Context:
Given that $7^{-1} \equiv 55 \pmod{78}$, find $49^{-1} \pmod{78}$, as a residue modulo 78. (Give an answer between 0 and 77, inclusive.)
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Context:
Given that $P$ is a point on the circle $(x-3)^{2}+(y-4)^{2}=4$, then the maximum distance from point $Q(\cos \alpha ,\sin \alpha )$ to point $P$ is ______.
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Context:
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The first train is 100 m long. They will be clear of each other in 10.999120070394369 seconds from the moment they meet. What is the length of the second train?
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Context:
Determine how many roots of the equation

$$
4 \sin 2x + 3 \cos 2x - 2 \sin x - 4 \cos x + 1 = 0
$$

are located in the interval \(\left[10^{2014!} \pi, 10^{2014!+2015} \pi\right]\). In the answer, write down the sum of all the digits of the found number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive integers $ m,n $  that satisfy the equation  \[ 3.2^m +1 = n^2 \]
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Context:
Dylan has a \( 100 \times 100 \) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?
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Context:
Given two circles with radii $R$ and $r(R>r)$, having internal tangency. Find the radius of the third circle that is tangent to the first two circles and their common diameter.

#
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Given the set $A=\{1,2, \cdots, 6\}$, the mapping $f: A \rightarrow A$ is a one-to-one mapping,
$$
\begin{array}{l}
f^{(2)}(x)=f(f(x)), \\
f^{(n+1)}(x)=f\left(f^{(n)}(x)\right)(x \geqslant 1) .
\end{array}
$$

Then the number of mappings $f$ that satisfy $f^{(12)}(x)=x$ for any $x \in A$ is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
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Context:
There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
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Context:
Positive integers  $a,b,c$  exist such that  $a+b+c+1$ ,  $a^2+b^2+c^2 +1$ ,  $a^3+b^3+c^3+1,$  and  $a^4+b^4+c^4+7459$  are all multiples of  $p$  for some prime  $p$ . Find the sum of all possible values of  $p$  less than  $1000$ .
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Context:
In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in the case of a tie, the order is arbitrary). It is noted that each participant won half of their points against the last ten ranked players. How many people participated in the tournament?
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Context:
Given a convex pentagon  $ABCDE$  with  $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$ . Find the area of  $ABCDE$ .
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Context:
Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number $i \in \{0, 1, . . . , 5\}$, some coin has exactly $i$ dots on it.
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Context:
The bases of three identical cones are located in the same plane and touch each other. The axial cross-section of each cone is an equilateral triangle with side length $a$. Find the radius of a sphere that touches the lateral surface of each cone and the plane in which their bases are located.
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Context:
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 100 residents of this island lined up, and each of them said one of the following phrases:

- "To the left of me there are as many liars as knights."
- "To the left of me there is 1 more liar than knights."
- "To the left of me there are 2 more liars than knights."
- "To the left of me there are 99 more liars than knights."

It is known that each phrase was said by exactly one person. What is the minimum number of liars that can be among these 100 residents?
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Context:
The product $20! \cdot 21! \cdot 22! \cdot \ldots \cdot 28!$ can be expressed in the form $m \cdot n^3$, where $m$ and $n$ are positive integers, and $m$ is not divisible by the cube of any prime. Find $m$.
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Context:
Four cars $A, B, C$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counter-clockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. When will all the cars meet for the first time after the race starts?
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. For the sake of secrecy, he will not visit the same village twice, and he will not visit villages in a row that are directly connected by a road. How many villages can he guarantee to fly over? He can start from any village.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The real number sequence \( x_0, x_1, x_2, \cdots \), is defined as \( x_0 = 1989 \) and \( x_n = -\frac{1989}{n} \sum_{k=0}^{n-1} x_k \) for \( n \geq 1 \). Calculate the value of \( \sum_{n=0}^{1988} 2^n x_n \).
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Context:
Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian Ji's bottom-tier horse being worse than King Qi's bottom-tier horse, determine the probability that Tian Ji's selected horse wins the race.
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Context:
Find all positive integer pairs \((a, b)\) such that the equation \(5^b + 2 = 7^c\) is satisfied.
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Context:
The terminal side of angle $α$ passes through point $P(x,4)$, and $\sin α= \frac {4}{5}$, then $x=$ _____ .
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Context:
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?
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Context:
Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence?
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Context:
 Brandon wants to maximize $\frac{\Box}{\Box} +\Box$ by placing the numbers $1$, $2$, and $3$ in the boxes. If each number may only be used once, what is the maximum value attainable?
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Context:
Find the number of tuples \((k_1, k_2, \ldots, k_{26})\) where each \(k_i\) is either 0, 1, or 3, such that the sum \(k_1 + k_2 + \cdots + k_{26} = 15\). Express this as finding the coefficient of \(x^{15}\) in the expansion of \((1 + x + x^3)^{26}\), and calculate this coefficient directly.
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Context:
Evaluate the limit: \[ \lim_{x\rightarrow a}\frac{a^{a^x}-a^{x^a}}{a^x-x^a} \] without using l'Hospital's rule.
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Context:
Evaluate the limit: $$ \lim_{n\to\infty} n\int_{1}^{2} \frac{1}{x^2(1+x^n)}dx $$.
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Context:
Find the maximum value of the expression \( T \) for real numbers \( x, y, z \geq 0 \):
\[ T = \frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3} + \frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3} + \frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3} \]
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Context:
Evaluate the integral \( \int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate $$\lim_{n\rightarrow \infty}\sum_{k=1}^{n}\cos\left(\frac{2\pi k}{2n+1}\right)$$
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Context:
Find the locus of the center of a circle that cuts the family of circles given by \(x^2+y^2+4x(\lambda-\frac{3}{2})+3y(\lambda-\frac{4}{3})-6(\lambda+2)=0\) orthogonally.
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Context:
The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$.
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Context:
A particle starts moving, and from the second mile onwards, its speed is inversely proportional to the distance traveled in miles. For each subsequent mile, the speed is constant. If it takes 2 hours to travel the second mile, then what is the time (in hours) required to travel the nth mile?
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Context:
Express the function $f$ in the form $f(e^t, t^2) = c$, where $c$ is a constant, given that the directional derivative of $f$ in the direction of $(e^t, t^2)$ is zero.
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Context:
Two white socks cost 25 cents more than a single brown sock. If two white socks cost 45 cents, how much would you pay for 15 brown socks?
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Context:
Determine the values of \( \alpha \) for which the matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ \alpha-3 & 6-2\alpha & 6-3\alpha \\ 2-\alpha & 2\alpha -4 & 3\alpha-4 \\ \end{pmatrix} \) is diagonalizable.
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Context:
The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon, Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.
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Context:
In triangle $ABC$, the area of the triangle is $70$. Points $E$ and $D$ are on side $AC$ such that $BE = ED = DA$, and points $F$ and $G$ are on side $AB$ such that $BF = FG = GC$. Quadrilateral $HIJK$ is formed within the triangle. What is the area of quadrilateral $HIJK$?
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Context:
Given \( x, y, z \geq 0 \) with at most one of them being zero, find the minimum value of
\[ f(x, y, z) = \sum \sqrt{\frac{x^{2}+256 y z}{y^{2}+z^{2}}} \]
where \(\sum\) represents a cyclic sum.
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Context:
Determine the radius of convergence for the series \(\sum_{n=1}^\infty \frac{(2z-3i)^n}{(3n-2i)^2}.\)
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Context:
计算下列实变函数定积分．

