Context:
Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{\text{max}}$ and $P_{\text{min}}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{\text{max}}$ and $P_{\text{min}}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$.
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Context:
Determine the number of functions \( \phi: G \times G \rightarrow G \) that define a group structure on the set \( G = \{a_1, a_2, a_3, a_4, a_5, a_6\} \) with \( a_1 \) as the neutral element. Consider the groups \( \mathbb{Z}_6 \) and \( S_3 \) up to isomorphism.
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Context:
In a dark room drawer, there are 100 red socks, 80 green socks, 60 blue socks, and 40 black socks. A young person picks out one sock at a time without seeing its color. To ensure that at least 10 pairs of socks are obtained, what is the minimum number of socks they must pick out?
(Assume that two socks of the same color make a pair, and a single sock cannot be used in more than one pair)

(37th American High School Mathematics Examination, 1986)
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Context:
Determine all positive integers \(k\) that are coprime to every term in the sequence \(a_n = 2^n + 3^n + 6^n - 1\), where \(n\) is a positive integer.
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Context:
In the $3 	imes 3$ grid shown, the central square contains the integer 5. The remaining eight squares contain $a, b, c, d, e, f, g, h$, which are each to be replaced with an integer from 1 to 9, inclusive. Integers can be repeated. There are $N$ ways to complete the grid so that the sums of the integers along each row, along each column, and along the two main diagonals are all divisible by 5. What are the rightmost two digits of $N$?
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Context:
Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$ is a point on the $y$-axis. The minimum area of $\triangle APF$ is $\_\_\_\_\_\_$.
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Context:
Calculate the Riemann-Stieltjes integral \( \int_0^4 x^2 \, d\alpha(x) \), where \( \alpha(t) = n^2 \) for \( t \in [n, n+1) \).
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Context:
Given a semi-simple Lie Algebra $L$ with dimension $d$ over $\mathbb{R}$, determine the dimension of its complexification $L_{\mathbb{C}} = L \otimes_{\mathbb{R}} \mathbb{C}$.
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Context:
Determine the number of zeros of the polynomial \( f(z) = z^{87} + 36z^{57} + 71z^4 + z^3 - z + 1 \) in the annular region \( 1 \leq |z| \leq 2 \).
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Context:
Determine a bound on the difference between $\sin x$ and its degree 5 Taylor polynomial for $x \in [0,1]$. Provide your answer as a numerical value or expression.
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Context:
Find the remainder when \( \displaystyle \sum^{2014}_{r=0}\sum^{r}_{k=0}(-1)^k(k+1)(k+2)\binom{2019}{r-k} \) is divided by 64.
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Context:
The hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) has its right focus at F2. A line passing through F with an inclination angle of 30 degrees intersects the right branch of the curve at point M. If MF is perpendicular to the x-axis, find the eccentricity of the hyperbola.
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Context:
Peter bought a scooter for a certain sum of money. He spent 5% of the cost on the first round of repairs, another 10% on the second round of repairs, and 7% on the third round of repairs. After this, he had to pay a 12% tax on the original cost. Also, he offered a 15% holiday discount on the scooter's selling price. Despite the discount, he still managed to make a profit of $2000. How much did he spend in total, including repairs, tax, and discount if his profit percentage was 30%?
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Context:
Evaluate the limit \( \lim_{n \to \infty} a_n \) where the sequence \( a_n \) is defined as follows for any natural number \( n \):
\[
a_n = \sqrt{4+\sqrt{4+3\sqrt{4+5\sqrt{4+7\sqrt{\cdots\sqrt{4+(2n-1)\sqrt{4+(2n+1)}}}}}}}
\]
Determine the value of this limit.
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Context:
Find the number of integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leading 0 s (i.e. no 0 s at the very left), its rightmost digit is strictly greater than its leftmost digit.
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Context:
Determine the asymptotic behavior of the iterated sum function \( f(x) \) as \( x \to \infty \), where:
\[
f(x)=\sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1
\]
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Context:
The total number of Democrats and Republicans in the House of Representatives was 434. There were some more Republicans than Democrats. There were 202 Democrats in the Senate. How many more Republicans than Democrats were there in the House of Representatives?
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Context:
A die is rolled 100 times, and the sum of the numbers rolled is recorded as X. A coin is tossed 600 times, and the number of heads is recorded as Y. Calculate the probability that X is greater than Y, denoted as P(X > Y).
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Context:
Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, determine the minimal number of connected components of the graph $G/(X \cup Y)$. Provide your answer as a single integer.
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Context:
Determine the values of $p > 0$ for which the sequence of functions $f_n(x) = \frac{nx}{1+n^2x^p}$ on $[0,1]$ converges uniformly to its pointwise limit $f(x) = 0$. Consider the expression \( \sup_{x \in [0,1]} \left| \frac{nx}{1+n^2x^p} \right| \) to analyze the uniform convergence.
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Context:
A man weighs 90 kg. If he takes no calories for 14 days, his mass reduces by 20%. How long will it take him to reduce his mass by the same amount if he consumes 3/4 of the calories required to maintain his mass at 90 kg? Assume the daily calorie intake is a fixed quantity \( C \) and is modeled by the differential equation \( \frac{dt}{dm} = \frac{1}{a(C-bm)} \), where \( a \) and \( b \) are constants. Calculate the time in days.
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Context:
The inscribed circle of triangle $DEF$ is tangent to $\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$.
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Context:
Determine how the prime number 2 splits in the ring of integers of the splitting field of the polynomial $f(X) = X^3 - 3X + 1$ over $\mathbb{Q}$.
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Context:
Two cubes with an edge length of $a$ share a segment that connects the centers of two opposite faces. One cube is rotated by $45^{\circ}$ relative to the other. Find the volume of the intersection of these cubes.
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Context:
Evaluate the integral \( \int\int\int_{\text{entire space}} 3(\vec{r}+2\vec{a})^2\delta^3(\vec{r}-\vec{a})d\tau, \) where \( \vec{a} \) is a constant vector and \( \delta^3(\vec{r}-\vec{a}) \) is the Dirac delta function in three dimensions.
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Context:
In a two-player game, each player picks an integer between $1$ and $100$ inclusive and reveals it simultaneously. The player who picks the higher number has their number halved. The player with the lower number (after halving) pays the other player an amount equal to the difference. What is the optimal number to choose to maximize your earnings, assuming you have no information about the other player's choice?
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Context:
Let $S$ be a set of 100 points in the plane. The distance between every pair of points in $S$ is different, with the largest distance being 30. Let $A$ be one of the points in $S$, let $B$ be the point in $S$ farthest from $A$, and let $C$ be the point in $S$ farthest from $B$. Let $d$ be the distance between $B$ and $C$ rounded to the nearest integer. What is the smallest possible value of $d$?
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Context:
Embed \(\mathbb{R}^2\) in the projective plane \(\mathbb{RP}^2\) by the map \((x,y)\rightarrow [1,x,y]\). Find the point of intersection in \(\mathbb{RP}^2\) of the projective lines corresponding to the parallel lines \(y = mx\) and \(y = mx+c\) in \(\mathbb{R}^2\).
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Context:
Determine the radius of convergence of the Taylor series at zero for the holomorphic function $g(z)$, given that $g^4 + 2g^2 - 5 = z$ and $g(0) = 1$. Provide your answer as a single numerical value.
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Context:
Evaluate the limit as \( x \) approaches infinity:
\[ \lim_{x\to\infty} x^{7/2} \left[\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\left(1-\frac1{8x^2}\right)\right] \]
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Context:
Five girls have a little store that opens from Monday through Friday. Since two people are always enough for taking care of it, they decide to do a work plan for the week, specifying who will work each day, and fulfilling the following conditions:Each girl will work exactly two days a weekThe 5 assigned couples for the week must be different
In how many ways can the girls do the work plan?
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Context:
Calculate the value of \( 718^{143} \mod 1260 \).
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Context:
Determine the number of distinct ways an $n$-cycle can be expressed as the product of a $2$-cycle and an $(n-1)$-cycle in cycle notation.
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Context:
The incircle of triangle \( ABC \) with center \( O \) touches the sides \( AB \), \( BC \), and \( AC \) at points \( M \), \( N \), and \( K \) respectively. It is given that angle \( AOC \) is four times larger than angle \( MKN \). Find angle \( B \).
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Context:
Find a monic polynomial over \(\mathbb{Q}\) whose Galois group is \(V_4 = \langle (12)(34), (13)(24) \rangle \subset S_4\).
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Context:
Determine the genus of the graph $G$ with vertex set $V(G)=\{u_1,\cdots,u_8,v_1,\cdots,v_5\}$ and edge set $E(G)=\{u_1u_3,u_1u_4,u_1u_5,u_1u_6,u_1u_7,u_1u_8,u_1v_2,u_1v_3,u_2u_3,u_2u_4,u_2u_5,u_2u_6,u_2u_7,u_2u_8,u_2v_2,u_2v_3,u_3u_5,u_3u_6,u_3u_7,u_3u_8,u_3v_4,u_3v_5,u_4u_5,u_4u_6,u_4u_7,u_4u_8,u_4v_4,u_4v_5,u_5v_1,u_6v_1,u_7v_1,u_8v_1.\}$.
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Context:
An interior angle of a regular polygon is equal to $900^{\circ}$, then it is a regular ______ polygon.
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Context:
Calculate the number of prime factors, including multiplicities, of the number \(\frac{20! \times 22!}{16! \times 11!}\). Let this number be \(N\). What is the value of \(N(N-2)\)?
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Context:
Find the isomorphism between \( \mathrm{Hom}(\mathbb{Z}_n, \mathbb{Q}/\mathbb{Z}) \) and \( \mathbb{Z}_n \).
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Context:
Let $\Gamma$ and $\Gamma_1$ be two circles internally tangent at $A$, with centers $O$ and $O_1$ and radii $r$ and $r_1$, respectively ($r>r_1$). $B$ is a point diametrically opposed to $A$ in $\Gamma$, and $C$ is a point on $\Gamma$ such that $BC$ is tangent to $\Gamma_1$ at $P$. Let $A'$ the midpoint of $BC$. Given that $O_1A'$ is parallel to $AP$, find the ratio $r/r_1$.
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Context:
For the numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\), the last two digits are discarded. How many of the first terms in the resulting sequence form an arithmetic progression?
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Context:
Calculate the series $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+k)}$, where $k\in \mathbb{Z}_+$.
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Context:
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal?
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Context:
Evaluate the limit $\lim_{t \rightarrow \frac{\pi}{4}^-}\frac{\sin^{\alpha}(t)-\cos^{\alpha}(t)}{(\sin(t)-\cos(t))^{\alpha}}$ for $\alpha \in (0,1]$.
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Context:
In triangle $ABC$ where $AB = 3 \cdot BC$, point $M$ is the midpoint of side $AB$ and $BD$ is the angle bisector. Find the angle $MDB$.
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Context:
Bernardo randomly picks $3$ distinct numbers from the set $\{1,2,3,...,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks $3$ distinct numbers from the set $\{1,2,3,...,6,7,8\}$ and also arranges them in descending order to form a $3$-digit number. Find the probability that Bernardo's number is larger than Silvia's number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?
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Context:
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = 1$ and $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the projection of the vector $2\overrightarrow{b} - \overrightarrow{a}$ in the direction of vector $\overrightarrow{a}$.
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Context:
将一枚均匀的硬币连续抛掷 $n$ 次，以 $p_{n}$ 表示未出现连续 3 次正面的概率. 求数列 $\left\{p_{n}\right\}$ 的其极限.
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Context:
For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?
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Context:
Find the index of the subgroup $\langle ba\rangle$ in the free product $\Bbb{Z_{2}*Z_{2}}$. Express your answer as an integer.
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Context:
若空间向量 $\boldsymbol{a} ,\boldsymbol{b} ,\boldsymbol{c}$ 满足 $|\boldsymbol{a}| \leqslant 1,|\boldsymbol{b}| \leqslant 1, \mid \boldsymbol{a}+2 \boldsymbol{b}+$ $\boldsymbol{c}|=|\boldsymbol{a}-2 \boldsymbol{b}|$ ，求 $| \boldsymbol{c} \mid$ 的最大值。答案的形式为 $k\sqrt{m}$，请给出 $k+m$ 的值。
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Context:
Given positive integers $k, m, n$ such that $1 \leq  k \leq  m \leq  n$. Evaluate

