Context:
Identical regular pentagons are arranged in a ring. Each of the regular pentagons has a perimeter of 65. The regular polygon formed as the inner boundary of the ring has a perimeter of \( P \). What is the value of \( P \)?
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Context:
Consider the complex function $f(z) = \frac{1}{(z-\alpha)^2(z-1)}$ where $\alpha \in \mathbb{C}$ with $\operatorname{Im}(\alpha) > 0$. Determine the principal value of the integral
$$P.V. \int_{-\infty}^{+\infty} \frac{\operatorname d x}{(x-\alpha)^2(x-1)}$$
using the definition of Cauchy's principal value.
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Context:
Find the upper bound for the index of the core of a subgroup $H$ of finite index in a group $G$.
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Context:
Given that the line \( y = -2x + 1 \) is tangent to the curve \( f(x, y) = 2 \) at the point \( (1, 3) \), and \( f_x(1, 3) = 4 \), find the maximum directional derivative of \( f(x, y) \) in the direction of the vector \( (1, 3) \).
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Context:
Given a regular tetrahedron ABCD with edge length a (a > 3), points E, F, G are on edges AB, AC, AD respectively. Calculate the number of triangles EFG that satisfy EF = EG = 3 and FG = 2. Express your answer as a whole number.
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Context:
Find the value of $n$ such that the cyclotomic polynomial $\Phi_n(x)$ is given by $f(x) = (x^4 - x^2 + 1)(x^2 + 1)$. What is $n$?
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Context:
Given that $G/Z(G) \cong \mathbb{Z}_2 \times \mathbb{Z}_2$, determine the structure of the derived subgroup $G'$.
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Context:
Find the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \) where \( f(x) = |1 - 2x| \) and \( x \in [0, 1] \).
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Context:
Given the numbers from 1 to 20, players A and B take turns placing either a "+" or "-" sign in front of each number. The order of placing the signs is arbitrary. After all numbers have been assigned a sign, the absolute value of the resulting algebraic sum \( S \) is calculated. Player A aims to minimize \( S \), while Player B aims to maximize \( S \). What is the maximum value of \( S \) that Player B can guarantee?
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Context:
Find the ratio of the areas of the largest and smallest equilateral triangles that can be inscribed in a unit square, with vertices touching the sides.
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Context:
In triangle $ABC$, $AB = 42$, $AC = 39$, and $BC = 45$. Points $E$ and $F$ are on sides $AC$ and $AB$ such that $AF = 21$ and $AE = 13$. Lines $CF$ and $BE$ intersect at $P$, and ray $AP$ meets $BC$ at $D$. Let $O$ be the circumcenter of triangle $DEF$, and $R$ its circumradius. Calculate $CO^2 - R^2$.
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Context:
Let $x,$ $y,$ and $z$ be positive real numbers.  Find the maximum value of
\[\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Consider a connected graph $G = (V, E)$ with $n$ vertices, where all edges are part of $3$-cycles. The graph contains exactly $n-2$ $3$-cycles, and every vertex is part of at least one $3$-cycle. The minimum degree of the graph is $2$. Determine the maximum number of vertices in $G$ that have degree $2$. Provide your answer as a function of $n$.
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Context:
Fill in the three blanks in the denominators of the equation $$\frac {1}{(   )} + \frac {4}{(   )} + \frac {9}{(    )} = 1$$ with three positive integers so that the equation holds. The minimum sum of these three positive integers is \_\_\_\_\_\_.
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Context:
For any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers.
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Context:
Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?
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Context:
Let \( x \) be an irrational number such that \( 0 < x < 1 \). The decimal expansion of \( x \) is denoted as \( 0.c_1c_2c_3\ldots \). Define \( B(x) \) as the set of all subsequences of 6 consecutive digits from \( c_1c_2c_3\ldots \). For example, if \( x = \frac{1}{22} \), then \( B(x) = \{045454, 454545, 545454\} \). Determine the minimum number of elements in \( B(x) \) as \( x \) varies among all irrational numbers with \( 0 < x < 1 \).
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Context:
How many fractions between \(\frac{1}{6}\) and \(\frac{1}{3}\) inclusive can be written with a denominator of 15?
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Context:
Calculate the expected number of consecutive random number inputs required to unlock a lock with the combination 5556, given that the lock checks for the sequence without needing a reset between inputs.
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Context:
Carol starts with 7,125 blocks. She loses 45% of her blocks and then gives away ⅔ of the remaining blocks. How many blocks does Carol end up with?
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Context:
Paul made two bank transfers of $90 and $60 respectively. A service charge was added to each transaction. The second transaction was reversed (without the service charge). His account balance is now $307 if it was $400 before he made any of the transfers. What was the percentage of the service charge added to each transaction?
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Context:
Find the maximum value of the function \( F(x_1, x_2, \ldots, x_N) = - \sum_{i=1}^N x_i \log(x_i) \) subject to the constraint \( \sum_{i=1}^N x_i = 1 \) and \( x_i > 0 \) for all \( i \).
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Context:
Given the numbers 1 through 5, determine the total number of permutations that respect the order constraints specified by the ordered sets \(s_1 = (1, 2, 3)\) and \(s_2 = (5, 2)\).
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Context:
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
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Context:
Calculate the expected value of the area under the trajectory of a Wiener process, $W_t$, above the line $y=t$. Provide your answer as a single numerical value.
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Context:
Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.
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Context:
Given that \( x \) and \( y \) are real numbers greater than 10, the leading digit of \( \lg x \) is \( a \) and the trailing digit is \( b \); the leading digit of \( \lg y \) is \( c \) and the trailing digit is \( d \). Additionally, it is known that \( |1 - a| + \sqrt{c - 4} = 1 \) and \( b + d = 1 \). Find the value of \( x \times y \).
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Context:
Find the largest natural number in which each non-edge digit is less than the arithmetic mean of the adjacent digits.

#
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Context:
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.
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Context:
Evaluate the double integral \(\int\int (x-y)^2 \,dx\,dy\) over the region enclosed by the circle \(x^2 + y^2 = 4\) and the line \(y=x\).
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Context:
Given an acute-angled triangle \(ABC\). Point \(M\) is the intersection point of its altitudes. Find angle \(A\) if it is known that \(AM = BC\).
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Context:
Evaluate
$\frac{\log_{10}20^2 \cdot \log_{20}30^2 \cdot \log_{30}40^2 \cdot \cdot \cdot \log_{990}1000^2}{\log_{10}11^2 \cdot \log_{11}12^2 \cdot \log_{12}13^2 \cdot \cdot \cdot \log_{99}100^2}$
.
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Context:
\left.\begin{array}{l}\text { The Law of Sines } \\ \text { [Midline of a Triangle] }\end{array}\right]