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } d x } { ( x ^ { 2 } + 9 ) ( x ^ { 2 } + 4 ) ^ { 2 } } ,$
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Context:
Let $\mathbb{N}$ denote the set of positive integers. For how many positive integers $k\le 2018$ do there exist a function $f: \mathbb{N}\to \mathbb{N}$ such that $f(f(n))=2n$ for all $n\in \mathbb{N}$ and $f(k)=2018$?
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Context:
Let $f:[0,\infty) \rightarrow[0,\infty]$ be a continuous function on $[0,\infty]$ and differentiable on $(0,\infty)$. Given that $f(x)=\int_{0}^{x}\sqrt{f(t)}dt$, determine the range of possible values of $f(6)$. \( \text{Provide your answer as an interval or set of values.} \)
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Context:
Will initially had twenty-eight bottles of water. He drank seven bottles each day, shared three bottles with his friends every two days, and bought an extra five bottles every three days. How many days would the bottles last him?
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Context:
Find the number of ways to put a number in every unit square of a $3 \times 3$ square such that any number is divisible by the number directly to the top and the number directly to the left of it, and the top-left number is $1$ and the bottom right number is $2013$.
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Context:
Evaluate the integral \( I = \int_0^{2\pi} \log\left|re^{it} - a\right| \, dt \) where \( a \) is a complex number and \( 0 < r < |a| \).
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Context:
Determine the order of the automorphism group of the group $A = \mathbb{Z} \oplus \mathbb{Z}_3$. What is $|\text{Aut}(A)|$?
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Context:
Assuming the twin prime conjecture, calculate the value of \( \mathcal{P}_\infty \), where \( \mathcal{P}_n \) is defined as follows: \[ \mathcal{P}_n = \frac{\#\text{twin primes} \le n:\min\{|b_1-a_1|,|b-a|,|b_2-a_2|\}\neq|b-a|}{\#\text{twin primes} \le n} \] Here, \( p_1 \) and \( p_2 \) are twin primes, \( p_1-1=a_1\times b_1 \) and \( p_2+1=a_2\times b_2 \) are factorizations such that \( |b_1-a_1| \) and \( |b_2-a_2| \) are minimized, and \( p_1+1=p_2-1=a\times b \) is a factorization such that \( |b-a| \) is minimized. Note that by convention the pair \((2,3)\) are not twin primes.
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Context:
Given that \(\cot \alpha + \cot \beta + \cot \gamma = -\frac{4}{5}\), \(\tan \alpha + \tan \beta + \tan \gamma = \frac{17}{6}\), and \(\cot \alpha \cot \beta + \cot \beta \cot \gamma + \cot \gamma \cot \alpha = -\frac{17}{5}\), find the value of \(\tan (\alpha + \beta + \gamma)\).
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Context:
Calculate the expected value of the following series:
\[ \sum_{i=4}^\infty 500 \frac{e^{-4} 4^i}{i!} \]
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Context:
Let $n = p^e m$, where $p^e$ is the largest power of the prime $p$ dividing $n$, and $p$ does not divide $m$. Consider a set $S$ with $|S| = n$. Compute the number of subsets of $S$ with exactly $p^e$ elements, denoted as $N$. Find $N \mod p$.
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Context:
Consider a $12 \times 12$ matrix $B$ where each diagonal entry is $2$ and all off-diagonal entries are $-1$. Calculate the sum of all entries of the inverse matrix $B^{-1}$. Provide your answer as a single number.
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Context:
Determine the values of $z$ on the unit circle $|z|=1$ for which the series \( \sum_{n=1}^\infty \frac{z^n}{n} \) converges.
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Context:
Determine the order of the projective general linear group $PGL_2(F_p)$, where $F_p$ is a finite field with $p$ elements.
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Context:
A function \( f \) is defined on the set of positive integers with the properties that for any positive integer \( n \), \( f(f(n)) = 2n \) and \( f(4n + 1) = 4n + 3 \). Determine the last three digits of \( f(2016) \).
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Context:
Calculate the integral: $$\int_{\mathbb{R}^5}\frac{e^{-x^2-y^2-z^2}}{1+w^2+s^2}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}w\,\mathrm{d}s$$
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Context:
Find the maximum value of $y$ for a point $(x,y)$ on the curve defined by the equation $2(x^3+y)^4+(x^3+y)^2=2x^3+y$. Provide your answer as a single number.
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Context:
Let $F = \{E_1, E_2, \ldots, E_n\}$ be a collection of $n$ subsets of a set $X$, where $n$ is a positive integer. Determine the number of distinct sets in the sigma-algebra $\sigma(F)$ generated by $F$. Provide your answer as a single integer.
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Context:
While visiting a small city in the United States, I lost my overcoat in a city bus. When I reported the matter to the bus company, I was asked to provide the bus number. Unfortunately, I couldn't remember the exact number, but I did recall certain peculiarities about it. The number plate showed the bus number as a perfect square, and even if the plate was turned upside down, the number remained a perfect square. Also, this number was a palindrome, reading the same backward and forward. There was no leading 0 in the number. I discovered that the bus company operated 2,000 buses, numbered sequentially from 1 to 2,000. Using this information, I was able to work out the bus number. Now your task is to find out which number was on the bus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a real vector space $V$ of dimension $2m$ with a non-singular quadratic form $Q$, if $Q$ is hyperbolic, determine the dimension of the subspace on which $Q$ vanishes.
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Context:
Minimize the function \( xy + 49xz + 7yz \) subject to the constraint \( xyz = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the total area covered by a circle of radius \( r \) whose center moves along the path \( x = t, y = \sin(t) \) for non-negative \( t \). Consider any overlaps in the area covered by the circle during its motion.
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Context:
Evaluate the double integral \(\int_{0}^{9}\int_{\frac{y-3}{2}}^{\sqrt{y}} x \, dx \, dy\) by reversing the order of integration.
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Context:
If x + y = 250 and x - y equals a certain number, for integers of x and y, y equals 225. What is the difference between x and y?
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Context:
过椭圆的左焦点 \(F\) 作倾斜角为 \(45^{\circ}\) 的直线 \(l\) 与该椭圆交于 \(A 、 B\) 两点。若 \(|B F|=2|A F|\)，则该椭圆的离心率是多少？答案应为\(\frac{\sqrt{n}}{m}\)的形式，请计算n+m的值。
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Context:
Find the norm of the Volterra operator $A$ acting on the space $L_2(0,1)$, where $Ax(t) = \int_0^t x(s) \, ds$. \( \boxed{} \)
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Context:
Calculate the cross product \( \nabla \times (\mathbf{i} \wedge \mathbf{r}) \), where \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), and \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the standard unit vectors in three-dimensional space.
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Context:
Let \( S = \{1, 2, 3, 4\} \) and let the sequence \( a_{1}, a_{2}, \cdots, a_{n} \) have the following property: For any non-empty subset \( B \) of \( S \) (with the number of elements in set \( B \) denoted as \( |B| \)), there exist \( |B| \) consecutive terms in the sequence that exactly form the set \( B \). Find the smallest possible value of \( n \).
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Context:
How many planes are equidistant from four points that do not lie in the same plane?
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Context:
A coin is tossed until two consecutive heads appear. The probability of the coin landing heads, denoted by \( p \), follows a beta distribution with parameters \( p \sim \beta(6, 8) \). What is the expected number of coin tosses required to achieve two consecutive heads?
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Context:
Let $V$ be a finite-dimensional vector space over the complex numbers $\mathbb{C}$, and let $T: V \rightarrow V$ be a linear transformation with a minimal polynomial of $x^{13}(x-1)$. Determine the minimal polynomial of $T^4$.
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Context:
Find the number of solutions to the equation $x^d = e$ in a cyclic group of order $n$, where $d = \gcd(m,n)$ and $m$ does not divide $n$. Provide your answer as a single integer.
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Context:
In a classroom with 10 girls and 5 boys, each of the 15 kids plays against every other kid exactly once, with a single winner in each game. The probability of winning for any participant is 0.5. Let X represent the total number of wins by the boys. Determine the minimum possible value of the variance, Var(X), considering potential dependencies between game outcomes.
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Context:
On a test, the boys in the class averaged 90 points and the girls in the class averaged 96 points. If the overall class average was 94 points, and there were 3 students with missing scores whose total average was 92 points, what is the ratio of boys to girls in the class, considering the students with missing scores?
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Context:
Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube.
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Context:
Calculate the determinant of the matrix \( J = A \left( A^\intercal W A \right)^{-1} A^\intercal W \), where \( A \) is a non-square matrix, and \( W \), \( J \), and \( A^\intercal W A \) are square matrices.
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Context:
129. In a parallelogram, there are two circles of radius 1, each touching the other and three sides of the parallelogram. It is also known that one of the segments of a side of the parallelogram from a vertex to the point of tangency is $\sqrt{3}$. Find the area of the parallelogram.
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Context:
Estimate the smallest integer $n$ such that $\frac{10^n}{n!} < 0.001$ using Stirling's approximation.
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Context:
How many positive integers between 10 and 2016 are divisible by 3 and have all of their digits the same?
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Context:
Find the number of ordered integer solutions \((A, B, C, D, E, F)\) such that \( A \times B \times C \times D \times E \times F = 7 \times 10^7 \).
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Context:
At 1 minute to midnight, 10 apples fall into a sack. The same happens at half a minute to midnight, then at a quarter minute to midnight, and so on. At each such event, you remove an apple randomly from the ones still present in the sack. What is the probability that the sack will be empty at midnight?
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Context:
Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______.
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Context:
In a math contest, Riley and Ofelia are on one team. Out of the 35 questions, Riley got some mistakes while Ofelia got 5 more than half the score of Riley. Their team got 17 incorrect answers. How many mistakes did Riley make?
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Context:
A sphere with radius 1 has its center at point \( O \). From a point \( A \), located outside the sphere, four rays are drawn. The first ray intersects the surface of the sphere at points \( B_1 \) and \( C_1 \) sequentially, the second ray at points \( B_2 \) and \( C_2 \), the third ray at points \( B_3 \) and \( C_3 \), and the fourth ray at points \( B_4 \) and \( C_4 \). Lines \( B_1B_2 \) and \( C_1C_2 \) intersect at point \( E \), and lines \( B_3B_4 \) and \( C_3C_4 \) intersect at point \( F \). Given that \( AO = 2 \), \( EO = FO = 3 \), and the angle between faces \( AOE \) and \( AOF \) is \( 30^\circ \), find the volume of pyramid \( OAEF \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group \(\mathrm{Gal}(\mathbb{Q}(\sqrt[5]{3})/\mathbb{Q})\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functional $f(x_1, x_2, x_3) = x_1 - 3x_2 + x_3$ and the dual basis $\mathcal{A}^* = \{g_1, g_2, g_3\}$ where $g_1(x_1, x_2, x_3) = x_1 - x_3$ and $g_2(x_1, x_2, x_3) = x_1 - 2x_2$, find the functional $g_3$ such that the third coordinate of $f$ in the basis $\mathcal{A}^*$ is 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Apple Street begins at Peach Street and runs directly east for some kilometers until it ends when it meets Cherry Street. Apple Street is intersected every 200 meters by a perpendicular street, and each of those streets other than Peach Street and Cherry Street is given a number. The highest-numbered street that intersects Apple Street is 15. How long is Apple Street in kilometers?
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Context:
We consider sports tournaments with  $n \ge 4$  participating teams and where every pair of teams plays against one another at most one time. We call such a tournament *balanced* if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the largest value of  $n$  for which a balanced tournament with  $n$  teams exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
当s、t取遍所有实数时，则$( s + 5 - 3 \vert \cos t \vert ) ^ { 2 } + ( s - 2 \vert \sin t \vert ) ^ { 2 }$所能达到的最小值是
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the probability that a standard Brownian motion hits 1 before hitting -1, and then hits -1 before hitting 2. Formally, find \( P(T_1 < T_{-1} < T_2) \), where \( T_a = \inf\{t \geq 0 \mid B_t = a\} \).
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Context:
对任意区间 $I$, 用 $M_{I}$ 表示函数 $y=\sin x$ 在 $I$ 上的最大值. 若正实数 $a$ 满足 $M_{[0, a]}= 2 M_{[a, 2 a]}$, 请提供所有情况中m+n的值的总和，其中原始答案的形式为\frac{m\pi}{n}。
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Context:
Let \( P(x) \) be a polynomial of degree 3 such that \( P(-1) = 10 \), \( P(1) = -6 \), and \( P(x) \) has a local maximum at \( x = -1 \). Additionally, \( P'(x) \) has a local minimum at \( x = 1 \). Find the distance between the local maximum and local minimum of the curve \( P(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression: \[ \sum_{n=1}^{64}(-1)^n \left\lfloor \dfrac{64}{n} \right\rfloor \varphi(n) \] where \(\varphi(n)\) denotes Euler's totient function. Use the identity \( \sum_{k=1}^n\left\lfloor\dfrac{n}{k}\right\rfloor \varphi(k) = \dfrac{n(n+1)}{2} \) and the fact that 64 is a power of 2 to simplify the sum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a convex \( n \)-gon \( A_1 A_2 \ldots A_n \) with area \( A \) and a point \( P \), we rotate \( P \) through an angle \( x \) about \( A_i \) to get the point \( P_i \). Find the area of the polygon \( P_1 P_2 \ldots P_n \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{4 \sin^2 \left(\frac{x}{2}\right) \cdot A}


