Context:
In rectangle $JKLM$, $P$ is a point on $LM$ so that $\angle JPL=90^{\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$. [asy]
size(7cm);defaultpen(fontsize(9));
real vd = 7/9 * 12;
path extend(pair a, pair b) {return a--(10 * (b - a));}

// Rectangle
pair j = (0, 0); pair l = (0, 16); pair m = (24 + vd, 0); pair k = (m.x, l.y);
draw(j--l--k--m--cycle);
label("$J$", j, SW);label("$L$", l, NW);label("$K$", k, NE);label("$M$", m, SE);

// Extra points and lines
pair q = (24, 7); pair v = (q.x, 0); pair u = (q.x, l.y);
pair r = IP(k--m, extend(j, q));
pair p = (12, l.y);
draw(q--j--p--m--r--cycle);draw(u--v);
label("$R$", r, E); label("$P$", p, N);label("$Q$", q, 1.2 * NE + 0.2 * N);label("$V$", v, S); label("$U$", u, N);

// Right angles and tick marks
markscalefactor = 0.1;
draw(rightanglemark(j, l, p)); draw(rightanglemark(p, u, v)); draw(rightanglemark(q, v, m));draw(rightanglemark(j, p, q));
add(pathticks(l--p, 2, spacing=3.4, s=10));add(pathticks(p--u, 2, spacing=3.5, s=10));

// Number labels
label("$16$", midpoint(j--l), W); label("$25$", midpoint(j--p), NW); label("$15$", midpoint(p--q), NE);
label("$20$", midpoint(j--q), 0.8 * S + E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}+b^{2}+4 \sqrt {2}=c^{2}$ and $ab=4$, find the minimum value of $\frac {\sin C}{\tan ^{2}A\cdot \sin 2B}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere. The base of the prism is \( A B C \), and the lateral edges are \( A A_{1}, B B_{1}, C C_{1} \). Segment \( C D \) is the diameter of the sphere. Points \( K \) and \( L \) are the midpoints of edges \( A A_{1} \) and \( A B \) respectively. If \( D L = \sqrt{2} \) and \( D K = \sqrt{3} \), find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any positive integer, if the number of $2$'s in its digits is greater than the number of $3$'s in its digits, we call that a **good** number. Conversely, if the number of $3$'s in its digits is greater than the number of $2$'s, we call that a **bad** number. For example, the number $2023$ has two $2$'s and one $3$, so $2023$ is a good number. However, in the number $123$, both the number of $2$'s and $3$'s are one, so $123$ is neither a good number nor a bad number.

Find the difference between the number of good numbers and bad numbers among the positive integers not greater than $2023$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral: $$\int_{0}^{1}\int_{0}^{1}e^{\max(x^{2},y^{2})}dxdy$$
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Context:
Let $B_t$ be a standard Brownian motion, and let $L(x, t)$ denote the local time at $x$ for time $t$. Define $L_t = L(0, t)$. Determine whether the following holds with probability one: $$\limsup_{t \to 0} \frac{L_t}{\sqrt{t}} = \infty.$$
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Context:
Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$.
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Context:
Given the function $f(x) = 3^{x} + \frac{1}{2}x - 2$, identify the interval where the zero point of the function lies. Express your answer in the form (a, b).
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Context:
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 16 and \( \sin \alpha = \frac{\sqrt{55}}{8} \).
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Context:
The circle touches the extensions of two sides $AB$ and $AD$ of the square $ABCD$ with a side length of $\sqrt{2+\sqrt{2}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is $45^\circ$, and it is known that $\sin 22.5^\circ = \frac{\sqrt{2 - \sqrt{2}}}{2}$.
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Context:
Determine the asymptotic behavior of the ratio \( \frac{{2n \choose k}}{{n \choose k}} \) as \( n \to \infty \), given that \( k = O(f(n)) \) for some function \( f(n) \).
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Context:
Given an ellipse and a hyperbola,\[\frac{x^2}{16} - \frac{y^2}{25} = 1\]and \[\frac{x^2}{K} + \frac{y^2}{25} = 1\], have the same asymptotes. Find the value of $K$.
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Context:
Calculate the rate of heat flow inward across the cylindrical surface defined by \(y^2 + z^2 = 324\) for \(-10 \le x \le 3\), given the temperature distribution \(u(x,y,z) = 4y^2 + 4z^2\) and the conductivity constant \(K = 5\). Use the formula \(\iint_S F \cdot dS = -K \iint_S \nabla u \cdot dS\) to find the rate.
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Context:
Let $a, b$, and $c$ be real numbers such that $3^a = 125$,  $5^b = 49$,and $7^c = 8$1.
Find the product $abc$.
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Context:
棱长为 $\sqrt{2}$ 的正四面体 $A B C D$ 中, 已知 $\overrightarrow{A P}=\frac{1}{2} \overrightarrow{A B}, \overrightarrow{A Q}=\frac{1}{3} \overrightarrow{A C}, \overrightarrow{A R}=\frac{1}{4} \overrightarrow{A D}$, 若点 $K$ 为 $\triangle B C D$ 的重心, 则四面体 $K P Q R$ 的体积为 $\qquad$.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:
设函数 $f_{0}(x)=|x|, f_{1}(x)=\left|f_{0}(x)-1\right|, f_{2}(x)=\left|f_{1}(x)-2\right|$ ，则函数 $f_{2}(x)$ 的图像与 $x$ 轴所围成图形中的封闭部分的面积是 $\qquad$.
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Context:
Harry is joining a marathon and diversifies his workouts with different activities, periods of rest, and varying terrain. On Monday, he ran 10 meters per hour on a flat terrain. On Tuesday, he spent half the day strength training, which made him run 30% slower on a slightly inclined terrain. On Wednesday, after a whole day rest on Tuesday, he ran 50% faster than he did on Monday on a hilly terrain, causing him to exert 10% more effort. On Thursday, he ran on a mixed terrain at the same speed as adjusted for Wednesday despite a downhill stretch reducing his effort by 5%. Then, after an intensive yoga session on Friday morning, which improved his stamina, Harry ran 60% faster than he ran on Thursday on a continuous uphill incline, decreasing his running efficiency by 15%. How fast did Harry run on Friday?
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Context:
The world-renowned Marxist theorist *Joric* is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer  $n$ , he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the *defect* of the number  $n$ . Determine the average value of the defect (over all positive integers), that is, if we denote by  $\delta(n)$  the defect of  $n$ , compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\]
*Iurie Boreico*
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Context:
Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction.
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Context:
Mr. Thompson, a traditionalist, enjoys spending his afternoons reading classic books in his quiet living room. He has a collection of 72 books on his wooden bookshelf. Every month, he decides to read 3 new books that he carefully selects from his local library, avoiding any digital versions. However, he also gives away 1 book each month to his neighbor, who shares his love for timeless stories. If Mr. Thompson continues this pattern for 6 months, how many books will he have on his shelf at the end of that period?
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Context:
A barrel of water is filled to a maximum height of 90 cm. After it stops raining, the water leaks out at a rate proportional to the square root of the water depth. If the water level drops from 90 cm to 85 cm in one hour, how long will it take for the barrel to become empty?
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Context:
$64$ teams with distinct skill levels participate in a knockout tournament. In each of the $6$ rounds, teams are paired into match-ups and compete; the winning team moves on to the next round and the losing team is eliminated. After the second-to-last round, winners compete for first and second and losers compete for third and fourth. Assume that the team with higher skill level always wins. What is the probability that the first, second, and third place teams have the highest, second highest, and third highest skill levels, respectively?

[i]2019 CCA Math Bonanza Lightning Round #3.3[/i]
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Context:
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?
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Context:
In a certain brick wall, each row of bricks above the bottom row contains one less brick than the row just below it. There are some rows in all and a total of 100 bricks in the wall. The bottom row contains 18 bricks. How many rows are there in the wall?
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Context:
设 $w=\cos \frac{2 \pi}{5}+\mathrm{i} \sin \frac{2 \pi}{5}, P(x)=x^{2}+x+2$, 求 $P(w) P\left(w^{2}\right) P\left(w^{3}\right) P\left(w^{4}\right)$ 的值。
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Context:
Find the degree of the field extension \(\mathbb{Q}[a]\) over \(\mathbb{Q}[\sqrt[3]{2}]\), where \(a=\sqrt{ \sqrt[3]{2}-1}\) and \(a\) is a root of the polynomial \(X^6 + 3X^4+3X^2-1\).
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Context:
Let $\{a_n\}$ be a positive sequence such that $a_{n+1} = a_n + \frac{n^4}{a_n}$. Determine the rate of growth of $a_n$. Provide your answer in terms of asymptotic notation.
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Context:
Determine the value of \(a\) such that the equation \(f(x) = a\) has a solution for \(x \in (0,1]\), where \(f: (0,\infty) \to \mathbb{R}\) is defined by \(f(x) = x^n - n\ln x + 1\) and \(n \neq 0\), \(n \in \mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the sum of all fractions with a denominator of 2014, where each fraction $\frac{1}{2014}, \frac{2}{2014}, \cdots, \frac{2012}{2014}, \frac{2013}{2014}$ is reduced to its simplest form?
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Context:
In a 3x3x3 tic-tac-toe game with three players A, B, and C taking turns in order, how many unique board states are possible after 6 moves? Assume that each player places their mark in an empty cell during their turn.
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Context:
What time is it 2017 minutes after 20:17?
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Context:
Determine the number of matrices similar to \( A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\ \end{pmatrix} \) with entries from \( \mathbb{Z}/3\mathbb{Z} \).
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Context:
Given real numbers $\alpha$ and $\beta$ where $1 < \alpha < \beta$, find the largest positive integer $r$ such that any coloring of the positive integers with $r$ colors ensures there exist two positive integers $x$ and $y$ of the same color satisfying $\alpha \leqslant \frac{x}{y} \leqslant \beta$.
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\boxed{\left\lceil \log_\alpha \beta \right\rceil}