Two chords, equal to 10 and 12, are drawn from one point on a circle. Find the radius of the circle if the distance from the midpoint of the smaller chord to the larger one is 4.
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Context:
A mother rabbit bought seven drums of different sizes and seven pairs of drumsticks of different lengths for her seven bunnies. If a bunny sees that both its drum is bigger and its drumsticks are longer compared to those of any of its siblings, it starts drumming loudly. What is the maximum number of bunnies that can start drumming?
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Context:
Evaluate the limit: $$\lim_{n \rightarrow \infty} \int^{n}_{0} \left(1+\frac{x}{n}\right)^{-n} \log\left(2+ \cos\left(\frac{x}{n}\right)\right)\ \mathrm{d}x.$$
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Context:
There is one black ball and one white ball in a bag. Each time a ball is taken from the bag, it is not returned, but instead, another black ball is added. What is the probability of drawing a black ball on the $n$-th draw?
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Context:
If 80% of a certain number is greater than 70% of 60.00000000000001 by 30, what is the number?
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Context:
Determine a tight upper bound for the expression \(-p \log p\) in terms of the probability \(p\), where \(0 < p < 1\), especially when \(p\) is very close to 0, such as \(p < 1/e\).
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Context:
Consider a $3 \times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $33$. Determine the value of the center square, labeled $Y$.
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Context:
For a permutation \( p = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{9}}\right) \) of the digits \( 1,2,\ldots ,9 \) ,let \( s\left( p\right) \) denote the sum of the three3-digit numbers \( {a}_{1}{a}_{2}{a}_{3} \) ,
 \( {a}_{4}{a}_{5}{a}_{6} \) ,and \( {a}_{7}{a}_{8}{a}_{9} \) . Let \( m \) be the minimum value of \( s\left( p\right) \) subject to the condition that the units digit of \( s\left( p\right) \) is 0 . Let \( n \) denote the number of permutations \( p \) with \( s\left( p\right)  = m \) . Find \( \left| {m - n}\right| \) .
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Context:
Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.
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Context:
Evaluate the integral \( \frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz \) where \(\mathcal{C}\) is the unit circle.
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Context:
Compute the limit of the integral \( I = \lim_{n \to +\infty} \int_0^n \frac{x^n \text{e}^{-x}}{n!} \text{d}x \).
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Context:
Given $a$, $b$, $c \in \mathbb{R}$, if $|a\cos^2x + b\sin x + c| \leq 1$ holds for $x \in \mathbb{R}$, then the maximum value of $|a\sin x + b|$ is.
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Context:
Simplify the expression \(\sum_{k=0}^{10} k \binom{10}{k} \binom{20}{10-k}\) using the binomial identity \(\sum_{k=i}^{n-j}\binom{k}{i}\binom{n-k}{j}=\binom{n+1}{j+1}\).
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Context:
Example 4.23 Make all permutations of 5 distinct elements $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$, where $a_{1}$ is not in the 1st or 2nd position, $a_{2}$ is not in the 2nd or 3rd position, $a_{3}$ is not in the 5th position, $a_{4}$ is not in the 4th or 5th position, and $a_{5}$ is not in the 3rd or 4th position. How many different permutations can be made?
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Context:
Given that \(\alpha\) and \(\beta\) are continuous and differentiable functions satisfying the following conditions:
\[\alpha'(x) = \beta(x),\]
\[\beta(x)^3 = (\alpha(x) - 1)^2,\]
\[\alpha(a) = 0,\]
\[\alpha(b) = 2,\]
compute the definite integral:
\[\int_a^b{\alpha(t)^2 \beta(t)^4 \,dt}.\]
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Context:
Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\triangle AFD$ to the area of square $ABCE$.
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Context:
Inside a truncated cone with a height of 8, there is a sphere O1 with a radius of 2. The center of sphere O1 lies on the axis of the truncated cone, and sphere O1 is tangent to both the upper base and the lateral surface of the truncated cone. Another sphere O2 with a radius of 3 can be placed inside the truncated cone such that sphere O2 is tangent to sphere O1, the lower base, and the lateral surface of the truncated cone, each at a single point. Excluding sphere O2, what is the maximum number of additional spheres with a radius of 3 that can be placed inside the truncated cone? Express your answer as a single integer.
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Context:
Using the digits 1, 2, 5, 6, and 9 exactly once, form five-digit even numbers in which no digit is repeated. Calculate the difference between the largest and smallest of these numbers. Express your answer as a numerical value.
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Context:
已知 $O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点, $\triangle A O B, \triangle B O C$, $\triangle C O D, \triangle D O A$ 的周长相等. 若 $\triangle A O B, \triangle B O C, \triangle C O D$ 的内切圆半径分别为 3, 4, 6, 则 $\triangle D O A$ 的内切圆半径的答案为\(\frac{k}{m}\)的形式，请给出k+m的值。
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Context:
Define the sequence of decimal integers \( a_n \) as follows: \( a_1 = 0 \); \( a_2 = 1 \); \( a_{n+2} \) is obtained by writing the digits of \( a_{n+1} \) immediately followed by those of \( a_n \). When is \( a_n \) a multiple of 11?
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Context:
You have $20,000 to invest in 5 different stocks, with the condition that you can only invest in multiples of $1,000 and not all the money can be put into one stock. How many different ways can you invest the $20,000?
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Context:
For what values of $m$ is the function \[ f\left(\begin{array}{c} x\\ y \end{array}\right)=\begin{cases} \frac{y^{m}\left(1-\cos x\right)}{x^{4}+2y^{2}} & \left(\begin{array}{c} x\\ y \end{array}\right)\neq\left(\begin{array}{c} 0\\ 0 \end{array}\right)\\ 0 & \left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} 0\\ 0 \end{array}\right) \end{cases} \] continuous at the point \(\left(\begin{array}{c} 0\\ 0 \end{array}\right)\)?
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Context:
Determine the values of \( p > 0 \) for which the integral \[ T = \sqrt{\frac{1}{2g}}\int_{0}^{b}\frac{\sqrt{1+y'(x)^2}}{\sqrt{-y(x)}} \, dx \] is convergent, given that \( y(x) = -2x^p \) and \( g \) is a constant.
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Context:
Let $X_1, X_2, \ldots, X_n$ be a random sample from a continuous distribution with unknown mean $\mu$ and known standard deviation $\sigma$. If $[X_{\min}, X_{\max}]$ is used as a confidence interval for $\mu$, determine the confidence level in terms of the standard normal distribution.
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Context:
Find the set $T$ of complex numbers defined by \[ T = \left\{ z \in \mathbb{C} \mid \left|\frac{z - i}{z + i}\right| = 1 \right\}. \]
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Context:
Find the residue of the sum \( S = 1^{965}+2^{965}+\cdots+2018^{965} \) when divided by \(2021\). Express your answer as \(S \equiv x \ (\text{mod}\ 2021)\). What is the value of \(x\)?
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Context:
Determine the completion of the metric space \(c_{00}\), defined as \(c_{00} = \{(x_n) \in \mathbb{R}: x_n=0, n>n_0\}\), with the metric induced from \(l_{\infty}\).
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Context:
Determine the degree of nilpotency of the dihedral group $G = D_{16}$ using only the upper central series.
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Context:
If the 26 letters of the alphabet are arranged in a circle such that no two vowels are adjacent, what is the probability that the letters 'a' and 'b' are next to each other?
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Context:
Consider a string of \( {n7’}\mathrm{\;s},{7777}\cdots {77} \) ,into which+signs are inserted to produce an arithmetic expression. For example,
 \( 7 + {77} + {777} + 7 + 7 = {875} \) could be obtained from eight 7’s in this way. For how many values of \( n \) is it possible to insert+
 signs so that the resulting expression has value 7000 ?
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Context:
Consider a string of $n$ 1's. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times, and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.
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Context:
Calculate the residue of the function \( \frac{z^{2}}{(z+i)^{2}(z-i)^{2}} \) at the pole \( z = i \).
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Context:
What is the maximum number of distinct integers in a row such that the sum of any 11 consequent integers is either 100 or 101 ?
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Context:
There is a pile of 10,000,000 matchsticks. Two players take turns in a game where on each turn, the current player can take away \( P^n \) matchsticks from the pile, where \( P \) is a prime number and \( n = 0,1,2,3, \cdots \) (for example, the first player might take 25 matchsticks, the second player 8 matchsticks, then the first player 1 matchstick, the second player 5 matchsticks, and so on). The player who takes the last matchstick wins the game. In a game played correctly, who will win?
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Context:
Given the cumulative distribution function (CDF) of a random variable X:

\[ F(x) = \begin{cases} 0, & \text{if } x < 0 \\ \frac{1}{4}, & \text{if } 0 \le x < 1 \\ \frac{2}{5}, & \text{if } 1 \le x < 2 \\ \frac{2x-3}{2}, & \text{if } 2 \le x < 2.5 \\ 1, & \text{if } x \ge 2.5 \end{cases} \]

