). Determine the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $k$ such that the equation \( \left(\binom{n}{2}-\binom{k+1}{3}\right)x=\binom{k+1}{2}-1 \) has no solution for \( x \in \mathbb{N} \), where \( n \in \mathbb{N} \) and \( n > 4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the $2$-adic valuation of $x^2 - 3$ in the field extension $\mathbb{Q}[x]/(x^3 - 3)$, given that $x$ has $2$-adic valuation $0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic order of the function defined by the recurrence relation:
\[ T(n) = n + \sum_{k=1}^n [T(n-k) + T(k)] \]
with the initial condition \( T(1) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the expression \[ c = \frac{\sqrt{n-1}\Gamma\left(\frac{n-1}{2}\right)}{\sqrt{2}\Gamma\left(\frac{n}{2}\right)} \] as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \rho \in \mathbb{R} \) for which the function \( f(x) = x + e^{\rho|x|} \) is convex on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $k=\mathbb{Q}(X_1,X_2,X_3,\ldots,X_{n^2})$ and $M=\mathcal{M}_{n,n}(k)$ denote the $k$-algebra of $n \times n$ matrices with coefficients in $k$. Let $A \in M$ be defined by \[ A=\begin{pmatrix} X_1 & X_2 & X_3 & \ldots & X_n \\ X_{n+1} & X_{n+2} & X_{n+3} & \ldots & X_{n+n} \\ X_{2n+1} & X_{2n+2} & X_{2n+3} & \ldots & X_{2n+n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X_{(n-1)n+1} & X_{(n-1)n+2} & X_{(n-1)n+3} & \ldots & X_{n^2} \end{pmatrix} \] and let $C=\{ B \in M \mid AB=BA \}$ be the commutant of $A$. Determine the dimension $d$ of $C$ as a $k$-vector space.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive constant \(c\) such that for any simple graph \(G = (V, E)\), if \(|E| \geq c|V|\), then \(G\) contains two cycles with no common vertex, and one of them contains a chord.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the exponential equation \(\frac{e^{2x}}{x+1}=\frac{4e^{\frac{5}{2}}}{5}\) without using the Lambert W function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be an integral domain with field of fractions $K$. If $M$ is a finitely-generated $A$-submodule of $K$ that is locally free of rank $r$, determine the dimension of the $K$-vector space $M \otimes_A K$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S z^2 \, dS \) where \( S \) is the surface of the cube defined by \([-1,1] \times [-1,1] \times [-1,1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. Find the minimum value of \( \sum_{k=1}^n \frac{x^k_k}{k} \), where $x_1, x_2, \ldots, x_n$ are positive real numbers such that \( \sum_{k=1}^n \frac{1}{x_k} = n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of continuous functions \( f: [0, 1] \to \mathbb{R} \) that satisfy the integral equation:
\[ \int_0^1 f(x) \, dx = \frac{1}{3} + \int_0^1 f^2(x^2) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(f(x)= \lim_{n\to \infty}\dfrac{\left(1- \cos \left(1- \tan \left(\dfrac {\pi}{4}-x\right)\right)\right)(x+1)^n+ \lambda\sin((n - \sqrt{n^2 -8n})x)}{x^2(x+1)^n+x}\) is continuous at \(x=0\), find the value of \(f(0) + 2\lambda\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of field homomorphisms from a finite field with nine elements to a finite field with four elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If six points are placed either along the boundary or inside of a unit square, such that the minimum distance of each of these points from the remaining five points is the same, what is the numerical value of this distance?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the approximate number of lattice points inside a regular hexagon with radius $r$ on a hexagonal lattice where the distance between adjacent points is 1 cm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the Kleisli category of the monad \( \mathbb{T} : \mathbf{Set} \to \mathbf{Set} \) associated with the Lawvere theory for the category of sets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a Markov chain modeling a population's genotype distribution, two individuals with genotype Aa reproduce, resulting in various states: (AA, AA), (AA, Aa), (Aa, Aa), (AA, aa), (Aa, aa), and (aa, aa), with the first and last states being absorbing. If the process starts in the state (Aa, Aa), calculate the probability that it will eventually reach the state (AA, AA). Assume each child receives one allele from each parent with a 50% chance of getting either allele.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $X$ and $Y$ are two independent non-negative random variables such that $E[X] = E[Y] = 1$. Determine the value of $\sup_X \{ \inf_Y \{P(X > Y)\}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Susan is presented with six boxes \( B_1, \ldots, B_6 \), each of which is initially empty, and two identical coins of denomination \( 2^k \) for each \( k = 0, \ldots, 5 \). Compute the number of ways for Susan to place the coins in the boxes such that each box \( B_k \) contains coins of total value \( 2^k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the real numbers \(a\) for which the series \(\sum \frac{\sin(ka)}{\log(k)}\) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence defined by \(S_n= n\,p^n \sum_{k=n}^{+\infty} \frac{1}{k}\, \binom{k-1}{n-1} \, (1-p)^{k-n}\) where \(0<p<1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the Newton-Raphson method, determine the maximum number of iterations required to find the root of the equation $g(x) = 0$ within the interval $(1, 2)$ to a precision of 6 decimal places after rounding. The function $g(x)$ satisfies $|g'(x)| \ge 4$ and $|g''(x)| \le 3$ for all $x$ in the interval, and the method converges for all initial approximations in $(1, 2)$. What is the maximum number of iterations needed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the weight of the topological space $X = (0,1) \times (0,1)$ with the dictionary order topology.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three bells begin to ring simultaneously. The intervals between strikes for these bells are,respectively, \( \frac{4}{3} \) seconds, \( \frac{5}{3} \) second and
 2 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 1 minute? (Include first and
 last.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of isomorphism classes of function fields of degree 3 over the rational function field $F = \mathbb{C}(t)$, given that they are ramified only at the points 1 and -1.
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Context:
Determine the rank of the set of functions from \(\omega\) to \(\omega\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} dk \left[ \Im\left( \frac{1}{(\omega+i\eta+k)(\omega+i\eta+k+q)} \right) \right] \), where \( \eta \) is an infinitesimal parameter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the function \( f(z) = z^5 + 2z^3 - z^2 + 2z + 5 \) within the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \) using the Argument Principle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume bounded by the surface \( \sqrt{x} + \sqrt{2y} + \sqrt{3z} = 1 \) in the region where \( x, y, z \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The 83 trainees from the Animath workshop each choose an activity for the free afternoon from 5 activities offered. The activities are: Beach, Laser Tag, Movie, Romanesque Church, and Shopping. The following conditions apply:

- Shopping was at least as popular as Laser Tag.
- Movie tickets are sold in batches of 6.
- At most 5 students go to the beach.
- At most one student visited a Romanesque church.

When writing the list of activities in alphabetical order, the number of students corresponding to each activity is written. How many different lists of numbers can be written?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a basis for $\Bbb R^d$ that includes the vector with the first $k$ coordinates equal to $1$ and the vectors with $1$ as the first component and $-1$ as the $i$th component for $2 \leq i \leq k$, determine the number of linearly independent vectors such that the sum of their first $k$ components is zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the closure of a given set $B$ in $\mathbb{R}^n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G = A_5$ and $H = \langle (1,2,3,4,5) \rangle$. If $g \in N_G(H)$ is an element of order $5$, compute the order of the group $H \langle g \rangle$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a nilpotent matrix $B$ with a minimal polynomial $\lambda^6$ and the largest Jordan block of size 6, determine the rank of $B^4$. It is known that the difference between the nullity of $B^k$ and $B^{k-1}$ indicates that there are 6 Jordan blocks of size at least $k \times k$. What is the rank of $B^4$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve for \(x\) in the equation: \[ \frac{60_{10}!}{12_2!} = 5^x \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
$ \lim\limits_{h\to0}\frac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^{4}}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint \frac{y^3}{x^2} \, dA \) over the region bounded by the curves \( x=2y,\; x=y,\; x=y^2,\; x=2y^2 \) using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(\mathbb{P}(\limsup_n A_n)=0.7\) and \(\mathbb{P}(\liminf_n B_n)=0\), find \(\mathbb{P}(\limsup_n(A_n\cap B_n))\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the period of the function $g(x)$ that satisfies the condition $g(m-x) = g(n+x)$ for any positive integers $m$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $2 \times 2019$ table, fill each cell with a number such that the top row contains 2019 distinct real numbers, and the bottom row contains the same numbers in a different order. Each column must contain two different numbers whose sum is rational. What is the maximum number of irrational numbers that can be in the first row?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the abstract algebraic concept that generalizes the idea of combining vectors end-to-start while preserving their origins.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{{n\to\infty}}\frac{\int_{0}^{1}\left(x^2-x-2\right)^n\,dx}{\int_{0}^{1}\left(4x^2-2x-2\right)^n\,dx} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} \left[x^n + (1-x)^n \right]^{1/n} \, \mathrm{d}x \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( k \) such that the limit \( \lim_{x\to 0} \frac{\sqrt[5]{x+\sqrt[3]{x}}-\sqrt[3]{x+\sqrt[5]{x}}}{x^{k}} = A \) exists, where \( A \) is a non-zero constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the spectrum of the operator $T = S + S^*$, where $S$ is the unilateral shift operator on $\mathcal{l}^2$ and $S^*$ is its adjoint.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a group $G$, if $ba = a^6b$, simplify the expression $a^ib^ja^kb^lb^{-j}a^{-i}b^{-l}a^{-k}$. What is the simplified form?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $ABC$ has integer side lengths, with $BC = 100 \times 7 - 4$. Given that $\angle ABC$ is a right angle, let $r$ and $s$ denote the inradius and semiperimeter of $\triangle ABC$, respectively. Find the perimeter of the triangle $ABC$ that minimizes $\frac{s}{r}$. \(T = 7\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of lines that are both normal and tangent to the curve \( y^2 = x^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$.\nIf $AO=AH$ find the angle $\\hat{BAC}$ of that triangle.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The angle \( \hat{BAC} \) is \( 60^\circ \).

}


Context:
What is the maximum number of distinct prime factors that a number less than $2^{64}$ can have? In other words, find the largest value of $\omega(n)$ for $n < 2^{64}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A car starts at a position \( b \), where \( b \) is normally distributed as \( N(0, 6) \). The car then drives a distance \( d \), which is normally distributed as \( N(0, 5) \). If \( d \) is positive, the car moves up; otherwise, it moves down. Given that \( b \) and \( d \) are statistically independent and the car's final position is \( c = 6 \), find the most likely initial position \( b \) of the car.
Please reason step by step, and put your final answer within \boxed{}.


Context:
175 schoolchildren arrived at a health camp. Some children know each other, and some do not. It is known that any six schoolchildren can be accommodated in two three-person rooms such that all the schoolchildren in one room will know each other. What is the minimum number of pairs of acquainted schoolchildren that could have come to the camp?
Please reason step by step, and put your final answer within \boxed{}.