Context:
A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$.
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Context:
Determine the interval in which the function $f(x)={(\dfrac{1}{2})}^{\sqrt{x-{{x}^{2}}}}$ is monotonically decreasing.
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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 second train is 280 m long. From the moment they meet, they will be clear of each other in 24.998 seconds. What is the length of the first train?
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Context:
1. Given the quadratic function
$$
\begin{aligned}
f(x)= & a(3 a+2 c) x^{2}-2 b(2 a+c) x+ \\
& b^{2}+(c+a)^{2}(a, b, c \in \mathbf{R}).
\end{aligned}
$$

Assume that for any $x \in \mathbf{R}$, we have $f(x) \leqslant 1$. Find the maximum value of $|a b|$. (Yang Xiaoming, Zhao Bin)
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Context:
对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有
$f(X)=\frac{1}{|X|} \sum_{a \in X} a$.
集合对 $(A, B)$ 满足
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$,
且 $1 \leqslant|A| \leqslant 98$.
任取 $p \in B$, 令
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$.
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$, 求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Ali and Veli go hunting. The probability that each will successfully hit a duck is \(\frac{1}{2}\) on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. What is the probability that Veli hits more ducks than Ali? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form, and find the value of k + m.
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Context:
Use the bisection method to find an approximate zero of the function $f(x) = \log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\log 2.5 \approx 0.398$, $\log 2.75 \approx 0.439$, and $\log 2.5625 \approx 0.409$.
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Context:
In \(\triangle QRS\), point \(T\) is on \(QS\) with \(\angle QRT = \angle SRT\). Suppose that \(QT = m\) and \(TS = n\) for some integers \(m\) and \(n\) with \(n > m\) and for which \(n + m\) is a multiple of \(n - m\). Suppose also that the perimeter of \(\triangle QRS\) is \(p\) and that the number of possible integer values for \(p\) is \(m^2 + 2m - 1\). What is the value of \(n - m\)?
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Context:
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?
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Context:
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. What is the area of the region of the plane that is covered by the union of the five triangular regions? Express your answer in the form of $a\sqrt{b}$, where $a$ and $b$ are integers. Please provide the value of a + b.
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Context:
Five soccer teams held a tournament - each team played with each other once. For a win, 3 points were awarded, for a draw - 1 point, and no points were given for a loss. Four teams scored 1, 2, 5, and 7 points respectively. How many points did the fifth team score?
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Context:
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). Rectangle \( MNLK \) is inscribed in square \( ABCD \) in such a way that one of its vertices is at point \( M \), and the other three vertices are located on different sides of the base square. Rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane of the upper face \( A_1B_1C_1D_1 \). The ratio of the side lengths \( MK_1 \) and \( MN \) of quadrilateral \( MK_1L_1N \) is \( \sqrt{54}:8 \). Find the ratio \( AM:MB \).
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Context:
Stella works at a pharmaceutical company and gets paid on a varying monthly income due to performance-based bonuses. Her base salary is 3500 dollars a month. Last year, she had the following bonuses in different months:

January: 1200 dollars
March: 600 dollars
May: 1500 dollars
August: 900 dollars
October: 1200 dollars