Calculate the expected value E(X).
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Context:
Determine the missing term in the equation \( z=\sqrt{r^2-x^2-y^2}\sin(?) \), where the equation represents a 3D standing electron wave with 2 periods of oscillation per rotation.
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Context:
Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$.
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Context:
You have a four-sided die numbered 1 to 4 and are playing a game. In the first round, you roll the die once. If you roll a 1, you lose the die. If you roll a 2, you keep the die. If you roll a 3 or 4, you gain an additional identical die. In the second round, you roll each die you have, applying the same rules to each die. You continue this process in subsequent rounds. If you ever have zero dice, you lose the game. What is the probability that you never lose the game over infinite rounds?
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Context:
The smaller leg of a rectangular trapezoid is equal to 3, and the larger leg forms an angle of $30^{\circ}$ with one of the bases. Find this base if the point of intersection of the angle bisectors at the other base lies on it.
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Context:
若 $\triangle A B C$ 的角 $A, C$ 满足 $5(\cos A+\cos C)+4(\cos A \cos C+1)=0$, 则 $\tan \frac{A}{2} \cdot \tan \frac{C}{2}=$ $\qquad$.
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Context:
Suppose \( \alpha ,\beta ,\gamma  \in  \{  - 2,3\} \) are chosen such that
 \( M = \mathop{\max }\limits_{{x \in  \mathbb{R}}}\mathop{\min }\limits_{{y \in  {\mathbb{R}}_{ \geq  0}}}{\alpha x} + {\beta y} + {\gamma xy} \)
 is finite and positive (note: \( {\mathbb{R}}_{ \ge  0} \) is the set of nonnegative real numbers).
 What is the sum of the possible values of \( M \) ?
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Context:
Given the function \( V(x(t)) = \frac{x''}{x} \), where the prime denotes differentiation with respect to time, find \( \frac{dV(x(t))}{dx} \).
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \left[ \ln\left(1 - \frac{1}{1^2}\right) + \ln\left(1 - \frac{1}{2^2}\right) + \cdots + \ln\left(1 - \frac{1}{n^2}\right) \right] \]
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Context:
Find the fundamental group of the space obtained by removing all points with integer coordinates from three-dimensional Euclidean space.
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Context:
At the Bertolli Farm, they grow 2073 tomatoes, 4112 cobs of corn, 985 onions, and 6250 carrots. The growth rates for each crop are as follows: 12% for tomatoes, 15% for cobs of corn, 8% for onions, and 10% for carrots. Calculate the difference in total yield between the highest and lowest yielding crops after accounting for their respective growth rates.
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Context:
Evaluate the integral \( \int_{0}^{1}\frac{x^3-1}{\log(x)} \,\mathrm{d}x \) using a method other than the Feynman integration method.
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Context:
Evaluate the path integral of the vector field \( F = \langle \arccos x, xy - e^y \rangle \) over the triangular path from \((0,0)\) to \((2,3)\) to \((2,0)\) and back to \((0,0)\).
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Context:
Determine the maximal number of elements in a sigma-algebra generated by a collection of N subsets of a set X.
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Context:
$PQ$ is a diameter of a circle. $PR$ and $QS$ are chords with intersection at $T$. If $\angle PTQ= \theta$, determine the ratio of the area of $\triangle QTP$ to the area of $\triangle SRT$ (i.e. area of $\triangle QTP$/area of $\triangle SRT$) in terms of trigonometric functions of $\theta$
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Context:
Compute the limit: $$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$
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Context:
It is winter, and the white rabbit has stored 180 carrots, while the gray rabbit has stored 120 cabbages. The gray rabbit trades some number of cabbages for some of the white rabbit's carrots so that they end up with an equal amount of food. How many carrots can be exchanged for one cabbage?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all the digits of the smallest positive integer \( n \) such that the product \( P_2 \times P_3 \times \ldots \times P_n \) exceeds 2010, where \( P_k = 1 + \frac{1}{k} - \frac{1}{k^2} - \frac{1}{k^3} \) for a positive integer \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let's determine the value of the following expression:

$$
X=\sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \cdots}}}}}}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of values of $k$ for which the polynomial equation \(3x^{4}+4x^3-12x^2+k=0\) has four real roots.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_0^\pi \log(\sin(t)) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the mixing time of a discrete random walk on an N-cycle, where the walk moves to the right or left with equal probability of \( \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{x \to 0} x^2\left(1+2+3+\ldots+\left[ \frac{1}{|x|} \right] \right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{+\infty} \frac{\cos(mx)}{x^2+1} \, dx \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of positive values of \( p \) for which the improper integral \( \int_1^{\infty}\frac{e^{-px}}{\ln{x}}dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Convex quadrilateral $ABCD$ has $AB = 18, \angle{A} = 60 \\textdegree$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of connected components of the set \( S = \{ A \in SL(2, \mathbb{R}) \mid A^T \operatorname{diag}(1, -1) A = \operatorname{diag}(1, -1) \} \) in the induced topology, where \( SL(2, \mathbb{R}) \) is the set of real \( 2 \times 2 \) matrices with determinant 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a square matrix $A = [a_{ij}] \in \mathbb{R}^{n \times n}$, where each entry is either $1$ or $-1$. If the columns of $A$ are balanced, meaning that the sum of entries in each column is zero, $\sum_{i=1}^n a_{ij} = 0$, determine the maximum possible value of the spectral norm of $A$, denoted as $\|A\|$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In their base $10$ representation, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. Find the sum of the digits of the base 10 representation of $9ab$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The curves \(y=mx\) and \(y=\frac{x}{x^2+1}\) enclose a region for \(0<m<1\). Find the area of this region in terms of \(m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a magician's hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 of the reds are drawn or until both green chips are drawn, calculate the probability that all 4 red chips are drawn before both green chips are drawn.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the value of $a$ such that the existence and uniqueness theorem applies to the initial value problem: \( x' = \frac{3}{2}|x|^{1/3}, \ x(0) = a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $X$ and $Y$ are points on tJhe boundary of the triangular region $ABC$ such that the segment $XY$ divides the region into two parts of equal area. If $XY$ is the shortest such segment and $AB = 5$, $BC = 4$, $AC = 3$ calculate the length of $XY$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, \( a_i = a_{i-1}^2 - 1 \) if $a_{i-1}$ is odd and \( a_i = \frac{a_{i-1}}{2} \) if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya constructed a robot, but due to some programming mistake, the robot makes random movements, which still somehow lead it to the target point.

A move is a shift of one kilometer either north (N), east (E), south (S), or west (W).

The robot reached its target in 22 moves represented by the sequence:

EENENNWNENESEEEEESSSWN.

How many moves would the robot need if it did not make any unnecessary movements?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \epsilon^{\mu \nu \alpha \beta}\epsilon_{\mu \nu \alpha \beta} \) in four dimensions.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, the ant travels from one vertex to an adjacent vertex every minute. After 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Determine the number of possible routes the ant could have taken.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint\limits_D (x^2-xy-1) \, dx \, dy \), where \( D \) is the triangular region bounded by the lines \( x-2y+2=0, x+3y-3=0 \) and \( y=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bulb manufactured by Factory A?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \(d(x_1 x_2)\) given the stochastic differential equations \(dx_1 = a_1 dt + b_1 dw_1(t)\) and \(dx_2 = a_2 dt + b_2 dw_2(t)\), where \(w_1\) and \(w_2\) are independent Wiener processes. Specifically, determine the value of \(d(x_1)d(x_2)\) in the context of Ito calculus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of all values of \( p \) such that for every \( q > 0 \), the inequality \( \frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q \) holds. Express your answer in interval notation in decimal form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Some vertices of a regular $2024$-gon are marked such that for any regular polygon, all of whose vertices are vertices of the $2024$-gon, at least one of its vertices is marked. Find the minimal possible number of marked vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim\limits_{n\to\infty}\int_0^a\frac{x^n} {x^n+1}\,\mathrm dx \), where \( a>0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the correct solution to the integral equation $$f(x) = a - \int^x_b (x-t)f(t)dt$$ where $a$ and $b$ are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose we have \(n\) independent variables from \(U[0,1+\theta]\) and let \(\lambda=1+\theta\). The estimator of \(\theta\) is given by \(\bar{\theta}=2 \bar{X} - 1\). Find the bias of \(\bar{\theta}\) given that the distribution of \(\bar{X}\) for \(n\) variables from \(U[0,1]\) is the Bates distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a $10 \times 10$ matrix $A$. Each element $a_{ij}$ of the matrix is multiplied by $\frac{i}{j}$, where $1 \leq i,j \leq 10$. Determine how the determinant of the matrix changes after this transformation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three. (50 points) Given a square table with four legs $L_{1}, L_{2}, L_{3}, L_{4}$, each of length $n$, where $n$ is a positive integer. How many ordered quadruples $\left(k_{1}, k_{2}, k_{3}, k_{4}\right)$ exist such that after sawing off a segment of length $k_{i}$ from the end of leg $L_{i}$ $(i=1,2,3,4)$, the table remains stable? Here, $k_{1}, k_{2}, k_{3}, k_{4}$ are non-negative integers (the table is considered stable if all four legs touch the ground, and it is allowed to saw off a length of 0).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] given that all of the roots are real and nonnegative.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on the constants \(a, b, c\) such that the function \(f(ax+by, cz)\) is harmonic in \(\mathbb{R}^3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".

What is the maximum number of coins the captain can receive under these rules?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of all possible values of the series \( \sum_{n\geq 1}\frac{1}{x_n} \) where \( (x_n)_{n\geq 1} \) is an increasing sequence of positive natural numbers and the series converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 450 meter long train crosses a man standing on the platform in 8 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( V \) and \( W \) be linear subspaces of \( \mathbb{R}^4 \) defined as follows:

\[ V=\{ \vec{x} \in \mathbb{R}^4 \mid x-y+2z-t=0 \} \]
\[ W=\text{span}(1,1,1,1)^T \]

Find the dimension of the linear subspace \( F=\{ f \in \text{End}(\mathbb{R}^4) \mid f(V) \subseteq W\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the digits 1, 2, 3, 4, and 5 without repetition, how many distinct five-digit "wave numbers" can be formed such that the tens digit and the thousands digit are each larger than their adjacent digits?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the linear operator L such that \( Lu = 0 \) for the function \( u = 2t + e^{-t} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the alternating group $A_5$, calculate the probability that a randomly selected pair of distinct elements generates a cyclic subgroup.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of all functions from the set of natural numbers \( \mathbb{N} \) to the set of real numbers \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five people are seated around a circular table. Let f represent the number of people sitting next to at least one female and m represent the number of people sitting next to at least one male, where f and m are non-negative integers. Calculate the number of unique ordered pairs (f, m).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tom decides to open a theater with a capacity of 500 seats. He knows it will cost $5 per square foot of space and he needs 12 square feet for every seat in his theater.	In addition to the cost of land and construction, he also needs to consider infrastructure costs, like electricity and plumbing, which add an additional 15% to the total cost of land and construction. Moreover, the government imposes a 7% tax on the total cost. Tom also plans to set aside 3% of the total cost for annual maintenance fees. Tom's partner will contribute 40% towards the overall cost. Calculate the total amount Tom will need to spend after considering all expenses and the contribution from his partner.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integer points (x, y, z) satisfy the equation \(35x + 55y + 77z = 1\) and are contained within a cube of side length 30 centered at the origin (0, 0, 0)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring isomorphisms from \( \mathbb{Z}^n \) to \( \mathbb{Z}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are playing a game in which you have  $3$  envelopes, each containing a uniformly random amount of money between  $0$  and  $1000$  dollars. (That is, for any real  $0 \leq a < b \leq 1000$ , the probability that the amount of money in a given envelope is between  $a$  and  $b$  is  $\frac{b-a}{1000}$ .) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be  $E$ . What is  $\lfloor E\rfloor,$  the greatest integer less than or equal to  $E$ ?