Context:
Given a matrix $A \in \mathbb{R}^{m \times n}$, find a matrix $X \in \mathbb{R}^{m \times r}$ with orthonormal columns such that the Frobenius norm $\|X^T A\|$ is maximized. Express the solution in terms of the singular value decomposition of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an integer triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer.
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Context:
Consider a collection of 12 sets, each containing 8 elements. Every pair of sets shares 6 elements, any collection of 3 sets shares 4 elements, and no collection of 4 sets shares any elements. Determine the total number of distinct elements in this collection.
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Context:
Given points $A, B, C,$ and $D$ such that segments $AC$ and $BD$ intersect at point $E$. Segment $AE$ is 1 cm shorter than segment $AB$, $AE = DC$, and $AD = BE$. Additionally, $\angle ADC = \angle DEC$. Find the length of $EC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be the set of the first $n$ positive integers, where $n \geq 3$. The group $\mathcal{G}$ consists of all bijections of $X$, with composition as the group operation. $\mathcal{G}$ acts on $X$ in the standard way: for $\varphi \in \mathcal{G}$ and $i \in X$, $\varphi \cdot i = \varphi(i)$. Consider the induced action of $\mathcal{G}$ on $X \times X \times X$ defined by $\varphi \cdot (i, j, k) = (\varphi(i), \varphi(j), \varphi(k))$. Determine the number of orbits of this action on $X \times X \times X$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
The diagonal of a trapezoid divides its area in the ratio of 3:7. In what ratio will the area of this trapezoid be divided if a line parallel to a lateral side is drawn from the end of the shorter base?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An insurance company receives claims of two types. Let $N_i(t)$ represent the number of type $i$ claims by time $t$, where $\\{N_1(t), t \ge 0\\}$ and $\\{N_2(t), t \ge 0\\}$ are independent Poisson processes with rates $\lambda_1 = 10$ and $\lambda_2 = 1$, respectively. Type 1 claim amounts are independent exponential random variables with a mean of 1000 dollars, and type 2 claim amounts are independent exponential random variables with a mean of 5000 dollars. Given a claim of 4000 dollars, calculate the probability that it is a type 1 claim.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(x) = e^{2x}\cos(3x) \), find the minimum number of correct digits required in an approximation of an irrational number \( x^* \) within the interval \([0, \pi]\) such that the computed value \( f(x^*) \) has at least four correct digits compared to the exact value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute the following sum:
\[
\sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right).
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \triangle ABC \), the interior angles \( A, B, C \) satisfy \( \sin A = \cos B = \tan C \). Determine the value of \( \cos^3 A + \cos^2 A - \cos A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two two-digit numbers have a total of 79. If we use the same digits to form four two-digit numbers, with each digit used twice, what is the total of these four numbers? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
When Tim was Jim’s age, Kim was twice as old as Jim.  When Kim was Tim’s age, Jim was 30.  When Jim becomes Kim’s age, Tim will be 88.  When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the polynomial \( p(z) = z^{2015} + 8z^{12} + 1 \) that lie in the first quadrant of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \( \lim\limits_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{(\sin(x))^{n}}{1-\sin{(x)}}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the order of the group of rigid motions of a tetrahedron. Use the hint that involves determining the number of places to which a given face may be sent and the number of positions to which a vertex on that face may be sent.
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Context:
Determine the cardinality of the set of injective functions from the set of natural numbers, \(\mathbb{N}\), to its power set, \(\mathcal{P} \mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral \( \int_{0}^{\pi}\lfloor\cot x\rfloor \, dx \), where \( \lfloor a \rfloor \) denotes the greatest integer less than or equal to \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A large vase has a square base with a side length of \(6 \text{ cm}\), and its flat sides slope outwards at an angle of \(120^{\circ}\) with the base. Water flows into the vase at a rate of \(12 \text{ cm}^3/\text{s}\). Calculate, to three significant figures, the rate at which the height of the water is rising when the water has been flowing in for \(3\) seconds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. The numbers $x$ and $y$ are such that the equalities $\sin y + \cos x = \sin 3x$ and $\sin 2y - \sin 2x = \cos 4x - \cos 2x$ hold. What is the smallest value that the sum $\cos y + \sin x$ can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct arrangements of the letters \(A, A, A, A, A, B, B, B, C, C, C, D, E, E, F\) such that no two \(C\)'s are adjacent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an n-dimensional normal distribution with a mean vector of 0 and an identity covariance matrix. Determine the distribution of the distance from the mean for points drawn from this distribution.
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Context:
Find the value of $x$ that satisfies the equation $x^{x^3} = 3$.