In addition, Stella took an unpaid leave for 2 months and had a 5% annual tax deduction on her total income. Calculate Stella's annual income after the tax deduction last year.
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Context:
A vat of orange juice contains 502.75 pints. If you wanted to pour the vat into 21 glasses with the same amount in each glass, how many pints would be in each glass?
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Context:
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the perimeter of $\triangle ADE$.
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Context:
13. How many polynomials $P$ with integer coefficients and degree at most 5 satisfy $0 \leq P(x)<120$ for all $x \in\{0,1,2,3,4,5\} ?$
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Context:
Using single variable calculus, find the volume of the tetrahedron formed by the tangent plane to the hyperboloid $$\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ at a point $(x_0, y_0, z_0)$ and the coordinate axis planes.
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Context:
A school is getting ready to open for the year and the sports class is organizing the equipment they have been donated. In total, they have 300 donations to organize. 60 of these were basketball hoops, half of which also had basketballs included as part of the donation. 120 pool floats were donated, but a quarter of these were damaged and were discarded before the sports class arrived. There were 50 footballs, 40 tennis balls, and the remaining donations were basketballs. In total, how many basketballs have been donated?
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Context:
Given the parabola $y^{2}=2px\left(x \gt 0\right)$, point $P\left(2,1\right)$ lies inside the parabola. A line $l:y=2x+m$ that does not pass through point $P$ intersects the parabola at points $A$ and $B$. Lines $AP$ and $BP$ intersect the parabola at points $C$ and $D$, respectively. If for any line $l$, there exists a $\lambda$ such that $\overrightarrow{AP}=λ\overrightarrow{PC}$ and $\overrightarrow{BP}=λ\overrightarrow{PD}(\lambda\ \ \gt 0,\lambda \neq 1)$ hold, then $p=\_\_\_\_\_\_$.
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Context:
Using the digits $0$, $1$, $2$, $3$, $4$ to form a five-digit number without repeating any digit, the probability that the number is even and the digits $1$, $2$ are adjacent is ______.
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Context:
On the side of a triangle, a point is taken such that an angle equals another angle. What is the smallest possible distance between the centers of the circles circumscribed around triangles, if \( BC = 1 \)?
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Context:
Consider the sequence of real numbers $S_n$ defined as $\sum_{i=0}^n \frac{(\log(n+i)-\log n)^2}{n+i}$. Determine if the limit $\lim_{n\to \infty}S_n$ exists, and if so, calculate its value.
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Context:
Given a cyclic quadrilateral \(ABCD\). Rays \(AB\) and \(DC\) intersect at point \(E\), and rays \(DA\) and \(CB\) intersect at point \(F\). Ray \(BA\) intersects the circumcircle of triangle \(DEF\) at point \(L\), and ray \(BC\) intersects the same circle at point \(K\). The length of segment \(LK\) is 5, and \(\angle EBC = 15^\circ\). Find the radius of the circumcircle of triangle \(EFK\).
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Context:
What is the square root of 16? Express your answer as a numerical value, including a ± symbol if both positive and negative roots are applicable.
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Context:
There are 13.5 multiples of 8 between 100 and a certain number (both are inclusive). What is the ending number of the range?
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Context:
An alarm clock runs 4 minutes slow every hour. It was set right $3 \frac{1}{2}$ hours ago. Now another clock, which is correct, shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
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Context:
Add $36_7 + 274_7.$ Express your answer in base 7.
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Context:
Determine the minimum of the expression

$$
\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}
$$

under the conditions that \(ab + bc + ca > 0\), \(a + b + c = 1\), and \(a, b, c\) are distinct.
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Context:
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
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Context:
Call a permutation $a_1, a_2, \cdots, a_n$ of the integers $1, 2, \cdots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1, 2, \cdots, 7$.
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Context:
Gina's snake is an interesting creature. In the first year of its life, it eats one mouse every 4 weeks. As it grows in the second year, it starts to eat one mouse every 3 weeks. For the rest of its life, it eats a mouse every 2 weeks. Given that a non-leap year has 52 weeks and a leap year has 52 weeks and 1 day, how many mice will Gina's snake eat over a decade assuming the decade starts in a non-leap year?
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Context:
In the regular triangular prism ABC-A1B1C1, there are six diagonals on the sides: AB1, A1B, BC1, B1C, C1A, CA1. Given that AB1 ⊥ BC1, how many other pairs of these diagonals are also perpendicular to each other? Express your answer as a whole number.
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Context:
Given the function $f(x)=x^{3}$, find the derivative of $f(-2)$, denoted as $[f(-2)]'$.
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Context:
Consider triangle $A B C$ with $\angle A=2 \angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{A B}$ at $E$. If $\frac{D E}{D C}=\frac{1}{3}$, compute $\frac{A B}{A C}$.
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Context:
Consider a modified sequence rule:
1) If a number is 30 or less, triple the number.
2) If a number is more than 30, subtract 15 from it.