*Author: Alex Zhu*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the abscissa of convergence for the Dirichlet series: $$\sum_{m,n\in\mathbb{Z}}'\frac{m^6+n^6}{(m^2+n^2)^s}$$ where the prime on the summation symbol indicates that the term $(0,0)$ is omitted.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the ring to which \( R[X,Y,Z]/(X-Y^3,Z-Y^9) \) is isomorphic, given that substitution for \( X \) yields \( R[x,x^{1/3},x^3] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by some percentage while buying and by 20% while selling. His percentage profit is 50%. What is the percentage by which he cheats while buying?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), side \(BC = 28\). The angle bisector \(BL\) is divided by the intersection point of the angle bisectors of the triangle in the ratio \(4:3\) from the vertex. Find the radius of the circumscribed circle around triangle \(ABC\) if the radius of the inscribed circle is 12.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $y = f(x)$, where $x \in \mathbb{R}$, it satisfies $f(-x + 2) = f(-x)$. For $x \in [-1, 1]$, $f(x) = |x|$. Find the number of intersection points between $y = f(x)$ and $y = \log_7 x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_1 = 4, x_2 = 6$ and define $x_n$ for $n ≥ 3$ to be the least non-prime greater than $2x_{n−1} − x_{n−2}.$ Find $x_{1000}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of pairwise non-isomorphic 2-connected graphs with 20 vertices that do not contain a $K_4^-$ minor.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest possible integer $n$ such that there exist two bi-infinite sequences $A$ and $B$ with the following properties: sequence $A$ has a period of 1995, and any segment of length $n$ from sequence $B$ is contained in $A$. However, sequence $B$ is not periodic with period 1995. The sequences can consist of arbitrary symbols.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}.
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence is recursively defined as follows: \( t_{1} = 1 \). For \( n > 1 \), if \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \); if \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \). Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From point \( A \), a secant and a tangent are drawn to a circle with radius \( R \). Let \( B \) be the point of tangency, and \( D \) and \( C \) be the points where the secant intersects the circle, with point \( D \) lying between \( A \) and \( C \). It is known that \( BD \) is the angle bisector of angle \( B \) in triangle \( ABC \) and its length is equal to \( R \). Find the distance from point \( A \) to the center of the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of incongruent integers of order $2^{10}$ modulo $3 \cdot 2^{12}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the distance \(|PQ|\) where point \(P\) moves on the circle \(x^{2}+(y-4)^{2}=1\) and point \(Q\) moves on the ellipse \(\frac{x^{2}}{9} + y^{2} = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider $\mathbb{R}$ with the lower limit topology, and $Y=(0, 1] \subset \mathbb{R}$ with the subspace topology. Determine the closure $\overline{Y}$ in this topological space.
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Context:
A circle $C: (x-3)^2+(y+1)^2=2$ has two tangent lines passing through a point $P$ on the line $l: y=3x$. If the two tangent lines are symmetric with respect to line $l$, then the distance from point $P$ to the center of circle $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the monoid of natural transformations from the identity functor to itself in the category of abelian groups, \( \text{Id}_{\mathfrak{Ab}} \rightarrow \text{Id}_{\mathfrak{Ab}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( A D E \), point \( C \) is the midpoint of side \( A D \). Points \( B \) and \( C \) are marked on side \( A D \) such that triangles \( A B E \) and \( B C E \) have the same area, and triangle \( B D E \) is isosceles with base \( B E \). The perimeter of triangle \( A B E \) is 6 cm greater than the length of the broken line \( E B C \), and the perimeter of triangle \( A C E \) is 2 cm greater than the perimeter of triangle \( C D E \). Calculate the perimeter of triangle \( A D E \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \(10\sqrt{5}\,\alpha\), where \(\alpha = \lim_{n\to\infty}\sum_{m = n^2}^{2n^2}\dfrac{1}{\sqrt{5n^4 + n^3 + m}}.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group of 21 students, there are 5 excellent students, 10 good students, and 6 poorly performing students. In the upcoming exam, excellent students can only receive excellent grades. Good students can receive either good or excellent grades with equal probability. Poorly performing students can receive good, satisfactory, or unsatisfactory grades with equal probability. A student is randomly selected to take the exam. Find the probability that the student will receive a good or excellent grade (event \( A \)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, and \( \log_{m} n \) is rational, with the condition:

\[ k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}} \]

Determine the minimum possible value of the sum \( k + 5m + n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \int_0^1 \int_x^{\sqrt{x}} \frac{e^y}{y} \, dy \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the limit of the sequence defined by the following terms as a function of $n$ and $k$:
\[ a_1 = \text{LOG }n \]
\[ a_2 = \text{LOG }k \cdot \text{LOG }\text{LOG }n \]
\[ a_3 = \text{LOG }k \cdot \text{LOG }\text{LOG }k \cdot \text{LOG }\text{LOG }\text{LOG } n \]
\[ a_4 = \text{LOG }k \cdot \text{LOG }\text{LOG } k \cdot \text{LOG }\text{LOG }\text{LOG }k \cdot \text{LOG }\text{LOG }\text{LOG }\text{LOG }n \]
...
Express the limit in terms of big-Oh notation, if applicable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter with a radius of $3\text{ cm}$ at the origin of the coordinate plane. Her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the pyramid \( S A B C D \) is a quadrilateral \( A B C D \). The edge \( S D \) is the height of the pyramid. Find the volume of the pyramid, given that \( |A B| = |B C| = \sqrt{5} \), \( |A D| = |D C| = \sqrt{2} \), \( |A C| = 2 \), and \( |S A| + |S B| = 2 + \sqrt{5} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let's suppose that $x^2$ is inversely proportional to $y^4$. If $x = 6$ when $y = 2$, determine the value of $x^2$ when $y = 4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the period of the cycle of numbers that are coprime to 9 when you repeatedly take the digit sum of their powers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a trapezoid $ABCD$ and a point $M$ on the lateral side $AB$ such that $DM \perp AB$. It turns out that $MC = CD$. Find the length of the upper base $BC$ if $AD = d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The little sprite wrote down consecutive natural numbers starting from 1 in a row:

From the 30th digit, 2021 (the four digits marked in red) first appears. The little sprite continues to write, and from the $\qquad$th digit, 2022 first appears.
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 \( \int_0^{90} f(x) \, dx \), where the function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases}
(x-3k)^2 & \text{if } 3k-1 \le x < 3k+1 \\
1 & \text{if } 3k+1 \le x < 3k+2
\end{cases}
\]
for \( k \in \mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate: \( 7 \frac{4480}{8333} \div \frac{21934}{25909} \div 1 \frac{18556}{35255} = \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of times the argument of the complex function \( f = \ln[z(z+1)] \) increases when traversing a closed contour that encircles the branch cut \([-1,0]\) on the real axis in the counter-clockwise direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设函数 f(x)=\df{1-2x}{(1-x+x^2)^2}，则 \left[\df{f^{(2022)}(0)}{2022!}\right]=__________.
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Context:
Evaluate the polynomial \( P_{2010}(x) \), which satisfies \( P_{2010}(n) = 1^{2010} + 2^{2010} + \cdots + n^{2010} \) for positive integers \( n \), at \( x = -\frac{1}{2} \). Find the value of \( P_{2010}\left(-\frac{1}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{a\to1^+}\int_0^{\pi/2}\frac{dx}{(1+\cos2x)^a} \). Determine if the integral converges and find its value if it does.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I_C = \frac{1}{2\pi}\int_{0}^{2\pi}\left(\frac{d}{d\theta}\phi(\theta)\right) d\theta \), where \( \phi(\theta) = \arctan\left[\frac{3\cos(\theta)}{4(\cos(\theta)+\sin(\theta))}\right] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse Laplace transform of \( \frac{2ks}{(s^2 + k^2)^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the number of pieces originally in the first box.
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Context:
On a purchase of $110, a store offered a payment plan consisting of a certain down payment and 12 monthly payments of $10 each. The customer paid 9.090909090909092% of the purchase price in interest by using this plan. What was the down payment amount?
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Context:
How many elements of order p are there in GL(2,Zp)?
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Context:
Find the maximum value of \(x_{1}^{2}+x_{2}^{2} + x_{3}^{2}\) subject to the constraints \(x_{1}-x_{2}\leq0\), \(x_{2}-x_{3}\leq0\), \((x_{1} - x_{2})^{2} + (x_{2} - x_{3})^{2} = \delta\), and \(x_{1} + x_{2} + x_{3} = 0\), where \(\delta\) is a positive constant.
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Context:
Find a differentiable function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(n) = O(g(n))$ as $n \to \infty$, where $f(n) = \exp\left(2\log\log\omega(n) + O(1)\right)$ and $\omega(n)$ is the number of prime divisors of $n$. Provide your answer in terms of a function $g(n)$. \( \omega(n) \) is the number of distinct prime factors of \( n \).
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Context:
Let $a_{1}, a_{2}, \ldots, a_{2018}$ be non-negative real numbers whose sum is 1. Determine the maximum possible value of