\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]
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Context:
A class has 50 students, and their scores in a math test $\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students scoring above 110.
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Context:
In the right-angled triangle $\mathrm{ABC}$, the angle at vertex $B$ is $30^{\circ}$. The center of the square constructed outward on the hypotenuse $\mathrm{ABC}$ is $D$. What is the measure of the angle $A D B$?
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Context:
Determine the function represented by the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},$$ where $|x|<1$.
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Context:
## Task A-4.3.

On the edge of a square, a total of $4 n$ points are marked: all four vertices of the square and an additional $n-1$ points on each side of the square. Determine the number of all (non-degenerate) triangles for which the marked points are vertices.
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Context:
Five students participate in three competitions: singing, dancing, and playing chess. Each competition is attended by at least one student. Given that student A cannot participate in the dance competition, how many different participation arrangements are there?
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Context:
Let  $k>1$  be a positive integer and  $n>2018$  an odd positive integer. The non-zero rational numbers  $x_1,x_2,\ldots,x_n$  are not all equal and:  $$ x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1} $$  Find the minimum value of  $k$ , such that the above relations hold.
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Context:
A triangle with vertices at \((1003,0), (1004,3),\) and \((1005,1)\) in the \(xy\)-plane is revolved all the way around the \(y\)-axis. Find the volume of the solid thus obtained.
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Context:
10 camels cost as much as 24 horses , 26 horses cost as much as 4 oxen and 6 oxen as much as 4 elephants . if the cost of 10 elephants is rs . 170000 , find the cost of a camel ?
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Context:
A sequence $y_1,y_2,\dots,y_k$ of real numbers is called \emph{zigzag} if $k=1$, or if $y_2-y_1, y_3-y_2, \dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\dots,X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1,i_2,\dots,i_k$ such that $X_{i_1},X_{i_2},\dots,X_{i_k}$ is zigzag. Find the expected value of $a(X_1,X_2,\dots,X_n)$ for $n \geq 2$.
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Context:
Evaluate the sum: $\sum_{n=1}^{\infty}\frac{H_{n+2}}{n(n+1)(n+2)}$, where $H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$.
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Context:
设正三棱雉 $P-A B C$ 的底面边长为 1 、高为 $\sqrt{2}$ ，过底边 $B C$ 作此三棱雉的截面。则截面面积的最小值为多少？原始答案为$\frac{m\sqrt{n}}{k}$的形式，请给出$m + n + k$的值。
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Context:
Suppose \( d \) is the number of positive factors of \( 500 + c \). Find the value of \( d \).
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Context:
Determine in which quadrant the angle $-510°$ lies.
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Context:
Find the first three figures after the decimal point in the decimal expression of the number \(\frac{0.123456789101112\ldots495051}{0.515049\ldots121110987654321}\). Please provide the value of the first three digits after the decimal point as a single integer.
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Context:
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1, 2, \ldots, 2015$.
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Context:
Let $X$ be a set of $n + 1$ elements, where $n \geq 2$. An ordered $n$-tuple $(a_1, \ldots, a_n)$ is formed from distinct elements of $X$. Two $n$-tuples are called *disjoint* if there exist distinct indices $1 \leq i \neq j \leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples that can be formed from $X$. \( \text{(Express your answer as a single number.)} \)
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Context:
You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done.

Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron are all different for the purpose of the coloring . In this way, two colorings are the same only if the painted edges they are the same.
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Context:
Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.
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Context:
Let $F=\mathbb{Z}_3$ and $V=F^4$. Define the subspaces $U=\text{span}\{(1,0,0,0),(1,0,1,0),(0,1,1,1)\}$ and $W=\text{span}\{(0,0,1,0),(-1,1,0,1),(1,1,1,1)\}$. Find the dimension of the intersection $U \cap W$. \[ \text{dim}(U \cap W) = ? \]
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Context:
Let \( p = \operatorname{exp} : \mathbb{C} \to \mathbb{C}^* \) be a covering map and \( (U,z) \) a chart of \( \mathbb{C}^* \) with \( z = x + iy \). Given the one-form \( \omega = \frac{dz}{z} \) on \( U \), find the pullback \( p^*\omega \).
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Context:
Given that the coordinates of a direction vector of line $l$ are $({-1,\sqrt{3}})$, the inclination angle of line $l$ is ____.
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Context:
Find a product of the Pauli matrices X, Y, and Z that results in the matrix \( B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \).
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Context:
Consider a Markov chain with a finite state space of size $k$ and a transition matrix where all non-diagonal entries are equal to $1/k$. Determine the steady-state probability distribution for this Markov chain.
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Context:
For a fixed complex number $c = a+bi$, where $c \neq 0, \pm1, \pm2, \dots$, determine the condition on $c$ such that the magnitude of the multi-valued expression $|i^c|$ remains constant.
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Context:
Determine the number of points at which the function \( f(x) = \begin{cases} x^2+1 & x \in \mathbb{Q}^c \\ \tan x & x \in \mathbb{Q} \end{cases} \) is continuous on the interval \([0, 4\pi]\).
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Context:
In an examination consisting of 20 questions, each question can be scored as -1, 0, or 4. Determine the number of distinct total scores a student can achieve.
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Context:
Simplify the following complexity expression into a polynomial in terms of \( n \):
\[ O\left (\sum_{i=1}^n\sum_{j=1}^{i^2}\sum_{k=1}^{j^2}k\right) \]
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Context:
Evaluate the path integral \(\int_\mathcal{C} (x^2+y^2) e^{-(x^2+y^2)} (1-2y^2) \,\mathrm{d}x + (x^2+y^2) e^{-(x^2+y^2)} (1-2x^2) \,\mathrm{d}y\) where \(\mathcal{C}\) is the lower semi-circle from \(A=(-1,0)\) to \(B=(1,0)\) with \(x,y\leq0\). Use the substitution \(y=\sin(t)\) when solving \(\int\frac{1-2y^2}{\sqrt{1-y^2}}\).
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Context:
Find the determinant of the $n \times n$ matrix given by:
\[
D_n = \left|
 \begin{matrix}
  n & -1 & -3 &  0 &  0 & \cdots & 0 & 0 &  0 \\
  n &  1 &  2 & -3 &  0 & \cdots & 0 & 0 &  0 \\
  n &  0 &  1 &  2 & -3 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  1 &  2 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  0 &  1 & \cdots & 0 & 0 &  0 \\
  \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\
  n &  0 &  0 &  0 &  0 & \cdots & 1 & 2 & -3 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 1 &  2 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 0 &  1 \\
 \end{matrix}
\right|
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( N \) be the number of ordered triple(A,B,C)of integers satisfying the conditions (a) \( 0 \leq  A < B < C \leq  {99} \) , (b) there exist integers \( a,b \) , and \( c \) , and prime \( p \) where \( 0 \leq  b < a < c < p \) , (c) \( p \) divides \( A - a \) , \( B - b \) , and \( C - c \) , and (d) each ordered triple(A,B,C)and each ordered triple(b,a,c)form arithmetic sequences. Find \( N \) ,
Please reason step by step, and put your final answer within \boxed{}.


Context:
A \( 3 \times  3 \) square is partitioned into9unit squares. Each unit square is painted either
 white or black with each color being equally likely, chosen independently and at rando
 The square is the rotated \( {90}^{ \circ  } \) clockwise about its center, and every white square in a
 position formerly occupied by a black square is painted black. The colors of all other
 squares are left unchanged. What is the probability that the grid is now entirely black?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the second-order linear homogeneous differential equation \(2x^2y''-(x+x^2)y'+(x^2-2)y=0\) with two linearly independent solutions \(\phi_1(x)\) and \(\phi_2(x)\), where \(\phi_1(0)=0\), find the smallest positive integer \(n\) such that \(\displaystyle \lim_{x \to 0}x^n\frac{\phi_2(x)}{\phi_1(x)}=0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the rate of convergence with respect to \(\epsilon\) for the norm \(\|f_\epsilon-f\|_{L^1}\), where \(f_\epsilon = \phi_\epsilon * f\) and \(\phi_\epsilon = (1/\epsilon)\phi(x/\epsilon)\) is a mollifier.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the moduli space of lines on a general quadric hypersurface in \(\mathbb{P}^n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the points of discontinuity for the function \( f(x) \), where \[ f(x)=\lim_{a\to x}\left(\frac{\sin a}{\sin x}\right)^{\frac{x}{\sin a-\sin x}}. \] If \( \sin x \neq 0 \), the function evaluates to \( f(x)=e^{\frac{x}{\sin x}} \). Analyze the behavior of \( f(x) \) at points where \( \sin x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function $$f(a,b)= \left(\sqrt{\frac{a^2b}{ab^2+1}}+\sqrt{\frac{ab^2}{a^2b+1}}+\sqrt{\frac{1}{ab(a+b)}}\right)\sqrt{\frac{ab(a+b+15)+1}{ab}}$$ for positive values of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the topological space that corresponds to the quotient $SO(n+1)/O(n)$, where $SO(n)$ and $O(n)$ are considered as topological groups.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the probability of 3 events occurring in a 90-minute interval is given by the Poisson distribution formula:

$$\Pr(X=3) = e^{-3\lambda}\frac{(3\lambda)^{3}}{3!}$$

where \(\lambda\) is the rate of events per 30 minutes, calculate the probability that all 3 events occur in the first 30 minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(X_1, X_2, \ldots, X_n \sim \text{Unif}[0, \beta]\). Define the estimators \(\hat{\beta_1} = (n+1)X_{\min}\) and \(\hat{\beta_2} = \frac{n+1}{n} X_{\max}\). Compute the ratio of the mean squared errors of \(\hat{\beta_1}\) and \(\hat{\beta_2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\iint_{S} \frac{\mathbf{x}}{||\mathbf{x}||} \cdot d\mathbf{S}\), where \(S\) is the surface defined by \(x^3 + y^3 + z^3 = a^3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F$ be a field, $K$ an extension field of $F$, and $a \in K$ such that the minimal polynomial of $a$ over $F$ is $x^n - a$, where $n$ is irreducible. Determine the degree of the minimal polynomial of $b^m$ over $F$, where $b$ is a root of $x^n - a$ and $m$ is a divisor of $n$. What is the degree of this minimal polynomial?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral of $\operatorname{sgn}(\sin\pi/x)$ over the interval $(0,1)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and $x^{20}+y^{20}=20$.  Find the sum of all possible values of $x^2+y^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the first six terms of the geometric series $3 + \left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + \left(\frac{1}{3}\right)^4 + \left(\frac{1}{3}\right)^5$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( x, y \in G \) and \( xyx^{-1} = y^{17} \), find the order of \( y \) if \( |x| = 2 \), the order of \( y \) is odd, and \( y \neq e \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \( \iint _S (y^2+x^2)\, dA \) by using a change of variables, where \( S \) is the region in the first quadrant bounded by the curves \( xy=1 \), \( xy=3 \), \( x^2-y^2=1 \), and \( x^2-y^2=4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the inverse Laplace transform of the function $$\mathcal{L}\{f(t)\} = \ln\left(\frac{s^2+a^2}{s^2+b^2}\right)?$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to +\infty} \frac{x^{\sqrt{\ln x}}\cdot(\sqrt{\ln x})^x}{(\sqrt{x})^{\ln x}\cdot(\ln x)^{\sqrt{x}}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a number $n$ randomly selected from the first 100 positive integers. Let $f(n)$ denote the product of all the positive divisors of $n$, and let $\varphi(n)$ denote the sum of all the positive divisors of $n$. What is the probability that $f(n) - \varphi(n)$ is an even number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a binary number with W bits, initially set to all zeros, determine the number of random bit toggles required to reduce the average value of any bit to approximately 0.499.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \((2023)^{2024} \mod{2025}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( S = \int_{-1}^{1} \int_{0}^{2\pi} \sqrt{\frac{1}{16}v^{2}} \,du \,dv \). What is the value of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an equilateral triangular grid  $G$  with  $20$  points on a side, where each row consists of points spaced  $1$  unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and  $20$  points in the last row, for a total of  $210$  points. Let  $S$  be a closed non-self-intersecting polygon which has  $210$  vertices, using each point in  $G$  exactly once. Find the sum of all possible values of the area of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximal \( \delta > 0 \) such that \( |f(x)| < \frac{1}{2} \) for all \( x \in (1 - \delta, 1 + \delta) \), where \( f(x) = e \cdot \ln(x) - x + 1 \). If a maximal \( \delta \) does not exist, find any such \( \delta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served $a$ years more, his pension would have been $p$ dollars greater, whereas had he served $b$ years more $(b\ne a)$, his pension would have been $q$ dollars greater than the original annual pension. Find his annual pension in terms of $a,b,p$ and $q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the cardinality of the set of all tagged partitions of all closed intervals in \( \mathbb{R} \), denoted as \( \bigcup_{a \leq b \in \mathbb{R}} M_{a,b} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A particle is moving randomly around a plane. It starts at $(0,0)$. Every second, it moves one unit randomly in a direction parallel to the $x$ or $y$ axis. At some time in the first hour, the particle was at the point $(2023,23)$. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \lim _{n \rightarrow \infty} \mathbb{E}\left\{\left|1-X_{n}\right|\right\} \), given that \( X_{1}, X_{2}, \ldots \) are nonnegative random variables such that \( X_{n} \stackrel{\text{a.s.}}{\rightarrow} 0 \) and \( \lim_{n \rightarrow \infty} \mathbb{E}\{X_{n}\} = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: \[ \lim_{n \to \infty} \frac{(2n)!}{(2^n \cdot n!)^2} \] using bounding techniques or simplification methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In space, there are 3 planes and a sphere. How many distinct ways can a second sphere be placed in space so that it touches the three given planes and the first sphere? (In this problem, sphere touching is considered, i.e., it is not assumed that the spheres can only touch externally.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the mixed partial derivative \( u_{yx} \) of the function \( u(x,y) = \frac{xy(x^2-y^2)}{(x^2+y^2)} \) at the point \((0,0)\), given that \( u(0,0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)$ be a cubic polynomial with rational coefficients. Determine the possible values of the degree of the field extension $[\mathbb{Q}(a_1,a_2,a_3):\mathbb{Q}]$, where $a_1, a_2, a_3$ are the roots of $f(x)$. Provide your answer as a set of possible values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum value of the expression \(2m + 7n\) given that there exist distinct positive integers \(x_i\) \((1 \leq i \leq m)\) and \(y_j\) \((1 \leq j \leq n)\) such that the \(x_i\)'s are even, the \(y_j\)'s are odd, and \(\sum_{i = 1}^m x_i + \sum_{j=1}^n y_j = 1986\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We project one face of a regular octahedron perpendicularly onto the plane of the opposite face. What fraction of the area of the opposite face is covered by the projection?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the indicial equation for the differential equation \(xy'' - 4y' + 5xy = 0\) at the regular singular point \(x=0\).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Exercise 6. A set of $\mathrm{n}$ cells in an $\mathrm{n} \times \mathrm{n}$ grid is said to be distributed if it never includes two cells in the same row or column. In how many ways can one color some (possibly none) cells of an $\mathrm{n} \times \mathrm{n}$ grid so that all distributed sets contain the same number of colored cells?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the order of the element \(g^{-2}h^{-2}fh^{2}g^{2}\) in the symmetric group \(S_9\), where \(f=(9\ 6\ 3\ 5\ 1\ 4)(2\ 3\ 4\ 5\ 7\ 8)\), \(g=(4\ 6\ 7\ 3\ 1)(1\ 4\ 2\ 6\ 5\ 7\ 8)\), and \(h=(1\ 3\ 5\ 2)(2\ 3\ 6\ 7)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a point P on the circle C: $x^2+y^2-4ax-2by-5=0$ (where $a>0$, $b>0$), if the symmetric point of P with respect to the line $x+2y-1=0$ is still on the circle C, then the minimum value of $\frac{4}{a} + \frac{1}{b}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many functions $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ have the property that $f(\{1,2,3\})$ and $f(f(\{1,2,3\}))$ are disjoint?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence for the series \( 1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty} a_n (x-3)^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the function \( f(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^n \sqrt{k(n-k)} \binom{n}{k} x^k (1-x)^{n-k} \) on the interval \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \frac{\sum_{k=1}^{n} \binom{n}{2k-1}(n-(2k-1))^3}{n^2(n+3)2^n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \int_{\partial F} 2xy\, dx + x^2 \, dy + (1+x-z) \, dz \) for the intersection of the surfaces \( z=x^2+y^2 \) and \( 2x+2y+z=7 \). The curve should be traversed clockwise with respect to the origin.
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 $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and $ \frac{1024}{2047}$-good set of size $s$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A captain has 8 crew members and wants to secure a safe such that it can only be opened by a group of at least 5 crew members. He distributes keys to the crew members and locks the safe with several locks. Determine the minimum number of locks required so that any group of 5 or more crew members can open the safe, but no group of 4 or fewer crew members can open it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are three committees of bureaucrats. It is known that for each pair of bureaucrats from different committees, among the members of the remaining committee, there are exactly 10 bureaucrats who know both of them, and exactly 10 bureaucrats who do not know both of them. Find the total number of bureaucrats in the committees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \alpha > 0 \) for which the series \[ \sum_{n_{1}=1}^{\infty}\cdots\sum_{n_{k}=1}^{\infty}\frac{1}{\left(n_{1}^{2}+\dots+n_{k}^{2}\right)^{\alpha/k}} \] converges, where \( k > 0 \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function f(x) is defined on (0, +∞) and is an increasing function, and it satisfies the equation f(x) * f(f(x) + 1/x) = 1, find the exact value of f(1).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设$P ( x )$为2n次多项式，$P ( 0 ) = P ( 2 ) = \cdots = P ( 2 n ) = 0 , P ( 1 ) =P ( 3 ) = \cdots = P ( 2 n - 1 ) = 2 , P ( 2 n + 1 ) = - 3 0$，求n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the second partial derivative of the expression \( A^{ij}y^ix^j + B^{ij}x^iy^j \) with respect to \( \bf x \) and \( \bf y \). Here, \( i,j \) follow the Einstein summation convention, \( A, B \) are constants, \( \bf x = [x^1, x^2, \dots, x^n]^T \), and \( \bf y = [y^1, y^2, \dots, y^n]^T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. A school has 3 teachers who can teach English, 2 teachers who can teach Japanese, and 4 teachers who can teach both English and Japanese. Now, 3 English teachers and 3 Japanese teachers are to be selected to participate in off-campus tutoring during the holiday. How many ways are there to select them?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exist? If it does, what is the value?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can 20 people, labeled A to T, be divided into 4 identical non-empty groups such that A and B are in the same group, and C and D are not in the same group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct normal subgroups $H$ in the free group of rank 2, $F_2$, such that the quotient group $F_2/H$ is isomorphic to the Klein four-group $V_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \(\int_C{\omega}\) where \(\omega=\frac{dx \wedge dy}{x^2+y^2}\) and \(C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - \text{{(0,0)}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sequence of independent flips of a fair coin, determine the minimum number of flips required to have at least a 90% probability of obtaining three consecutive heads.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using a power series approach: $$\lim_{x \to 0} \left[\frac{2+x}{2x*(e^x-1)}-\frac{1}{x^2}\right]$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the accurate rate of convergence of $\bar{X}_n - \mu$ to 0, expressed as $o_p(n^\alpha)$, where $X \sim N(\mu, \sigma^2)$ and $X_1, \cdots, X_n$ are samples from $X$. Express your answer in terms of $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of \( x \in \mathbb{R} \) is the integral \( \int_0^\infty e^{-t}t^{x-1} \, dt \) convergent?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of $X^Ta^TaX$ with respect to $X$, where $X$ is a matrix and $a$ is a vector.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the domain of the function $f(x^2-1)$ is $[4,9]$, determine the domain of the function $f(\log_2x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:

$$\lim_{t \to 0} \frac{f(t^2, \cosh t, e^t)}{f(t, \cos t, \cosh t)}$$

for a continuously differentiable function $f: \mathbb{R}^3 \to \mathbb{R}$ with the following conditions at the point $(0,1,1)$:

$$f(0,1,1) = 0, \quad \frac{\partial f}{\partial x}(0,1,1) = 1, \quad \frac{\partial f}{\partial y}(0,1,1) = 2, \quad \frac{\partial f}{\partial z}(0,1,1) = 3.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{ a_n \}$, where $S_n$ denotes the sum of the first $n$ terms, $S_4 = \pi$ (where $\pi$ represents the value of pi), and $a_4 = 2a_2$. Now, if we randomly select an element from the first 30 terms of this sequence, what is the probability that the cosine of this element is negative?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radical of the ideal $J = (ab, ac-bc)$. Express your answer in terms of generators.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $0 < A < 90^{\circ}$ and \( \sin A + \cos A + \tan A + \sec A + \operatorname{cosec} A + \cot A = 7 \), and that \( \sin A \) and \( \cos A \) are roots of the quadratic equation \( 4x^2 + 3x + k = 0 \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a convex quadrilateral $ABCD$, determine the number of points in the plane that return to their original positions after being reflected successively across the lines $AB$, $BC$, $CD$, and $DA$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation: \[ T(n) = T(n/3) + T(n/2) + n^3 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the divergence theorem to compute \( \int \int \limits_S F \cdot n \,dS \) where \( n \) is the outward normal. The surface \( S \) is defined by \( x^2 + y^2 = z^2 \) with \( z \in [0,1] \), and the vector field is \( F(x,y,z) = (x, xy, z) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a\) be a real number in the interval \([0,314]\) such that \(\displaystyle \int^{3\pi+a}_{\pi+a}|x-a-\pi|\sin \frac{x}{2}dx=-16.\) Determine the number of such real values of \(a.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest possible value of \(\alpha\) such that a grid graph \(M_{n,n}\) may not have an \(\alpha\)-spanning tree.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( D \subset \mathbb{R}^3 \) be the set \( D = \{(x, y, z) \mid x^2 + y^2 = 1~\text{and}~z = 0\} \). Let \( T :  \mathbb{R}^3 \to \mathbb{R}^3 \) be the linear transformation defined by \( T(x, y, z) = (2x − y + z, x − y + 10z, 3x − 7y + 2z) \). Find the area of \( T(D) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{i=1}^{2021} \frac{z_i^3}{z_i+1} \), where \( z_1, z_2, \ldots, z_{2021} \) are the roots of the polynomial \( z^{2021}+z-1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $x$ and $y$ are positive integers such that $x^y = y^x$. If the prime factorization of $x$ includes a prime $p$ with exponent $m$, determine the exponent of $p$ in the prime factorization of $y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $y_1$ and $y_2$ are a fundamental set of solutions of the differential equation $(\mathrm{e}^x y')' + y' + y = 0$ for $x > 0$, and the Wronskian $W(y_1, y_2)(0) = 1$, find the value of $W(y_1, y_2)(-1)$. Use Abel's formula to determine your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section, the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$              .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the spectrum of an operator $S: C([0,2]) \to C([0,2])$ defined by $(Sf)(t) = \int_0^{t+1} f(x) dx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya can draw only 4 things: a sun, a ball, a tomato, and a banana. Today he drew several things, including exactly 15 yellow items, 18 round items, and 13 edible items. What is the maximum number of balls he could have drawn?

Petya believes that all tomatoes are round and red, all balls are round and can be of any color, and all bananas are yellow and not round.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set \( A = \{0, 1, 2, 3, 4, 5, 6, 7\} \), how many mappings \( f \) from \( A \) to \( A \) satisfy the following conditions?
1. For all \( i, j \in A \) with \( i \neq j \), \( f(i) \neq f(j) \).
2. For all \( i, j \in A \) with \( i + j = 7 \), \( f(i) + f(j) = 7 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the eccentricity is \( \frac{\sqrt{17}}{3} \). Let \( F \) be the right focus, and points \( A \) and \( B \) lie on the right branch of the hyperbola. Let \( D \) be the point symmetric to \( A \) with respect to the origin \( O \), with \( D F \perp A B \). If \( \overrightarrow{A F} = \lambda \overrightarrow{F B} \), find \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of the parameter \( a \) for which the system of equations \( \left\{\begin{array}{l} |x-6-y|+|x-6+y|=12 \\ (|x|-6)^{2}+(|y|-8)^{2}=a \end{array}\right. \) has exactly two solutions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n \) such that the function \( g(x) = \ln \ln \ldots \ln (|x| + 3) \), where the natural logarithm is applied \( n \) times, is surjective onto \((-\infty, \infty)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a three-person game involving the following three types of fair six-sided dice.
- Dice of type $A$ have faces labelled 2, 2, 4, 4, 9, 9.
- Dice of type $B$ have faces labelled $1, 1, 6, 6, 8, 8$.
- Dice of type $C$ have faces labelled $3, 3, 5, 5, 7, 7$.
All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$'s roll (and hence is either 0, 1, or 2). Assuming all three players play optimally, what is the expected score of a particular player?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$L=\lim_{ x\to \infty} x^2 \times \log \left(x \cot^{-1}x\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied:

(i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too.

(ii) If two balls marked by different numbers $a$ and $b$ are colored green and $a+b<13$, then the ball marked by the number $a+b$ is also colored green.

How many ways are there of coloring the balls?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Wendy takes Honors Biology at school, a small class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday, none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $\frac{m}{n}$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. On a chessboard of size $2015 \times 2015$, dominoes are placed. It is known that in every row and every column there is a cell covered by a domino. What is the minimum number of dominoes required for this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mary typed a six-digit number, but the two  $1$ s she typed didn't show.  What appeared was  $2002$ .  How many different six-digit numbers could she have typed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jack walked a total distance of 8 miles in 1 hour and 15 minutes. The route included an elevation gain of 600 feet, and he walked at varying speeds during this duration. Initially, he walked at a speed of 4 miles per hour for the first 30 minutes, then increased his speed to 6 miles per hour for the next 20 minutes. Finally, he slowed down his pace and walked the remaining distance at a rate of 3 miles per hour. What was his average speed for the entire walk, in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.


Context:
求和$\sum _ { n = 1 } ^ { 2 0 0 0 } ( - 1 ) ^ { n } \frac { n ^ { 2 } + n + 1 } { n ! }$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sample consists of 1, 3, 2, 2, a, b, c. It is known that the mode of this sample is 3, and the mean is 2. Calculate the variance of this sample. Express your answer as a fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $g(x) = x^3 - x$, and $f(x)$ be a polynomial of degree at most 100. If $f(x)$ and $g(x)$ have no common factor, and \( \frac{d^{100}}{dx^{100}}\left[\frac{f(x)}{g(x)}\right] = \frac{p(x)}{q(x)} \), determine the degrees of the polynomials $p(x)$ and $q(x)$. Provide your answer as a pair of integers representing the degrees of $p(x)$ and $q(x)$. \( \boxed{(\text{degree of } p(x), \text{degree of } q(x))} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( ABC \) be a triangle, and let \( D, E \), and \( F \) be the midpoints of sides \( BC, CA \), and \( AB \), respectively. Let the angle bisectors of \( \angle FDE \) and \( \angle FBD \) meet at \( P \). Given that \( \angle BAC = 37^\circ \) and \( \angle CBA = 85^\circ \), determine the degree measure of \( \angle BPD \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( x f(x) - y f(y) = (x - y) f(x + y) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example - (2004 American Invitational Mathematics Examination) A circle of radius 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect with the diagonal $AC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$. Express your answer as a numerical value in terms of pi if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two matrices A and B, where S denotes the column space of A and T denotes the column space of B, find the matrix M whose column space is the sum of S and T. Assume both A and B have columns in \( \mathbb{R}^m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
[asy]
size(130);
pair A = (2, 2.4), C = (0, 0), B = (4.3, 0),
E = 0.7*A, F = 0.57*A + 0.43*B, D = (2.4, 0);
draw(A--B--C--cycle);
draw(E--D--F);
label("$A$", A, N);
label("$B$", B, E);
label("$C$", C, W);
label("$D$", D, S);
label("$E$", E, NW);
label("$F$", F, NE);
//Credit to MSTang for the diagram[/asy]
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the first site, higher-class equipment was used, and at the second site, first-class equipment was used. There was less higher-class equipment than first-class equipment. Initially, 40% of the equipment from the first site was transferred to the second site. Then, 20% of the equipment at the second site was transferred back to the first site, with half of the transferred equipment being first-class. After these transfers, the amount of higher-class equipment at the first site was 26 units more than at the second site, and the total amount of equipment at the second site increased by more than 5% compared to the original amount. Determine the total amount of first-class equipment.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum number of prime factors that a negative square-free even integer must have to ensure that the class number of \( \mathbb{Q}(\sqrt{n}) \) is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Question 104, Given complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ satisfy the following conditions: $\left|\mathrm{z}_{1}\right|=2,\left|\mathrm{z}_{2}\right|=3,3 \mathrm{z}_{1}-2 \mathrm{z}_{2}=2-\mathrm{i}$, then $\mathrm{z}_{1} \mathrm{z}_{2}=$ $\qquad$ -
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the length of the module $A = \mathbb{R}[X] / (X^3 - X^2 + X -1)$ as an $\mathbb{R}[X]$-module.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 4-state Markov Chain with state space \(\{1,2,3,0\}\), represented by the transition matrix:
\[
\begin{pmatrix}
0.2 & 0.5 & 0.2 & 0.1 \\
0.2 & 0.6 & 0.1 & 0.1 \\
0.2 & 0.3 & 0.3 & 0.2 \\
0 & 0 & 0 & 0.1 \\
\end{pmatrix}
\]
calculate the probability that state 2 is never reached, meaning the process is absorbed into state 0 without transitioning to state 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when \(7^{2023}\) is divided by 1001, given the following congruences:

\[7^{2023} \equiv 3 \pmod{7}\]
\[7^{2023} \equiv 4 \pmod{11}\]
\[7^{2023} \equiv 8 \pmod{13}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the genus of the graph with vertex set \( V(G) = \{u_1, \cdots, u_7, v_1, \cdots, v_9, w_1, \cdots, w_5\} \) and edge set \( E(G) = \bigcup_{i=1}^{11} E_i \), where:

\[ E_1 = \{u_1u_j \mid 2 \leq j \leq 7\} \cup \{u_1v_j \mid j=1,3,5,6,8,9\} \cup \{u_1w_3, u_1w_4\} \; ; \]
\[ E_2 = \{u_2u_j \mid j=3,4,6,7\} \cup \{u_2v_j \mid j=4,5,6,7\} \cup \{u_2w_5\} \; ; \]
\[ E_3 = \{u_3u_j \mid j=4,5,7\} \cup \{u_3v_j \mid j=1,7,8,9\} \cup \{u_3w_2\} \; ; \]
\[ E_4 = \{u_4u_j \mid j=5,6\} \cup \{u_4v_j \mid j=1,2,3,4\} \cup \{u_4w_1\} \; ; \]
\[ E_5 = \{u_5u_j \mid j=6,7\} \cup \{u_5v_j \mid j=4,5,6,7\} \cup \{u_5w_5\} \; ; \]
\[ E_6 = \{u_6u_7\} \cup \{u_6v_j \mid j=1,7,8,9\} \cup \{u_6w_2\} \; ; \]
\[ E_7 = \{u_7v_j \mid j=1,2,3,4\} \cup \{u_7w_1\} \; ; \]
\[ E_8 = \{v_1v_5, v_1v_6\} \; ; \]
\[ E_9 = \{v_2v_7\} \; ; \]
\[ E_{10} = \{v_3v_7\} \; ; \]
\[ E_{11} = \{v_4v_8, v_4v_9\} \; . \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi} \sin(a\sin \theta - b \theta)\,d\theta \) where \(a \in \mathbb{R}\) and \(b \geq 0\) is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $C$ be an arbitrary point on the circle $k$ with diameter $AB$, different from points $A$ and $B$. Draw the perpendicular from point $C$ to the diameter $AB$, with the foot of the perpendicular being point $D$ on segment $AB$, and let the second intersection of this perpendicular with circle $k$ be point $E$. The circle with center $C$ and radius $CD$ intersects the circle $k$ at points $P$ and $Q$. Let $M$ be the intersection point of segments $CE$ and $PQ$. Determine the value of $\frac{P M}{P E}+\frac{Q M}{Q E}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an approximation for the constant coefficient in the asymptotic expansion of $k\log(x)$ as $k$ tends to infinity, where $x$ is the solution to the equation $p_K(x)=\frac{1}{2}$. Provide your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit \( \lim_{x\to e}\frac{x^{e^x}-e^{x^e}}{x-e} \) without using L'Hospital's rule.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A continuous-time Markov chain $X_t$ with state space $\{1,2,3\}$ has an infinitesimal generator matrix:
\[ A= \begin{bmatrix}-6 & 2 & 4\\2 &-5 &3 \\2 & 3& -5\end{bmatrix} \]
If the chain starts in state 2, what is the expected total time spent in state 2 before the chain first enters state 3?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a blackboard, two three-digit numbers are multiplied together. If the multiplication sign is replaced with a 0, the resulting seven-digit number is an integer multiple of the product of the two numbers. Determine the exact factor by which the seven-digit number is a multiple of the product.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of \(2 \times 2\) matrices \(A\) with elements from the set \(\{-1, 0, 1\}\) such that \(A^2 = I\), where \(I\) is the identity matrix of order 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equations $x^{5} y^{2} + y^{5} z^{2} + z^{5} x^{2} = q^{3}$, $xy = yz = zx = r$, and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{q}$, find $\sqrt{x} + \sqrt{y} + \sqrt{z}$ in terms of $q$ and $r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of crossings required for a non-alternating knot.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{|z|=1}\frac{z^2-z+1}{z-1}dz \) using Cauchy's Integral Formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An equilateral cone contains an equilateral cylinder. Determine the ratio of the volume of the cone to the volume of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer  $n>1$ . In the cells of an  $n\times n$  board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following sum
$$\frac{1}{\log_2{\frac{1}{7}}}+\frac{1}{\log_3{\frac{1}{7}}}+\frac{1}{\log_4{\frac{1}{7}}}+\frac{1}{\log_5{\frac{1}{7}}}+\frac{1}{\log_6{\frac{1}{7}}}-\frac{1}{\log_7{\frac{1}{7}}}-\frac{1}{\log_8{\frac{1}{7}}}-\frac{1}{\log_9{\frac{1}{7}}}-\frac{1}{\log_{10}{\frac{1}{7}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that triangle XYZ is a right triangle with two altitudes of lengths 6 and 18, determine the largest possible integer length for the third altitude.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a study of product lifespan, consider the hypothesis test:

\[H_0: \mu = 2\]
\[H_1: \mu < 2\]

where \(\mu\) is the expected lifetime of the product, assumed to follow an exponential distribution with parameter \(\lambda\). The probability density function is:

\[f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{otherwise} \end{cases}\]

Given that \(E[X] = \mu = \frac{1}{\lambda}\), a sample of 10 products yields a sample mean \(\bar{X}\). Define \(c > 0\) such that if \(W \geq c\), the null hypothesis is accepted; otherwise, it is rejected. Determine the critical value \(c\) for a significance level of 95%.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim\limits_{x\rightarrow 0}\frac{\ln\left( 3+4x\right) -3{\ln\left( 3+x\right)} }{x^3} \) using the standard result \( \lim\limits_{x\rightarrow 0}\frac{\ln(1+x)}{x}=1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample of 7 elements with an inter-quartile range (IQR) of 4, find the minimum possible value of the variance of the sample.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a Poisson process with a rate of 0.1 observations per minute, calculate the expected time of the 2nd arrival, given that the 2nd arrival occurs within the first 2.5 minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 10 dozen mangoes in a box, with each box of mangoes having a varying price per dozen. The prices per dozen are as follows: $4, $5, $6, $7, $8, $9, $10, $11, $12, and $13. If there are 36 such boxes, what is the total cost of all the mangoes in all the boxes together?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the spectrum of the weighted right shift operator \(T\) on \(\ell^{\infty}(\mathbb{N})\), defined by:
\[ T(a_1,a_2,a_3,...) = \left(0, \frac{a_1}{1}, \frac{a_2}{2}, \frac{a_3}{3}, ...\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit without using L'Hopital's rule: \[ \lim_{x\rightarrow b}\frac{b^{x^b}-b^{b^x}}{x^b-b^x} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \(x \geq 1\) and \(y \geq 1\), for real number \(k\), it holds that
\[
\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+(x-1)(y-1) \geq k x y
\]
Find the maximum value of \(k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values of \( b \) for which the solution \( u(t) \) of the initial value problem \( u'(t) = Bu(t), \; t > 0 \) and \( u(0) = u_0 \), where \( u_0 \) is a given vector in \( \mathbb{R}^2 \) and \( B = \begin{bmatrix} -1 & 2 \\ -1 & b \end{bmatrix} \), satisfies \( |u(t)| \to 0 \) as \( t \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). Given that the system of equations \(2x + y = 2019\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution, find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( {ABC} \) be a right triangle with \( m\left( \widehat{A}\right)  = {90}^{ \circ  } \) . Let \( {APQR} \) be a square
 with area 9 such that \( P \in  \left\lbrack  {AC}\right\rbrack  ,Q \in  \left\lbrack  {BC}\right\rbrack  ,R \in  \left\lbrack  {AB}\right\rbrack \) . Let \( {KLMN} \) be
 square with area 8 such that \( N,K \in  \left\lbrack  {BC}\right\rbrack  ,M \in  \left\lbrack  {AB}\right\rbrack \) ,and \( L \in  \left\lbrack  {AC}\right\rbrack \) .
 What is \( \left| {AB}\right|  + \left| {AC}\right| \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A game is played by two players, A and B. They first agree on an integer N, and then A starts. They take turns filling one of the six boxes with any of the digits 1, 2, 3, 4, 5, 6, 7, 8, or 9, where each box can hold only one digit and digits can repeat, forming a six-digit number. If this six-digit number is divisible by N, B wins; otherwise, A wins. When N is less than 15, what is the value of N that guarantees a winning strategy for B? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the Riemann-Stieltjes integral \( \int_2^7 t^2 \, d\alpha(t) \) where \( \alpha(t) = -t + k \) for \( k \le t < k+1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \( \sin x = 1 - x \) for \( x \) without using graphical methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distribution of the ratio $Y/X$ where $X$ and $Y$ are random variables uniformly distributed over the region $A = \{(x,y) \in \mathbb{R^2}: 0 \leq y \leq x \leq 1\}$ with joint density $d\mu_{(X,Y)}(x,y) = 2\mathbf{1}_A(x,y)dxdy$. Provide the distribution of $Y/X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
对每个正整数 $n$ ，定义集合
$P_{n}=\left\{n^{k} \mid k=0,1, \cdots\right\}$ 。
对于正整数 $a 、 b 、 c$, 若存在某个正整数 $m$ ，使得 $a-1 、 a b-12 、 a b c-2015$ 这三个数
（不必两两不等）均属于集合 $P_{m}$ ，则称正整数组 $(a, b, c)$ 为"幸运的". 求所有幸运的正整数组的个数.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest common divisor (GCD) of the complex numbers \(5-3i\) and \(7+i\) in the ring of Gaussian integers \(\mathbb{Z}[i]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measure the base of a spiral staircase. The castle walls form a rectangle, and the center of the staircase is:
- 9 inches from one corner,
- 15 inches from another corner,
- 16 inches from a third corner,
- and some whole number of inches from the furthest corner.

Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four distances are measured on the same plane (the ground).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $x$ that satisfies the equation $2^{x^2-3}=x^{-1/3}$ using a numerical approximation method.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Embed $\mathbb{R}^2$ in the projective plane $\mathbb{RP}^2$ using the map $(x, y) \mapsto [1, x, y]$. Determine the point of intersection in $\mathbb{RP}^2$ of the projective lines corresponding to the parallel lines $y = mx$ and $y = mx + c$ in $\mathbb{R}^2$, where $c \neq 0$. Provide your answer as a point in homogeneous coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( F(\Lambda) = \int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy \), where \( \Lambda \) is a positive real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a group $H$ with elements $x$ and $y$ such that $|x|=|y|=4$, $x^2=y^2$, $xy=x^{-1}y$, $x\neq y$, and $H=\langle x,y\rangle$. Determine the order of the group $H$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the vector space $\Bbb Q[x,y]/J$ over $\Bbb Q$, where $J=\langle(x-1)^3,(x-1)(y+2),(y+2)^2\rangle$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Integrate the function \( f(x,y,z) = 14xz \) over the region in the first octant \((x, y, z \ge 0)\) that lies above the parabolic cylinder \( z = y^2 \) and below the paraboloid \( z = 8 - 2x^2 - y^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be a finite-dimensional vector space over the field $\mathbb{F}_q$ with $q$ elements, and let $U$ be a subspace of $V$. Determine the number of subspaces $W$ of $V$ such that $W \cap U = \{0\}$ and $V = W \oplus U$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the joint mass density function \( f_{X,Y}(x,y) = \frac{{e}^{-\left(y + \frac{x}{y}\right)}}{y} \) for \( x > 0 \) and \( y > 0 \), calculate the covariance \( \text{Cov}[X,Y] = E[XY] - E[X]E[Y] \). It is known that \( Y \) follows an exponential distribution with rate parameter \( \lambda = 1 \), hence \( E[Y] = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a non-constant polynomial \( p(x) \) in \( \mathbb{Z}_4[x] \) such that \( p(x) \) is a unit and has a degree strictly between 2 and 6.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider three mutually interpenetrating solids: two right circular pyramids and one right circular prism. The base of each pyramid and both bases of the prism are sealed by flat circular surfaces. Determine the maximum number of bounded volumes that can be formed, considering only the surfaces of the solids as boundaries and counting only volumes that are not further subdivided.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a positive non-constant geometric progression, the arithmetic mean of the second, seventh, and ninth terms is equal to some term of this progression. What is the smallest possible index of this term?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vectors \(\mathbf{v}_1 = (1, 2, 0, 2)\), \(\mathbf{v}_2 = (1, 1, 1, 0)\), and \(\mathbf{v}_3 = (2, 0, 1, 3)\), find a fourth vector \(\mathbf{v}_4\) such that the set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4 \}\) forms a basis for \(\mathbb{R}^4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( \bigcup_n \mathcal{P} (B-A_n) \), where \( n \in \mathbb{N} - \{0,1\} \), \( A_n=[0,1/n] \), and \( B=\{1/k | k \in \mathbb{N} - \{0\}\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the subset of $2 \times 2$ matrices $(c_1, c_2)$ that generate a 2-dimensional Lie algebra, given by the condition $[c_1, c_2] = k_1 c_1 + k_2 c_2$, where $[a,b]$ denotes the commutator $ab - ba$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of unique ways to color a triangular array of 10 balls, arranged in rows of 1, 2, 3, and 4 balls respectively, using 2 colors. Consider colorings that are equivalent under rotation to be the same.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a grid where each cell contains a number representing the height of a mountain, moving from a mountain of height \( a \) to a mountain of height \( b \) takes \( (b-a)^{2} \) time. Starting from the mountain of height 1, determine the minimum amount of time required to reach the mountain of height 49, moving only up, down, left, or right.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an analytic function with a Laurent series expansion that converges on an annulus. Determine the maximum possible radius of convergence for a power series expansion about a point within the annulus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $u_1=(1,1,1)$, $u_2=(2,0,0)$, and $u_3=(0,3,0)$ be a basis of $\mathbb{R}^3$. Calculate the coordinates of $w=2e^{*2}+3e^{*3}$ in the dual basis $\{u^{*1}, u^{*2}, u^{*3}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Lipschitz constant for a solution $x(t)$ to the differential equation $y'(t) = \sin(3\pi(y(t)+2t))$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $P, Q, R, S, T, U, V,$ and $W$ lie, in that order, on line $\overline{PW}$, dividing it into seven equal segments, each of length 1. Point $X$ is not on line $PW$. Points $Y$ and $Z$ lie on line segments $\overline{XR}$ and $\overline{XW}$ respectively. The line segments $\overline{YQ}, \overline{ZT},$ and $\overline{PX}$ are parallel. Determine the ratio $\frac{YQ}{ZT}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A large bucket is filled with water. You want to divide this water into 7 smaller buckets of the same shape and size. If you divide it by 900 grams (g), 200 grams (g) remain. How many kilograms (kg) of water is in the large bucket?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series \( \sum_{n=0}^\infty b_n x^n \), where the coefficients \( b_n \) are defined by the recurrence relation:

\[ \begin{aligned} b_{n+2} &= \frac{(n+2)(n+1) b_{n+1} - b_n}{(n+2)(n+1)}, \quad n\ge 1 \\
 b_2 &= -b_0/2 \end{aligned} \]

with arbitrary initial values \( b_0 \) and \( b_1 \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the maximum number of points in general position in a finite projective plane of order $n$, where $n$ is an odd integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the units digit of the sum:
\[2^{2^2} + 4^{4^4} + 6^{6^6} + \cdots + 2016^{2016^{2016}} + 2018^{2018^{2018}}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \oint_{|z| = 3} \frac{\mathrm{d}z}{\sqrt{z^2 - 1}} \) using a branch of the function that is analytic outside the interval \([-1, 1]\) and positive on \((1, \infty)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fundamental group of the space \( \mathbb{C}^2 \setminus \Delta \), where \( \Delta = \{(z,z) : z \in \mathbb{C}\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the intersection point of the functions $x^{0.999}$ and $\frac{x}{\log(x)}$ for $0 < p < 1$. Assume $p = 0.999$. Determine the value of $x$ where these two functions are equal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( N \) such that the difference \( |s - s_N| \) is less than \( 10^{-6} \), where \( s = \sum_{n=1}^\infty \frac{2}{2+e^n} \) and \( s_N = \sum_{n=1}^N \frac{2}{2+e^n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a finite arithmetic progression, the sum of the absolute values of its terms is 100. If each term is increased by 1 or by 2, the sum of the absolute values of the terms remains 100 in both cases. Determine the possible values of the expression \( n^2 d \), where \( n \) is the number of terms and \( d \) is the common difference of the progression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an infinite sequence of independent and fair coin tosses. Let $H_i$ denote the event that the $i$th coin lands heads (where $H_i = 1$ for heads and $0$ for tails). Compute the probability of the event:

$$\mathbb{P}\left(\bigcap_{i=1}^{\log_2(n)} H_{n+i} \text{ occurs infinitely often}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{n \to \infty }\left[\frac{\rm f\left(1 + 1/n\right)}{\rm f\left(1\right)}\right]^{1/n} $$ where ${\rm f}$ is differentiable at $1$ and ${\rm f}\left(1\right) > 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a real number $a$, for each $n \ge 0$, let $a_n$ be the distance from $a$ to the nearest rational number of the form $\frac{m}{2^n}$, where $m$ is an integer. Determine the maximum possible value of $\sum\limits_{n=0}^{\infty} a_n$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the indexes of the singular points other than zero of the vector field:
\[ z\overline{z}^2 + z^4 + 2\overline{z}^4 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,
\[\sqrt{\frac{x}{y + z + w}} + \sqrt{\frac{y}{x + z + w}} + \sqrt{\frac{z}{x + y + w}} + \sqrt{\frac{w}{x + y + z}} < M.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular sphere has a surface area given by \( A = 4\pi r^2 \). Determine the distance from the bottom of a hemisphere such that the surface area above and below this point is equal, excluding the sliced area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In \(\triangle ABC\), \(AB = 2\), \(BC = \sqrt{3}\), and \(\angle ABC = 150^\circ\). \(P\) is a point on the plane such that \(\angle APB = 45^\circ\) and \(\angle BPC = 120^\circ\). Find \(BP\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \[ \int_0^{\infty} \frac{2 \sin(x) \cos^2(x)}{x e^{x \sqrt{3}}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when $40!$ is divided by $5^{10}$. Express your answer as $40! \mod 5^{10}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the solutions to the complex equation \( \frac{1}{4}\ln(z)=i\pi+8k\pi \), where \( k \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Minimize the function \( c = 5x^2 - 80x + y^2 - 32y \) subject to the constraints \( x, y \geq 0 \) and \( x + y \geq 20 \) using the Karush-Kuhn-Tucker (KKT) conditions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game, a sequence of numbers is generated as follows: the first number is chosen randomly from the interval (0, 1), and each subsequent number is chosen randomly from the interval between 0 and the previous number. The game ends when a number less than a target value \( y \) (where \( 0 < y < 1 \)) is generated. What is the expected length of the sequence in terms of \( y \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sequence of $n$ integers, the sum of any three consecutive numbers is positive, and the sum of any five consecutive numbers is negative. Determine the maximum possible value of $n$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $x^3[f(x+1)-f(x)]=1$ and $f$ is continuous, determine $\lim_{x\rightarrow \infty}f(x)$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( \{ (x,y) \in \mathbb{R}^2 \ | \ y > x > 0 , x^x = y^y \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area enclosed by the polar curve \( r = 1 + \sin\theta \) and outside the curve \( r = 2 \sin\theta \) using double integrals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An equilateral triangle \(ABC\) is placed on the floor. Three nails are driven into the floor, one near each side of the triangle, preventing it from rotating without being lifted. The first nail divides side \(AB\) in the ratio \(1:3\) from vertex \(A\), and the second nail divides side \(BC\) in the ratio \(2:1\) from vertex \(B\). Determine the ratio in which the third nail divides side \(AC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the minute hand is moved back by 5 minutes, the number of radians it has turned is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the homotopy group \( \pi_3(S^2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of convergence of the series \( S_N(x) = \sum_{n=1}^{N}\frac{\cos(x n)}{n} \) to its limit \( -\log\left(2\sin\left(\frac{x}{2}\right)\right) \) as \( N \to \infty \). Find a function \( g(N) \) such that:
\[ S_N(x) = -\log\left(2\sin\left(\frac{x}{2}\right)\right) + \mathcal{O}(g(N)), \]
or equivalently,
\[ \left|S_N(x) + \log\left(2\sin\left(\frac{x}{2}\right)\right)\right| \leq C g(N), \]
for some constant \( C > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{1} \frac{x^{t}-1}{\ln x} \, dx \) for \( \text{Re}(t) > -1 \) without using Feynman's technique.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the expressions a = 2ln 1.01, b = ln 1.02, and c = sqrt(1.04) - 1, determine the correct ordering of a, b, and c. Express your answer as a sequence of variables using standard comparison operators (>, <, =).
Please reason step by step, and put your final answer within \boxed{}.


Context:
You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is good if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice.
Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a good coloring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a polynomial $F\in \mathbb{Z}\left [ X_{1},X_{2},X_{3},X_{4},X_{5} \right ]$ defined as $F=(X_{1}+X_{2}+X_{3})^{2}X_{4}X_{5}+X_{1}X_{2}X_{3}(X_{4}+X_{5})$, and a set $A = \{ a_1,a_2,a_3,a_4,a_5 \} \subseteq \mathbb{R}$ with $|A| = 5$, find the maximum number of elements of the set:
$$\{F(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)},a_{\sigma(4)},a_{\sigma(5)}) \} \mid \sigma \in S_5\}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
从 1,2,\cdots,2024 中任取两个数 a,b(a\leq b)，则 3^a+7^b 的值中，个位数字为 8 的数有__________个.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( g(x) = \int_0^x f(y)(y-x)dy \) be a function that is three times continuously differentiable. Determine the greatest integer \( n \) such that the function \( f \) must be \( n \) times continuously differentiable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If there are two points on the parabola \( y = ax^2 - 1 \) that are symmetric with respect to the line \( x + y = 0 \), find the range of values for \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Georgia is working on a math test with 75 problems on it. After 20 minutes, she has completed 10 problems. After another 20 minutes, she has completed twice as many problems. She has some time left to complete the rest of the test and has 45 problems left to solve. How many minutes does she have left to complete the test?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using Taylor expansions: $$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function\[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\]It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the given pattern of numbers, each row \( n \) contains the number \( n \) multiple times. Determine \( a(n) \), the number of times the number \( n \) appears in the \( n \)-th row of the pattern.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \(\left\lfloor \sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+\sqrt[2009]{\frac{2009}{2008}} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can you transform the number 0 into the number 100 in exactly 4 seconds, if each second you can add or subtract any factorial number (e.g., 1!, 2!, 3!, etc.) to your current number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the mean of some numbers including x, 50, 78, and 104 is 62, and the mean of 48, 62, 98, 124 and x is 76.4, what is the missing number in the first set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f: D_{18} \to GL(2, \mathbb{R}) \) be a homomorphism where \( |r| = 18 \) and \( f(r) = \begin{pmatrix} 1 & 1 \\ -1 & 0 \end{pmatrix} \). Determine the order of the kernel of \( f \), denoted as \( |\ker(f)| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \) across the surface \( \sigma \), which is the portion of the cone \( z = \sqrt{x^2 + y^2} \) between the planes \( z = 3 \) and \( z = 6 \), oriented by downward unit normals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of epimorphisms from the free group with 2 generators, $F_2$, to the cyclic group $\mathbb{Z}_5$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of $n$ is the graph $G_n$ with vertex set $V_n = \{1, 2, ..., n\}$, where two distinct vertices $x$ and $y$ are adjacent if $x$ is a divisor of $y$ or $y$ is a divisor of $x$, a planar graph?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region in the first quadrant bounded by the parabolas $y=x^2$, $y=2x^2$, $x=y^2$, and $x=2y^2$ using a change of coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles are given: $\Gamma_1$ with radius $4$ and center $(0,0)$, and $\Gamma_2$ with radius $14$ and center $(25,0)$. Let $A$ and $B$ be variable points on $\Gamma_1$ and $\Gamma_2$, respectively, and $M$ be the midpoint of $\overline{AB}$. Find the area of the region $S$ formed by all possible locations of $M$. \( \text{Express your answer in square units.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
If in the sequence $\{a_n\}$, the sum of any three consecutive terms is positive, and the sum of any four consecutive terms is negative, then the maximum value of the term number $n$ is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( f(x) = \mathbb{1}_{(0,1)}(x)x^\alpha \) belongs to the space \( L^p(\mathbb{R}) \), where \( p \in [1, \infty) \) and \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( -\frac{1}{p}, \infty \right)}


Context:
If the inequality \(4x^2 + y^2 + 1 \geq kx(y+1)\) holds for any integers \(x\) and \(y\), find the maximum constant \(k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series \( f(x) = \sum_{k=1}^{\infty} \frac{x^{2k}}{(4+(-1)^k)^{3k}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the polyhedron P defined by the following inequalities:

$$\begin{align} x_1 + x_2 &\leq 5 \\ x_1 - x_2 &\geq 1 \\ x_2 &\leq 3 \\ x_1 &\geq 0 \\ x_2 &\geq 0 \end{align}$$

Determine the face F induced by the equations $x_1 + x_2 = 5$ and $x_1 - x_2 = 1$. Find an inequality $c x \leq \gamma$ that is valid for P and such that $F = \{x \in P \mid c x = \gamma\}$. Provide the inequality $c x \leq \gamma$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Write down an irrational number that is greater than $\sqrt{3}$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a value of $\epsilon$ such that for any $N$, there exists an $n > N$ such that the sequence $$a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$$ satisfies $|a_{n} + \frac{3n}{4}| \geq \epsilon$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.