Express your answer as a decimal rounded to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of functions \( f: \mathbb{R} \setminus \{0, 1\} \to \mathbb{R} \) that satisfy the functional equation: \[ f\left(\frac{x-1}{x}\right) + f\left(\frac{1}{1-x}\right) = 2 - 2x. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Susie Q has $2000 to invest. She invests part of the money in Alpha Bank, which compounds annually at 4 percent, and the remainder in Beta Bank, which compounds annually at 6 percent. After three years, Susie's total amount is $\$2436.29$. Determine how much Susie originally invested in Alpha Bank.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers $n$, let $f(n)$ denote the number of integers $1 \leq a \leq 130$ for which there exists some integer $b$ such that $a^b - n$ is divisible by $131$. Let $g(n)$ denote the sum of all such $a$. Find the remainder when \( \sum_{n = 1}^{130} [f(n) \cdot g(n)] \) is divided by $131$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint (x^2 + y^2) \, dx \, dy \) over the region enclosed by the curves \( y = 8x \), \( x + y = 6 \), \( y = 4 \), and \( y = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the reflections of points \( A, B, \) and \( C \) over a line \( e \) are \( A_1, B_1, \) and \( C_1 \), respectively, and their reflections over a point \( P \) are \( A_2, B_2, \) and \( C_2 \), find the reflection of \( P \) over the line \( e \) if \( \overrightarrow{A_1 A_2} + \overrightarrow{B_1 B_2} + \overrightarrow{C_1 C_2} = \overrightarrow{0} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the form of the function $f(x)$ that satisfies the equation $f(x^2 + yf(x)) = f(0)$ for all real numbers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the decimal expansion of \sqrt{\pi}$. Your score will be \min (23, k)$, where $k$ is the number of consecutive correct digits immediately following the decimal point in your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an irreducible polynomial in \(\mathbb{Z}[x]\) that is reducible modulo 2, 3, and 5.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x,$ $y,$ $z$ be nonzero real numbers such that $x + y + z = 0,$ and $xy + xz + yz \neq 0.$  Find all possible values of
\[
\frac{x^7 + y^7 + z^7}{xyz (xy + xz + yz)}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the least ordinal \( \beta \) such that it is provable in \( \mathsf{ZFC} \) that \( 2^{\aleph_0} \leq \aleph_{\aleph_\beta} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the coefficient of \(x^{33}\) in the expansion of \((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\). Use the identity \((1-x)^{-k} = \binom{k+n-1}{k-1}x^n\) to assist in your calculation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the dimension of the vector space \( \mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,s^2t-st+s-1) \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many different ways are there to select two numbers from the integers 1 through 20 such that their difference is greater than 10?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I_{n} = \int_{-1}^{1} \frac{1}{x} P_{n}(x) P_{n-1}(x) \, dx \) using the recurrence relation \((n+1)I_{n+1} + nI_{n} = 2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a square $\left\{(x,y): 0 \leq x,y \leq 1\right\}$ divided into $n^2$ smaller squares by the lines $x = \frac{i}{n}$ and $y = \frac{j}{n}$, where $1 \leq i,j \leq n$. Define $x_i = \frac{i}{n}$ and $d_i$ as:
$$d_i = \min_{0 \leq j \leq n} \left| \sqrt{1 - x_i^2} - \frac{j}{n}\right|.$$
Determine the limit:
$$\lim_{n\to\infty} \sum_{i=1}^n d_i$$
if it exists. This problem was posed in the American Mathematical Monthly in 1994 and has a connection to Weyl's equidistribution theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimal order of a homogeneous linear ordinary differential equation with constant coefficients that admits the solution \( y(t) = t^2 e^{-t/3} \cos(2t) \)?
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Context:
Let $a$ and $b$ be randomly selected three-digit integers and suppose $a > b$.
We say that $a$ is clearly bigger than $b$ if each digit of $a$ is larger than the corresponding digit of $b$.
If the probability that $a$ is clearly bigger than $b$ is $\tfrac mn$, where $m$ and $n$ are relatively prime integers,
compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Admit 4 students into 3 universities, with each university admitting at least one student. Find the total number of different admission methods. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the function $y=(\alpha - 1)x^{-4\alpha - 2}$ is a power function, then the value of the real number $\alpha$ is \_\_\_\_\_\_ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest degree of a monic polynomial $p(x)$ with integer coefficients such that $p(n)$ is divisible by 2021 for every positive integer $n$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1$, $X_2$, and $X_3$ be independent, identically distributed continuous random variables from an Exponential Distribution. Find the probability that the median of $X_1$, $X_2$, and $X_3$ is closer to the smallest value $X_{(1)}$ than it is to the largest value $X_{(3)}$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the degree of the angle between the hour and minute hands at 8:30 on a clock?
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Context:
How many pairs (x, y) of non-negative integers with 0 ≤ x ≤ y satisfy the equation 5x^2 - 4xy + 2x + y^2 = 624?
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Context:
Determine the dimensions $n$ for which the following statement holds: "For any norm $N$ in $\mathbb{R}^n$ and for any vectors $x, y \in \mathbb{R}^n$ such that $|x_i| \leq |y_i|$ for all $i$, we have $N(x) \leq N(y)$."
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Context:
Let $f \in L^1(\mathbb{R}) \cap C(\mathbb{R})$, meaning $f$ is integrable and continuous. For $z \in \mathbb{C}$ with $\text{Im}(z) \neq 0$, define
\[ g(z) = \int_{-\infty}^\infty \frac{f(t)}{t-z} \, dt. \]
Find \( \lim_{\text{Im}(z) \to 0} \left( g(z) - g(\bar{z}) \right) \).
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Context:
Christy and her friend Tanya go to Target to buy some face moisturizer and body lotions. Tanya pays 50$ for two face moisturizers each and 60$ per body lotion, buying four of them. Christy buys three face moisturizers and five body lotions, spending twice as much as Tanya. If there is a 10% discount on face moisturizers and a 15% discount on body lotions, how much money did they spend together in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability of $X > Y$ using integration, where $X$ and $Y$ are independent and uniformly distributed between $300$ and $700$, and $500$ and $800$, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrix $A = (c_1,...,c_n)$ where $c_i$ are the column vectors of $A$ and $c_i \in \mathbb{R}^m$, with $c_1 = (1,...,1)$ and $c_2 = c_3 = ... = c_m = (0,...,0)$, compute the 2-norm of the matrix $A$, denoted as $||A||_2 = \sup_{x \in \mathbb{R}^m} \frac{||Ax||_2}{||x||_2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways can 4 managers and 3 employees be arranged around a round table such that no 3 managers sit together?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a $4 \times 4$ square grid with 25 points. Determine the number of distinct lines passing through at least 3 of these points.
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Context:
Evaluate the limit: $$\lim_{n\to\infty} \frac{e^n}{1+n+\frac{n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}$$
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Context:
3. On the surface of the cube $A B C D E F G H$, a broken line composed of four identical segments is constructed in the faces $A B F E, B C G F, C D H G$ and $G H E F$, which starts from vertex $A$ and ends at vertex $E$. Determine in what ratio this broken line divides the edge $C G$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular triangle is constructed on the diameter of a semicircle such that its sides are equal to the diameter. Determine the ratio of the areas of the parts of the triangle lying outside and inside the semicircle.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\dfrac{3\sqrt{3} - \pi}{3\sqrt{3} + \pi}}


Context:
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the *product* of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are  $8^6=262144$  participants, and no two get the same score on every problem. Find the score of the participant whose rank was  $7^6 = 117649$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the polynomial \( f(x) = x^4 - 2x^3 - x^2 + 17x + 6 \) with roots \( x_1, x_2, x_3, x_4 \). Let \( g(x) = x^5 - 3x^4 + 22x^2 - 18x - 2 \). Find the value of \( \sum_{k=1}^{4} g(x_k) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an integer  $n\ge 2$ , a function  $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$  is called *good*, if for any integer  $k,1\le k\le n-1$  there exists an integer  $j(k)$  such that for every integer  $m$  we have
\[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \]
Find the number of *good* functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A singer earns a profit of $0.75 per concert. After each concert, there is a 50% chance the singer falls into a bad mood. To recover the singer's mood, the producer can send flowers costing $x. The probability of the singer recovering from the bad mood due to flowers is $\sqrt{x}$. If the singer's career lasts for $n$ days and all profits go to the producer, determine the value of $x$ that maximizes the producer's expected profit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A conical part has a base radius of 5 cm, and its height is three times the radius of the base. What is the volume of this part in cubic centimeters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABCD$  be a tetrahedron with  $AB=CD=1300$ ,  $BC=AD=1400$ , and  $CA=BD=1500$ . Let  $O$  and  $I$  be the centers of the circumscribed sphere and inscribed sphere of  $ABCD$ , respectively. Compute the smallest integer greater than the length of  $OI$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the directional derivative of the function \( f(x, y) \) at the point \((0, 0)\) in the direction of the vector \( \vec{v} = \langle \frac{4}{5}, \frac{3}{5} \rangle \), where the function is defined as:

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & \text{if } x^2 - y^2 \neq 0 \\ 0, & \text{if } x^2 - y^2 = 0 \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to arrange the numbers 1 through 9 in a 3x3 grid such that the sum of the numbers in each row and each column is the same. Note that the grid cannot be rotated or reflected.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John decides to get gym memberships so he can get in shape. The gym close to his house is close but doesn't have everything he wants, so he gets two different gym memberships. The cheap one costs $10 a month and has a sign-up fee of $50. The second gym is 3 times more expensive and it has a sign-up fee of 4 months membership. John also has to pay an annual maintenance fee of $30 at the cheap gym and $60 at the expensive gym. John refers one friend to each gym, receiving a 10% discount on the sign-up fees.