$$
S=\sum_{i \neq j, i \mid j} a_{i} a_{j}
$$

where $i \mid j$ denotes that $i$ divides $j$.
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Context:
Evaluate the limit: $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + \frac{1}{3}\log{\left(1+3x^3\right)}-1}{2\sin{\left(3x^2\right)}-3\arctan{\left(2x^2\right)}}}$$ without using L'Hôpital's rule.
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Context:
At least how many days must a festival last if 11 theatrical groups participate, and each group must see at least one performance of every other group during their days off?
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Context:
Find the degree of the polynomial \( p(x) \) that satisfies the equation \( (x)p(x - 1) = (x - 15) p(x) \).
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Context:
The number of the members of a club is more than 20 and less than 50. When 5 people sit at a table, other people are exactly divided into 6-people groups or 7-people groups. If the members are divided into groups of a certain number of people, 7 people will be left. How many people are in this new group size?
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Context:
Determine the bases $b$ for which the recursive sequence defined by $a_0 = 1$ and $a_{n+1} = \log_{b} (1 + a_n)$ results in the series $\sum_{n=0}^\infty a_n$ having a finite sum.
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Context:
In an isosceles triangle, the angle at the base is $\alpha$. The altitude drawn to the base is greater than the radius of the inscribed circle by $m$. Find the radius of the circumscribed circle.
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Context:
In Möbius geometry, at how many points do two distinct lines intersect?
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Context:
Find \( \lim_{\varepsilon\rightarrow 0^+} f_\varepsilon \) in \( D'(\mathbb{R}) \), where \( f_\varepsilon(x)=\frac{\sin{\varepsilon x}}{x} \).
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Context:
Determine the radius of convergence for the Taylor series expansion of \( f(z) = \frac{1}{1+e^z} \) around \( z = \pi i \).
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Context:
Find the exact y-coordinate of a circle with radius \( \frac{1}{4} \) that makes it tangent to the function \( y = e^{-x^2} \).
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Context:
Evaluate the limit: \[ \lim _{p\to \infty }\left(p\ln \left(e\left(1+\frac{1}{p}\right)^{1-p}\right)\right) \]
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Context:
Evaluate the double integral \( \iint_R 8(x^2+y^2)(x^2-y^2) \, dA \), where \( R \) is the region in the first quadrant of the \( xy \)-plane bounded by the circles \( x^2+y^2=1 \), \( x^2+y^2=4 \), and the lines \( y=x+1 \) and \( y=x-1 \).
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Context:
Find the gradient of the function \( f(x) = \frac{1}{2}\|G(x)\|_2^2 \) in terms of \( G(x) \) and \( G'(x) \), where \( G(x): \mathbb{R}^n \rightarrow \mathbb{R}^n \) is a twice continuously differentiable function.
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Context:
Find all natural numbers \( m \) such that \( 5^{m-2} - 2 \equiv 0 \pmod{m} \).
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Context:
Determine the number of iterations required to reach an error of $10^{-6}$ when approximating Dottie's number using the iterative formula $\epsilon^{(k)}\approx(-\sin D)^k\epsilon$, where $D$ is Dottie's number.
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Context:
Evaluate the limit: $$\lim_{x\to\infty}\left(\sum_{n=1}^{\infty}\left(\frac{x}{n}\right)^n \right)^{1/x}$$
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Context:
A car starts its trip in town A and travels towards town B. During the first 4 hours, the car averages 75 mph. The car then slows down to an average speed of 60 mph for the next 3 hours and finally averages 50 mph for the remaining 2 hours before reaching town B. Upon arrival, the car stops for a 30 minute break, then heads towards town C, covering a distance with varying average speeds of 40 mph for the first hour, 55 mph for the second hour and 65 mph for the next 1.5 hours. The car stops again in town C for a 45 minute break and finally goes back to town A with a consistent average speed of 70 mph. Considering the entire trip from town A to town B, town B to town C, and town C back to town A, determine the total time it takes for the complete journey including breaks.
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Context:
From point \( K \) on side \( AC \) of triangle \( ABC \), perpendiculars \( KL_1 \) and \( KM_1 \) are dropped to sides \( AB \) and \( BC \) respectively. From point \( L_1 \), perpendicular \( L_1 L_2 \) is dropped to \( BC \), and from point \( M_1 \), perpendicular \( M_1 M_2 \) is dropped to \( AB \).

It turns out that triangles \( BL_1M_1 \) and \( BL_2M_2 \) are similar (point \( L_1 \) in the first triangle corresponds to point \( M_2 \) in the second). Additionally, \( BL_2 = 6 \) and \( L_2M_1 = 4 \). Find \( L_1L_2 \).
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Context:
You start with a single piece of chalk of length 1. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have 8 pieces of chalk. What is the probability that they all have length $\frac{1}{8}$?
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Context:
How many lines pass through point A (1, 2) and have intercepts on the two coordinate axes with equal absolute values?
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Context:
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(\overline{BC}\).
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Context:
Given a function $f(x)$ which is an odd function defined on $(-\infty,0) \cup (0,+\infty)$ and is monotonically decreasing on $(0,+\infty)$, with the condition $f(4)=0$. If $f(x-3) \leqslant 0$, then the range of $x$ is ______.
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Context:
Let $x$ be a real number, $P=e^x+e^{-x}$, $Q=(\sin x+\cos x)^2$. Determine the relationship between $P$ and $Q$, expressing your answer as an inequality or equality relationship between $P$ and $Q$.
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Context:
Find the positive integer $k$ such that the function \( f(19, k) = 20 \), where \( a_{n} = 4 - 3n \) for all integers \( n \geq 1 \) and \( f(x, y) = x + \sum_{i=1}^{\infty}\left [\left(\frac{\prod_{j=1}^{i}a_{j}}{3^i\cdot i!} \right )x^{a_{i+1}}y^{i}\right] \) for all real \( x \) and \( y \).
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Context:
Determine the cardinality of the subset of the interval [0,1] consisting of infinite decimal expansions that use only the digits 2 and 5.
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Context:
Two numbers are chosen from the interval $[0,1]$. The probability that the sum of their squares is greater than $l$ is $\_\_\_\_\_\_$.
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Context:
A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column. Please provide the value when n = 1.
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Context:
In a cyclic quadrilateral $PQRS$, the sides are given as $PQ = RS = x$, $QR = 2x$, and $PS = 1$. Determine the value of $x$.
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Context:
Find the sum of all integers \( n \) such that \( n^2+2n+2 \) divides \( n^3 + 4n^2 + 4n - 14 \).
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Context:
With one mighty blow, Maria cracked open the pinata, and countless candies spilled all over the floor. There were 50 red candies, 35 less than three times as many yellow candies as red candies, and a third as many blue candies as twice the number of yellow candies. Additionally, there were 20 green candies, which was twice the number of purple candies. Finally, 10 silver candies rolled to the side, completing the assortment.

