Context:
Two kings are placed on the same square of a 3x3 chessboard. Each king moves independently and randomly to any adjacent square (including diagonally) with equal probability. Determine the probability that, after a large number of moves, both kings will be on the same square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int\limits_0^\infty e^{-1/x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alexander is eating sliced olives for dinner. Olives are sliced into thirds so that each olive is composed of two indistinguishable “end“ pieces and one “middle“ piece. For each olive slice that Alexander puts on his plate, there is a $\frac23$ chance that it is an end piece and a $\frac13$ chance that it is a middle piece. After placing a randomly selected olive slice on his plate, he checks to see if he can rearrange the slices on his plate so that he has at least one whole olive, which consists of any middle piece and any two end pieces. He can successfully rearrange the olive slices on his plate into at least one whole olive after randomly selecting $n$ olive slices with probability of at least $0.8$. What is the minimum n for which this is true?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Initially, several different natural numbers are written on a board. The average magic of a set of numbers is defined as the ratio of their sum to their product. After erasing the smallest number from the board, the average magic of the remaining numbers increased threefold. Determine the original set of numbers on the board.
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Context:
Given skew lines \( a \) and \( b \), with \( AB \) as the common perpendicular segment where \( A \) is on line \( a \) and \( B \) is on line \( b \), and \( AB = 2 \). The angle between the skew lines is \( 30^{\circ} \). If \( AP = 4 \) on line \( a \), find the distance from point \( P \) to line \( b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a general formula for the denominator of the coefficients in the Taylor expansion of \( \sqrt{1-x} \), given by \( \sum_{n=0}^{\infty} x^n (-1)^n \binom{1/2}{n} \). The denominators are of the form \( 2^m \). Determine \( m \) in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the supremum of the function \( \frac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}} \) within the unit ball centered at the origin, where \( x \neq y \) and \( x, y \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an integer $x \in \mathbb{Z}[i]$ such that $(1+2i)x \equiv 1 \pmod{3+3i}$. Provide your answer in the form of a Gaussian integer.
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Context:
Determine the condition on the coefficients \(a\) and \(b\) such that the cubic curve \(bx^3 + y^3 = 3axy\) has a node at the origin with tangent directions along the coordinate axes.
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Context:
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
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Context:
In a truth table with $n^2$ entries, an algorithm takes two rows, $a$ and $b$, and performs $s$ logical operations, resulting in Boolean values. If each iteration returns $c$ TRUE values, where $c > b - a + 1$, determine the number of additional operations required before selecting a new pair of rows.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a non-abelian group of order $21$. Determine the number of automorphisms of $G$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many polynomials of the form \( z^3 + az^2 + bz + c \), where \( a \), \( b \), and \( c \) are integers, have all roots \( z \) satisfying either \( \left\lvert z \right\rvert = 10 \) or \( \left\lvert z \right\rvert = 15 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \sum_{cyc} \frac{a^4}{(a-b)(a-c)} \) where \( a = -\sqrt{3} + \sqrt{5} + \sqrt{7} \), \( b = \sqrt{3} - \sqrt{5} + \sqrt{7} \), and \( c = \sqrt{3} + \sqrt{5} - \sqrt{7} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of vertices a planar graph and its complement can have while ensuring both graphs remain planar.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) for the vector field \( \mathbf{F}(x,y,z) = xi + yj + zk \), where \( S \) is the plane surface bounded by the triangle with vertices at \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The unit normal \( \mathbf{n} \) to \( S \) has a nonnegative z-component. Use the vector representation \( \mathbf{r}(u,v) = (u+v)i + (u-v)j + (1-2u)k \) to perform the evaluation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the points where the curve $x^y = y^x$ intersects itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An $8 \times 10$ rectangular board is to be completely covered without overlapping using a combination of $S$-shaped tiles and $T$-shaped tiles, where each tile consists of 4 unit squares. Both types of tiles can be rotated or flipped. Determine the minimum number of $T$-shaped tiles needed to cover the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the transition matrix for a Markov chain:
\[
P = \begin{bmatrix}
0 & 0.3 & 0 & 0.7 \\
0 & 0.8 & 0 & 0.2 \\
0.4 & 0 & 0.6 & 0 \\
0.4 & 0 & 0.6 & 0
\end{bmatrix}
\]
with states 0, 1, 2, 3 representing (win, lose), (lose, lose), (win, win), and (lose, win), respectively, determine the probability that Jones won the game on Sunday given that he won on Monday.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the genus of a coffee mug with two tubes going through the middle, where one tube goes through the other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smaller one in terms of the golden ratio?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( L = \lim_{x \to 0} \frac{2 + \tan(e^x - \cos x) - e^x - \cosh x}{x(\sqrt{1+2x} - \sqrt[3]{1+3x})} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct non-isomorphic graphs can be constructed using a set with 7 elements?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \( \lim_{n\rightarrow\infty}\frac{\int_{0}^{1}f^n(x)\ln(x+2)dx}{\int_{0}^{1}f^n(x)dx} \) where \( f(x)=1-x^2+x^3 \) and \( f^n(x) \) is the function \( f(x) \) raised to the power of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ADE$, $\measuredangle ADE=140^\circ$, points $B$ and $C$ lie on sides $AD$ and $AE$, respectively, and points $A,~B,~C,~D,~E$ are distinct.$^\star $  If lengths $AB,~BC,~CD,$ and $DE$ are all equal, then the measure of $\measuredangle EAD$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct bracelets can be formed using 8 red beads, 6 blue beads, and 4 yellow beads, given that all beads must be used and two bracelets are considered identical if one can be obtained from the other through rotation or reflection? Beads of the same color are indistinguishable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be an $11\times 11$ diagonal matrix with the characteristic polynomial $(x-3)^4(x-4)(x-7)^4(x-9)^2$. Find the dimension of the space of $11\times 11$ matrices $V$ such that $AB=BA$ for all $B \in V$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the probability density function for the y-coordinates of a random point uniformly sampled on the triangle with vertices A = (1,0,0), B = (0,1,0), and C = (0,0,1).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three different prime numbers \( p, q, r \) such that \( pqr = 1899\ldots962 \) (where the number contains \( n \) nines), define \( f(n) \) as the sum of the digits of the natural number \( n \). Calculate the value of \( f(p) + f(q) + f(r) - f(pqr) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the measure of the set \( A = \{ (x, x): x \in [0, 1] \} \) under the product measure \( \mu_1 \times \mu_2 \), where \( \mu_1 \) is the Lebesgue measure and \( \mu_2 \) is the counting measure on \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = \lim_{n\rightarrow \infty}\frac{x^2+2(x+1)^{2n}}{(x+1)^{2n+1}+x^2+1} \), where \( n \in \mathbb{N} \). Define \( g(x) = \tan \left(\frac{1}{2}\arcsin\left(\frac{2f(x)}{1+(f(x))^2}\right)\right) \). Determine the number of points where \( g(x) \) is not differentiable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f: \mathbb{C}^5 \rightarrow \mathbb{C}^7$ be a linear function such that $f(2i e_1 + e_3) = f(e_2)$ and $\mathbb{C}^7 = X \oplus \text{Im}(f)$. Determine the dimension of $X$. \( \text{Im}(f) \) denotes the image of $f$. \( e_1, e_2, e_3 \) are standard basis vectors in \( \mathbb{C}^5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of positive roots that the equation \( \sum_{i=1}^N \frac{a_i}{x+b_i} = 0 \) can have, where each \( b_i \) is a positive constant and \( a_i \) can be either positive or negative.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a set $D \subset \mathbb{R}$ and a point $c \in D$. Determine the number of sequences $f: \mathbb{N} \rightarrow D$ such that $\lim\limits_{n \rightarrow \infty} f(n) = c$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ be the integer determined by \(\frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{23}=\frac{a}{23!}\). Find the remainder when $a$ is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$\int_{0}^{\infty}\frac{\cos(x)-\cos(2x)}{x}dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $a$ for which the integral \[ \int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050} \] holds, given that $a^{2}\geq0$. \(a\) is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence of the power series: $$\sum_{n=1}^\infty \frac{x^n}{\cos\left(\frac{n\pi}2\right)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the differential equation \((xy^3 + x^2y^7)\frac{dy}{dx} = 1\) with the initial condition \(y \left(\frac{1}{4}\right)=1\), find the value of \(\frac{dy}{dx}\) when \(y = -1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a 3x3 invertible matrix $A$ with $\operatorname{tr}(A) = \operatorname{tr}(A^2) = 0$ and $\det(A) = 1$, determine the characteristic polynomial of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=4} \frac{\sin^2(z)}{(z-\frac{\pi}{6})^2(z+\frac{\pi}{6})} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a discrete-time Markov chain defined by the equation $X(n+1) = X(n) - 1 + A(n)$, where $A(n)$ is a random variable taking values $0, 1, 2$ with probabilities $p_0$, $p_1$, and $p_2$, respectively, and $X(n) = n$ with $X(n) \geq 1$, find the relationship between $p_0$, $p_1$, and $p_2$ that ensures the stability of the Markov chain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Doppler shift channel where the input signal is $x(t)$ with Fourier transform $X(f)$, and the output is $Y(f) = X(f-f_1)$, find the channel response $H(f) = \frac{Y(f)}{X(f)} = \frac{X(f-f_1)}{X(f)}$. Determine the inverse Fourier transform of $H(f)$, denoted as $h(t)$. Provide your answer as $h(t)$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the group $G=\mathbb{Z}_6 \times \mathbb{Z}_3$, the element $a=(4,1)$, the subgroup $H=\langle a\rangle$, and the quotient group $G_1=G/H$, determine $(k+\ell) \mod{5}$, where $k$ is the order of the class of $(3,1)$ in $G_1$ and $\ell$ is the order of the class of $(2,2)$ in $G_1$. What is the value of $(k+\ell) \mod{5}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A customer wants to purchase exactly four items from a store, with a total cost of 1000 rubles, where the cheapest item costs at least 99 rubles. The store offers two special deals:
1) Buy at least three items and get one item for free, with the free item's cost not exceeding the minimum price of the paid items.
2) Buy exactly one item costing at least $N$ rubles and receive a 20% discount on the next purchase.