Let $G$ be the first number in a sequence generated by the new rule. $G$ is a "magic number" if 18 is not a term in the sequence that starts with $G$. Determine how many of the whole numbers 1 through 60 are "magic numbers".
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Context:
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. India was first to finish, followed by Gertrude, Fiona, Janice, and lastly Hannah. What is the lowest total number of points that could have been awarded?
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Context:
A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering.
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Context:
Tim has a bag of 36 orange-flavoured sweets and Peter has a bag of some grape-flavoured sweets. They have to divide up the sweets into small trays with equal number of sweets; each tray containing either orange-flavoured or grape-flavoured sweets only. If the largest possible number of sweets in each tray without any remainder is 4, how many grape-flavoured sweets does Peter have?
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Context:
Example 5 (2005 National High School Mathematics League Test Question) Define the function $f(k)=\left\{\begin{array}{l}0, k \text { is a perfect square } \\ {\left[\frac{1}{\{\sqrt{k}\}}\right], k \text { is not a perfect square, }}\end{array}\right.$ Find $\sum_{k=1}^{240} f(k)$.
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Context:
Arrange 8 spheres, each with a radius of 2, in two layers within a cylindrical container such that each sphere is tangent to its four neighboring spheres, as well as to one base and the side surface of the cylinder. Find the height of the cylinder.
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Context:
在 $\triangle A B C$ 中, $A=90^{\circ}, A B=1, A C=\sqrt{3}$, 点 P 满足 $\frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}$。如果答案是 $x^\circ$, 求 $x$ 的值。
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Context:
The angle bisector \( A L \) and the median \( B M \) of triangle \( A B C \) intersect at point \( X \). Line \( C X \) intersects side \( A B \) at point \( Y \). Find the area of triangle \( C Y L \) given that \( \angle B A C = 60^{\circ} \) and \( A L = x \).
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Context:
Three planes intersect pairwise, and their three lines of intersection are parallel to each other. Determine the number of distinct parts the space is divided into as a result of these intersections. Express your answer as a single integer.
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Context:
Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs:
$1$. Andy tells the students to arrange themselves in a line in arbitrary order.
$2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$ , where $j \equiv 3i + 1$ (mod $8$).
$3$. Andy tells each student $s_i$ to move to the current spot of the student who was in the $i$th position of the line after step $1$.
How many possible orders can the students be in now?
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Context:
For a natural number \( N \), if at least five of the natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "five-rule number." What is the smallest "five-rule number" greater than 2000?
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Context:
Let \(\mathcal{P}\) be a parabola with focus \(F\) and directrix \(\ell\). A line through \(F\) intersects \(\mathcal{P}\) at two points \(A\) and \(B\). Let \(D\) and \(C\) be the feet of the altitudes from \(A\) and \(B\) onto \(\ell\), respectively. Given that \(AB = 20\) and \(CD = 14\), compute the area of \(ABCD\).
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Context:
A real number \(x\) is chosen uniformly at random from the interval \([0,1000]\). Find the probability that \(\left\lfloor\frac{\left\lfloor\frac{x}{2.5}\right\rfloor}{2.5}\right\rfloor=\left\lfloor\frac{x}{6.25}\right\rfloor\).
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Context:
in a group of cheerleaders , the group made of 10 males and 16 females . every cheerleader gets to choose between a coke or malt . the number of those that choose malt is double the number that chose coke , if all the 6 males choose malt , how many females chose malt ?
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Context:
In Fifi's closet, she hangs all her clothes on colored plastic hangers. She has clothes hanging on:

- 7 pink hangers
- 4 green hangers
- One less blue hanger than there are green hangers
- One less yellow hanger than there are blue hangers
- Twice the number of orange hangers as the sum of pink and green hangers
- Three more purple hangers than the difference between blue and yellow hangers
- The arithmetic mean of pink, green, and blue hangers as the number of red hangers
- One more brown hanger than triple the number of red hangers
- 3 gray hangers for every 5 purple hangers