If Carlos ate all of the yellow candies and half of the green candies, how many candies remained?
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Context:
Evaluate the limit: $$\lim_{T\to \infty}\frac{\int_0^T\sin^2(s) \exp(-s)\int_0^s \sin(\sin a)\exp(a)\,\mathrm{d}a\,\mathrm{d}s}{T}$$
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Context:
Compute the number of nonempty subsets $S \subseteq \{-10, -9, -8, \ldots, 8, 9, 10\}$ that satisfy $|S| + \min(S) \cdot \max(S) = 0$.
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Context:
A Markov chain $X_0,X_1,X_2,\ldots$ has the transition probability matrix:
$$P=\begin{Vmatrix}
0.3 & 0.2 & 0.5\\
0.5 & 0.1 & 0.4 \\
0 & 0 & 1
\end{Vmatrix}$$
and starts in state $X_0=0$. Eventually, the process will end up in state 2. What is the probability that the time $T=\text{min}\{n\geq 0;X_n=2\}$ is an odd number?
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Context:
Suppose a farmer's cow gives birth to a heifer at the age of two. Subsequently, both the original cow and each heifer will produce one heifer per year starting at the age of two. Given this scenario, determine the total offspring of the original cow after 25 years.

The time is counted from the birth of the first cow, and no cattle are slaughtered within this period.
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Context:
If the function $y=f(x)$ ($x \in \mathbb{R}$) satisfies $f(x+2)=f(x)$, and when $x \in [-1,1]$, $f(x)=1-x^2$, and the function $g(x)=\lg|x|$, then the number of intersection points between the graphs of $y=f(x)$ and $y=g(x)$ in the interval $[-5,5]$ is.
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Context:
An ant starts out at \((0,0)\). Each second, if it is currently at the square \((x, y)\), it can move to \((x-1, y-1)\), \((x-1, y+1)\), \((x+1, y-1)\), or \((x+1, y+1)\). In how many ways can it end up at \((2010,2010)\) after 4020 seconds?
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Context:
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega.$ Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A,$ $\omega_B,$ and $\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\sqrt{a} - \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C, W, WA, WB, WC, X, Y, Z; A = 18*dir(90); B = 18*dir(210); C = 18*dir(330); W = (0,0); WA = 6*dir(270); WB = 6*dir(30); WC = 6*dir(150); X = (sqrt(117)-3)*dir(270); Y = (sqrt(117)-3)*dir(30); Z = (sqrt(117)-3)*dir(150); filldraw(X--Y--Z--cycle,green,dashed); draw(Circle(WA,12)^^Circle(WB,12)^^Circle(WC,12),blue); draw(Circle(W,18)^^A--B--C--cycle); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$\omega$",W,1.5*dir(270),linewidth(4)); dot("$\omega_A$",WA,1.5*dir(-WA),linewidth(4)); dot("$\omega_B$",WB,1.5*dir(-WB),linewidth(4)); dot("$\omega_C$",WC,1.5*dir(-WC),linewidth(4)); [/asy] ~MRENTHUSIASM ~ihatemath123
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Context:
Let's call a number "elegant" if all the digits in its decimal representation are distinct and in ascending order. Are there more four-digit elegant numbers or five-digit elegant numbers?
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Context:
Let  $ABC$  be a triangle with  $AB = 13$ ,  $BC = 14$ ,  $CA = 15$ . Let  $H$  be the orthocenter of  $ABC$ . Find the radius of the circle with nonzero radius tangent to the circumcircles of  $AHB$ ,  $BHC$ ,  $CHA$ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{The radius of the circle is \(\frac{65}{16}\).}


Context:
You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut off all of its claws and legs. In how many ways can you defeat the giant enemy crab?
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Context:
Given that \( S \) is a set consisting of \( n \) (\( n \geq 3 \)) positive numbers. If three different elements in \( S \) can form the sides of a triangle, then \( S \) is called a "triangular set." Consider the set of consecutive positive integers \(\{4,5, \cdots, m\}\). All of its 10-element subsets are triangular sets. What is the maximum possible value of \( m \)?
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Context:
一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 $p(0 \leq p \leq 1)$ 的概率消失，有 $\frac{1-p}{3}$的概率保持不变，有 $\frac{1-p}{3}$ 的概率分裂成两个，有 $\frac{1-p}{3}$ 的概率分裂成三个。对所有新产生的生物每天也会发生上述事件。假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 $\frac{1}{2}$ ，则 $p$ 至多为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Three mutually perpendicular lines intersect at a point \( O \). Points on one of these lines that are at a distance of 1 unit from \( O \) are denoted by \( X_{1} \) and \( X_{1}^{\prime} \), and points 2 units from \( O \) are denoted by \( X_{2} \) and \( X_{2}^{\prime} \). Similarly, points on the other two lines are labeled \( Y_{1}, Y_{1}^{\prime}, Y_{2}, Y_{2}^{\prime} \) and \( Z_{1}, Z_{1}^{\prime}, Z_{2}, Z_{2}^{\prime} \). Planes are formed with every possible combination of a point with a 1-index from one line and a 2-index point from each of the other two lines, for example, passing through \( X_{1}^{\prime} \), \( Y_{2} \), and \( Z_{2}^{\prime} \).

Determine the volume of the solid enclosed by these planes.
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Context:
A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest distance that the grasshopper can be away from the point $2010$.
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Context:
Given a signal x(t) with a Laplace transform X(s) that has four distinct poles, and knowing that x(t) includes an impulse at t = 0, determine the minimum number of zeros that X(s) must have.
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Context:
Points R, S, and T are vertices of an equilateral triangle, and points X, Y, and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? Express your answer as a single integer.
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Context:
Find the equation of the tangent line to the curve given by $$(x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2$$ at the point $(1, 1)$.
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Context:
Use the 0.618 method to find the optimal addition amount for an experiment. If the current optimal range is $[628, 774]$, and the good point is 718, then the addition point value to be tested at this time is.
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Context:
Evaluate the nested limit: $$\lim_{m\to \infty }\left[\lim_{y\to \,0}\left(1+\cos^2y+\cos^22y+\ldots+\cos^2my\right)^{1/(m^3y^2)}\right]$$
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Context:
Let $p, q, r$ be real numbers satisfying the equations \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] and \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] If $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

*Author: Alex Zhu*
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Context:
Evaluate the integral \(\int_{1}^{10} [2x] \{x/2\} \, dx\), where \([2x]\) is the greatest integer function and \(\{x/2\}\) is the fractional part function.
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Context:
[ Equilateral (Regular) Triangle ]

In an equilateral triangle $D E F$, a circle of radius $r$ is inscribed. This circle is externally tangent to three other circles of the same radius at the points of tangency with the sides of the triangle. The centers of the external circles are $O_{1}, O_{2}, O_{3}$, respectively. Find the area of the hexagon formed by the intersection of triangles $D E F$ and $O_{1}, O_{2}, O_{3}$.
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Context:
A cyclist is walking at a constant rate of 15 miles per hour. A car traveling in the same direction along the same path at 60 miles per hour passes the cyclist. The car stops to wait for the cyclist for some time after passing her, while the cyclist continues to go at her constant rate. The car must wait 18 minutes until the cyclist catches up. How many minutes does the car initially wait for the cyclist after passing her?
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Context:
Suppose the sequence $\{a\_n\}$ satisfies the following condition: for any $n\in\mathbb{N}^*$, there are only finitely many positive integers $m$ such that $a\_m < n$ holds. Let $b\_n$ denote the number of such $m$'s. This gives rise to a new sequence $\{b\_n\}$. For example, if the sequence $\{a\_n\}$ is $1$, $2$, $3$, $...$, $n...$, then the sequence $\{b\_n\}$ is $0$, $1$, $2$, $...$, $n-1$, $...$ Now, it is known that the sequence $\{a\_n\}$ is a geometric sequence with $a\_2=2$, $a\_5=16$. Find the number of positive integers $i$ such that $b\_i=2016$ in the sequence $\{b\_n\}$.
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Context:
Consider a staircase of height \( n \), which consists of all the cells of an \( n \times n \) square that lie above the diagonal. Determine the number of ways to divide this staircase into several rectangles, such that the sides of the rectangles lie along the grid lines and all the rectangles have pairwise distinct areas.
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Context:
Determine the number of five-digit sequences \(d_1d_2d_3d_4d_5\), where \(d_i\in\{0,1,\ldots,9\}\), that do not contain any palindromic subsequence of length 3 or greater.
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Context:
Given the hyperbola $C: \frac{x^2}{a^2}-\frac{y^2}{a^2}=1(a>0)$, with left and right vertices $A_{1}$ and $A_{2}$ respectively. Point $P$ is a point on the hyperbola $C$ other than the vertices, and $\angle A_{1}PA_{2}=3\angle PA_{1}A_{2}$. Find $\cos 5\angle PA_{1}A_{2}$.
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Context:
Evaluate the limit: $$\lim_{x\to-\infty} x\log(-e^x + 1)+\operatorname{Li}_2(e^x)-\frac12x^2$$
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Context:
Determine the value of \(a\) such that the range of the function \(f(x) = \frac{1}{3x^4 - 8ax^3 + 12a^2 + x^2 + a}\) is completely contained within the interval \([0, \frac{1}{3}]\).
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Context:
Determine the number of unique deal combinations in the Klondike solitaire card game, taking into account both the order of the cards and the game's rules.
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Context:
Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a standard normal distribution $N(0,1)$. Define $T = \sum_{i=1}^5 (X_i - \bar{X})^2$, where $\bar{X}$ is the sample mean. Calculate $E(T^2 \bar{X}^2)$. Provide your answer as a single value.
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Context:
In the convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) are equal. Moreover, \(\angle BAC = \angle ADB\) and \(\angle CAD + \angle ADC = \angle ABD\). Find the angle \(BAD\).
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Context:
Find the flux of the vector field \( \mathbf{F} = 3\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) across the part of the plane \( 1x + 3y + z = 3 \) that lies in the first octant, oriented upwards.
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Context:
Calculate the limit of the following expression as $n$ approaches infinity: $$\lim_{n\rightarrow\infty}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)\sqrt{n}}$$
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Context:
A standard deck contains 52 cards numbered from 1 to 52. You randomly select 26 cards without replacement. Determine the number of ways to select these 26 cards such that the sum of the reciprocals of their numbers is exactly 2.
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Context:
Suppose \( x_{1}, x_{2}, \ldots, x_{2011} \) are positive integers satisfying
\[ x_{1} + x_{2} + \cdots + x_{2011} = x_{1} x_{2} \cdots x_{2011} \]