In addition, John also books personal training sessions at each gym. At the first gym, he pays $25 per session for the first 10 sessions and gets a 20% discount for the 11th to 20th sessions. At the second gym, he pays $45 per session for the first 5 sessions and then gets a 15% discount for the 6th to 15th sessions. John attends a total of 20 personal training sessions at the first gym and 15 sessions at the second gym in the first year.

How much total did he pay in the first year for gym memberships, personal training sessions, and annual maintenance fees?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya, Vasya, Kolya, and Tolya counted their catch after fishing. Tolya caught more than Kolya. Petya and Vasya together caught as much fish as Tolya and Kolya did together. Petya and Kolya together caught less fish than Vasya and Kolya. Which fisherman could have caught the most fish?
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task A-3.7.

We have ten white, and one red, blue, green, yellow, and purple card each. The white cards are indistinguishable from each other. On exactly one side of each card is the symbol $X$. In how many ways can we stack these cards one on top of the other so that no two cards are facing each other with the side that has the $X$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the mathematical expression:

$$\left(\left(\frac{a \sqrt[3]{b}}{b \sqrt{a^{3}}}\right)^{3 / 2}+\left(\frac{\sqrt{a}}{a \sqrt[8]{b^{3}}}\right)^{2}\right):\left(a^{1 / 4}+b^{1 / 4}\right)$$
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Context:
Determine the values of \(a \in \mathbb{R}_+\) for which the series \(\sum_{n>0} \frac{a(a+1)(a+2)\cdots(a+n-1)}{n!\cdot n^2}\) is convergent.
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Context:
Consider an equilateral triangle $T$ of side length 12. Matthew cuts $T$ into $N$ smaller equilateral triangles, each of which has side length 1,3, or 8. Compute the minimum possible value of $N$.
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Context:
Circle \( X \) has a radius of \( \pi \) . Circle \( Y \) has a circumference
 of \( {8\pi } \) . Circle \( Z \) has an area of \( {9\pi } \) . List the circles in order
 from smallest to largest radius.
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Context:
15 From the five numbers $-1, 0, 1, 2, 3$, select three (without repetition) to form the coefficients of the quadratic function $y=a x^{2}+b x+c$. The number of different parabolas that intersect the negative direction of the $x$-axis at least once is
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Context:
Determine the minimum number of equations required to ensure that an $n \times n$ matrix has a rank of at most $n-3$. Provide your answer as a single integer.
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Context:
What is the least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{ n : n3^n + (2n+1)5^n \equiv 0 \pmod{7} \}$?
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Context:
Determine if there exists a real number \( N \) such that the double sum \[ C_N\sum_{r_1\in \mathbb{Z}}\sum_{x_1\in \mathbb{Z}}\left(1+\frac{1}{2}\left(|x_1+r_1|+|x_1|\right)\right)^{-N} \] is convergent. If so, find the range of \( N \) for which the sum converges.
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Context:
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$,  it is true that at least one of the following numbers: $$a, b,\\frac{5}{a^2}+\\frac{6}{b^3}$$is less than or equal to $k$.
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Context:
Evaluate the triple integral: \[ \int_0^1 \int_0^{z\pi} \int_{\frac{y}{\pi}}^z \sin\frac{y}{x} \, dx \, dy \, dz \]
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Context:
Petya, Sasha, and Misha are playing an elimination tennis game. An elimination game means that in each match, two players compete while the third waits. The loser of the match gives up their place to the waiting player and becomes the waiting player for the next match. Petya played a total of 12 matches, Sasha played 7 matches, and Misha played 11 matches. How many times did Petya win against Sasha?
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Context:
Compute the sum of  $x^2+y^2$  over all four ordered pairs  $(x,y)$  of real numbers satisfying  $x=y^2-20$  and  $y=x^2+x-21$ .
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Context:
Evaluate the infinite series \( \sum_{k=1}^{\infty}\frac{18[(k-1)!]^2}{(2k)!} \).
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Context:
On a river with a current speed of $5 \text{ km/h}$, there are docks $A$, $B$, and $C$ positioned in the direction of the current, with $B$ located halfway between $A$ and $C$. A raft, which moves with the current towards dock $C$, and a motorboat, which heads towards dock $A$ with a speed of $v \text{ km/h}$ in still water, simultaneously depart from dock $B$. After reaching dock $A$, the motorboat turns around and moves towards dock $C$. Find all values of $v$ for which the motorboat arrives at $C$ later than the raft.
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Context:
Determine the conditional expectation \( \mathrm{E}(A|B) \) where \( B \) is the number of bats leaving a cave at the time of a nuclear explosion, following a geometric distribution with mean \( \lambda \). Each bat contributes an independent exponentially distributed amount of wind movement with mean \( \frac{1}{\mu} \). Let \( A \) be the total amount of wind movement at the time of the explosion. Assume no wind movement is accumulated if no bats are flying.
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Context:
In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$.
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Context:
Let $r, s,$ and $t$ be the distinct roots of $x^3- 2022x^2 + 2022x + 2022.$ Compute $$\frac{1}{1-r^2} + \frac{1}{1-s^2} + \frac{1}{1-t^2}.$$
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Context:
A workshop has 11 workers, of which 5 are fitters, 4 are turners, and the remaining 2 master workers can act as both fitters and turners. If we need to select 4 fitters and 4 turners to repair a lathe from these 11 workers, there are __ different methods for selection.
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Context:
Assume $a_n > 0$ and $\lim\limits_{n \to \infty}\dfrac{a_1+a_2+\cdots+a_n}{n}=a<+\infty$. Find the limit \( \lim\limits_{n \to \infty}\dfrac{a_1^p+a_2^p+\cdots+a_n^p}{n^p} \) where $p > 1$. \( \boxed{} \)
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Context:
There is an oil tank in the shape of a right frustum that can hold 190L of oil. If the lengths of its two bases are 60cm and 40cm respectively, find its depth.
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Context:
Find the maximum value of the expression \[|1+x|+|1+xy|+|1+xy^2|+\cdots+|1+xy^{2013}|-1007|1+y|\] given that \(|x| = |y| = 1\) for complex numbers \(x\) and \(y\).
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Context:
What is the minimal Hamming distance of the concatenated word $R$, where $R$ is formed by concatenating a word $M$ of size $n^2$ bits (organized into an $n \times n$ matrix with $n > 3$) and the encoded main diagonal $B$ using a Hamming code with a minimal distance of 3, given that $M$ is unencoded with a Hamming distance of 1?
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Context:
已知$A = \{ x \vert x ^ { 2 } - 4 x + 3 < 0 , x \in R \}$,

$B = \{ x \vert 2 ^ { 1 - x } + a \leq 0 , x ^ { 2 } - 2 ( a + 7 ) x + 5 \leq 0 , x \in R \}$,