Context:
Given that for a function \( g(t) \), \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2} \) for any real value \( \omega \), and \( y(t)=\int_{-\infty}^{t}g(\tau)d\tau \), calculate the value of \( \int_{-\infty}^{\infty} y(t)dt \).
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Context:
Given the presence of hereditary traits determined by paired genes $A$ and $a$ in peas, where $A$ is dominant and $a$ is recessive, and the offspring $Aa$ is yellow, determine the probability that the offspring will produce yellow peas after self-crossing.
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Context:
Given that \( \int \frac{f(x)}{x^2(x+1)^3}\,dx \) is a rational function and \( f(x) \) is a quadratic function with \( f(0) = 1 \), find \( f'(0) \).
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Context:
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen, and on one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\) if the radius of the circle is 8, and \(\cos \alpha = \frac{3}{4}\).
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Context:
Find the value of: ${9}^{\frac{1}{2}\log_{3}10}+(\frac{1}{81})^{\frac{1}{4}}-0.027^{-\frac{1}{3}}+\log_{6}2+\frac{1}{1+\log_{3}2}=\_\_\_\_\_\_$.
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Context:
Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_9 = 42\) where each variable satisfies \(0 \le x_n \le 9\) for \(1 \le n \le 9\).
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Context:
Task B-4.8. A circle is described around the focus of the parabola $y^{2}=16 x$ which touches the directrix of the parabola. At what angle do these two curves intersect?
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Context:
The natural numbers \(a, x\) and \(y\), each greater than 100, satisfy the equation \(y^{2} - 1 = a^{2}(x^{2} - 1)\). What is the smallest value that the fraction \(\frac{\alpha}{x}\) can take?
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Context:
Given the functions \( f(x) \) and \( g(x) \) as defined in Theorem 1, with \( T_1 = 1 \) and \( T_2 = \frac{1}{m} \) where \( m \in \mathbb{N} \) and \( m > 1 \), determine the smallest positive period of the function \( h(x) = f(x) + g(x) \). The period is \( \frac{1}{k} \), where \( k = 1 \) or \( k \in \mathbb{N} \), \( k \) is not a multiple of \( m \), and \( m \) is not a multiple of \( k \).
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Context:
Determine the minimum number of coin flips, $n$, required to ensure that the estimated probability $\hat{p} = X/n$ is within 5% of the true probability $p$, given that $0.4 \leq p \leq 0.6$, with a confidence level of at least 0.95. That is, find $n$ such that \( \mathbb{P}[0.95p \leq \hat{p} \leq 1.05p] \geq 0.95 \).
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Context:
In the Intense Goblet competition, 10 teams compete in a "round robin" format: that is, each team plays every other team exactly once. A team gets 4 points for a win, 2 points for a tie (a draw), and 1 point for a loss. What is the difference between the maximum total points and the minimum total points that can be gained by all teams (added together) in the Intense Goblet competition?
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Context:
Consider distinct natural numbers $a, b, c \neq 0$. Express them in the form $3^a + 3^b + 3^c$. When these numbers are ordered in increasing order, what is the sum $a + b + c$ for the 101st number in this sequence?
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Context:
[ Combinatorics (miscellaneous) $]$  $[$ Estimation + example ]

In a pond, 30 pikes were released, which gradually eat each other. A pike is considered full if it has eaten at least three pikes (full or hungry). What is the maximum number of pikes that can become full?
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Context:
Given sequences \( a_n \) and \( b_n \) of positive numbers such that \( \lim_{n\to\infty}a_n^n = a \) and \( \lim_{n\to\infty}b_n^n = b \), where \( a, b \in (0, \infty) \), and constants \( p \) and \( q \) such that \( p + q = 1 \), find the limit:
\[ \lim_{n\to\infty}(pa_n + qb_n)^n. \]
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Context:
Given the average highest monthly temperatures (in degrees Celsius) in our city in 2022 were $12$, $11$, $10$, $20$, $23$, $28$, $36$, $36$, $31$, $24$, $23$, $19$, find the 60th percentile of this data set. Express your answer as a single number in degrees Celsius.
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Context:
A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?
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Context:
Assume that the failure rate of each airplane engine during flight is $1 - P$, and the events of each engine failing are independent. The airplane can successfully fly if at least $50\%$ of the engines are running normally. What is the range for the value of $P$ to ensure that a four-engine airplane is safer than a two-engine airplane?
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Context:
Given a function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and when $x \in [-1, 1]$, $f(x) = |x|$. The function $y=g(x)$ is defined on $\mathbb{R}$ as an odd function, and when $x \in (0, +\infty)$, $g(x) = \log_{3}x$. Find the number of intersection points between the graphs of $y=f(x)$ and $y=g(x)$.
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Context:
Compute the partial derivative \( \frac{\partial\alpha}{\partial x} \) for \( \alpha = x^TAx \), where \( x \in \mathbf{R}^{n \times 1} \) and \( A \in \mathbf{R}^{n \times n} \), using the product rule without expressing \( A \) in terms of its coordinates.
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Context:
Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red packets are collected in any order before the fourth click.
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Context:
Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum_{z \in S}|z|^2$. For a given $m \in \mathbb{N}$, determine a rough upper bound for the size of $\varphi^{-1}(m)$, the set of all finite subsets of $\mathbb{Z}^d$ with sum of squares equal to $m$.
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Context:
Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$
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Context:
Suppose $A$ has $n$ elements, where $n \geq 2$, and $C$ is a 2-configuration of $A$ that is not $m$-separable for any $m<n$. What is (in terms of $n$) the smallest number of elements that $C$ can have?
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Context:
Calculate $E[\mu^T\Lambda]$ for a normal-wishart distribution, given the parameters $\mu_0$, $\lambda$, $W$, and $v$.
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Context:
Determine the values of \(a \in \mathbb{R}\) for which the function \(f_n(x, y, z) = \frac{\arctan^a(|x| + |y|)e^z}{n^4 + 4x^2 + y^2}\) is integrable over the domain \(D = \{(x, y, z): 1 < x^2 + y^2 + z < 2\}\) for all \(n \geq 1\).
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Context:
Evaluate the definite integral \( \frac{105}{19}\int^{\pi/2}_0 \frac{\sin 8x}{\sin x}\,dx \).
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Context:
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x\le 29$, a subset of $A$ sums to $x$?
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Context:
On the xOy coordinate plane, there is a Chinese chess "knight" at the origin (0,0). The "knight" needs to be moved to the point P(1991,1991) using the movement rules of the chess piece. What is the minimum number of moves required?
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Context:
In a right-angled triangle, the center $O$ of the inscribed circle is such that the half of the hypotenuse closer to $O$ is seen at a right angle from $O$. What is the ratio of the sides of the triangle?
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Context:
Calculate \( \frac{\partial T}{\partial x}\left(\frac{\pi}{2}, - \frac{\pi}{2}\right) \) for the function \( T(x,y) = \int_{x-y}^{x+y} \frac{\sin(t)}{t} \, dt \).
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Context:
A 270 meters long bullet train running at a certain speed crosses another bullet train running in opposite direction at the speed of 80 kmph in 9 seconds. The length of the other bullet train is 230.04 meters. What is the speed of the first bullet train?
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Context:
In  $\triangle ABC$ , points  $E$  and  $F$  lie on  $\overline{AC}, \overline{AB}$ , respectively. Denote by  $P$  the intersection of  $\overline{BE}$  and  $\overline{CF}$ . Compute the maximum possible area of  $\triangle ABC$  if  $PB = 14$ ,  $PC = 4$ ,  $PE = 7$ ,  $PF = 2$ .
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Context:
Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF.
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Context:
A puddle is inhabited by amoebas of three types: red, blue, and yellow. From time to time, any two amoebas of different types can merge into one amoeba of the third type. It is known that in the morning there were 47 red, 40 blue, and 53 yellow amoebas, and in the evening, only one amoeba remained. What color is it?
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Context:
Determine the rate of divergence of the series \( \sum_{m=1}^{\infty}J_{0}(mx) \) as \( x \to 0^{+} \).
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Context:
How many distinct trees with exactly 7 vertices are there? Here, a tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting \(n\) vertices with \(n-1\) edges.
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Context:
Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces

$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$
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Context:
Let proposition $p$: The equation $5x^{2}+my^{2}=1$ represents an ellipse with foci on the $x$-axis; proposition $q$: The equation $(m+1)x^{2}-my^{2}=1$ represents a hyperbola with foci on the $x$-axis. If $p \land q$ is false and $p \lor q$ is true, find the range of the real number $m$.
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Context:
Determine the number of distinct normal subgroups of a non-abelian group $G$ of order $21$, excluding the trivial subgroup $\{e\}$ and the entire group $G$ itself.
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Context:
设$x$，$y$为非负整数，使得$x+2y$是5的倍数，$x+y$是3的倍数，且$2x+y$≥99，求$7x+5y$的最小值.
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Context:
Let $a < 0$, and $|a| \cdot x \leq a$. Simplify: $|x+1| - |x-3| = \ ?$
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Context:
A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$  Two runners start at point A at the same time and run at the same speed.  The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track.  There is a first time after they begin running when their two positions are collinear with the point A.  At that time each runner will have run a distance of $\frac{m\pi}{n}$ where  $m$  and  $n$ are relatively prime positive integers.  Find  $m + n. $
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Context:
Determine the number of solutions to the equation \(|\ln|x|\;|=\sin(\pi x)\).
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Context:
Let \( x \) and \( y \) be positive integers such that \( \frac{100}{151} < \frac{y}{x} < \frac{200}{251} \). What is the minimum value of \( x \) ?
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Context:
Given the equation $|2z + 3i| = |z^2|$, where $z = a + ib$, find the range of values for $\|z\|$.
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Context:
Calculate the functional derivative of the functional \( J(y) = \int_{x_0}^{x_1} \left( y^6(x) + 3 y^2(x) \right) \, \mathrm{d}x \).
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Context:
When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts?
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Context:
Evaluate the limit \( \lim_{R\to\infty} \int_{C_R} \frac{e^{-iz}}{z^2} dz \), where \( C_R \) is the semi-circle of radius \( R \) in the lower half-plane, centered at the origin and oriented clockwise, using a direct approach without Jordan's Lemma.
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Context:
Find the norm of the operator \( \hat{C}: L^2[0,1] \to L^2[0,1] \) defined by \((\hat{C}f)(x) = \int_0^x f(t) \, dt\).
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Context:
Find the smallest positive period of the function \( y = f(x) \) given that its graph is symmetric about the points \( A\left(a, y_{0}\right) \) and \( B\left(b, y_{0}\right) \) on \( \mathbf{R} \), with no other symmetric points in the interval \( (a, b) \).
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Context:
Let  $L$  be the number formed by  $2022$  digits equal to  $1$ , that is,  $L=1111\dots 111$ .
Compute the sum of the digits of the number  $9L^2+2L$ .
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Context:
Given the function \( f(x) = \frac{x}{\sqrt{1+x^2}} \), find the value of \( f^{(99)}(1) \), where \( f^{(n)}(x) \) denotes the function \( f \) applied \( n \) times to \( x \).
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Context:
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) where \( C \) is the boundary of the square with vertices \((0,0), (2,1), (1,-2), (3,-1)\) oriented clockwise, and \( \mathbf{F}(x,y) = \langle xy - \sin(x), y^2 + \cos(y) \rangle \). Use Green's theorem to solve this.
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Context:
Suppose we have matrices $A_1, A_2, \ldots, A_m$ and $B_1, B_2, \ldots, B_m$ such that $A_i, B_i \in \mathbb{R}^{p_i \times p_{i+1}}$ and $\|A_i - B_i\|_F \leq \varepsilon$ for $1 \leq i \leq m$, where $\|\cdot\|_F$ denotes the Frobenius norm. Determine the upper bound, in terms of $\varepsilon$, for the quantity $\|A_1 \cdot A_2 \cdot \ldots \cdot A_m - B_1 \cdot B_2 \cdot \ldots \cdot B_m\|_\infty$, where $\|\cdot\|_\infty$ is the maximum absolute value of the entries of a matrix.
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Context:
Suppose that the least common multiple of the first $25$ positive integers is equal to $26A7114B4C0$. Find $100 \times A + 10 \times B + C$.
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Context:
Let the polynomial \( p_{k}(x)=\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2} \), where \( k \) is any given positive integer. Find the coefficient of \( x^{2} \) in \( p_{k}(x) \).
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Context:
Find the sum a + b + c for the ordered triplet (a, b, c) of real numbers that maximizes the expression \((1 - x)(2 - y)(3 - z)(x + \frac{y}{2} + \frac{z}{3})\) under the conditions \(x < 1\), \(y < 2\), \(z < 3\), and \(x + \frac{y}{2} + \frac{z}{3} > 0\).
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Context:
Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones.
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Context:
The incircle of triangle $A B C$ touches its sides $A B$, $B C$, $C A$ at points $C_{1}$, $A_{1}$, $B_{1}$ respectively. The segment $B B_{1}$ intersects the circle again at point $K$. It is known that $A B = B C = 5$, $A C = 6$. Find $B K$.
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Context:
Given the random variables $X, X_1, X_2, I_1,$ and $I_2$ which are independent, where $X, X_1,$ and $X_2$ have the same distribution and $I_1, I_2 \sim \text{Bernoulli}\left(\frac{1}{2}\right)$, and the relationship $X = 2 + I_1X_1 + I_2X_2$, determine the correct sign in the quadratic solution for the probability-generating function $M_X(s)$ of $X$ given by:

$$M_X(s) = \frac{2-s^2 \pm \sqrt{1-s^2}}{s^2}.$$
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Context:
If $X_1, X_2 \stackrel{i.i.d}{\sim} N(0,1)$ and $Y = \min(X_1, X_2)$, what is the distribution of $Y^2$?
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Context:
## Task A-4.5.

Given are 100 points arranged in 10 rows and 10 columns. Each point is either red or blue. A point is in a corner if it is at the end of both a row and a column, and on the edge if it is at the end of a row or a column, but not both. Two points are adjacent if they are in the same row or column, with no other points between them. Each pair of adjacent points is connected by a line of red, blue, or green color, such that each red line connects two red points, each blue line connects two blue points, and each green line connects one red and one blue point. It is known that there are 52 red points in total, of which 16 are on the edge and two are in the corners. It is also known that there are a total of 98 green lines.

Determine the total number of blue lines.
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Context:
What is the largest number for which the $3x+1$ conjecture has been verified to be true?
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Context:
A triangle moves in space such that its center of gravity remains stationary, while one of its vertices describes a circle, and the plane of the triangle remains constantly perpendicular to the plane of the circle. What paths are traced by the vertices of the triangle?
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Context:
Angelina walked 150 meters from her home to the grocery store at a constant speed. She then walked 200 meters to the gym at double her initial speed. Then, she strolled another 250 meters to the park at half the speed she used from her home to the grocery store. After that, she rushed 300 meters to the library at thrice the speed from the grocery to the gym. She spent 10 seconds less on her way from the grocery to the gym than on her way from home to the grocery, but spent 20 seconds more on her way to the park than on her way to the library. What was Angelina ' s speed , in meters per second , from the park to the library?
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Context:
Calculate \( \lim_{n\to\infty}\sqrt[n]{|1-z^n|} \) where \( z \in \mathbb{C} \) and \( |z| \neq 1 \).
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Context:
Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then how many students are at the table?
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Context:
In the school, bananas were to be distributed among students on a 5-day schedule (Monday through Friday). However, varying numbers of students were absent each day, leading to the remaining students receiving extra bananas. Here are the details:

- On Monday, the goal was 3 bananas per student. Due to 370 students being absent, each remaining student received 2 extra bananas.
- On Tuesday, the goal was 4 bananas per student. With 150 students absent, each remaining student got 3 extra bananas.
- On Wednesday, the goal was 2 bananas per student. Given that 490 students were absent, each remaining student received 1 extra banana.
- On Thursday, the goal was 5 bananas per student. When 250 students were absent, each remaining student got 2 extra bananas.
- On Friday, the goal was 1 banana per student. With 340 students absent, each remaining student received 3 extra bananas.