Determine the maximum value of $N$ for which the second offer is more advantageous than the first.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a subspace $W$ of the vector space $V$ with maximal dimension such that for all vectors $w$ in $W$, the quadratic form $$ q(x,y,z) = 2x^2 - \frac{1}{2}y^2 + 4z^2 $$ satisfies $q(w) \geq 0$. Determine the dimension of this subspace.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a random walk in $\mathbb{Z}^2$ with increments having finite support and an expectation vector on the horizontal axis, determine whether the probability of first intersecting a vertical line at a given height scales with $t$ or $\sqrt{t}$ as $t$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three coins are initially showing "Heads" on a table. In each round, Danny turns one coin upside down: the left coin with a probability of \( \frac{1}{2} \), the middle coin with a probability of \( \frac{1}{3} \), and the right coin with a probability of \( \frac{1}{3} \). Let \( X_n \) represent the number of coins showing "Heads" after \( n \) rounds. Determine \( \lim_{n \to \infty} \Pr(X_n = 0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ben chooses random numbers from the set \( \{0, 1, \ldots, 9\} \) independently and with uniform probability, arranging them in a sequence in the order they are picked. What is the expected number of distinct numbers that appear between the first two occurrences of the number 0?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $X = h(Z, H)$, where $h$ is a measurable function, and $Z$ is a random variable independent of $H$, and knowing that $\mathbb{E}[X | H = s] = f(s)$, find the value of $\mathbb{E}[h(Z, s)]$. Express your answer in terms of $f(s)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimal number of vertices required for a triangulation of the Klein bottle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), \(\angle A = 45^\circ\) and \(M\) is the midpoint of \(\overline{BC}\). The line \(\overline{AM}\) intersects the circumcircle of \(\triangle ABC\) at a second point \(D\), such that \(AM = 2MD\). Determine \(\cos \angle AOD\), where \(O\) is the circumcenter of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of inflection points that a cubic planar curve, given by the equation $F(x,y)=0$ where $\deg F=3$, can have. Consider the real roots of the determinant $F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x$ in your analysis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence for the Taylor series of the function \( f(z) = \frac{z}{\sin(z^2 + z)} \) at \( z = 0 \), given that \( f(0) = 1 \) to make \( f \) analytic at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of $X_k 1_{\{|X_k| > n\}}$ for $x \geq e$, where $X_1, X_2, \ldots$ are independent and identically distributed random variables with $P(X_i > x) = \frac{e}{x \log(x)}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of abelian groups \( A \) of order \( p^4 \), where \( p \) is a prime number, such that there exists a subgroup \( N \) with \( A/N \cong \mathbb{Z}/p^3\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that \( h: \mathbb{R} \to \mathbb{R} \) is a function with a fixed point at \( x^* \). If \( h'(x^*) = 0 \) and \( |h''(x)| < 2 \), determine the smallest value of \( n \) such that the sequence \( x_0, x_1, x_2, \ldots \) defined by \( x_n = h(x_{n-1}) \) satisfies \( |x_n - x^*| < 10^{-50} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). How many ways can this be done so that all friends are included at least once?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of unique numbers \(\{1, 2, 3, \ldots, n\}\), determine the number of binary search trees (BSTs) that can be formed with a height not exceeding a given height \(h\). For instance, if \(n = 4\) and \(h = 2\), calculate the number of BSTs with height at most 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Farmer James sends his favorite chickens, Hen Hao and PEAcock, to compete at the Fermi Estimation All Star Tournament (FEAST). The first problem at the FEAST requires the chickens to estimate the number of boarding students at Eggs-Eater Academy given the number of dorms $D$ and the average number of students per dorm $A$. Hen Hao rounds both $D$ and $A$ down to the nearest multiple of $10$ and multiplies them, getting an estimate of $1200$ students. PEAcock rounds both $D$ and $A$ up to the nearest multiple of $10$ and multiplies them, getting an estimate of $N$ students. What is the maximum possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given five distinct positive integers $x_1, x_2, x_3, x_4, x_5$ such that $x_1 + x_2 + x_3 + x_4 + x_5 = 100$, find the maximum value of the expression:

\[
\frac{(x_2 x_5 + 1)(x_3 x_5 + 1)(x_4 x_5 + 1)}{(x_2 - x_1)(x_3 - x_1)(x_4 - x_1)} \\
+ \frac{(x_1 x_5 + 1)(x_3 x_5 + 1)(x_4 x_5 + 1)}{(x_1 - x_2)(x_3 - x_2)(x_4 - x_2)} \\
+ \frac{(x_1 x_5 + 1)(x_2 x_5 + 1)(x_4 x_5 + 1)}{(x_1 - x_3)(x_2 - x_3)(x_4 - x_3)} \\
+ \frac{(x_1 x_5 + 1)(x_2 x_5 + 1)(x_3 x_5 + 1)}{(x_1 - x_4)(x_2 - x_4)(x_3 - x_4)}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a graph with $n$ vertices where each edge is chosen with probability $\frac{1}{2}$. A graph $G$ is separated if we can divide the vertices into two equal-sized sets with no edges between them. It is almost-separated if we can divide the vertices into two equal-sized sets with fewer than $10n$ edges between them. Determine the probability that $G$ is either separated or almost-separated.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the operation \( C_{ij} = B_{ik} B_{jl} A_{kl} \) in terms of standard matrix multiplication using Einstein's notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tower is constructed with a square base and square horizontal cross-sections. Viewed from any direction perpendicular to a side, the tower has base $y = 0$ and profile lines $y = (x - 1)^2$ and $y = (x+1)^2$. Find the volume of the solid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(m\) be a positive integer, and let \(T\) denote the set of all subsets of \(\{1, 2, \ldots, m\}\). A subset \(S\) of \(T\) is called \(\delta\)-good if for all distinct \(s_1, s_2 \in S\), the symmetric difference \(|\Delta(s_1, s_2)| \geq \delta m\). Find the largest possible integer \(s\) such that there exists an integer \(m\) and a \(\frac{1024}{2047}\)-good set of size \(s\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the pole of the function \(\frac{\mathrm{Log}(z)}{(z-1)^4}\) at \(z=1\), where \(\mathrm{Log}(z)\) is the principal branch of the logarithm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). If $p$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $m$. \( \text{(Proposed by Ahaan Rungta)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a solution to the differential equation \( ty'' - (t+1)y' + y = 0 \) that is not a constant multiple of the solution \( y(t) = e^t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Minimize the determinant of the matrix \( A = (VV^T + \lambda I)^{-1} \), where \( V \) is an \( m \times n \) matrix and \( \lambda \) is a very small positive scalar.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of points with integer coordinates that lie inside the region bounded by the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) and the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Nick is mixing two solutions. He has $100$ mL of a solution that is $30\%$ $X$ and $400$ mL of a solution that is $10\%$ $X$. If he combines the two, what percent $X$ is the final solution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to select four 3-element subsets from a 6-element set such that each pair of subsets shares exactly one common element, considering distinct arrangements only (ignoring rotations and reflections).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A graph has 20 vertices and does not contain a $K_7$. If each edge is colored either green or yellow such that there are no monochromatic triangles, determine the maximum number of edges this graph can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G$ be a group of order 30 with a center $Z(G)$ of order 5. Determine the isomorphism class of the quotient group $G/Z(G)$. To which commonly known group is $G/Z(G)$ isomorphic?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a specific solution to the equation $x = \log(x) + c$ without using numerical methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function f is defined on the set of integers and satisfies
$\bullet$ $f(n) = n - 2$, if $n \ge 2005$
$\bullet$ $f(n) = f(f(n+7))$, if $n <  2005$.
Find $f(3)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} \frac{1}{n}\int_{0}^{n\pmod {2\pi}} \arcsin(\sin(t))dt \), where \( n\pmod {2\pi} \) represents the remainder when \( n \) is divided by \( 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the conditional covariance of $B_s$ and $B_t$ given $B_1$, where $B$ is a one-dimensional Brownian motion and $s, t \in [0,1]$. Provide your answer as a function of $s$ and $t$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p_1, \dots, p_r$ be $r$ distinct odd primes. Define the group \( G = \prod_{i=1}^r \mathrm{SL}_3(\mathbb{F}_{p_i}) \). Determine the maximum possible rank of a subgroup of \( G \), denoted as \( \max \{d(H) : H \leq G\} \), where \( d(K) \) is the smallest cardinality of a generating set of a group \( K \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the roots of the polynomial $Y^3 - Y + 1$ in the ring $\mathbb{F}_3[Y]/(Y^3 - Y^2 + 1)$. Provide your answer as a set of elements in this ring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be a Von Neumann algebra and $\phi$ be a completely bounded (c.b.) map on $M$. Consider the Hilbert space $H$ and the map $Id\otimes \phi$ on $B(H)\overline{\otimes} M$, where $B(H)$ is the algebra of bounded linear operators on $H$ and $\overline{\otimes}$ denotes the minimal tensor product of operator spaces. Determine if $Id\otimes \phi$ is a completely bounded map, and if so, find its completely bounded norm $\|Id\otimes \phi\|_{cb}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest value of \( c \) such that the series \( \sum_{n=1}^{\infty}\frac{a_n}{S_n^c} \) converges, where \( a_n > 0 \), \( S_n = \sum_{k=1}^{n}a_k \), and \( \lim_{n \rightarrow \infty} S_n = +\infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ appears in $W$ if there are indices $k\le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called special if $Ua, Ub, aU, bU$ all appear in $W$. (The empty word is considered special) You are given that there are no special words of length greater than 2015.Let $N$ be the minimum possible number of special words. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the surface integral \( \iint_S{\mathbf{F} \cdot \mathbf{n}} \, dS \) for the hemisphere \( S \) defined by \( x^2 + y^2 + z^2 = 1 \) with \( z \ge 0 \), where \( \mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} \) and \( \mathbf{n} \) is the unit outward normal to \( S \). Use the parameterization \( \mathbf{r}(u, v) = \sin u \cos v \mathbf{i} + \sin u \sin v \mathbf{j} + \cos u \mathbf{k} \) to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \beta \) for which the series \( \sum_{n=1}^{\infty} \left( \sqrt{1+\frac{(-1)^n}{n^\beta}} - 1 \right) \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int^{(2,2,2)}_{(1,1,1)}5x^4y^{15}z^{15}dx+15x^5y^{14}z^{15}dy+15x^5y^{15}z^{14}dz$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the function \( f(x) = \sum_{n=0}^{+\infty} x^{n^2} \) as \( x \to -1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the average distance between two randomly chosen points from the Cantor set.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A young baseball fan wants to collect a complete set of 262 baseball cards. The cards are distributed randomly, one per package of chewing gum. The fan buys two packages of chewing gum each day. On average, how many days will it take the fan to collect the complete set of cards?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all possible values of $b$ in the equation $p(x)=x^3+ax+b$ if $r$ and $s$ are the roots of $p(x)$ and $r+4$, $s-3$ are the roots of $q(x)=x^3+ax+b+240$. Express your answer in the form of $|\frac{k}{5}|$ where $|.|$ stands for the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the best known upper bound on the index of an abelian subgroup in a discrete subgroup of the Euclidean group of isometries, \( \isom(\mathbb{R}^n) \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of points at which a line in \(\mathbb{CP}^n\) (a copy of \(\mathbb{CP}^1\)) intersects with \(\mathbb{CP}^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{\lambda} \frac{dz}{(z^2-1)^2} \), where \( \lambda \) is a path in \( \mathbb{R}^2 - \{1,-1\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A uniform container of height $6m$ is full of water and has three identical drainage pipes $A$, $B$, and $C$. Each pipe can independently empty the container (the part above it) in $T$ minutes. Pipe $A$ is fixed at the base, while pipe $B$ is fixed at a height of $5m$. Pipe $C$ is fixed somewhere between them. If all three pipes together can empty the container in $\frac{2T}{3}$ minutes, determine the height at which pipe $C$ is fixed. Assume no variations due to pressure differences at different heights.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set $S(R)$ of all representations $\rho : \mathbb{Z}/2\mathbb{Z} \to GL(2, R)$ such that $\rho(0+2\mathbb{Z}) \neq \rho(1+2\mathbb{Z})$, where $R$ is a ring and $GL(2, R)$ is the group of 2x2 invertible matrices with entries from $R$. Let $R = \mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime. Determine whether the following property $(P(\rho, R))$ holds for every, some, or no $\rho \in S(R)$: There exist $v, w \in R^2$ such that for all $g \in \mathbb{Z}/2\mathbb{Z}$, $\rho(g)(R\cdot v) = R\cdot v$, $\rho(g)(R\cdot w) = R\cdot w$, and $R\cdot v + R\cdot w = R^2$. Provide your answer as "every", "some", or "no".
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function $g(x)$ such that for all positive integers $k$, the following conditions hold: $g^{\underline{2k}}(x) = x$ and $g^{\underline{2k + 1}}(x) = g(x)$. Here, $g^{\underline m}(x)$ denotes the composition of $g(x)$ with itself $m$ times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the order of the Weyl group $E_6$ using elementary group theory.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a $202 \times 202$ matrix $A$ with $||A||_2 = 100$ and $||A||_F = 101$, find the sharpest lower bound on the 2-norm condition number $\kappa(A) = ||A||_2 \, ||A^{-1}||_2$. Use the theorem that states $||A||_2 = \sigma_1$ (the largest singular value), $||A^{-1}||_2 = \sigma_r^{-1}$ (where $\sigma_r$ is the smallest singular value), and $||A||_F = \sqrt{\sigma_1^2 + \ldots + \sigma_r^2}$, with $r = \text{rank}(A)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair die has two faces each of brown, black, and white colors. Let Y be the random variable representing the number of tosses needed for all three colors to appear at least once, given that two of the three colors appear in the first k-1 tosses and all colors appear in the first k tosses. Calculate the expected value of Y, E[Y].
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest integer \( n \) such that \( 64n\log_2 n < 8n^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum \( \sum_{n=1}^{2023} (-1)^{\Omega(n)} \left\lfloor \frac{2023}{n} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence of independent and identically distributed (i.i.d.) uniform random variables $X_1, X_2, \dots$ on the interval $(0,1)$, let $N$ be the minimum value of $n \geq 2$ for which $X_n < X_{n-1}$. Calculate the expected value of $N$, denoted as $E(N)$. \( \text{Express your answer in terms of } e. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $f: D \Rightarrow \mathbb{C}$ is analytic in a domain $D$ and $a$ is a singularity of $f$, with the condition that there exist $r > 0$ and positive numbers $M$ and $N$ such that:
\[
N|z-a|^{-\frac{5}{2}} \leq |f(z)| \leq M|z-a|^{-\frac{7}{2}},
\]
for $0 < |z-a| < r$, determine the order of the pole at $a$. The order is known to be higher than 2 and less than or equal to 4. Determine whether the order is 3 or 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a small town, there are $n\times n$ houses indexed by $(i, j)$ for$1\leq i,j\leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c\leq\frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[\/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters?
$\text{A house indexed by }(i,j)\text{is a neighbor of a house indexed by}$
$(k,l)\mathrm{if}|i-k|+|j-l|=1.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 15 passengers, each with an assigned seat on a flight. Each passenger can either take their correct seat or a different one. Calculate the probability that at least one passenger sits in their assigned seat.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 25\) subject to the constraints:
\(-2 \le x_1 \le 5\), \(1 \le x_2 \le 4\), \(-2 \le x_3 \le 6\), \(3 \le x_4 \le 7\), \(2 \le x_5 \le 8\), and \(3 \le x_6 \le 7\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the function:
\[ f(y) = \max_{\mu>0} \left[ \exp\left(\frac{-n\mu^{2}}{\sigma^{2}}\right) \exp\left(\frac{2\mu}{\sigma^{2}}\sum_{k=1}^{n}y_{k}\right) \right] \]