若A⊆B，则实数a的取值范围是
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Context:
Typhoon "Haima" moves northward at a speed of 25 km/h. The observation station is located at point A on the sea. At 9 a.m., the typhoon's center was observed to be at point B, southeast of the station. At 10 a.m., the typhoon's center was observed to be at point C, in a south-southeast direction 75° from the station. At this time, the distance between the observation station and the typhoon's center, AC, is __________ km.
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Context:
Evaluate the limit \( \lim_{n \to \infty} n\left(\left(2x^{\frac{1}{n}}-1\right)^{n}-x^2\right) \) for \( x \ge 1 \).
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Context:
已知集合 A=\{1,2,\cdots,6\}，映射 f:A\rightarrow A 为一一映射. 则满足对于任意的 x\in A，f^{(12)}(x)=x 的映射 f 的个数为__________.
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Context:
Given the basis \(B = \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4\}\) for \(\mathbb{R}^4\), where \(\mathbf{e}_1 = (1, 0, 0, 0)\), \(\mathbf{e}_2 = (1, 1, 0, 0)\), \(\mathbf{e}_3 = (1, 1, 1, 0)\), and \(\mathbf{e}_4 = (1, 1, 1, 1)\), and the dual basis \(B^\perp = \{\mathbf{e}^1, \mathbf{e}^2, \mathbf{e}^3, \mathbf{e}^4\}\), where \(\mathbf{e}^1 = (1, -1, 0, 0)\), \(\mathbf{e}^2 = (0, 1, -1, 0)\), \(\mathbf{e}^3 = (0, 0, 1, -1)\), and \(\mathbf{e}^4 = (0, 0, 0, 1)\), find the dot product \(\mathbf{u} \cdot \mathbf{v}\) where \(\mathbf{u} = \mathbf{e}^1 - \mathbf{e}^2 + \mathbf{e}^3 - \mathbf{e}^4\) and \(\mathbf{v} = (1, 2, 3, 4)\). Use the covariant components of \(\mathbf{v}\) and the contravariant components of \(\mathbf{u}\) to compute the dot product.
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Context:
There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ unit apart. Find the maximum possible value of $I$.
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Context:
A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$ and $S(m, 1) = 1$, along with the recurrence relation $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m \geq 2$ and $n \geq 2$. Determine the largest integer $k$ such that $2^k$ divides $S(7, 7)$. \( \text{Provide your answer as an integer.} \)
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Context:
Calculate the remainder when \(787^{777}\) is divided by \(100\). Use modular arithmetic to determine the tens digit efficiently.
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Context:
Calculate the line integral \( E = \int_\gamma \frac{-y \cdot dx + x \cdot dy}{x^2 + y^2} \) for the curve \( \gamma \), which is the parabola \( y = 2x^2 - 1 \) from \((1, 1)\) to \(\left(\frac{1}{2}, -\frac{1}{2}\right)\).
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Context:
Calculate the line integral of the vector field \( F(x,y) = (\sin(x) + 3x^2, 2x - e^{-y^2}) \) along the path \( \gamma_1(t) = (\cos(t), \sin(t)) \) for \( 0 \leq t \leq \pi \). The path is the upper half of the unit circle from \((1,0)\) to \((-1,0)\).
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Context:
How many ways are there to fill in a three by three grid of cells with $0$’s and $2$’s, one number in each cell, such that each two by two contiguous subgrid contains exactly three $2$’s and one $0$?
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Context:
Compute the integral \( \int_S \omega \) over the set \( S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1, z\geq 0\} \), where the differential form \( \omega \) is defined as follows: \( \omega = \frac{dydz}{x} \) when \( x \neq 0 \), \( \omega = \frac{dzdx}{y} \) when \( y \neq 0 \), and \( \omega = \frac{dxdy}{z} \) when \( z \neq 0 \).
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Context:
Evaluate \( \lim_{x\rightarrow 0}\frac{(1+x)^{\frac{1}{x}}+(1+2x)^{\frac{1}{2x}}+(1+3x)^{\frac{1}{3x}}-3e}{9x} \) using the fact that \( \lim_{x\to 0}\left(\frac{(1+kx)^{\frac{1}{kx}}-e}{x}\right)= \frac{-ke}{2} \).
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Context:
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)
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Context:
Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
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Context:
6. Three identical cones with vertex $A$ and vertex angle $\frac{\pi}{3}$ touch each other externally. Each of them touches internally a fourth cone with vertex at point $A$. Find the vertex angle of the fourth cone. (The vertex angle of a cone is the angle between its generators in the axial section.)
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Context:
In how many ways can the nine digits from 1 to 9 be placed in a $3 \times 3$ grid so that the sums of the rows and the sums of the columns are all equal?
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Context:
Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'?
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Context:
Given a triangle $ABC$ with $O$ as the incenter. Find the angle $A$ if the circumradii of triangles $ABC$ and $BOC$ are equal.
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Context:
An object with an initial temperature $T_0$ is placed in a room with a constant temperature of 30°C. After 10 minutes, the object's temperature is -10°C, and after another 10 minutes, it is -5°C. Using Newton's law of cooling, determine the initial temperature $T_0$ of the object.
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Context:
Let $g_n(x) = \frac{\sin^2(nx)}{n \sin x}$. Find the limit:

$$\lim_{n\to\infty} \max_{x\in R} g_n(x)$$
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Context:
Write out the interval where the function $y = -(x - 1)^2$ is monotonically increasing.
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Context:
Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \mathbb{F}_2[x]/(1+x+x^2+x^3) \).
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Context:
实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 1 }$满足$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$，令$y _ { k } = \frac { 1 } { k } ( x _ { 1 } + x _ { 2 } + \cdots +$ $x _ { k } )$ k=1,2,⋯,2001.求$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 }$ 的最大可能值.
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Context:
Given natural numbers \(A\) and \(B\) such that \(A \le B\), you repeatedly toss a fair coin to decide the next step. If the result is heads, decrement \(A\) by 1. If the result is tails and \(A < B\), increment \(A\) by 1; otherwise, do nothing. Determine the expected number of coin flips required for \(A\) to reach 1.
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Context:
The number 52f3b is divisible by 7, 13, and 89. What is the value of f * 2 * 3 * b?
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Context:
Evaluate the limit: \[ \lim_{x \to 0}\left[\frac{1}{\log (x+ \sqrt{1+x^2})}-\frac{1}{\log (1-x)}\right] \]
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Context:
Determine the asymptotic complexity of the recurrence relation \( T(n) = T\left(\frac{3}{5} n + 1\right) + n^2 \) using Big O notation.
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Context:
Given that $f(x)= \lim\limits_{n\to\infty} n(x^{1/n} - 1)$, find the value of $f(xy)$.
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Context:
Three projective planes $\pi_1$, $\pi_2$, and $\pi_3$ intersect such that $\pi_1 \cap \pi_2 \cap \pi_3$ is a point, $\dim\,(\pi_1 \cap \pi_2) = 0$, $\dim\,(\pi_1 \cap \pi_3) = 1$, and $\dim (\pi_2 \cap \pi_3) =1$. Determine the dimension of $\pi_1 \lor \pi_2\lor \pi_3$. \(\boxed{\ }\)
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Context:
Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).
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Context:
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that$$f^{f^{f(x)}(y)}(z)=x+y+z+1$$for all $x,y,z \in \mathbb{N}$.
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Context:
A random number selector can only select one of the nine integers $1, 2, ..., 9$, and it makes these selections with equal probability. Determine the probability that after $n$ selections ($n>1$), the product of the $n$ numbers selected will be divisible by $10$. Please calculate the probability when n = 2 and give the result in the form of \frac{m}{n}, and then provide the value of m + n.
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Context:
Find the mass percentage of O in Al2(CO3)3. What is the mass percentage of O in this compound?
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Context:
Immediately after harvest, the water content in 1 ton of tomatoes was $99 \%$. By the time of sale, the water content decreased by $4 \%$ (after drying). What is the total weight of the tomatoes now?
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Context:
A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105?
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Context:
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once, and the European teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?
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Context:
What is the order of the symmetry group of a cube?
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Context:
Find a solution to the differential equation \((x + 1 + f(−x)) \cdot (1 − f′(x)) = x + 1\) for \(-1 \leq x \leq 1\).
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Context:
Let $L \ge 2$ be an integer, with $a_0 = 10$, $a_L = 2$, and integers $a_1, \ldots, a_{L-1} \ge 2$ such that their sum is 36. Find the maximum value of $\sum_{i=0}^{L-1} a_i a_{i+1}$ over all possible values of $L$ and $a_1, \ldots, a_{L-1}$.
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Context:
Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$.Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$
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\boxed{
The maximum value of |M| is 21.

}


Context:
Task 4.5. Write the largest eight-digit number in which all even digits are present. (Even digits: $0,2,4,6,8$.)
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Context:
It is known that for the inequality involving \( x \)
$$
\frac{\mathrm{e}^{x}}{x^{3}} - x - a \ln x \geqslant 1
$$
to hold for any \( x \in (1, +\infty) \), determine the range of values for the real number \( a \).
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Context:
Determine the set of values of $x$ for which the infinite series $\sum_{k=0}^{\infty} E_k(x)$, where $E_0(x) = x$ and $E_{n+1}(x) = e^{E_n(x)}$, converges.
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Context:
Given a positive arithmetic geometric sequence {a_n} where the sum of the first n terms is S_n, and a_{n+1}/a_n < 1, if a_3 + a_5 = 20 and a_2 * a_6 = 64, calculate the value of S_6. Express your answer as a single numerical value.
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Context:
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). Find the perimeter of \(\triangle P S T\). Express your answer as a whole number.
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Context:
The coordinates of the vertices of a triangle are $A(p, q), B(q, r), C(r, p)$, where $p<q<r$. What are the bounds of the angles at each vertex?
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\boxed{45^\circ < \alpha, \beta < 90^\circ}