Find the actual total number of students in the school.
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Context:
Given that $α$ is an angle in the second quadrant and $P(x, \sqrt {5})$ is a point on its terminal side with $\cos α= \frac{ \sqrt {2}}{4}x$, find the value of $\sin α$.
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Context:
Given the sets \( A = \{ x \mid x^2 - 3x + 2 \leq 0 \} \) and \( B = \left\{ x \left\lvert\, \frac{1}{x-3} < a \right. \right\} \), if \( A \subseteq B \), then the range of values for the real number \( a \) is.
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Context:
Which is larger: $Ackermann(G_{64}, G_{64})$ or $G_{G_{64}}$? Compare the two expressions and determine which one represents a larger number.
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Context:
There are 2 boys and 3 girls standing in a row. If boy A does not stand at either end, and among the 3 girls, exactly 2 girls stand next to each other, then the number of different arrangements is \_\_\_\_\_\_\_\_.
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Context:
For  $n > 1$ , let  $a_n$  be the number of zeroes that  $n!$  ends with when written in base  $n$ . Find the maximum value of  $\frac{a_n}{n}$ .
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Context:
The length of the upper base of a trapezoid is $L$, and the length of its midline is $m$. The length of the segment connecting the midpoints of the two diagonals is ______.
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Context:
Find the splitting field of the polynomial \( f(x)g(x) \) over \( \mathbb{Z}_2 \), where \( f(x) = x^3 + x + 1 \) and \( g(x) = x^3 + x^2 + 1 \).
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Context:
On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\).
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Context:
Determine the total length of intervals for which the inequality \( f(a) \leq 5 \) holds, where \( f(a) \) is the number of distinct solutions to the equation:

\[ \sin \frac{a \pi x}{x^{2}+1}+\cos \frac{\pi\left(x^{2}+4 a x+1\right)}{4 x^{2}+4}=\sqrt{2-\sqrt{2}} \]

Round your answer to two decimal places. If there are no solutions, write 0; if the length is infinite, write 9999.
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Context:
Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-kx-4=0$, and ${x}_{1}^{2}+{x}_{2}^{2}+x_{1}x_{2}=6$, find the value of $k$.
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Context:
Given a partial order \( \langle P, \preceq \rangle \) where \( P = \{ \langle k, a_k \rangle : k = 1, \ldots, n^2 + 1 \} \) and \( \langle k, a_k \rangle \preceq \langle \ell, a_\ell \rangle \) if and only if \( k \le \ell \) and \( a_k \le a_\ell \), determine what can be concluded about the sequence \( a_1, a_2, \ldots, a_{n^2+1} \) given that each antichain in \( P \) has length at most \( n \).
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Context:
If the unfolded diagram of a cone is a sector with a central angle of $\frac{2\pi}{3}$ and a radius of $2$, then the surface area of this cone is ______.
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Context:
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.
Rule $1$: If the integer is less than $10$, multiply it by $9$.
Rule $2$: If the integer is even and greater than 9, divide it by $2$.
Rule $3$: If the integer is odd and greater than $9$, subtract $5$ from it.
$A$ sample sequence: $23, 18, 9, 81, 76, \cdots$.
Find the value of the $98^\text{th}$ term of the sequence that begins $98, 49, \cdots$.
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Context:
As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot Y moves clockwise, and robots J and B move counterclockwise. After 12 seconds, robot J reaches point B. After another 9 seconds, when robot J first catches up with robot B, it also meets robot Y for the first time. How many seconds after B first reaches point A will Y first reach point B?
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Context:
At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers?
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Context:
A group of some adults and 70 children go for trekking. There is meal for either 70 adults or 90 children and if 21 adults have their meal, the total number of children that can be catered with the remaining food is approximately 63. How many adults went for trekking?
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Context:
Calculate the area of the figure bounded by the lines

$$
y = x^3, \quad y = -1, \quad y = -3, \quad \text{and} \quad x = 0.
$$
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Context:
If \((2x + 4)^{2n} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}\) \((n \in \mathbf{N}^{*})\), then the remainder when \(a_2 + a_4 + \cdots + a_{2n}\) is divided by 3 is \(\quad\)
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Context:
What is the least number of moves it takes a knight to get from one corner of an $n \times n$ chessboard, where $n \geqslant 4$, to the diagonally opposite corner?
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Context:
A kite is inscribed in a circle with center  $O$  and radius  $60$ . The diagonals of the kite meet at a point  $P$ , and  $OP$  is an integer. The minimum possible area of the kite can be expressed in the form  $a\sqrt{b}$ , where  $a$  and  $b$  are positive integers and  $b$  is squarefree. Find  $a+b$ .
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Context:
Given that the odd function $f(x)$ defined on $\mathbb{R}$ is monotonically increasing on $(0,+\infty)$, and $f(-1)=2$, determine the solution set of the inequality $f(x-1)+2\leqslant 0$ on $(0,+\infty)$.
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Context:
On a rectangular piece of paper, a picture in the form of a "cross" is drawn using two rectangles \(ABCD\) and \(EFGH\), the sides of which are parallel to the edges of the sheet. It is known that \(AB=9\), \(BC=5\), \(EF=3\), and \(FG=10\). Find the area of the quadrilateral \(AFCH\).
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Context:
Determine the number of terms required in the Maclaurin series for $\ln(1+x)$ to estimate $\ln(1.4)$ with an error less than $0.001$. Provide your answer as the number of terms needed.
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Context:
How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row?
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Context:
This year, the Second Son of the Gods’ age is 4 times the age of Chen Xiang. Eight years later, the Second Son’s age is 8 years less than three times Chen Xiang's age. How old was Chen Xiang when he saved his mother?
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Context:
Determine the number of distinct ways to color a 3x4 rectangle using exactly 6 of each of two colors, considering that two colorings are equivalent if they can be obtained from each other by permuting rows or by cyclically permuting columns.
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Context:
7. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
\cdots \cdots \cdots \cdots \cdots \\
a_{n 1} a_{n 2} a_{n 3} a_{n 4} \cdots a_{n n}
\end{array}
$$

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+a_{23}+\cdots+a_{n n}$.
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Context:
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$?
[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6;  path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0)); [/asy]
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Context:
Given that $a, b \in \mathbb{R}$ and $a \neq 0$, when the polynomial $x^2 - 2ax + a^4$ is divided by $x + b$, the remainder is 1. Additionally, the polynomial $bx^2 + x + 1$ has $ax - 1$ as a factor. Determine the value of $b$. \( \text{(Express your answer as a real number.)} \)
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Context:
Evaluate the surface integral \( \iint_S k \cdot \mathrm{d}S \), where \( S \) is the part of the cylinder \( z = \sqrt{1-x^2} \) that lies above the square with vertices \((-1,-1), (1,-1), (-1,1), (1,1)\) and has an upward orientation.
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Context:
Let $T:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be a linear function defined as $T(x)=b_{1}x_{1}+\dots +b_{n}x_{n}$. Let $S=\left \{ \overline{x}\in \mathbb{R}^{n}:||x||_{1}\leq1 \right \}$. Determine the range of $T(S)$.
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Context:
Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $d$.
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Context:
Determine the values of $a \in \mathbb{R}$ for which the function \( \frac{1}{n^2+x^{a}+2y^{a}+3z^{a}} \) is integrable over \([0,+\infty)^3\) for all \(n \in \mathbb{N} \cup \{0\}\).
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Context:
Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$.

[i]Proposed by Tristan Shin[/i]
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Context:
Calculate the line integral \(\int_\gamma \mathbf{f} \cdot d\mathbf{x}\) for the vector field \(\mathbf{f}(x_1, x_2, x_3) = (x_1^2 + 5x_2 + 3x_2x_3, 5x_1 + 3x_1x_3 - 2, 3x_1x_2 - 4x_3)\), where \(\gamma\) is the helix with radius 1 and pitch 1 from \((0, 1, 0)\) to \((0, 1, 2\pi)\). Use the parametrization \(\gamma(t) = (\cos{t}, \sin{t}, t)\) for \(t\) in the interval \([0, 2\pi]\).
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Context:
There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \( k \) societies. Suppose that the following conditions hold:
1. Each pair of students are in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with \( 2m + 1 \) students (where \( m \) is a positive integer) is in exactly \( m \) societies.