where \( y_{k} \) is a random variable with a normal distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle \(ABC\), the altitudes \(AD\) and \(BE\) intersect at point \(H\). The circumcircle of triangle \(ABH\) intersects sides \(AC\) and \(BC\) at points \(F\) and \(G\) respectively. If \(DE = 5\) cm, find the length of \(FG\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \(\alpha > 0\) for which the function \(f(z) = |z|^\alpha z\) belongs to the class \(C^m(\mathbb{C}, \mathbb{C})\), where \(m\) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the number of positive integers \( n < 180 \) such that \((n^3 + 3n + 1)^{180} \equiv 1 \pmod{180}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{\epsilon \to 0} \left(\sup_{z \in \partial B(x,\epsilon)} |G(x,z)|\right) \) for \( n \geq 3 \), where \( G(x,z) = \phi (z-x) - w^{x}(z) \) and the fundamental solution for \( n \geq 3 \) is given by \( \phi(z-x) = \frac{1}{(n-2)w_{n}} \cdot \frac{1}{|z-x|^{n-2}} = \frac{1}{(n-2)w_{n}} \cdot \frac{1}{\epsilon^{n-2}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\\triangle ABC$ be a triangle such that the angle bisector of $\\triangle BAC,$ the median from $B$ to side $AC,$ and the perpendicular bisector of $AB$ intersect at a single point $X.$ If $AX = 5$ and $AC = 12,$ compute $a+b$ where $BC^2=\\tfrac{a}{b}$ and $a,b$ are coprime positive integers.\n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a 3-fold covering $X \to Y$ of a manifold $Y$ ramified over $m$ disjoint submanifolds, and knowing that the Euler characteristic of $Y$ is $\chi(Y) = 1$, determine the Euler characteristic of $X$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are an equal number of red, yellow, green, and blue cards. One card is placed in a box, and the rest are randomly selected. It is observed that red cards are most frequently selected, followed by yellow, green, and blue. What is the most probable color of the card in the box?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_0^1{\dfrac{\left(2x^3-3x^2\right)f'(x)}{f(x)}}\,dx \] given that the function $f(x)$ is differentiable on $[0, 1]$ and satisfies the conditions $f(1) = 1$ and $f(x)f(1-x) = e^{x^2 - x}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A family buys two insurance policies with independent losses uniformly distributed on the interval from $0$ to $10$. One policy has a deductible of $1$, and the other has a deductible of $2$. The family experiences exactly one loss under each policy. Calculate the probability that the total benefit paid to the family does not exceed $5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the values of $k$ for which the function $f(x) = x^k|x|$ is thrice differentiable at $x = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $n$ such that \( \int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the directional derivative of the function \( f(x,y,z) = \sqrt{x^2 + y^2 + z^2} \) at the point \((0,0,0)\) in the direction of the vector \( v = \left(\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the splitting field of the polynomial \(x^4 + x + 1\) over the finite field \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Transform the quadratic form \( F(x) = x_1^2 - 3x_3^2 - 2x_1x_2 + 2x_1x_3 - 6x_2x_3 \) into its normal form over the real number field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all possible values of \(|z^z|\) using the polar form of the complex number \(z\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the points of discontinuity of the function \( f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor 2rx \rfloor}{n^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of values for $p$ such that the integral \( \int_1^{\infty} \frac{\log{x}}{x^p} \,dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Markov chain model for rolling a fair six-sided die. What is the probability that the sequence "1 followed by 2" occurs before the sequence "two consecutive sixes"?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the simplified form of the sum \(\sum_{k=r}^{n} {k\choose r} p_{n,k}\), where \(p_{n,k}\) represents the number of permutations of \([n]\) with exactly \(k\) fixed points, and is given by \(p_{n,k}={n\choose k}!(n-k)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( n \) that makes the equation \( \binom{n}{n/2} \approx 2^n\sqrt{\frac{2}{0.02 \pi}} \) true.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of integer solutions to the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\) within the interval \([-25, 25]\).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fundamental group of the complex projective space \( \mathbb{P}^n(\mathbb{C}) \) for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider eight light bulbs placed on the eight lattice points in space that are $\sqrt{3}$ units away from the origin. Each light bulb can be either on or off. If two light bulbs that are at most 2 units apart are both on, they explode. Determine the number of configurations where no explosions occur.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. Suppose that the diophantine equation$\\z^n=8x^{2009}+23y^{2009}\\$ uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the characteristic polynomial of a linear operator \( S \) as \( p(z) = (z-1)^2(z-2)^2 \), determine the minimal polynomial of \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct values of \(z^{\frac{m}{n}}\) when \(m\) and \(n\) are integers such that \(n \neq 0\) and \(m\) and \(n\) are relatively prime. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square number in the decimal system has the form: $\overline{a b 1 a b}$. Which is this square number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)\cdots (x-k) \), let \( S \) be the sum of all real roots of \( f \) and \( T \) be the sum of all real roots of \( f' \). Find the value of \( S-T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $W$ be a finite-dimensional vector space over $\mathbb{R}$. Let $S:W\to W$ be a linear transformation with minimal polynomial $x^6(x+2)^2$. Determine the minimal polynomial of $S^3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of unique seating plans possible for 15 people in a row of 15 chairs, where two plans are considered identical if they share the same adjacent quadruples. For instance, if the people are seated left to right as 1, 2, 3, 4, ..., 15, then the quadruples 1234, 2345, ..., 12131415 cannot appear in any other plan. What is this maximum number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the expression: $$\frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all real numbers \( x \) and \( y \), the equation \( f(f(x) + y) = 2x + f(f(y) - x) \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the symmetric group $S_4$ to the cyclic group $\mathbb{Z}_6$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \alpha \) that minimizes the condition number \( k(A) = \Vert A \Vert \Vert A^{-1} \Vert \) for the matrix \( A = \begin{bmatrix} 0.1\alpha & 0.1\alpha \\ 1 & 1.5 \end{bmatrix} \), assuming the \( \infty \)-norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the transition matrix of a Markov chain:
\[
\begin{bmatrix}
1&0&0&0\\
0.1&0.6&0.1&0.2\\
0.2&0.3&0.4&0.1\\
0&0&0&1
\end{bmatrix}
\]
with states \( \{0, 1, 2, 3\} \), calculate the probability that the Markov chain, starting from state 1, will end in state 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume a remailer reorders messages and maintains a pool of \( n-1 \) messages at all times. When the \( n \)th message arrives, one of the \( n \) messages is selected at random and forwarded. An attacker sends enough messages to force the \( n-1 \) messages in the original pool to be sent. Assuming the message to be sent is chosen according to a uniform random distribution, what is the expected number of messages the attacker would have to send to achieve this goal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), let \( A_1 \) be the midpoint of \( BC \), and \( G \) be the centroid of the triangle. Squares \( GBKL \) and \( GCMN \) are constructed on the left with respect to rays \( GB \) and \( GC \), respectively. Let \( A_2 \) be the midpoint of the segment connecting the centers of these squares. The circumcircle of triangle \( \triangle A_1A_2G \) intersects \( BC \) at points \( A_1 \) and \( X \). Find the ratio \( \frac{A_1X}{XH} \), where \( H \) is the foot of the altitude from \( A \) to \( BC \) in triangle \( \triangle ABC \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Find the Fréchet derivative of the function \( F[u] = u^2 \) at a point \( u \) in a commutative Banach algebra.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a function \( f(z) \) with an expansion of the form \( f(z) = \sum_{n\geq 1} a_n {\rm e}^{2\pi i n z} \). If \( f(z) \) is bounded on the upper half-plane, determine the decay rate of the terms \( a_n {\rm e}^{2\pi i n z} \) in the series as the imaginary part of \( z \) tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the best known bound on the index \(m\) of an abelian subgroup \(\Gamma'\) in a discrete subgroup \(\Gamma\) of the group of isometries in \(\mathbb{R}^n\), where \([\Gamma : \Gamma']\) denotes the index of \(\Gamma'\) in \(\Gamma\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the integral \( \int_{-1}^1 x^n f(x) \, dx \) for every \( n \)-th degree polynomial \( f \) with integer coefficients. Let \( \alpha_n \) denote the smallest positive real number that such an integral can give. Determine the limit value \( \lim_{n\to \infty} \frac{\log \alpha_n}{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the cardinality of the set of all non-decreasing functions defined on the interval \([0,1]\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work done by the force field \( \mathbf{F}(x, y, z) = (x^2 + y)\mathbf{i} + (y^2 + z)\mathbf{j} + (z^2 + x)\mathbf{k} \) along the curve parametrized by \( \mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k} \) from \( t = 0 \) to \( t = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the number of distinct Cayley tables that can be constructed for the set \( \{0, 1, \ldots, n-1\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the product \( 24^{a} \cdot 25^{b} \cdot 26^{c} \cdot 27^{d} \cdot 28^{e} \cdot 29^{f} \cdot 30^{g} \), the exponents \(a, b, c, d, e, f, g\) are assigned the numbers \(1, 2, 3, 5, 8, 10, 11\) in some order. Determine the maximum number of zeros that can appear at the end of the decimal representation of this product.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two independent random variables $A$ and $B$ are uniformly distributed over $[0,1]$, representing the moments two bulbs are ignited. Two other independent random variables $S$ and $T$ are uniformly distributed over $[0,2]$, representing the durations for which the bulbs remain lit. Calculate the probability that both bulbs are lit simultaneously.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a $5 \times 7$ matrix $B$, a basis for $\operatorname{Nul} B$ is given by $$\mathcal{B_N} = \left\{\begin{pmatrix} 1 \\ 2 \\ -1 \\ 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} , \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 \\ -1 \\ 1 \\ 0 \end{pmatrix}\right\},$$ and $\vec c\in\operatorname{Col}B$. If a particular solution to $\operatorname{proj}_{\operatorname{Col}B}\vec c = \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}$ is given by $\vec y = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}$, find the sum of the elements in $\vec c$. Express your answer as a single number.
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Context:
In a first-price auction with two bidders, each bidder may have either a low valuation $V_L$ or a high valuation $V_H$. Bidders do not know the valuations of others. Determine the minimum bid (i.e., the lowest bid in the support $[\underline{b_i},\overline{b_i}]$) for a bidder with a low valuation $V_L$. Assume that strategies involve a probability distribution over the domain $[\underline{b_i},\overline{b_i}]$. What is $\underline{b_i}$ for a $V_L$ type bidder?
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Context:
Determine the least real number $L$ such that$$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.
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Context:
Let real numbers \(x\) and \(y\) be such that \(x-y\), \(x^2-y^2\), and \(x^3-y^3\) are all prime numbers. Determine the value of \(x-y\).
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Context:
Find the possible minimal polynomials of a matrix $B$ such that $B^3 = I$ and $B$ has only $-1$ as an eigenvalue.
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Context:
Evaluate the integral \( \int_{1}^{2} \frac{e^x}{x} \, dx \) to five decimal places using the Taylor series.
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Context:
Given 10 subsets of a set X, determine the maximum number of unique subsets that can be generated using the operations of union, intersection, complement, and symmetric difference, even if some subsets might be empty.
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Context:
Let \( X \) be a normally distributed random variable with mean \( \mu \) and standard deviation \( \sigma = \mu \). Given that \( P(-2 < X < 17) = 0.4375 \), find the value of \( \mu \).
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Context:
In a $2023 \times 2023$ square grid, each cell is colored with one of $n$ colors. It is known that for any six cells of the same color in a single row, there are no cells of the same color above the leftmost or below the rightmost of these six cells. Determine the smallest possible value of $n$ for which this configuration is achievable.
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Context:
Let $V$ be the vector space of all $2\times 2$ matrices over the field of real numbers and let \(B=\begin{pmatrix}2&-2\\-1&1\end{pmatrix}.\) Let $W$ be the subspace of $V$ consisting of all $A$ such that $AB=0.$ Let $f$ be a linear functional on $V$ which is in the annihilator of $W.$ Suppose that $f(I)=0$ and $f(C)=3,$ where $I$ is the $2\times 2$ identity matrix and \(C=\begin{pmatrix}0&0\\0&1\end{pmatrix}.\) Find $f(B).$
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Context:
Find the KKT point for the following optimization problem:

Minimize \( f(x_1, x_2) = (x_1 - 3)^2 + (x_2 - 2)^2 \)

Subject to:

\( x_1^2 + x_2^2 \le 5 \)

\( x_1 + 2x_2 = 4 \)

\( x_1 \ge 0, x_2 \ge 0 \)
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Context:
Determine the value of $k$ for which 100 arcs can be placed on a circle such that each arc intersects exactly $k$ other arcs.
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Context:
Find the maximum value of the function \( A(x,y) = \frac{1}{2}(x(12-x) + y(13-y)) \) subject to the constraint \( x^2 + (12-x)^2 - y^2 - (13-y)^2 = 0 \) using Lagrange multipliers.
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Context:
Determine the possible values of the integral \( \frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz \), where \( \gamma \) is a closed continuously differentiable path in the upper half-plane that does not pass through \( i \).
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Context:
Determine the number of points \( c \) such that for the function \( f(x) = x^3 + 3x + 2 \), the condition \((b-a)f'(c) \neq f(b) - f(a)\) holds for any two real numbers \( a \) and \( b \).
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Context:
Calculate the volume of the tetrahedron with vertices at $A_{1}(-1, 2, -3)$, $A_{2}(4, -1, 0)$, $A_{3}(2, 1, -2)$, and $A_{4}(3, 4, 5)$. The height is dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$. Provide your answer as a single numerical value.
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Context:
The sequence \( x_1, x_2, x_3, \ldots \) is defined by \( x_1 = 2022 \) and \( x_{n+1} = 7x_n + 5 \) for all positive integers \( n \). Determine the maximum positive integer \( m \) such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of 7 for any positive integer \( n \).
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Context:
Let $\Omega = D(0,2)/\{\frac{1}{2}\}$, where $D(0,2)$ is a disc, and let $f$ be holomorphic in $\Omega$. Given that $\frac{1}{2}$ is a simple pole for $f$ with residue 1, calculate the limit:
\[ \lim_{n \to \infty} \frac{f^{(n)}(0)}{2^{n}n!} \]
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Context:
Evaluate the expression \((\sin^{-1}a)^2-(\cos^{-1}b)^2+(\sec^{-1}c)^2-(\csc^{-1}d)^2\) given that:
\[(\sin^{-1}a)^2+(\cos^{-1}b)^2+(\sec^{-1}c)^2+(\csc^{-1}d)^2=\frac{5\pi^2}{2}\]
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Context:
Identify a Banach space that is isometric to the quotient space $C[0,1]/M$, where $M=\{f\in C[0,1]:f(0)=0\}$ and $C[0,1]$ is equipped with the supremum norm.
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Context:
Find the number of solutions to the equation \( \{f(f(x))\} = 0.5 \), where the function \( f(x) \) is defined as follows: \( f(x) = 4x - 2 \) for \( 0 \leq x \leq 1 \) and \( f(x) = -x^2 - 2x \) for \( -2 \leq x < 0 \). Here, \( \{x\} \) denotes the fractional part of \( x \).
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Context:
In a finite discrete probability space $(\Omega, P)$ with $|\Omega| = n$, determine the maximum number of mutually independent random variables that can exist on this space.
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Context:
Let $C = H \cap S$, where $S$ is a surface of general type canonically embedded in $\mathbb{P}^N$ for some $N > 0$, and $H$ is the divisor of a hyperplane in $\mathbb{P}^N$. Determine the degree of the divisor $H_{|C}$ on the curve $C$.
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Context:
In a particular water sample, ten bacteria are found, of which three are of type A. What is the probability of obtaining exactly six type A bacteria in a second independent water sample containing 12 bacteria in total?
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Context:
Given a Hamiltonian action of a torus on a symplectic 4-manifold \( M \) with a moment map \( \mu: M \to \mathbb{R}^2 \), where \( \mu = (\mu_1, \mu_2) \), determine the expression for the moment map \( \phi \) for the circle action on \( M \) induced by the homomorphism \( S^1 \to T^2 \) given by \( \theta \mapsto (2\theta, 3\theta) \). Express \( \phi \) in terms of \( \mu_1 \) and \( \mu_2 \).
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Context:
Determine the number of sets \( A \) containing 9 positive integers such that for any positive integer \( n \leq 500 \), there exists a subset \( B \subset A \) with \( \sum_{b \in B} b = n \).
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Context:
A right circular cone has base of radius 1 and height 3.  A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone.  What is the side-length of the cube?
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Context:
In a triangle divided into four smaller triangles and three quadrilaterals, each smaller triangle has an area of 1. Calculate the area of the quadrilateral \( C A_{0} C_{0} A_{1} \).
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Context:
Let $f$ be a continuous function defined on $[0,1]$ such that \( \int_{0}^1f^2(x)dx=\left(\int_{0}^1f(x)dx\right)^2 \). Determine the range of $f$.
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Context:
Evaluate the integral \( \int_0^1 \{2x\}\{3x\} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \).
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Context:
Determine the radius of convergence of the power series \(\sum_{n \geq 0}a_n z^{n!}\) given that the radius of convergence of \(\sum_{n \geq 0}a_n z^{n}\) is \(L\).
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Context:
Determine the number of equivalence classes under logical equivalence for the set of all formulas that can be constructed using the atoms \( \{A_1, A_2, \ldots, A_n\} \).
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Context:
Determine the cardinality of the set of all obtuse triangles on the Cartesian plane.
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Context:
Let $X_1, X_2, \ldots$ be a sequence of infinite independent and identically distributed random variables, where $X_i \sim U[0,1]$. Define $D_n = \sqrt{X_1^2 + X_2^2 + \ldots + X_n^2}$. Determine the value of the following cumulative distribution function at the point $\frac{1}{\sqrt{3}}$: $$\lim_{n\to \infty} F_{\frac{D_n}{\sqrt{n}}} \left(\frac{1}{\sqrt{3}}\right).$$
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Context:
Find the value of \( k \) such that \( \int^{\infty}_{0}\frac{\ln^2(x)}{(1-x)^2}dx + k\int^{1}_{0}\frac{\ln(1-x)}{x}dx = 0. \)
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Context:
Dave is shaped like two hemispeheres placed on either side of a vertical cylinder. If Dave is $4$ feet across in width (excluding arms) , and has total volume $20\pi$ feet$^3$ , then how tall is he? (excluding legs and hair)
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Context:
Calculate the determinant of the linear transformation \( T(X) = \frac{1}{2}(AX + XA) \), where \( V \) is the vector space of all \( 3 \times 3 \) matrices and \( A \) is the diagonal matrix given by:
\[ A = \begin{pmatrix} 1 & 0 & 0\\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
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Context:
Find the points of continuity for the function \( f(x,y) = \frac{\sin(xy)}{\sqrt{4-x^2-y^2}} \).
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Context:
Determine the values of \(a > 0\) for which the expected value of the random variable \(X_1\), with distribution \(\mathbb{P}(X_k \geq x) = \frac{1}{x^a}\) for \(x \in \mathbb{N}\), is finite.
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Context:
Find the function \( f(x) \) that satisfies the following conditions: \( f(ax) = \frac{1}{a}f(x) \) and \( \int_{a}^{ab} f(x) \, dx = \int_{1}^{b} f(x) \, dx \).
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Context:
In triangle $\Delta ABC$, let $O$ be a point inside the triangle. Extend $AO, BO, CO$ to meet $BC, CA, AB$ at $D, E, F$ respectively. Given that $AO = 30$, $FO = 20$, $BO = 60$, $DO = 10$, and $CO = 20$, find the length of $EO$. \(\boxed{\text{EO}}\)
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Context:
Find a function $g(n)$ such that $f(n) \in \Theta(g(n))$ for the given recurrence relation: $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\; \forall n \geq 0 $$.
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Context:
Find the value of the series \( \sum_{p \in \mathbb{P}} \frac{1}{p^2} \), where \( \mathbb{P} \) is the set of all prime numbers.
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Context:
What is the maximum number of teams that could have participated in a quiz with 50 questions, if the score for each correct answer can be adjusted such that the teams can be ranked in any order preferred by the experts?
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Context:
Evaluate the definite integral \( I = \int_0^\pi \frac{\sin^2(\theta) \, d\theta}{10-6\cos(\theta)}. \)
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Context:
Determine the minimum value of the parameter \( c \) such that the following system of equations has a unique solution:

\[
\begin{cases}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{cases}
\]
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Context:
Given that the numbers \( x, y, z \) form an arithmetic progression with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression, find the value of \( \cos^{2} y \).
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Context:
Draw cards one at a time, with replacement, from a standard deck of playing cards. What is the expected number of draws until four successive cards of the same suit appear?
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Context:
In a simple, connected planar graph where all regions have 8 edges, determine the minimum possible degree of a vertex.
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Context:
Inside the tetrahedron \(ABCD\), there is a point \(O\) such that line segments \(AO, BO, CO, DO\) intersect the faces \(BCD, ACD, ABD, ABC\) at points \(A_1, B_1, C_1, D_1\) respectively, with \(\frac{AO}{A_1O} = \frac{BO}{B_1O} = \frac{CO}{C_1O} = \frac{DO}{D_1O} = k\). Determine all possible values of \(k\).
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Context:
The Shannon formula C=Wlog2(1+S/N) indicates that the maximum information transmission rate C depends on the bandwidth W, the average power of the signal S in the channel, and the power of Gaussian noise N in the channel. Due to technological advancements, the bandwidth W increases by 20% from its original value, and the signal-to-noise ratio S/N increases from 1000 to 4000. Calculate the approximate percentage increase in C. Express your answer as a percentage, rounded to the nearest whole number.
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Context:
From the ten digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, select nine digits to form a two-digit number, a three-digit number, and a four-digit number such that the sum of these three numbers equals 2010. What is the digit that is not selected?
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Context:
If the lateral surface of a cone is unfolded into a semicircle, then the apex angle of the cone (the angle between two generatrices in the axial section of the cone) is __________.
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Context:
In a sports complex, stations $A, B, C, D, E$ formed the vertices of a regular pentagon. These stations were connected by straight paths. Additionally, on the path from $A$ to $B$ there was a fountain $F$, which was connected to station $C$ by a path perpendicular to the path from $B$ to $E$. Pat and Mat met at station $E$ and decided to sweep some of the paths. Pat swept the path from $E$ to $B$. Mat swept the path from $E$ to $A$ and also from $A$ to $F$.

Determine the difference in the lengths of the paths swept by Pat and Mat.
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Context:
已知 $a$ 为实数, 且对任意 $k \in[-1,1]$, 当 $x \in(0,6]$ 时, $6 \ln x+x^{2}-8 x+a \leqslant k x$ 恒成立, 则 $a$的最大值是$m-m \ln m$，求$m$的值。
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Context:
99 identical balls lie on a table. 50 balls are made of copper, and 49 balls are made of zinc. The assistant numbered the balls. Once a spectrometer test is applied to 2 balls, it allows determining whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What is the minimum number of tests required to determine the material of each ball if all the tests should be performed today?
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Context:
An isosceles triangle \(ABC\) has an inscribed rhombus \(DECF\) such that vertex \(E\) lies on side \(BC\), vertex \(F\) on side \(AC\), and vertex \(D\) on side \(AB\). Find the side length of the rhombus if \(AB = BC = 12\) and \(AC = 6\).
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Context:
Consider a Markov chain $Z_k$ with 6 states, depicted as a pyramid where each node has a probability of $1/2$ to move to a neighboring node. Starting from state $1$ with $P(Z_0=1)=1$, determine the expected time $E[T_B]$ to first reach the bottom states, which are $3$, $4$, or $5$.
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Context:
In triangle \( ABC \), it is known that \(\angle BAC = 75^\circ\), \( AB = 1 \), and \( AC = \sqrt{6} \). On the side \( BC \), a point \( M \) is chosen such that \(\angle BAM = 30^\circ\). The line \( AM \) intersects the circumcircle of triangle \( ABC \) at a point \( N \) different from \( A \). Find \( AN \).
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Context:
Find the least positive integer $n$ such that the polynomial $P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1$ has at least one root of modulus $1$.
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Context:
[ Motion problems ]

A cyclist rode from point A to point B, where he stayed for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and 2 hours later met him on the return trip. The pedestrian arrived in B at the same time the cyclist returned to A. How much time did it take the pedestrian to travel from A to B, if his speed is four times less than the cyclist's speed?
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Context:
Given the function $$f(x) = \int_{B(0,R)} \log\left(\frac{\left\vert\,x - y\,\right\vert} {\left\vert\,x - \frac{R^2}{\left\vert\left\vert\,y\,\right\vert\right\vert^2}y\,\right\vert}\right)\,dy$$ for $x \in B(0,R)$, find the minimum value of $f(x)$ within the ball $B(0,R)$, which depends only on the radius $R$.
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Context:
Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p) = h(x)(x-p)^2$ for all x \in R. Now, consider the polynomial $$f(x) = 12x^5 - 15x^4 - 40x^3 + 540x^2 - 2160x + 1,$$ and suppose that it’s double attractors are $a_1, a_2, ... , a_n$. If the sum $\sum^{n}_{i=1}|a_i|$ can be written as $\sqrt{a} +\sqrt{b}$, where $a, b$ are positive integers, find $a + b$.
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Context:
$f(u)=u^{2}+a u+b-2, u=x+\frac{1}{x}, f(u)$ 有零点, 则 $a^{2}+b^{2}$ 的最小值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
9 |  |

In a square table of $9 \times 9$ cells, 9 cells are marked at the intersections of the second, fifth, and eighth rows with the second, fifth, and eighth columns. How many ways are there to get from the bottom-left cell to the top-right cell, moving only through unmarked cells upwards or to the right?
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Context:
Let $Z_1,Z_2,Z_3,Z_4 \in\mathbb{C}$ such that $Z_1+Z_2+Z_3+Z_4=0$ and $|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2=1$. What is the minimum value of $|Z_1-Z_2|^2+|Z_2-Z_3|^2+|Z_3-Z_4|^2+|Z_4-Z_1|^2$?
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Context:
14. Teacher $D$ placed cards with the numbers $1, 2, \cdots, 13$ in a clockwise order on a circle. Initially, all cards are face up. Each time, one card can be flipped to face down (once a card is face down, it cannot be flipped again). Teacher $D$'s rule for flipping cards is: if a face-up card has the number $A$, and the card that is 2 positions away is also face up, then Teacher $D$ can flip the card with the number $A$. For example, as long as the cards with the numbers 9 or 2 are face up, Teacher $D$ can flip the card with the number 12 (of course, provided that the card with the number 12 is still face up). In the end, if Teacher $D$ can flip 12 cards to face down, then Teacher $D$ is successful. To achieve success, how many different sequences of flips can Teacher $D$ use?
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Context:
The lateral face of a regular quadrilateral pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the adjacent lateral faces.
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Context:
Calculate the areas of the figures bounded by the lines given in polar coordinates.

$$
r = 4 \cos 4 \phi
$$
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Context:
In a neighborhood having 90 households, 11 did not have either a car or a bike. If some households had both a car and a bike and 44 had a car, there were 35 households with a bike only. How many households had both a car and a bike?
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Context:
Evaluate the integral:

$$\int_{\tan x}^{\cot x} \cfrac{tdt}{1+t^2} + \int_{\cot x}^{\tan x} \cfrac{dt}{1+t^2}$$
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Context:
3. (17 points) In a $5 \times 5$ grid, 6 crosses need to be placed such that each row and each column contains at least one cross. How many ways can this be done?
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Context:
What is the depth of the quotient ring $R/(p_1\cap p_2)$, where $R=K[x_1,\dots,x_n]$ is a polynomial ring over a field $K$, $I$ is a squarefree monomial ideal in $R$, and $p_1, p_2$ are distinct minimal prime ideals of $I$, each generated by subsets of the variables $\{x_1,\dots,x_n\}$ such that $I = p_1 \cap p_2$?
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Context:
Given positive numbers $a$ and $b$ satisfying $\ln \frac{{2-b}}{a}=2a+2b-4$, find the minimum value of $\frac{1}{a}+\frac{2}{b}+\frac{2}{{ab}}$.
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Context:
Let  $x$ - minimal root of equation  $x^2-4x+2=0$ .
Find two first digits of number  $ \{x+x^2+....+x^{20} \}$  after  $0$ , where  $\{a\}$ - fractional part of  $a$ .
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Context:
The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land he had equally among his sons. If he had no sons, the land was transferred to the state. No other family members acquired or lost land by any other means. In total, there were 150 people in the family. What is the smallest possible fraction of the original plot that could be received by a family member?
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Context:
What is the tenth number in the row of Pascal's triangle that has 100 numbers?
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Context:
Given the complex polynomial \( p(z) = z^4 + z^3 + 4z^2 + 3z + 7 \), determine the change in the argument along the curve \( C_3 \) of a quarter circle of sufficiently large radius centered at the origin. Assume there are no real or imaginary zeros. Express the argument of \( p(iy) \) as \( \text{arg}(p(iy)) = \tan^{-1} \left\{ \frac{y(\sqrt{3} - y)(\sqrt{3} + y)}{(y^2 - 2 - \sqrt{3}i)(y^2 - 2 + \sqrt{3}i)} \right\} \). Calculate the change in this argument along \( C_3 \).
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Context:
Find the number of solutions in positive integers to the equation \(x_1 + x_2 + x_3 + x_4 = 19\) subject to the conditions \(x_2 \neq 2x_3\) and \(x_1 \neq x_2\).
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Context:
Express the conjugate of the function \( g(x) = \inf_z f(x,z) \) in terms of the conjugate \( f^*(y,s) \) of \( f(x,z) \), where \( f(x,z) \) is convex in \((x,z)\).
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Context:
The author of this question was born on April 24, 1977. What day of the week was that?
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Context:
Find the bounded solutions to the differential equation: $$\alpha'''(x) =\alpha'(x) x^2 + \alpha(x) x.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the coefficients in the expansion of \((x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7)^{11}\) where each variable has a degree of at least one.
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Context:
Determine the number of bases required to test in order to conclusively determine if any integer less than \(2^{63} - 1\) is prime using the Miller-Rabin test.
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Context:
Let $U_{\theta} \sim U[0,\theta]$ be independent for each $\theta \in [0,1]$. Define $X = \sup_{\theta \in [0,1]} U_{\theta}$. Find $\mathbb{P}(X \leq x)$.
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Context:
Find the equation of the curve that passes through the point (0,0) for the differential equation: \[ y'=\frac{\sin x + e^x (\sin x + \cos x) + x\cos x - y(1 + \cos x + e^x)}{e^x + \sin x + x - 2y} \]
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Context:
Given integers \(a, b > 5\) such that \(\operatorname{lcm}(a,b)=8160\) and \(\gcd(a,b)=5\), find the pair \((a',b')\) where \(a=da', b=db'\) and \((a',b')=1\).
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Context:
Determine the minimum number of weighings required to order 5 coins of different weights using a balance scale.
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Context:
A miner is trapped in a mine with three identical doors. The first door leads to safety in 2 hours. The second door leads back to the starting point in 3 hours. The third door leads to a maze, returning to the starting point in 5 hours on the first entry, and 1 hour on subsequent entries. The miner chooses each door with equal probability, independent of previous choices. What is the expected time until the miner reaches safety?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2 \pi} \int_{0}^{2 \pi} \ln (|r e^{it} - z|) \, dt \) for \( r > |z| \).
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Context:
Consider a linear transformation $x' = U x$ that preserves the inner product of vectors. If the transformation matrix $U$ has a determinant of $-1$, how many free parameters does $U$ have?
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Context:
Determine the largest value of $n$ as a function of $k$ such that a good order always exists, where a good order is defined as one in which each person either sees or hears both of their friends.
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Context:
Determine the omega limit set of a trajectory in the following three-dimensional dynamical system:

$$\dot x = -y$$

$$\dot y = x$$

$$\dot z = -x^2-y^2$$
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Context:
A cylinder has a fixed diameter of 50 mm and a length of 500 mm. Spherical balls have a diameter of 49.5 mm with a standard deviation of 0.1 mm. Estimate the probability that 12 of these balls will fit into the cylinder, considering both the width and length dimensions.
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Context:
Given two independent random variables $X$ and $Y$ with the following distributions: $F_X(x) = \frac{1}{2}x$ for $0 < x < 2$ and $f_Y(y) = \frac{2}{9}y$ for $0 < y < 3$, find the probability that the product $XY \ge 1$. Provide your answer as a probability value.
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Context:
In the quadrilateral $E-ABCD$, the base $ABCD$ is a square, and $AE$ is perpendicular to the plane $CDE$. Given that $AE = DE = 3$, and $F$ is a point on the line segment $DE$, the dihedral angle $E-BC-F$ is equal to the dihedral angle $F-BC-D$. Find the length of $DF$.
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Context:
Calculate the probability that exactly 2 out of 4 couples sit next to each other when seated randomly in a row of 8 seats. Assume each couple consists of two people, and the seating arrangement is completely random.
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Context:
Determine the minimum number of points in a set $S$ in the plane such that for any point $X$ in the plane, there exists a point $M$ in $S$ where the distance $|XM|$ is rational.
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Context:
Find the derivative of the expression $(Wx)^T A (Wx)$ with respect to the matrix $W$, where $W$ is an $m \times n$ matrix, $x$ is an $n \times 1$ vector, and $A$ is an $m \times m$ symmetric positive semi-definite matrix.
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Context:
Given the initial and final positions of two stars orbiting a black hole, determine the coordinates of the black hole. Star 1 moves from (12, 2) to (10, 8), and Star 2 moves from (5, 6) to (2, 7). Find the coordinates of the black hole.
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Context:
Calculate the sum of the Fourier series for the 2π-periodic function \(f(x)=e^{x}\) at \(x=4\), given that the function is defined for \(|x|\le \pi\) and its Fourier series is \[f(x)=\frac{e^{\pi}-e^{-\pi}}{2\pi} + \frac{e^{\pi}}{\pi}\sum_{n=-\infty}^\infty \frac{(-1)^{n}}{1+n^{2}}e^{inx} \]
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Context:
Find the smallest positive integer \( N \) such that the function \( f(x) = \sin \left(\frac{1}{x}\right) \) has exactly 2020 local extrema in the interval \( \left(\frac{1}{N}, 1\right) \).
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Context:
A trough is 8 feet long and 1 foot high, with a vertical cross-section parallel to an end shaped like the graph of \( y = x^2 \) from \( x = -1 \) to \( x = 1 \). If the trough is full of water, calculate the amount of work required to pump all the water over the top of the trough.
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Context:
Evaluate the determinant of the adjugate of an $n \times n$ matrix $A$, denoted as $\det(\text{adj}(A))$. Provide the correct expression for this determinant.
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Context:
Let $g:\mathbb{R} \to \mathbb{R}$ be a differentiable function such that $g'(x) < x^3$ for every $x > 0$. Determine the value of $$\lim_{x \to \infty} g(x).$$
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Context:
Calculate the sum of the series \( \sum_{n=1}^\infty \frac{f(n)}{n(n+1)} \), where \( f(n) \) represents the number of 1's in the binary representation of \( n \).
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Context:
Find the minimum value of the integral