Context:
Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid?
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Context:
a trailer carries 3 , 4 and 6 crates on a trip . each crate weighs no less than 120 kg . what is the maximum weight of the crates on a single trip ?
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Context:
A mouse has a wheel of cheese which is cut into  $2018$  slices. The mouse also has a  $2019$ -sided die, with faces labeled  $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on  $k$ , and the mouse has at least  $k$  slices of cheese remaining, then the mouse eats  $k$  slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
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Context:
A variable parabola with latus rectum \( l \) touches a fixed parabola \( y^2 = 4bx \), with the axes of the two parabolas being parallel. Determine the latus rectum of the parabola that is the locus of the vertex of the moving parabola.
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Context:
Find the area of an equiangular octagon with side lengths 1, 2, 2, 4, 1, 2, 2, 4, in that order.
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Context:
Find the minimum positive integer  $k$  such that there exists a function  $f$  from the set  $\Bbb{Z}$  of all integers to  $\{1, 2, \ldots k\}$  with the property that  $f(x) \neq f(y)$  whenever  $|x-y| \in \{5, 7, 12\}$ .
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Context:
Let \(a\) be a root of the polynomial \(x^3 + x^2 + 1\) in some extension of the field \(\mathbb{Z}_2\). Determine the multiplicative inverse of \(a+1\) in this extension.
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Context:
How many 6-tuples  $ (a_1,a_2,a_3,a_4,a_5,a_6)$  are there such that each of  $ a_1,a_2,a_3,a_4,a_5,a_6$  is from the set  $ \{1,2,3,4\}$  and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for  $ j \equal{} 1,2,3,4,5,6$  (where  $ a_7$  is to be taken as  $ a_1$ ) are all equal to one another?
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Context:
Let \( P(x) \) and \( Q(x) \) be distinct polynomials of degree 2020 with non-zero coefficients. Suppose they have \( r \) common real roots (counting multiplicity) and \( s \) common coefficients. Determine the maximum possible value of \( r + s \).
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Context:
The brothers found a hoard of gold and silver. They divided it in such a way that each received 100 kg. The eldest received the most gold - 30 kg - and a fifth of all the silver. How much gold was in the hoard?
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Context:
Given a positive integer \( n \) that is a multiple of 5, and the least common multiple of \( n \) and \( 5! \) equals 5 times the greatest common divisor of \( n \) and \( 10! \), find the number of \( n \) that satisfy this condition. Express your answer as a single integer count.
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Context:
12 chess players take part in a tournament. Every player plays twice with each of his opponents. How many games are to be played?
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Context:
 A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus?
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Context:
Find the minimum value of the expression \( \large \dfrac{y}{x + 2} + \dfrac{x}{y + 2} + \dfrac{y^2 + 3x^2 + 4}{8xy \cdot (xy + 2)} \) for \( 1 < x \le y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, the orthocenter is $H$. Given $AB = 84$, $AC = 32\sqrt{3}$, and $\angle BAC = 60^\circ$, points $D, E, F$ are on $AH, BH, CH$ respectively such that $7AD = AH$, $7BE = BH$, and $7CF = CH$. Calculate the value of $AB \cdot CF + BC \cdot AD + AC \cdot BE$. \(\boxed{\text{Your answer here}}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the prime numbers \( p \) for which there exist \( x, y \in \mathbb{N}^{*} \) such that

\[ x\left(y^{2} - p\right) + y\left(x^{2} - p\right) = 5 p \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A basic computer, printer, monitor, and keyboard have a total price of $2,500. If the same printer, monitor, and keyboard had been purchased with an enhanced computer, the total price would be $600 more than the price of the basic computer setup. Additionally, in the enhanced setup, the price of the printer would be 1/6 of the total, the price of the monitor would be 1/5 of the total, and the price of the keyboard would be 1/8 of the total. What was the price of the basic computer in the original setup?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A person can swim in still water at 4 km/h. If the speed of water is 2 km/h, how many hours will the man take to swim back against the current for a certain distance, if it takes him 4 hours to do so?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim\limits_{n \to \infty} \int_0^n \frac{1}{1 + n^2 \cos^2 x} \, dx$$.
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Context:
Let $G$ be the centroid of $\triangle ABC$. Determine the position of point $P$ in the plane of $\triangle ABC$ such that $PA \cdot AG + BP \cdot BG + CP \cdot CG$ is minimized. Express this minimum value in terms of the side lengths of $\triangle ABC$.
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Context:
A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
An inscribed circle with radius \( R \) is placed in a certain angle, and the length of the chord connecting the tangent points is \( a \). Two tangents are drawn parallel to this chord, forming a trapezoid. Find the area of this trapezoid.
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Context:
Find the smallest integer value of $n$ such that when a fair coin is tossed $n$ times, the probability that the proportion of heads is between 0.49 and 0.51 is greater than 0.95.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the expression \( N=\frac{1987^{4(\sqrt{3}+2)^{1987}+1988}}{1987^{(\sqrt{3}+2)^{1988}}+(\sqrt{3}+2)^{1988}} \), calculate the last digit of \( N \). Express your answer as a single digit (0-9).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the expected value of the piecewise function \( u(x,y) \) given two independent continuous random variables \( x \) and \( y \) with uniform distribution in the range \([0,b]\). The joint PDF is \( f(x,y)=\frac{1}{b^2} \). The piecewise function is defined as:

\[
u(x,y) = \begin{cases}
0 & \text{if } x,y < b/2 \\
b/2 & \text{if } (y<b/2 \text{ and } x > b/2) \text{ or } (x < b/2 \text{ and } y > b/2) \\
x & \text{if } y,x>b/2 \text{ and } y>x \\
y & \text{if } y,x>b/2 \text{ and } x>y
\end{cases}\]

Calculate \( E(u(x,y)) = \int_0^b\int_0^b \frac{u(x,y)}{b^2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle of 100 balls, 3 are red and 97 are blue. The balls are randomly placed in the circle. Determine the most probable minimum interval between the red balls.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$\lim_{n \to \infty } \int_0^{\infty} x^{\frac{n}{n+1}} e^{-x^2} \, dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that 2017 is a prime number, find the smallest positive integer \( n \) for which \( 2017^{{2017}^{2017}} \) divides \( n! \). Use the formula:

\[ v_{2017}(n!) = \frac{n - S_{2017}(n)}{2016} \]

where \( S_{p}(n) \) represents the sum of the standard base-\( p \) digits of \( n \). Determine this smallest value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel?\n\n[i]Proposed by Aaron Lin[\/i]
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Context:
Suppose $f(x,y)$ is differentiable at $(1,1)$, and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$. Find the value of $\frac{d}{dx}f(x,f(x,x))$ at $x=1$. \( \boxed{?} \)
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Context:
In a certain code, 'range' is coded as 12345 and 'random' is coded as a sequence of numbers. The code for the word 'rand' is 1236. What is the code for the word 'random'?
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Context:
Given a rectangle \(ABCD\). A line through point \(A\) intersects segment \(CD\) at point \(X\) such that the areas of the resulting shapes satisfy \(S_{AXD}: S_{ABCX} = 1:2\). A line through point \(X\) intersects segment \(AB\) at point \(Y\) such that \(S_{AXY}: S_{YBCX} = 1:2\). Finally, a line through point \(Y\) intersects segment \(XC\) at point \(Z\) such that \(S_{XYZ}: S_{YBCZ} = 1:2\).

Calculate the ratio of the areas \(S_{AXD}: S_{AXZY}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If  $x > 10$ , what is the greatest possible value of the expression
\[
  {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10.
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Context:
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is  $5:9$ . Compute the ratio of the secant line versus circle radius.
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Context:
Diagonals connecting the second and third neighboring vertices of a regular octagon determine two convex octagons. What is the ratio of their areas?
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Context:
Let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) = 0$. Consider the approximation of the integral $\int_0^1 f(x) \, dx$ by dividing $[0,1]$ into $n$ sub-intervals of equal length and using the maximum value of $f(x)$ on each sub-interval to form rectangles. Determine the maximum relative error of this approximation as a function of $n$.
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Context:
Given that  $a,b,c$  are integers with  $abc = 60$ , and that complex number  $\omega \neq 1$  satisfies  $\omega^3=1$ , find the minimum possible value of  $|a + b\omega + c\omega^2|$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
现有2个红球，3个黑球，5个白球（同色球完全相同），从中任取6个球，有多少种不同的取法？
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Context:
There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?
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Context:
Three spheres with radii $\sqrt{6}, 1$, and 1 touch each other pairwise. A plane $\gamma$ is passed through the line containing the centers $A$ and $B$ of the second and third spheres, such that the center $O$ of the first sphere is at a distance of 1 from this plane. Find the angle between the projections of the lines $OA$ and $OB$ on the plane $\gamma$ and compare it to $\arccos \frac{4}{5}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $n,k\to\infty$ such that $k/n^2\to a>0$. What is the limit of $\dfrac{{{k-1}\choose n}}{{{n+k-1}\choose n}}$ using the correct asymptotic analysis?
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Context:
The door of a rabbit hutch is made from a wooden frame and wire mesh with square openings. The wooden slats of the frame are $5 \mathrm{~cm}$ wide. Some nodal points of the mesh align with the inner edges of the frame as shown in the diagram. The inner (mesh) part of the door has an area of $432 \mathrm{~cm}^{2}$.