Find all possible values of \( k \).
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Context:
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. If ${a}^{2}+{b}^{2}-{c}^{2}=\sqrt{3}ab$, find $\angle C$.
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Context:
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum\limits_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$.
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Context:
Consider a deck of cards consisting only of the jack, queen, king, and ace of each suit (clubs, diamonds, hearts, and spades). You start with the four jacks in your hand. In each round, you randomly select one of the four cards in your hand and replace it with one of the three other cards of the same suit, each with equal probability. What is the expected number of swaps needed to return to having all four jacks in your hand?
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Context:
Determine the codimension of the set \( X = \{ A \in M_{m \times n}(\mathbb{R}) : \mathrm{rk}(A) = r \} \) as a submanifold of \( \mathbb{R}^{m \times n} \).
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Context:
Let $\gamma_1, \gamma_2, \gamma_3$ be three circles with radii $3, 4, 9,$ respectively, such that $\gamma_1$ and $\gamma_2$ are externally tangent at $C,$ and $\gamma_3$ is internally tangent to $\gamma_1$ and $\gamma_2$ at $A$ and $B,$ respectively. Suppose the tangents to $\gamma_3$ at $A$ and $B$ intersect at $X.$ The line through $X$ and $C$ intersect $\gamma_3$ at two points, $P$ and $Q.$ Compute the length of $PQ.$
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Context:
Determine the condition on \( k \) such that the polynomial \( f(x) = x^4 + 4x^3 - 2x^2 - 12x + k \) has four real roots.
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Context:
Let $z_1, z_2,$ and $z_3$ be complex numbers such that $z_1 + z_2 + z_3 = 2$, $z_1^2 + z_2^2 + z_3^2 = 3$, and $z_1z_2z_3 = 4$. Calculate the value of \( \sum_{i=1}^{3} \frac{1}{z_i^2 - z_i + 4} \).
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Context:
Calculate the average distance between two randomly chosen points on the circumference of a circle with a large number of vertices (N).
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Context:
Find the value of $$ \binom{2003}{1}+\binom{2003}{4}+\binom{2003}{7}+\cdots+\binom{2003}{2002} $$
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Context:
Given a Markov chain with transition matrix \( M \), calculate the probability of transitioning from state 1 to state 3 in exactly \( n \) steps.
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Context:
Given points A(0,30), B(20,10), and O(0,0), use Pick's theorem to calculate the number of lattice points inside triangle ABO. Pick's theorem states that the area of a polygon with lattice points as vertices is S = N + 1/2L - 1, where N and L represent the number of lattice points inside and on the boundary of the polygon, respectively. Express your answer as a whole number.
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Context:
Determine the smallest non-commutative ring with unity, where "smallest" refers to having the least cardinality.
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Context:
What is the sum of all possible \( \left( \begin{matrix} i \\  j \end{matrix}\right) \) subject to the restrictions that \( i \geq  {10},j \geq  0 \) ,and \( i + j \leq  {20} \) ? Count different \( i \) ,
 that yield the same value separately - for example,count both \( \left( \begin{matrix} {10} \\  1 \end{matrix}\right) \) and \( \left( \begin{matrix} {10} \\  9 \end{matrix}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of open subsets required to form a base for the topological space $(Y, u)$, where $Y=\{p,q,r,s,t\}$ and $u=\{\emptyset,Y,\{p\},\{q\},\{p,q\},\{p,q,r\}\}$.
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Context:
A man can row his boat with the stream at 16 km/h and against the stream at some speed. The man's rate is 4 km/h. What is his speed against the stream?
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Context:
Given two concentric circles with radii \(r\) and \(R\) (where \(R \geq r\)), forming an annulus, and \(n\) equidistant points \(p_0 = p_n, p_1, p_2, \dots, p_n\) on the outer circle, find \(n\) equidistant points \(q_1, \dots, q_n\) on the inner circle such that the triangles \(\Delta(p_i, p_{i+1}, q_i)\) for \(1 \leq i \leq n\) do not overlap and are adjacent to each other. Determine the angle \(\alpha\) that allows for this configuration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vertices of triangle $\triangle ABC$ as $A(2,0)$, $B(0,4)$, and $C(4,1)$, where $AE$ is the angle bisector of $\angle BAC$, calculate the coordinates of a directional vector of line $AE$. Express your answer as a vector in the form $(x, y)$.
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Context:
已知函数$f(x)=(2-x){{e}^{x}}-ax-a$，若不等式$f(x)>0$恰好存在两个正整数解，则实数$a$的取值范围是__________．
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Context:
Evaluate the triple integral \( \iiint\limits_R{4y^2ze^{xy}dxdydz} \) where \( R \) is the region bounded by \( x=0,\; y=1,\; y=x,\; z=0,\; z=1. \)
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Context:
Given the series $$ S=\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}(2k+1)}\ \ \ (*) $$ I need to find the smallest value of $n$ such that $|S-S_n|<\epsilon=10^{-4}$. I cannot use expansion series for any functions.
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Context:
Given the measurable space \((X, \mathcal{A}) = ([-1, 1], \mathcal{B}_{[-1, 1]})\) equipped with the Lebesgue measure \(m\), consider the functional \(\phi : \mathcal{L}^1(m) \to \mathbb{R}\) defined by \(\phi(f) = \int (x - \frac{1}{2}) f(x) \, dm(x)\). Determine the norm \(||\phi||\).
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Context:
Determine all values of $k$ for which the series \( \sum_{n=1}^{\infty} \frac{k^n}{n^k} \) converges.
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Context:
Determine the number of bijections \(f: \mathbb{N} \to \mathbb{N}\) such that \(\sum_{n=1}^{\infty} \frac{f(n)}{n^2} < \epsilon\).
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Context:
A sum is lent at compound interest at a certain rate per annum and amounts to Rs. 240 in 2 years. If the sum amounts to Rs. 217.68707482993196, what is the rate of interest per annum?
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Context:
Find the limit: \[ \lim_{n \to \infty} \frac{\gamma_{n-1}}{\gamma_{n}} \] where \( \gamma_{n} \) is the \( n \)-th Stieltjes Constant.
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Context:
$F_{1}, F_{2}$ 是粗圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的两个焦点， $P$ 为椭圆上的一点。如果 $\triangle P F_{1} F_{2}$ 的面积为 $1, \tan \angle P F_{1} F_{2}=\frac{1}{2}, \tan \angle P F_{2} F_{1}=-2$ 。请给出原始答案的形式 $a=\frac{\sqrt{m}}{n}$ 中的 $m + n$ 值。
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Context:
Determine the largest real number \( k \) such that the inequality \[ \frac{a^{2}}{b} + \frac{b^{2}}{a} \geq 2 \frac{a^{k+1} + b^{k+1}}{a^{k} + b^{k}} \] holds for all positive real numbers \( a \) and \( b \).
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Context:
In a certain kingdom, the workforce consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{dwarves}}^{S} = 1 + \frac{L}{3} \), and the aggregate labor supply of elves is \( w_{\text{elves}}^{S} = 3 + L \). The inverse function of aggregate labor demand for dwarves is \( w_{\text{dwarves}}^{D} = 10 - \frac{2L}{3} \), and the inverse function of aggregate labor demand for elves is \( w_{\text{elves}}^{D} = 18 - 2L \). Recently, a newly enthroned king became very concerned that the wage rates of his subjects are different, so he issued a law stating that the wages of elves and dwarves must be equal and that workers should not be discriminated against based on their clan affiliation. The king believes that regulatory intervention in wage rates will negatively impact the kingdom's economy overall and mandates that all his subjects behave in a perfectly competitive manner. By how many times will the wage of the group of workers whose wage was lower before the king's intervention increase if the firms in the kingdom are indifferent to hiring elves or dwarves?
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Context:
In the spring college entrance examination of Shanghai in 2011, there were 8 universities enrolling students. If exactly 3 students were admitted by 2 of these universities, the number of ways this could happen is ____.
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Context:
Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which\n\[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\]\nis satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.
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Context:
Given a non-constant sequence {a_i} that satisfies the recurrence relation a_{i+1}^2 - a_i a_{i+1} + a_i^2 = 0 and a_{i+1}!= a_{i-1} for i = 1, 2, 3,..., n, and the boundary conditions a_1 = a_{n+1} = 1, find the sum of the sequence from i = 1 to n. Express your answer as a single integer.
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Context:
Maria ordered a certain number of televisions for the stock of a large store, paying R\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered?
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Context:
Consider a vector space \( V \) over \( \mathbb{Z}_2 \) with dimension 4. Determine the number of linearly independent subsets of \( V \) that contain exactly 3 elements.
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Context:
A rectangle is divided into three smaller congruent rectangles, and each smaller rectangle is similar to the large rectangle. Find the ratio of the length of a longer side to that of a shorter side in each of the rectangles. Express your answer as a simplified ratio (e.g., x:y).
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Context:
Given a square matrix $A$ with real entries, consider a generalized eigenvector $v$ of $A$ with rank $r$, satisfying $(A - \lambda I)^r v = 0$ and $(A - \lambda I)^{r-1} v \neq 0$, where $v \neq 0$, $\lambda$ is an eigenvalue of $A$, and $I$ is the identity matrix. If $v$ is also a generalized eigenvector of $A^n$ with rank $s \leq r$, such that $(A^n - \lambda^n I)^r v = Q(A, \lambda)(A - \lambda I)^r v = 0$ for some polynomial $Q(X, \lambda)$, and $Q[A]$ is invertible on the generalized eigenspace for $\lambda$ in $A$, determine the relationship between $r$ and $s$ when $\lambda \neq 0$.
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Context:
A mathematics class took a test with 6 true or false questions. Each pair of students had at least three different answers. What is the maximum number of students that could be in the class?
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Context:
3.50. The radius of the circle inscribed in a right trapezoid is $r$, and the acute angle of the trapezoid is $\alpha$. This trapezoid rotates around the smaller lateral side. Find the lateral surface area of the solid of revolution.
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Context:
What fraction of the area of an isosceles trapezoid $KLMN (KL \parallel MN)$ is the area of triangle $ABC$, where $A$ is the midpoint of base $KL$, $B$ is the midpoint of base $MN$, and $C$ is the midpoint of leg $KN$?
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Context:
Kolya Vasin walked for five hours after school. First, he walked on a horizontal road, then climbed a hill, and finally returned to the starting point along the same route. His speed was 4 km/h on the horizontal section, 3 km/h when climbing the hill, and 6 km/h when descending the hill. What distance did Kolya Vasin cover?
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Context:
Consider all pairs of points $(a,b,c)$ and $(d,e,f)$ in the 3-D coordinate system with the condition $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?
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Context:
Find the fraction that equals $0.72\overline{6}$.
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Context:
Given an isosceles triangle \(ABC\) with \(\angle A = 30^\circ\) and \(AB = AC\). Point \(D\) is the midpoint of \(BC\). Point \(P\) is chosen on segment \(AD\), and point \(Q\) is chosen on side \(AB\) such that \(PB = PQ\). What is the measure of angle \(PQC\)?
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Context:
For what positive integer $A$ is the sequence $(a_k)_{k=0}^\infty$, defined by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k \ge 0$, eventually constant? Here, $S(n) = n - m^2$, where $m$ is the greatest integer such that $m^2 \le n$. Determine the positive integer $A$ for which the sequence is eventually constant.
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Context:
Find the multiplicative inverse of the element $x^2 + 1$ in the ring $\Bbb{F}_3[x]/(x^3 + 2x + 2)$. Express your answer as a polynomial in $\Bbb{F}_3[x]$ of degree less than 3.
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Context:
Determine the number of labeled trees on 7 vertices, where the vertex set is \( V = \{1, 2, 3, 4, 5, 6, 7\} \), and the vertices \( 1, 2, 3, \) and \( 4 \) are exactly the leaves of the tree.
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Context:
Calculate the expectation of the product of Brownian motion terms: \( E[B(U)B(U+V)B(U+V+W)B(U+V+W+x)] \), given that \( U + V + W > U + V > U \) and \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest cubic graph that contains a square-free 2-factor, where a square-free 2-factor is a 2-factor with no cycles of length 4.
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Context:
Find the smallest natural number \( n \) such that it is possible to tile the plane with squares whose side lengths are from the set \( \{1, 2, \ldots, n\} \) in such a way that no two squares with the same side length share a segment of an edge. Squares with the same side length may touch at a vertex, and not all side lengths from 1 to \( n \) need to be used in the tiling.
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Context:
Determine the class number $h(G)$ for the group $G = \textbf{PGL}_{2,K}$, where $K = F_q(x)$ is the rational function field.
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Context:
Consider the 800-digit integer
$$
234523452345 \cdots 2345 .
$$