What is the total number of colored hangers in Fifi's closet?
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Context:
Given any real numbers \(a > b > c > d > 0\), to ensure \(\log_{\frac{b}{a}} 2004 + \log_{\frac{c}{b}} 2004 + \log_{\frac{d}{c}} 2004 \geq m \log_{\frac{d}{a}} 2004\) always holds, find the minimum value of \(m\).
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Context:
Suppose we have an (infinite) cone $\mathcal{C}$ with apex $A$ and a plane $\pi$. The intersection of $\pi$ and $\mathcal{C}$ is an ellipse $\mathcal{E}$ with major axis $BC$, such that $B$ is closer to $A$ than $C$, and $BC=4, AC=5, AB=3$. Suppose we inscribe a sphere in each part of $\mathcal{C}$ cut up by $\mathcal{E}$ with both spheres tangent to $\mathcal{E}$. What is the ratio of the radii of the spheres (smaller to larger)?
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Context:
A trapezoid \(K L M N\) is inscribed in a circle with center \(O\), where \(K L\) is parallel to \(M N\), \(K L = 8\), \(M N = 2\), and the angle \(\angle N K L\) is \(45^{\circ}\). The chord \(M A\) of the circle intersects the segment \(K L\) at point \(B\) such that \(K B = 3\). Find the distance from point \(O\) to the line \(A K\).
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Context:
Seven schools organized a science fair competition and invited students from all the schools to develop ideas and present them before a panel of judges selected by all the schools competing in the science fair. There were 90 students from Quantum High School at the competition, 2/3 times as many students from Schrödinger High as Quantum High School, 4/9 times as many students from Einstein High as Schrödinger High, 5/12 times as many students from Newton High as Einstein High, 11/20 times as many students from Galileo High as Newton High, and 13/50 times as many students from Pascal High as Galileo High. From the first six schools, the combined total number of students was x. Faraday High School sent four times the number of students from the combined total of the other six schools to the competition. Calculate the total number of students at the competition.
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Context:
An error of 20% in excess is made while measuring the side of a square. Additionally, the equipment used for calculating the area has an error of 4%. What is the combined percentage of error in the calculated area of the square?
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Context:
Let $X_1, X_2, X_3, \ldots$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Calculate the value of \[ \lim_{n \to \infty} P\left(-\ln(1-X_1) - \ln(1-X_2) - \cdots - \ln(1-X_n) \geq n\right). \]
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Context:
the h.c.f. of two numbers is 20 and the other three factors of their l.c.m. are 3^2, 17 and 23. The larger of the two numbers is
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Context:
For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Let \( x = -\sqrt{2} + \sqrt{3} + \sqrt{5} \), \( y = \sqrt{2} - \sqrt{3} + \sqrt{5} \), and \( z = \sqrt{2} + \sqrt{3} - \sqrt{5} \). Evaluate the expression:
$$
\frac{x^{4}}{(x-y)(x-z)} + \frac{y^{4}}{(y-z)(y-x)} + \frac{z^{4}}{(z-x)(z-y)}
$$
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Context:
Solve the equation: \( 3 \operatorname{tg} 3x - 4 \operatorname{tg} 2x = \operatorname{tg}^{2} 2x \operatorname{tg} 3x \).
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Context:
Find a function $h(x)$ such that the power series expansion of $h(x)$ around $x=0$ has coefficients $b_k$ satisfying \( \sum_{k\geq 0} (k+1) b_k = 0 \).
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Context:
A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color?
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Context:
Given that the domain of the function y=f(2^x) is [1,2], determine the domain of the function y=f(x+1)/(x-1). Express your answer as an interval.
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Context:
Consider the integer，
$$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_{321 ~digits}.$$
Find the sum of the digits of $N$.
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Context:
The base of the pyramid \( PABCD \) is the parallelogram \( ABCD \). Point \( N \) is the midpoint of edge \( AP \), point \( K \) is the midpoint of the median \( PL \) of triangle \( BPC \), and point \( M \) lies on edge \( PB \) such that \( PM = 5MB \). In what ratio does the plane passing through points \( M, N, K \) divide the volume of the pyramid \( PABCD \)?
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Context:
Haley's old washing machine could only wash a combination of three shirts, two sweaters, and four pairs of socks at a time. Additionally, she cannot mix whites with any colored clothes in a single load. If she had to wash nine white shirts, twelve colored shirts, eighteen white sweaters, twenty colored sweaters, sixteen white pairs of socks, and twenty-four colored pairs of socks, how many loads would she have to do, considering both capacity and color restrictions?
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Context:
以 $\sqrt{2}$ 和 $1-\sqrt[3]{2}$ 为两个根的有理系数多项式的次数最低是____。
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Context:
Find the condition that must be satisfied by the positive real numbers \(a, b, c\) such that the iterative process of constructing a new tuple \((S - 2a, S - 2b, S - 2c)\), where \(S = a + b + c\), results in a sequence of tuples with all positive elements.
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Context:
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$.
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Context:
Evaluate the limit \( \lim_{n \to \infty} \dfrac{1}{n^2} \sum_{k=1}^{n-1} k \cdot \left \lfloor x + \dfrac{n-k-1}{n} \right \rfloor \) to three decimal places, where \( x = \sqrt{2015} \).
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Context:
The circle is divided by points \(A, B, C, D\) such that \(\sim AB: BC: CD: DA = 2: 3: 5: 6\). Chords \(AC\) and \(BD\) are drawn, intersecting at point \(M\).