Find the maximum value of \( x_{1} + x_{2} + \cdots + x_{2011} \).
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Context:
Two regular tetrahedrons  $A$  and  $B$  are made with the 8 vertices of a unit cube. (this way is unique)

What's the volume of  $A\cup B$ ?
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Context:
Calculate the residue of the function \(-\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s}\) at \(s = 1\).
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Context:
Find the smallest positive number $\lambda$, such that for any 12 points on the plane $P_1, P_2, \ldots, P_{12}$ (points can overlap), if the distance between any two of them does not exceed 1, then the inequality \( \sum_{1 \le i < j \le 12} |P_iP_j|^2 \le \lambda \) holds.
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Context:
Determine the number of non-ordered quadruples \((a, b, c, d)\) of non-negative integers that satisfy the equation \(a + b + c + d = 18\).
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Context:
Calculate the flux of the vector field \( \mathbb{v} = \operatorname{curl}\mathbb{u} \) through the surface of the solid bounded by the paraboloid \( z = 1 - x^2 - y^2 \) and the planes \( x = 0 \) and \( z = 0 \). The vector field \( \mathbb{u} \) is given by \( \mathbb{u} = (xy, xz^2, x^2y) \).
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Context:
Let \( f(x,y,z,u)=\frac{24}{(1+x+y+z+u)^5}, x>0,y>0,z>0,u>0 \) be a density function. Calculate the probability \( P(X+Y+Z+U \ge 1) \).
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Context:
Evaluate the double integral \( \int_{0}^{1}\mathrm{d}y\!\int_{0}^{1}\frac{x-y}{(x+y)^3}\mathrm{d}x. \)
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Context:
Determine the number of functions \( g \) from the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) to itself such that \( g(g(g(x))) \) is a constant function. Find the remainder when this number is divided by 1000.
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Context:
Let \( f(x) = ax^2 - bx + c \), where \( a, b, \) and \( c \) are positive integers. If \( f(p) = f(q) = 0 \) with \( 0 < p < q < 1 \), determine the smallest possible value of \( a \). The vertex of the parabola is at \( \left(-\frac{b}{2a}, \frac{b^2 - 4ac}{4a}\right) \), which lies in the interval \((0, 1)\). Additionally, \( f(0) > 0 \) and \( f(1) > 0 \), implying \( c > 0 \) and \( a - b + c > 0 \). Moreover, the discriminant \( b^2 - 4ac \) must be positive.
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Context:
Evaluate the double integral \( \iint_D \left(\sqrt{a^2-x^2-y^2}-\sqrt{x^2+y^2}~\right)dxdy \) where \( D_{xy} \) is the disk \( x^2+y^2\le a^2 \), using a change of variables.
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Context:
Given $\Delta ABC$ equilateral, with $X\in[A,B]$. Then we define unique points Y,Z so that $Y\in[B,C]$, $Z\in[A,C]$, $\Delta XYZ$ equilateral.

If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.
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Context:
Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n + 1/n} \) is convergent.
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Context:
Let  $P$  be a regular  $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of  $P$  into two parts, each composed of an odd number of sides of  $P$ . The sides of  $P$  are also called *good*.
Suppose  $P$  has been dissected into triangles by  $2003$  diagonals, no two of which have a common point in the interior of  $P$ . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
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Context:
Given the moment generating function of a multivariate normal random variable \((X, Y, Z)\):
\[ M(t_1, t_2, t_3) = \exp(t_1 - 2t_3 + t_1^2 + t_1t_2 + \frac{3}{2}t_2^2 - t_1t_3 + \frac{5}{2}t_3^2) \]
Find the value of \(a\) such that \(X\) and \(Y + aX\) are independent.
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Context:
Evaluate the integral \( \int_{|z|=1} \frac{\sin(z)}{z^3} \, dz \) over the unit circle \(|z|=1\) in the complex plane without using the residue theorem.
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Context:
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
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Context:
Find the $x^8$ term in the expansion of \((x+\frac{1}{\sqrt[3]{x}})^5(2x - \frac{1}{x^2})^7\) using Newton's binomial theorem.
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Context:
Let $p$, $q$, and $r$ be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if $x > 2$ or $3 \le x \le 5$. Given that $p < q$, find the value of $p + q + 2r$.
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Context:
Find the number of distinct ways to color the vertices of a 12-vertex cycle graph using 8 different colors, where two colorings are considered identical if one can be transformed into the other by a rotation or reflection of the cycle.
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Context:
Solve the equation: \[ 2^{1+\sin(3x)} - 8^{\sin(x)+\frac{1}{3}} + \sin(x) \cos(2x) = \sin(x) \]
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Context:
Let $V$ be the region defined by $\frac{1}{4} \leq x^2 + y^2 + z^2 \leq 1$ in $\mathbb{R}^3$. Consider the vector field $\vec{F} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2 + y^2 + z^2)^2}$ for $(x, y, z) \in V$. The surface $S$ is the part of the boundary of $V$ given by $x^2 + y^2 + z^2 = \frac{1}{4}$, with $\hat{n}$ denoting the outward unit normal vector. Compute the surface integral $\int\int_S \vec{F} \cdot \hat{n} \, dS$. \(\text{Express your answer as a single number.}\)
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Context:
Given a triangle ABC with points P, Q, R lying on the sides AB, BC, CA, respectively, and BP = PQ = QR = RC = 1, find the maximum possible area of triangle ABC. Express your answer as a single number.
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Context:
Find the tenth digit (from the right) of the sum \( \sum_{k=1}^{30} (k!) \) using modular arithmetic.
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Context:
Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly chosen inside the quadrilateral \( AA'B'B \), what is the probability that \( M \) lies inside the triangle \( FA'B' \)?
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Context:
Evaluate the contour integral \(\int_{|z|=1}\dfrac{\sin\pi z}{z^{2}\cos\pi z}dz\) by considering all the singularities of the function within the contour.
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Context:
Let $M$ be the real vector space of $2\times3$ matrices with real entries. Define the linear transformation $T:M\rightarrow M$ by \[ T\left(\begin{bmatrix} x_{1} & x_{2} & x_{3} \\\ x_{4} & x_{5} & x_{6}\end{bmatrix}\right)=\begin{bmatrix}-x_{6}& x_{4}& x_{1}\\ x_{3}& x_{5}& x_{2}\end{bmatrix}. \] Calculate the determinant of the linear transformation $T$. \(Note: Consider the transformation as a linear map on a vector space of dimension 6.\)
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Context:
Given that radioactive decay of an element occurs according to a Poisson process with a rate of 10,000 decays per second, calculate the approximate probability that the millionth decay occurs within 100.2 seconds.
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Context:
What least number should be added to 1024, so that the sum is completely divisible by the product of the following prime numbers: 17, 23, and 29?
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Context:
Consider two independent and identically distributed random variables $X_1$ and $X_2$ with a normal distribution $N(\theta, 1)$. Additionally, let $V$ be a Bernoulli random variable with $P(V=1) = \frac{1}{4}$, independent of $X_1$ and $X_2$. Define $X_3$ as follows:
- $X_3 = X_1$, if $V=0$
- $X_3 = X_2$, if $V=1$