The first \( m \) digits and the last \( n \) digits of the above integer are crossed out so that the sum of the remaining digits is 2345. Find the value of \( m+n \).
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Context:
Compute the minimal number of generators, \( \mu(mA_m) \), of the ideal \( mA_m \), where \( A=\mathbb{C}[x,y]/(x^3-y^2) \) and \( m=(\bar{x} -1 , \bar{y} -1) \) is a maximal ideal of \( A \).
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Context:
Determine the number of terms needed in the series expansion to approximate π to within $10^{-3}$ using the formula:

$$\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)$$

and the Maclaurin series for the arctangent function:

$$\arctan(x)=\sum_{i=1}^n\frac{(-1)^{i+1}x^{2i-1}}{2i-1}$$

Find the smallest integer $n$ such that the approximation error is less than $10^{-3}$. Use the inequality:

$$\frac{4}{2n+1} \left(\frac{4}{5^{2n+1}}-\frac{1}{239^{2n+1}}\right)<10^{-3}$$
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Context:
There are 10 cups on a table, 5 of them with the opening facing up and 5 with the opening facing down. Each move involves flipping 3 cups simultaneously. What is the minimum number of moves required to make all cup openings face the same direction?
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Context:
Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$. The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the smaller semicircles. The original answer is in the form \(\frac{k}{m}\pi - \frac{\sqrt{n}}{p}\). Please calculate the value of k + m + n + p.
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Context:
20. (MON 4) Find the least natural number $n$ such that if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two nonintersecting subsets, then one of the subsets contains three distinct numbers such that the product of two of them equals the third.
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Context:
Find the number of distinct subgroups that can be generated by two elements in the group \( \mathbb{Z}_q \oplus \mathbb{Z}_q \oplus \mathbb{Z}_q \), where \( q \) is a prime number.
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Context:
Evaluate the integral \( \int_0^\infty \ln(x)\,x^2e^{-x}\,\mathrm{d}x \).
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Context:
设二位数AB与三位数ABC的和仍为三位数BAD，其中A、B、C、D 为相异的数字；设BAD最大可能之值为x，则$1 0 0 0 - x = \underline { }$ .
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Context:
If 183 is divisible by 11, find the value of the smallest natural number n?
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Context:
Determine the number of roots of the polynomial \( Q(z) = 3z^5 - 2z^3 + z^2 - z + 1 \) in the region where \( \operatorname{Re}(z) > 0 \).
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Context:
Find the quadratic variation of the process \( Y_t = B_{2t} \), where \( (B_t)_{t \geq 0} \) is a standard Brownian motion.
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Context:
Evaluate the limit: \[ \lim_{x \to 0} \left(\frac{2f(x)-x}{x^2}\right) \] where \( f(x) = \lim_{n \to \infty} \frac{x}{n} \left(\frac{1}{1+e^{-x/n}}+\frac{1}{1+e^{-2x/n}}+\dots+\frac{1}{1+e^{-x}} \right) \).
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Context:
Find all sets of three distinct integers \(a, b, c\) such that \(W(a) = W(b) = W(c)\), where \(W(x) = x^4 - 3x^3 + 5x^2 - 9x\).
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Context:
In the Laurent series for \(f(z) = \frac{1}{z-4}\) centered at \(z=1\), find the coefficient of \((z-1)^{-2}\).
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Context:
An ancient civilization has a tribe of 12 members organized hierarchically. The tribe has one main chief, two supporting chiefs (Senior and Junior), and each supporting chief has three inferior officers. If the tribe has 12 members in total, in how many ways can the leadership structure of the tribe be formed under these restrictions?
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Context:
A ball is floating on the surface of a lake. After the lake freezes and the ball is removed, a hole with a diameter of 24cm and a depth of 8cm is left on the ice surface. Find the surface area of the ball. Express your answer in terms of π.
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Context:
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?
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Context:
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Calculate the value of \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\). Express your answer as a single integer.
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Context:
Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elements of $s$ are written as reduced fractions?
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Context:
Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3x \in S$ and $3x + 1 \in S$. Determine the number of non-negative integers in $S$ less than $2008$.
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Context:
To guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either a day, or a night shift, or work around the clock. In these cases the guard is given a leave of absence of not less than $1, 1.5$ or 2.5 full days, respectively. What is the least number of guards necessary to ensure the security of the installation?
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Context:
A ball is passed around a circular table with 10 seats, each occupied by a person. In the first run, the ball moves to the $1^{1} = 1$st position. In the second run, it moves an additional $2^{2} = 4$ positions, ending up in the 5th position. In the third run, it moves another $3^{3}$ positions, ending up in the 2nd position. Determine the position of the ball after the 2017th run.
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Context:
Determine the order of accuracy for numerically integrating a function \( f(x,y) \) over the square domain \([0,1]\times[0,1]\) using a two-dimensional Cartesian grid with equal spacing \( \Delta x = \Delta y \) and evaluating the function at the grid cell centers.
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Context:
Determine the value of \( k \) if the function \( f(x) = \begin{cases} ax(x-1)+b & ; x < 1 \\ x+2 & ; 1 \le x \le 3 \\ px^2+qx+2 & ; x > 3 \end{cases} \) is continuous for all \( x \in \mathbb{R} \) except \( x=1 \), but \( |f(x)| \) is differentiable everywhere, and \( f'(x) \) is continuous at \( x=3 \). Additionally, \( |a+p+b+q| = k \).
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Context:
 The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires?
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Context:
Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio
\[\frac{AB \cdot CD}{AC \cdot BD} \]
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Context:
Paul fills in a \( 7 \times  7 \) grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same
 thing again, to obtain two different random arrangements of the numbers in the grid. What is the expected number of pairs of
 numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?
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Context:
Given a $2\times2$ matrix $A$ with real entries that is invertible and has a determinant $\text{Det}(A) = k$. If $\text{Det}(A + k\text{adj}(A)) = 0$, where $\text{adj}(A)$ denotes the adjugate of $A$, find the value of $\text{Det}(A - k\text{adj}(A))$. \(\text{adj}(A)\) is the adjugate of \(A\).
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Context:
Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1}+f_{n-2}$. For each $i$, $1 \le i \le 200$, we calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. What is the sum of the distinct values of $g_i$?
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Context:
A $7 \times 7$ board is filled with $49$ distinct numbers, one in each cell. You can ask questions where each question reveals the set of numbers in any $3$ chosen cells. What is the minimal number of questions required to determine the number in each cell of the board?
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Context:
Determine the value of \(\alpha\) for which the series \(\sum \frac{\sqrt{a_{n}}}{n^{\alpha}}\) converges, given that \(\sum a_{n}\) converges and \(a_{n}\) are positive.
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Context:
Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAAA$, while $BBAB$ is not such a sequence. How many such sequences have length 14?
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Context:
Given that $X$ and $Y$ are independent central chi-square random variables with degrees of freedom $m \geq 3$ and $n \geq 3$, respectively, and that $E \left(\frac{X}{Y}\right) = 3$ with $m + n = 14$, find the expected value $E \left(\frac{Y}{X}\right)$. Express your answer as a single number.
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Context:
Evaluate the integral \( \int_0^y w^{a-1}(1-w)^{b-1} \, dw \).
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Context:
A convex polygon has at least one side with length 1. If all diagonals of the polygon have integer lengths, what is the maximum number of sides the polygon can have? Express your answer as a single integer.
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Context:
A cat is going up a stairwell with ten stairs. The cat can jump either two or three stairs at each step, or walk the last step if necessary. How many different ways can the cat go from the bottom to the top?
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Context:
Let \( f(x) \) be a real-valued function defined on the interval \((0,\infty)\). Determine the smallest possible value of \( \alpha \) such that \(|f'(x)| < |f(x)|\) for all \( x \in (\alpha,\infty)\).
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Context:
Given the permutations \(\alpha = (1\ 2\ 3\ 7\ 11\ 10\ 9\ 5)\), \(\beta = (2\ 3\ 4\ 8\ 12\ 11\ 10\ 6)\), \(\gamma = (5\ 6\ 7\ 11\ 15\ 14\ 13\ 9)\), and \(\delta = (6\ 7\ 8\ 12\ 16\ 13\ 14\ 10)\), determine the order of the permutation group generated by these cycles.
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Context:
Jackson buys a computer game for $66 which had a discount of 15% off the original price. He also buys three movie tickets, each originally $12, with a 10% entertainment tax added to the ticket price. Additionally, he spends $7 on snacks and $5 on transportation for each movie trip. What is the total amount he spent on entertainment, additional expenses, after incorporating the discount and tax?
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Context:
Find all continuous functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation \( g(x+y) = g(x) + g(y) \) for all real numbers \( x \) and \( y \).
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Context:
In an equilateral triangle $\triangle PRS$, if $QS=QT$ and $\angle QTS=40^\circ$, what is the value of $x$?
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Context:
There are 7 numbers from 9 to 79 that are exactly divisible by a certain divisor. What is the divisor?
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Context:
Given that $A(z_1)$, $B(z_2)$, and $C(z_3)$ are the vertices of an equilateral triangle $ABC$, determine the value of $\arg\left(\frac{z_2+z_3-2z_1}{z_3-z_2}\right)$. Provide your answer in radians.
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Context:
Given the differential equation \( \frac{dx}{dy} \cdot \frac{d^2y}{dx^2} = k \frac{dy}{dx} \), where \( k \) is a constant, find the value of the gradient \( \frac{dy}{dx} \) at the point where \( y = 2 \). It is known that the gradient \( \frac{dy}{dx} = 1 \) at the point \( (0,1) \).
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Context:
Chris gave his 10 friends 12 pieces of candy each. Then, he gave another 7 friends 1.5 times the amount of candy he gave the first 10 friends. Lastly, he gave the remaining 5 friends an equal share of 20% more candy than the total amount given to the first 17 friends combined. How many pieces of candy did Chris give overall?
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Context:
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was \( 6 : {57} \) AM,
 and the sunset as \( 8 : {15} \) PM. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun real
 set?
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Context:
Consider the initial value problem \(\frac{dy}{dx} = 3y^{\frac{2}{3}}\) with the initial condition \(y(0) = 0\). Determine the number of distinct solutions for this IVP, considering the possible definitions of \(y^{\frac{2}{3}}\) for \(y < 0\).
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Context:
Find the set of all real numbers $x$ such that $\sin x > \ln x$. Express your answer in interval notation.
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Context:
In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to 42: $42=\left(-8053873881207597 \_\right)^{3}+(80435758145817515)^{3}+(12602123297335631)^{3}$. One of the digits, labeled by an underscore, is missing. What is that digit?
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Context:
Calculate the value of the expression \( \sum_{r = 0}^{15}(2-3r)^{15}C_r \).
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Context:
Determine the disk of convergence for the power series \( \sum_{k=0}^\infty b_k\cdot z^k,\) where \(b_k = \begin{cases} 1 &, k = 2^n\\ 0 &, \bigl(\forall n\bigr)(k \neq 2^n).\end{cases}\)
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\{ z \in \mathbb{C} \mid |z| < 1 \}}