$$ \mathcal {I} [y] = \int_{- \infty}^{\infty} \dfrac{1}{2} y'^2 + 1 - \cos y \; \mathrm d x $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2 \pi$. \( \boxed{\text{Minimum value}} \)
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Context:
Find integers \((a, b)\) such that \(\gcd(a, b) = 13\) and they satisfy the equation \(49a - 125b = 169\).
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Context:
Determine the shortest sequence of letters using the alphabet A-E that guarantees the inclusion of any possible code of length up to 4. The code panel will only remember the last number of buttons pressed equal to the length of the code. For example, if the code is 2 letters long and you entered ABECD, the panel would only remember CD.
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Context:
Find the limit distribution of \( Z_n = \sqrt{n} \cdot \min\{Y_1, Y_2, \ldots, Y_n \} \) as \( n \to \infty \), where \( Y_1, Y_2, \ldots \) are independent random variables uniformly distributed on \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-4x+1=0\), find the value of
\[f(\alpha, \beta) =\frac {\beta^3}{2} \csc^2\left (\frac {1}{2}\arctan \frac {\beta}{\alpha}\right) +\frac {\alpha^3}{2} \sec^2\left (\frac {1}{2}\arctan \frac {\alpha}{\beta}\right)\]
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Context:
In a vector space $R$ of dimension 6, let $A$, $B$, and $C$ be subspaces each of dimension 4. The dimension of the intersection of any two subspaces (i.e., $A \cap B$, $A \cap C$, and $B \cap C$) is 3. Determine the possible dimensions for the intersection of all three subspaces, $A \cap B \cap C$. Provide your answer as a set of possible dimensions.
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Context:
Given a square matrix $B$ with spectral radius $\rho(B) > 1$, determine a matrix norm $||| \cdot |||$ such that $|||B||| > 1$ using the abstract method of defining an adapted norm.
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Context:
Let \( f(x) \) be a continuous function satisfying \( f(x^2+1) = \frac{x^2}{f(2^x)-1} \) and \( f(x) > 0 \) for all \( x \in \mathbb{R} \). Find \( \lim_{x\rightarrow 1}f(x) \).
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Context:
In how many ways can 9 distinct items be distributed into three boxes so that one box contains 3 items, another contains 2 items, and the third contains 4 items?
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Context:
Determine the convergence rate of the iterative method defined by:
\[ x_{k+1} = x_k - \frac{f(x_k)}{g(x_k)} , \quad g(x_k) = \frac{f(x_k + f(x_k)) - f(x_k)}{f(x_k)} \]
where \( f(x) \in C^2[a,b] \) and there exists a point \( x_0 \) such that \( f(x_0) = 0 \) and \( f'(x_0) \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $k$ such that there exists a constant $c < 1$ for which the following inequality holds for all continuous functions $F$ defined on the interval $[0, r]$, where $r \in \mathbb{R}^+$:
\[ \sup_{x \in [0, r]} \left\{ e^{kx} \int_0^x |F(s)| \, ds \right\} \leq \frac{c}{M} \sup_{s \in [0, r]} \left\{ e^{ks} |F(s)| \right\} \]
where $M > 0$ is fixed.
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Context:
Determine the smallest natural number \( n \) such that in any two-coloring of the edges of the complete graph \( K_n \), there are always two monochromatic triangles that share exactly one common vertex.
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Context:
Determine the holomorphic function \( f: \mathbb{C} \rightarrow \mathbb{C} \) such that for all \( x, y \in \mathbb{R} \), the imaginary part of \( f(x + iy) \) is equal to \( x^2 - y^2 \).
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Context:
Determine the minimum number of repetitions possible when superimposing two different Latin squares of the same order.
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Context:
Determine the limit distribution of the sequence of random variables $X_1, X_2, \ldots$, where $P(X_n = \frac{k}{n}) = \frac{1}{n}$ for $k = 1, 2, \ldots, n$, as $n$ approaches infinity.
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Context:
Evaluate the integral \( \int\limits_0^{\pi}\log (1-\cos x)\, dx \).
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Context:
Determine whether the sequence \( a_n = \sqrt[n]{|\cos n|} \) converges or diverges.
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Context:
If $f(x)$ is a function such that $|f(x)|\le 1, x \in R$, and $$g(x)=\frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}},$$ then what is the maximum value of $g(x)$ if it is guaranteed that $f(x)$ covers the entire range $[-1,1]$?
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Context:
If 5% more is gained by selling an article for a higher price than by selling it for Rs. 340, the cost of the article is Rs. 400. What is the higher selling price?
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Context:
a is twice as good a workman as b, and c is thrice as good as a. If they took 9 days together to do the work, and b worked 2 days less than a and c, how many days will b alone take to complete the work?
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Context:
A $5 \times 5$ grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
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Context:
Find all positive integers  $k$  such that there exists a positive integer  $n$ , for which  $2^n + 11$  is divisible by  $2^k - 1$ .
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Context:
The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to the nearest ten-thousandth of a square inch is:
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Context:
Determine the coefficient of $x^4$ in the Taylor series expansion of $\frac{\tan x}{x}$.
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Context:
Determine all real numbers \(x\) for which the series \(\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n}\) converges.
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Context:
Let $S(t)$, $t \geq 0$ be a geometric Brownian motion with drift parameter $\mu = 0.2$ and volatility parameter $\sigma = 0.2$. Calculate the probability $P(S(3) > S(1))$. Provide your answer as a single probability value.
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Context:
Find the volume of the union $A \cup B \cup C$ of the three subsets $A$, $B$, $C$ in $xyz$ space such that:

$$
 A = \{(x, y, z) \mid |x| \leq 1, \, y^2 + z^2 \leq 1\}
$$

$$
 B = \{(x, y, z) \mid |y| \leq 1, \, z^2 + x^2 \leq 1\}
$$

$$
 C = \{(x, y, z) \mid |z| \leq 1, \, x^2 + y^2 \leq 1\}
$$
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Context:
Let $X_1, X_2, X_3$ be independent random variables, each following a normal distribution with mean $0$ and variance $\sigma^2$. Determine the distribution of the random variable:

$$Y=\frac{1}{3}\left(\left(\frac{X_1}{\sigma}-\frac{X_2}{\sigma}\right)^2+\left(\frac{X_2}{\sigma}-\frac{X_3}{\sigma}\right)^2+\left(\frac{X_3}{\sigma}-\frac{X_1}{\sigma}\right)^2\right)$$
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Context:
Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \( \mathfrak{h}_n \) of dimension \( 2n+1 \).
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Context:
Evaluate the expression \( \left\lfloor \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}} \right\rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.
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Context:
Call a fraction \( \frac{a}{b} \) ,not necessarily in the simplest form special if \( a \) and \( b \) are positive integers whose sum is 15 . How many
 distinct integers can be written as the sum of two, not necessarily different, special fractions?
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Context:
Determine the smallest number of colors needed to color the edges of any directed simple graph with \( n \) vertices such that there is no monochromatic cycle.
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Context:
Given the polynomial \( P(x) = x^{3} - 2x^{2} - x + 1 \) with real zeros \( a > b > c \), find the value of the expression:
\[ a^{2} b + b^{2} c + c^{2} a. \]
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Context:
The 62 trainees at the Animath camp each choose an ice cream from 5 proposed flavors: bubblegum, tabasco, cactus, lemon basil, and foie gras. The following conditions are known:

- The bubblegum flavored ice cream is at least as popular as the tabasco flavored ice cream.
- The number of students who chose cactus flavored ice cream is a multiple of 6.
- At most 5 students chose lemon basil flavored ice cream.
- At most one student chose foie gras flavored ice cream.

Determine the number of different lists of numbers Igor could announce to the ice cream vendor, in alphabetical order of the flavors, representing the number of students who chose each flavor.
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Context:
Consider a sequence of independent and identically distributed (i.i.d.) random variables, $X_1, X_2, \ldots, X_n, \ldots$, where each $X_n$ has the following distribution: $P(X_n = 0) = P(X_n = 1) = P(X_n = 2) = P(X_n = 3) = \frac{1}{4}$. Define another sequence $\{Y_n\}$ recursively as: $Y_0 = 0$ and for all $n \in \mathbb{N}$, \[Y_n = \begin{cases} 3 & \text{if } X_n = 3, \\ \min\{Y_{n-1}, X_n\} & \text{if } X_n < 3. \end{cases}\] Determine the limit: \[\lim_{n \to +\infty} E[Y_nY_{n-1}].\]
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Context:
Evaluate the infinite series: $$\sum_{n=1}^{\infty}{\log{\frac{1}{\sqrt[6]{n}}}-\log{(\sin{\frac{1}{\sqrt[6]{n}}})}}$$
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Context:
Find the closure of the set \( K = \left\{ \frac{1}{n} \mid n \in \mathbb{N} \right\} \) under the finite complement upper limit topology generated by \( \beta = \{(-\infty, a)\} \).
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Context:
Find the orthogonal matrix that aligns the axes of a given rotation matrix with the standard coordinate axes.
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Context:
Evaluate the limit: \[ \lim _{n\to \infty }\left(n-n^2\int _0^{\frac{\pi }{4}}(\cos x-\sin x)^n dx\right) \]
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Context:
Determine the largest integer $n$ such that a $C^n$ approximate embedding of the hyperbolic plane $\Bbb H^2$ into $\Bbb R^3$ exists for all $\epsilon > 0$, where the geodesic distance on the embedded surface is within an arbitrarily small $\epsilon$ of the true hyperbolic distance.
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Context:
Given the permutation \( \alpha \in S_{k+l} \) defined by \( \alpha = \left(\begin{array}{cccccc}1&\cdots&k&k+1&\cdots&k+l\\l+1&\cdots&l+k&1&\cdots&l\end{array}\right) \), where \( k, l \in \mathbb{Z}^+ \), determine the number of adjacent transpositions required to express \( \alpha \) as a composition of adjacent transpositions.
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Context:
Determine the number of Sylow $3$-subgroups in a simple group of order $168$.
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Context:
A simple bipartite graph \( G \) has 20 vertices, where 18 vertices have degree 7 and the remaining 2 vertices have degree 6. Determine the chromatic number of the complement of \( G \).
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Context:
Evaluate the contour integral \( \int_C f(z) \, dz \), where \( f(z) = 2xy + i(-x^2 + y^2) \) and \( C \) consists of the arc \( y = x^2 \) from \((0,0)\) to \((1,1)\) and the line segment from \((1,1)\) to \((0,1)\).
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Context:
Evaluate the integral \( \int_{0}^{i\infty} e^{-x} x^{t-1} \, dx \).
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Context:
Let \( A = \begin{bmatrix} 1 & 3 & 4\\ 3 & 6 & 9\\ 1 & 6 & 4 \end{bmatrix} \) and \( B \) be a \( 3\times 3 \) matrix such that \( A \cdot A^{T} \cdot A +3B^{-1} =0 \). Find the value of \( \det( \operatorname{adj} (A^{-1}(B^{-1}){2B^{T}})) \).
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Context:
Find the inverse Z-transform of the function \( \frac{z}{(z-2)^2} \).
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Context:
Determine the number of Cayley tables that can be constructed for the set \( \{0, 1, \ldots, n-1\} \).
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Context:
Given a $3\times 3$ matrix $A$ with $\det A = 9$ and the equation $2A - A^2 = 3I_3$, where $I_3$ is the identity matrix, find the determinant of $4I_3 - 2A$. Use the observation that $4I_3 - 2A = (I_3 - A)(I_3 + A)$ to help calculate $\det(4I_3 - 2A)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.