We are testing the hypothesis $H_0: \theta = 0$ versus $H_1: \theta = 1$. Reject $H_0$ if $\frac{X_1 + X_2 + X_3}{3} > C$. Find the critical value $C$ such that the test size is 0.05.
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Context:
Determine the cardinality of the set of functions from the natural numbers \( \mathbb{N} \) to the real numbers \( \mathbb{R} \), assuming the generalized continuum hypothesis (GCH).
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Context:
Let $n$ be an integer greater than $1$ and $x$ be an integer between $1$ and $10^{12}$. Calculate the probability that $x^{2n+1}$ ends with the digit sequence $11$. Express your answer as a fraction.
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Context:
For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?
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Context:
Determine the number of elements of order 5 in a group of order 90.
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Context:
Calculate the flux of the vector field \( \vec{F}(x, y, z) = \langle x, y, z \rangle \) across the surface of the cone defined by \( z = \sqrt{x^2 + y^2} \) for \( 0 \le z \le 1 \) using the divergence theorem.
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Context:
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_5=3$ and $S_{10}=40$, then the minimum value of $nS_n$ is .
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Context:
Let \( p(x) \) be a polynomial of degree less than 1992 such that \( p(0) \), \( p(1) \), and \( p(-1) \) are all non-zero. The 1992nd derivative of \( \frac{p(x)}{x^3 - x} \) is expressed as \( \frac{f(x)}{g(x)} \) for polynomials \( f(x) \) and \( g(x) \). Determine the smallest possible degree of \( f(x) \).
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Context:
Find a matrix $T$ that transforms vectors in a 2D vector space with a weighted inner product defined by a positive-definite matrix $Q$ such that the distance between vectors in the transformed space matches the distance in the Euclidean space.
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Context:
In the parallelogram \(ABCD\), points \(K, L,\) and \(M\) are the midpoints of sides \(AB, BC,\) and \(CD\) respectively. Given that quadrilaterals \(KBLM\) and \(BCDK\) are inscribed, determine the ratio \(AC : AD\).
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Context:
Find the last two digits of the sum of the eighth powers of 100 consecutive integers.
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Context:
Let $A$ be a $7 \times 7$ matrix satisfying $2A^2 - A^4 = I$. If $A$ has two distinct eigenvalues and each eigenvalue has geometric multiplicity 3, determine the number of non-zero entries in the Jordan Canonical Form of $A$. \( \text{(Express your answer as a single integer.)} \)
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Context:
The left and right foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ are $F_1$ and $F_2$, respectively. The chord $AB$ passes through point $F_1$. If the circumference of the incircle of $\triangle ABF_2$ is $\pi$, and the coordinates of points $A$ and $B$ are $(x_1,y_1)$ and $(x_2,y_2)$, respectively, then find the value of $|y_1-y_2|=$ __________.
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Context:
Find the largest positive integer $N$ for which there exists a harmonious coloring for a regular $N$-gon, where each side and each diagonal is colored in one of 2018 different colors, and no two sides or diagonals are all the same color.
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Context:
In an equilateral triangular prism \( S-ABC \), the lateral edges are equal in length to the edges of the base. If \( E \) and \( F \) are the midpoints of \( SC \) and \( AB \) respectively, find the angle between the skew lines \( EF \) and \( SA \).
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Context:
Determine the minimum value of $d - n$ such that the probability that the images of $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$ remain linearly independent after projection onto a random $d$-dimensional subspace is at least $1 - \frac{1}{\mathrm{poly}(n)}$. Provide your answer in terms of $n$.
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Context:
Person A, Person B, and Person C compare the number of candies they have.

Person A says: "I have 13 candies, 3 less than Person B, and 1 more than Person C."
Person B says: "I do not have the least, Person C and I differ by 4 candies, Person A has 11 candies."
Person C says: "I have fewer candies than Person A, Person A has 10 candies, Person B has 2 more candies than Person A."

If each person has one false statement out of the three they made, how many candies does the person with the fewest candies have?
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Context:
The Fibonacci numbers are defined by $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exist unique positive integers $n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}$ such that $\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}$. Find $n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of coin tosses needed to obtain the sequence THH (T: tail, H: head) when flipping a fair coin indefinitely.
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Context:
Let $x, y, z$ be positive real numbers such that $(1 + x + y + z)(1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}) = 25$. Find the value of $x + y + z$.
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Context:
If a sequence $\{a_n\}$ satisfies $$\frac {1}{a_{n+1}}- \frac {1}{a_{n}}=d$$ (where $n\in\mathbb{N}^*$, $d$ is a constant), then the sequence $\{a_n\}$ is called a "harmonic sequence". Given that the sequence $\{\frac {1}{x_{n}}\}$ is a "harmonic sequence", and $x_1+x_2+\ldots+x_{20}=200$, the maximum value of $x_3x_{18}$ is \_\_\_\_\_\_.
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Context:
Given the function \( f(x) = ax^2 + bx + c \), where \( a, b, c \) are real numbers and \( a > 100 \), determine the maximum number of integers \( x \) such that \( |f(x)| \leq 50 \).
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Context:
Determine the projective dimension of the ring $A = \mathbb{Z}_l[T,T^{-1}]$, where $l$ is a prime number and $\mathbb{Z}_l$ is the ring of $l$-adic integers.
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Context:
Find the torque around the x-axis of the triangle with vertices (0, 0), (1, 4), and (1, 0), assuming the density is 1.
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Context:
In a life test, 100 items are tested simultaneously, each with a lifetime that is an independent exponential random variable with a mean of 200 hours. The test ends after 5 failures. Let T be the time at which the test concludes. Calculate the expected value of T, denoted as E[T].
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Context:
Let $S$ be the set of all ordered pairs $(x,y)$ of integer solutions to the equation $$6x^2+y^2+6x=3xy+6y+x^2y.$$  \$S$ contains a unique ordered pair $(a,b)$ with a maximal value of $b$. Compute $a+b$.
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Context:
Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$.
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Context:
Calculate the integral of the pullback of the form \( \omega = \frac{xdy-ydx}{x^2+y^2} \) over the interval \([0, 2\pi]\), where \( g(t) = (e^t \sin(17t), e^{t^2} \cos(17t)) \).
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Context:
Determine the cardinality of a sigma algebra generated by countably many sets.
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Context:
Two equilateral triangles are inscribed in a circle of radius \( R \) such that when they intersect, each side is divided into three equal segments. Find the area of the intersection of these triangles.
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Context:
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC=3\angle CAD$, $ AB=CD$, $ \angle ACD=\angle CBD$. Find angle $ \angle ACD$
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Context:
Find the simple closed curve that minimizes the line integral \( \int_C F \cdot d\mathbf{x} \), where \( F:\mathbb{R}^2 \to \mathbb{R}^2 \) is defined by \( F(x,y) =\begin{bmatrix} x^2 +y^2 +4y \\ x^3 +xy^2 +2xy\end{bmatrix} \).
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Context:
Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?
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Context:
Let $x, y, z > 0$ such that $x + y + z = 1$. Find the minimum value of the expression \[ T = \sqrt[5]{\frac{x}{1 - x}} + \sqrt[5]{\frac{y}{1 - y}} + \sqrt[5]{\frac{z}{1 - z}}. \]
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Context:
Determine the Galois group of the polynomial \((x^3 - 2)(x^2 + 3)\) over \(\mathbb{Q}\).
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Context:
Evaluate the line integral \( \int_\gamma y\,dx + z\,dy + x\,dz \) using Stokes' theorem, where \( \gamma = \{(x,y,z) : x^2 + y^2 + z^2 = 9, x + y + z = 0\} \) with positive orientation.
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Context:
Find the limit points of the set \( B = \left\{ \frac{m}{n} - \frac{n}{m} \mid m, n \in \mathbb{N} \right\} \subseteq \mathbb{R} \) where \( \mathbb{R} \) has the standard metric.
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Context:
已知 $x_{1}, x_{2}, x_{3}, x_{4}$ 是两两不等的实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 是 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意一个排列，若记 $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left(x_{i 3}, x_{i 4}\right\}\right\}, Y=$ $\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right\}$ ，则 $X>Y$ 的概率是$\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Determine the value of Arg(f(z)) after a complete counter-clockwise rotation of the point z about the origin, starting at z = 2 with the angle there being 0, where \( f(z) = \sqrt{z^2+2z-3} \).
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Context:
The sides of a rectangular park are in the ratio 3 : 2 and its area is 3750 sq m. The cost of fencing it at a certain price per meter is $100. What is the cost of fencing per meter in ps?
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Context:
Given non-negative real numbers \( a, b, c, x, y, z \) such that \( a+b+c = 1 \) and \( x+y+z = 1 \), determine the maximum value of the expression \( \left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right) \).
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Context:
Given the function $y=\log_{a}(x+3) (a > 0,a\neq 1)$, its graph passes through point $A$. If point $A$ also lies on the graph of the function $f(x)=3^{x}+b$, then $f(\log_{3}2)=$ _____ .
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Context:
Determine the values of \(x\) for which the infinite series \(\sum_{n=1}^{\infty} -\sin(nx)\) converges.
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