Context:
In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$.
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Context:
Find the angle between the asymptotes of the hyperbola given by the equation \(5x^2 - 2\sqrt{7}xy - y^2 - 2x + 1 = 0\).
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Context:
The sum \( A = \left\lfloor \frac{8}{9} \right\rfloor + \left\lfloor \frac{8^2}{9} \right\rfloor + \cdots + \left\lfloor \frac{8^{2014}}{9} \right\rfloor \) when divided by 63 leaves a remainder of \(\qquad\). (\(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to the real number \( x \)).
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Context:
Determine the proportion of nonplanar graphs with $v$ vertices that satisfy either $e > 3(v - 2)$ or $e > 2(v - 2)$ (if triangle-free) to the total number of nonplanar graphs with $v$ vertices.
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Context:
Find the largest positive integer  $n$  ( $n \ge 3$ ), so that there is a convex  $n$ -gon, the tangent of each interior angle is an integer.
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Context:
Stephen made 10 round trips up and down a 40,000-foot tall mountain. On each ascent, he reached 3/4 of the mountain's height, but on each descent, he only went down 2/3 of the height he had climbed due to an automated transportation system at certain heights. Throughout his trips, he stopped to rest at two different resting points on the way up, carrying an additional 50 lbs on every even trip. Furthermore, during two of his trips, he faced harsh weather conditions that forced him to reroute and cover an extra 10% of the ascent distance each time. Calculate the total distance he covered on his trips, taking into consideration the resting points, additional weight, and weather conditions.
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Context:
Find the greatest common divisor (GCD) of all the numbers in the set \(\{(n+2014)^{n+2014}+n^n \mid n \in \mathbb{N}, n > 2014^{2014}\}\).
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Context:
In approximate calculations, the number of significant figures can be used to measure the degree of approximation of an approximate number, simplifying the calculation. However, if not careful, it can also bring us trouble. For example, $2^{100} = 1267650600228229401496703205367$, in a certain calculation, $(2^{100} + 10^{10}) - 2^{100} = 0$, please point out the maximum number of significant figures taken in this approximate calculation.
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Context:
Determine the Lebesgue measure of the set \( A = \left \{x \in [0,1] \cap (\mathbb{R} - \mathbb{Q}) \mid \text{in the decimal expansion of } x, \text{ the first 1 (if it exists) is followed by 2} \right \} \).
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Context:
Determine the normalizer of the subgroup \(\langle(1,2,...,p)\rangle\) in the symmetric group \(S_p\).
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Context:
A point \( P \) is chosen randomly inside a triangle with sides 13, 20, and 21. Find the probability that the circle centered at \( P \) with radius 1 will intersect at least one of the sides of the triangle.
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Context:
Find the sum of the series: $$\sum_{k=0}^\infty\frac{(k+2)^3-3(k+2)^2+2(k+2)}{(k+4)!}$$
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Context:
Evaluate the limit: $$\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}-1}{y^3}dy}{\ln(1+x)}$$ as x approaches 0 from the right.
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Context:
Find a sufficient condition on \(\alpha\) that ensures the integral is finite.
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Context:
What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$?
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Context:
Determine all natural numbers $n$ such that $2^n$ divides $3^n - 1$. Provide your answer as a set of numbers.
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Context:
What is the probability that Sisyphus must labor forever, given that each rock he places on Mount Olympus is either vaporized with probability 10%, rolled back into the valley with probability 50%, or split into two rocks that roll back into the valley with probability 40%? Assume he starts with one rock in the valley and none on Mount Olympus.
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Context:
In how many ways can a commander issue a series of 8 orders to a row of 2018 soldiers, where each order instructs either all soldiers in even positions or all soldiers in odd positions to leave the formation, such that exactly 7 soldiers remain in formation?
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Context:
Out of 410 students of a school, some play football, 175 play cricket and 50 neither play football nor cricket. There are 140 students who play both football and cricket. How many students play football?
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Context:
Compute the flux of the vector field \( \nabla \ln \sqrt{x^2 + y^2 + z^2} \) across the surface of an icosahedron centered at the origin. Let \( S \) be the surface of the icosahedron, and \( f(x,y,z) = \ln \sqrt{x^2+y^2+z^2} \). Calculate the flux \( \iint_S (\nabla f \cdot n) \, d\sigma \), where \( n \) is the outward unit normal vector on \( S \).
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Context:
The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle.
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Context:
Calculate the flux of the vector field \( v = (\cos{2 \pi x}, y^3, z^3) \) through the unit sphere centered at the origin. The flux is given by the integral \( \iint_S(v \cdot n) \, ds \), where \( n \) is the outward unit normal vector to the surface \( S \), and \( ds \) is the area element on \( S \).
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Context:
Tokarev S.I.

Find the smallest natural number that can be represented as the sum of 2002 natural addends with the same sum of digits and as the sum of 2003 natural addends with the same sum of digits.