Determine the external dimensions (i.e., width and height) of the entire door.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{28 \text{ cm} \times 34 \text{ cm}}


Context:
Given the identity $S(n, n - 3) = a \binom n 4 + b \binom n 5 + c \binom n 6$, find $a, b, c$. $S(n, k)$ denotes a Stirling number of the second kind, i.e., the number of ways to place $n$ labeled balls into $k$ non-labeled non-empty boxes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Starting from the number $123456789$, at each step, we are swaping two adjacent numbers which are different from zero, and then decreasing the two numbers by $1$. What is the sum of digits of the least number that can be get after finite steps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sum of all values of the parameter \( a \) for which the equation \( f(x) = p(x) \), where \( f(x) = \left|\frac{2 x^{3}-x^{2}-18 x+9}{(1.5 x+1)^{2}-(0.5 x-2)^{2}}\right| \) and \( p(x) = |-2 x+2| + a \), has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
  [center]
        C,CA,CAM,CCAM,CCAMT.
    [/center]
   [center]
        A,AT,CAT,CAMT,CCAMT.
  [/center]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given matrices $A$ and $B$, compute the gradient $\nabla_X g$ of the scalar field $g(X) = \operatorname{tr}(AXB)$. Express your answer in terms of $A$ and $B$.
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Context:
In an acute triangle  $ABC$ , the segment  $CD$  is an altitude and  $H$  is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle  $DHB$ , determine all possible values of  $\angle CAB$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hemisphere with a radius of $10 \mathrm{~cm}$ protrudes from a horizontal plane (to the annoyance of drivers). A cylinder rolls on the plane and collides with the hemisphere with its side. Let $\alpha$ be the angle between the common tangent plane at the point of contact of the two bodies and the horizontal plane. What is the minimum radius of the cylinder if $\alpha \leq 30^{\circ}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
24. S2 (POL) $)^{1 \mathrm{MO}}$ The positive real numbers $x_{0}, x_{1}, \ldots, x_{1996}$ satisfy $x_{0}=$ $x_{1995}$ and
$$
x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}
$$
for $i=1,2, \ldots, 1995$. Find the maximum value that $x_{0}$ can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points \( M \) and \( N \) divide side \( AC \) of triangle \( ABC \) into three equal parts, each of which is 5, with \( AB \perp BM \) and \( BC \perp BN \). Find the area of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the norm of the projection operator \( P: c \to c_0 \) defined by \( P(x_0,x_1,x_2,\ldots) = (x_0-\lim\limits_{n\to\infty}x_n,x_1-\lim\limits_{n\to\infty}x_n,x_2-\lim\limits_{n\to\infty}x_n,\ldots) \) in the space of convergent sequences with respect to the supremum norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a=\sqrt{0.5}$, $b=\sqrt{0.3}$, and $c=\log_{0.3}{0.2}$. Determine the order of $a$, $b$, and $c$, expressing your answer as an inequality using the variables $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the molecular weight of 7 moles of KBrO3
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sum of a sequence and the number of iterations \( T \), determine the base number \( x \) such that the sequence \( x^2 \) repeated \( T \) times and summed results in the given sum. For example, if the sum is 126 and \( T = 6 \), find the value of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We call a positive integer  "special" if the number is divided by all of its digits (except the $0$s). At most how many consequtive special numbers are there?
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Context:
The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:

1. The Falcons won more games than the Hawks.
2. The Warriors won more games than the Knights but fewer than the Royals.
3. The Knights won more than 25 games.

How many games did the Warriors win? The wins recorded are 20, 26, 30, 35, 40, and 45.
Please reason step by step, and put your final answer within \boxed{}.


Context:
我从1,2，···，144中任取一个数，你每次取一个｛1,2，···，144｝的子集，问我取的数是否在该子集中，最终确定我取的数.如果每次回答，答案是Yes，则你需付出2元，答案是No，则你需付出1元．为保证能找出我取的数，你需要付出的钱的最小值为多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the parameter $a$ such that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \begin{cases} \sin\frac{1}{x} & \text{for } x \in \mathbb{R} \setminus \{0\} \\ a & \text{for } x = 0 \end{cases} \) has an antiderivative on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers from 1 to 10 were written in some order, resulting in the numbers \( a_{1}, a_{2}, a_{3}, \ldots, a_{10} \). Then the sums \( S_{1} = a_{1} \), \( S_{2} = a_{1} + a_{2} \), \( S_{3} = a_{1} + a_{2} + a_{3} \), \(\ldots\), \( S_{10} = a_{1} + a_{2} + a_{3} + \ldots + a_{10} \) were calculated. What is the maximum number of prime numbers that could appear among the numbers \( S_{1}, S_{2}, S_{3}, \ldots, S_{10} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval of uniform convergence for the series \(\sum_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^{np} e^{-nx}\).
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Context:
For any integer $n$, define $\lfloor n\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let $$f(n)=\lfloor n\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{3}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor.$$ For how many values of $n, 1 \leq n \leq 100$, is $f(n)$ odd?
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Context:
Fix an odd integer  $n > 1$ . For a permutation  $p$  of the set  $\{1,2,...,n\}$ , let S be the number of pairs of indices  $(i, j)$ ,  $1 \le i \le  j \le  n$ , for which  $p_i +p_{i+1} +...+p_j$  is divisible by  $n$ . Determine the maximum possible value of  $S$ .

Croatia
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Context:
Mom gave Vasya money for 30 pencils. It turned out that the pencil factory is running a promotional campaign in the store: in exchange for a receipt for the purchase of a set of 20 pencils, they return 25% of the cost of the set, and in exchange for a receipt for the purchase of a set of 5 pencils, they return 10%. What is the maximum number of pencils Vasya can buy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint_D \sin(x^2+y^2)~dA \) by transforming it to polar coordinates, where \( D \) is the region in the first quadrant bounded by the lines \( x=0 \) and \( y=\sqrt{3}\cdot x \) and the circles \( x^2+y^2=2\pi \) and \( x^2+y^2=3\pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a directed graph with  $2013$  vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach  $k$  vertices using at most two edges. Find the maximum value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a recent math contest, Evan was asked to find  $2^{2016} \pmod{p}$  for a given prime number  $p$  with  $100 < p < 500$ . Evan has forgotten what the prime  $p$  was, but still remembers how he solved it:

- Evan first tried taking  $2016$  modulo  $p - 1$ , but got a value  $e$  larger than  $100$ .
- However, Evan noted that  $e - \frac{1}{2}(p - 1) = 21$ , and then realized the answer was  $-2^{21} \pmod{p}$ .

What was the prime  $p$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $P$: $a < x < a+1$, and $q$: $x^{2}-7x+10\leqslant 0$, if $P$ is a necessary but not sufficient condition for $q$, find the range of values for $a$.
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Context:
Given the equation \(x^4 + 3\cos(ax^2 + bx +c) = 2(x^2-2)\) with parameters \(a, b, c \in (2,5)\), it has two solutions. Determine the maximum value of \(\frac{ac}{b^2}\).
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Context:
Find all integer values of $n$ such that $n-2$ divides $n^{2} + 3n + 27$.
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Context:
设 $x, y, z$ 是三个两两不等且都大于 1 的正整数, 若 $x y z$ 整除 $(x y-1) \cdot (x z-1)(y z-1)$, 求 $x$ 所有可能值之和。
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Context:
New this year at HMNT: the exciting game of $R N G$ baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number $n$ uniformly at random from \{0,1,2,3,4\}. Then, the following occurs: - If $n>0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by $n$ bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. - If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A roulette wheel can land on any number from 0 to 2007 with equal probability. The roulette wheel is spun repeatedly. Let \( P_{k} \) denote the probability that at some point the sum of the numbers that have come up in all the spins made equals \( k \). Determine which is greater: \( P_{2007} \) or \( P_{2008} \).
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Context:
Perimeter of triangle  $ABC$  is  $1$ . Circle  $\omega$  touches side  $BC$ , continuation of side  $AB$  at  $P$  and continuation of side  $AC$  in  $Q$ . Line through midpoints  $AB$  and  $AC$  intersects circumcircle of  $APQ$  at  $X$  and  $Y$ .
Find length of  $XY$ .
Please reason step by step, and put your final answer within \boxed{}.