#
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Context:
Determine the number of group homomorphisms from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_3$. Provide your answer as a single integer.
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Context:
What is the least positive integer \( m \) such that \( m \cdot  2! \cdot  3! \cdot  4! \cdot  5!\cdots {16}! \) is a perfect square?
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Context:
There are 37 people lined up in a row, and they are counting off one by one. The first person says 1, and each subsequent person says the number obtained by adding 3 to the previous person’s number. At one point, someone makes a mistake and subtracts 3 from the previous person's number instead. The sum of all the numbers reported by the 37 people is 2011. Which person made the mistake?
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Context:
Find the initial conditions for which $\lim_{t \rightarrow \infty}y = 4$ for the following differential equation: $$y' = y(y-4)(y+4)(y-6)$$
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Context:
Evaluate the double integral \( \int_0^1 \int_0^1 \frac{x-y}{(x+y)^3} \, dy \, dx \).
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Context:
Find the smallest positive solution of the infinite series equation \(\sum_{n=0}^{\infty}\frac{(-x)^n}{(n!)^2}=0\) accurate to four digits of the fractional part.
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Context:
Calculate the volume of the region $W$ in $\\mathbb{R}^3$ defined by $W=\\{ (x,y,z)\\in\\mathbb{R}^3 ∣ z≥2,\\, x^2+y^2≤6−z,\\, y≤x \\}$ using triple integration.
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Context:
Let \( N = {10}^{6} \) . For which integer \( a \) with \( 0 \leq  a \leq  N - 1 \) is the value of maximized?
 Proposed by Lewis Chen
 $\binom{N}{a+1}-\binom{N}{a}$
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=n}^{5n}{k-1\choose n-1}\left(\frac{1}{5}\right)^{n}\left(\frac{4}{5}\right)^{k-n} \]
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Context:
Find the degree of the field extension \(\mathbb{Q}(\sqrt[4]{3}, \omega_{12})\) over \(\mathbb{Q}\), where \(\omega_{12}\) is a primitive 12th root of unity.
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Context:
Find the volume of the solid bounded by the surfaces $x + z - 1 = 0$, $z = 1$, and $x^2 + y^2 = 4$ in $\mathbb{R}^3$.
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Context:
Evaluate the expression \( \int_1^a \sqrt[5]{x^5-1}\ dx + \int_0^b \sqrt[5]{x^5+1}\ dx \) given that \( a^5 - b^5 = 1 \).
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Context:
$\text{Suppose that}20^{21}=2^{n}5^{b}=4^{e}5^{d}=8^{e}5^{f}\text{for positive integers}a,b,c,d,e,\mathrm{and}f.\mathrm{Find}\frac{100bdf}{ace}.$
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Context:
Find the minimum possible number of distinct terms in the 6-tuple \((a, b, c, \alpha, \beta, \gamma)\) assigned to a scalene triangle, where \(a, b, c\) are the side lengths and \(\alpha, \beta, \gamma\) are the corresponding angles measured in radians.
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Context:
Evaluate the limit: $$ \lim_{n \to \infty} \sum_{k=0}^{3n} \binom{k+n-1}{n-1} \left(\frac{1}{4} \right)^{n} \left(\frac{3}{4}\right)^{k} $$
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Context:
How many polynomials \( P \) with integer coefficients and degree at most 5 satisfy the inequality \( 0 \le P(x) < 120 \) for all \( x \in \{0,1,2,3,4,5\} \)?
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Context:
Find the Fourier transform of the function $f(x) = e^{iax}$, where $a \in \mathbb{R}$, given that $F \delta_a = e^{-iax}$ and $\overline{F} \delta_a = f$, with $F$ denoting the Fourier transform and $\delta_a$ being the Dirac delta function centered at $a$. Assume $f \in S'(\mathbb{R})$. Provide your answer in terms of $a$ and $x$.
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Context:
Determine the dimension of the quotient ring $R/J$, where $J$ is the ideal generated by an arbitrary nonconstant monic polynomial $f(x)$ in $K[x]$. Provide your answer in terms of the degree of $f(x)$.
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Context:
Calculate the expected claim size for an insurance policy with the following distribution function: $$F(x)=\begin{cases}0&x\le0\\0.0002x&0<x<1000\\0.4&x=1000\\1-0.6e^{(x-1000)/2000}&x>1000\end{cases}$$
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Context:
Evaluate the sum \( \sum_{n=1}^{\infty} \arccos\left(\sqrt{\frac{n}{n+1}}\right) \).
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Context:
Calculate the average IQ of Mensa members, given that Mensa is a group of people with an IQ of at least 130. Assume the IQ is normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 15. Use the formula for the mean of a function over an interval to find the average IQ for this group.
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Context:
Evaluate the integral \( \int_{-\pi}^\pi \frac{\sin(13x)}{\sin x} \cdot \frac{1}{1+2^x} \, \mathrm{d}x \).
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Context:
What is the relationship between the dual space of the dual space, \(V^{**}\), and the original vector space \(V\)?
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Context:
What is an upper bound for the difference between the $n$-th moment and the $n$-th power of the expected value of a random variable $X$ constrained to the interval $[0,1]$, for $n \in \mathbb{Z}_+$?
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Context:
Determine all complex numbers \(a\) for which the function \(f(z) = \frac{z^2 - z +1}{z(z-1)^2}+\frac{a}{\sin z} = \frac{1}{z}+\frac{1}{(z-1)^2}+\frac{a}{\sin z}\) has an antiderivative in the domain \(U = \{z : |z|<2\} \setminus \{0,1\}\).
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Context:
Evaluate the contour integral \( \oint_C \sqrt{z^2 - 1} \, dz \) where \( C \) is the circle of radius \( R > 1 \), centered at the origin, in the complex plane. The integrand is defined using a straight branch cut connecting \( z = 1 \) and \( z = -1 \), and \( \sqrt{z^2 - 1} > 0 \) on the line \( y = 0 \) for \( x > 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive solution of the equation \(\sum_{n=0}^{\infty}\frac{(-x)^n}{(n!)^2}=0\) and provide the answer with four digits of the fractional part.
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Context:
Let \( X \) and \( Y \) be independent random variables with densities \( f_X(x) = 4x^2e^{-2x} \) for \( x > 0 \) and \( f_Y(y) = \frac{8}{3}y^3e^{-2y} \) for \( y > 0 \). Define \( V = \frac{X}{X+Y} \). Find the variance of \( V \), \( \text{Var}(V) \).
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Context:
Determine the limiting proportion of odd composite positive numbers that are divisible by a given prime number greater than 2.
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Context:
Let \( n \) be the answer to this problem. In acute triangle \( {ABC} \) , point \( D \) is located on side \( {BC} \) so that \( \angle {BAD} = \angle {DAC} \) and
 point \( E \) is located on \( {AC} \) so that \( {BE} \bot  {AC} \) . Segments \( {BE} \) and \( {AD} \) intersect at \( X \) such that \( \angle {BXD} = {n}^{o} \) : Given that
 \( \angle {XBA} = {16}^{o} \) , find the measure of \( \angle {BCA} \) .
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Context:
Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of $$a^abc + b^bca + c^cab.$$
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Context:
There are 2017 distinct points $A_{1}, A_{2}, \ldots, A_{2017}$ marked on a circle. All possible chords connecting these points in pairs are drawn. A line passing through point $A_{1}$, which does not pass through any of the other points $A_{2}, \ldots, A_{2017}$, is drawn. Determine the greatest possible number of chords that can intersect this line.
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Context:
A train 110 m long is running with a speed of 84 km/hr. In 4.399648028157747 seconds, it will pass a man who is running at a certain speed in the direction opposite to that in which the train is going. What is the speed of the man in km/hr?
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Context:
Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined. What is the maximum number of numbers that can be underlined? Express your answer as a whole number.
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Context:
It is known that there exists a natural number \( N \) such that \( (\sqrt{3}-1)^{N} = 4817152 - 2781184 \cdot \sqrt{3} \). Find \( N \).
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Context:
For a given positive integer \( k \), denote the square of the sum of its digits by \( f_1(k) \) and let \( f_{n+1}(k) = f_1(f_n(k)) \). Determine the value of \( f_{1001}(2^{1990}) \).
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Context:
Central Park originally had 8 more than half of the number of trash cans as in Veteran's Park, and Liberty Park had twice the number of trash cans as in Central Park. One night, someone took half of the trash cans from Central Park and put 1/3 of them in Veteran's Park and the remaining 2/3 in Liberty Park. Then, they took 1/4 of the original trash cans from Liberty Park and put them in Veteran's Park. If there were originally 24 trash cans in Veteran's Park, how many trash cans are now in Veteran's Park after the exchange?
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Context:
In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows:

1. The first contestant takes $\frac{1}{n}$ of the cake.
2. The second contestant takes $\frac{2}{n}$ of the remaining cake.
3. The third contestant takes $\frac{3}{n}$ of what remains after the first and second contestants have taken their portions.
4. This process continues until the last contestant, who takes all of the remaining cake.

Determine which competitor takes the largest piece of cake.
Please reason step by step, and put your final answer within \boxed{}.