Find the angle \(AMB\).
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Context:
Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$
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Context:
How many liters of pure hydrochloric acid must be added to a certain amount of solution that is 10% hydrochloric acid to get a solution that is 15% hydrochloric acid, if 3.52941176471 liters of pure hydrochloric acid is added?
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Context:
For all positive integers  $n$ , denote by  $\sigma(n)$  the sum of the positive divisors of  $n$  and  $\nu_p(n)$  the largest power of  $p$  which divides  $n$ .  Compute the largest positive integer  $k$  such that  $5^k$  divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where  $N=6^{1999}$ .
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Context:
The product of a million whole numbers is equal to million. What can be the greatest possible value of the sum of these numbers?
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Context:
21 Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. Find the maximum value of $m$ when
$$a_{1}^{2}+\cdots+a_{n}^{2}=1$$
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Context:
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win earned 3 points, a draw earned 1 point, and a loss earned 0 points.

After the tournament ended, it turned out that some 6 teams scored at least $N$ points each. What is the highest possible integer value for $N$?
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Context:
A dilation centered at $2 - 3i$ with a scale factor of $3$ maps which complex number to the point $1 + 2i$?
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Context:
Let $a = 111\ldots111$ (forty ones) and $b = 111\ldots111$ (twelve ones). Determine the greatest common divisor of $a$ and $b$.
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Context:
Example 8 Given $A \subseteq\{1,2, \cdots, 2000\}$, and the absolute difference between any two numbers in $A$ is not equal to 4 or 7, find the maximum value of $|A|$.
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Context:
在透明的球袋中有 1 个黑球和 2 个白球。接下来从袋中摸球，每次摸出一个球，然后放回袋中，并继续往袋中再放入 2 个与摸出球同色的球. 记 $S_{n}$ 为前 n 次中摸出黑球的次数. 已知第 2 次摸出的球是黑球，记 $S_{100}$ 的期望为 E ，则 $[E]=\qquad$.
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Context:
Given that $p$ is a sufficient condition for $q$, $s$ is a sufficient condition for $q$, $r$ is a necessary condition for $q$ and also a sufficient condition for $s$, what is the condition of $p$ for $s$?
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Context:
Problem 4. A circle is inscribed in trapezoid $ABCD$, touching the lateral side $AD$ at point $K$. Find the area of the trapezoid if $AK=16, DK=4$ and $CD=6$.
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Context:
Given the function $y = \log_a(x - 1) + 8$ $(a > 0$ and $a \neq 1)$, its graph always passes through the fixed point $P(P, P)$ on the graph of the power function $f(x)$. Find the value of $f(3)$.
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Context:
A line $l$ passes through the right focus of the hyperbola $x^2 - y^2 = 1$ and intersects the hyperbola at points $A$ and $B$. If the real number $\lambda$ makes $|AB| = \lambda$ and there are exactly 3 such lines $l$, then $\lambda = \quad$.
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Context:
给定大于 2004 的正整数 $n$ ，将 $1,2, \cdots$, $n^{2}$ 分别填入 $n \times n$ 棋盘（由 $n$ 行 $n$ 列方格构成）的方格中，使每个方格都有 1 个数。如果 1个方格中填的数大于它所在行至少 2004 个方格内所填的数，且大于它所在列至少 2004 个方格内所填的数，则称这个方格为"优格"。求棋盘中"优格"个数的最大值。请给出 n = 2010 时的结果。
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Context:
Let $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x$, where $\lfloor a \rfloor$ is the greatest integer not exceeding $a$.
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Context:
Given a function $f(x)=a\sin x+b\cos x$ where $x\in \mathbb{R}$ and $\frac{b}{a} \in (1,\sqrt{3}]$. If the function $f(x)$ attains its maximum value at $x=x_{0}$, then the range of $\tan 2x_{0}$ is ______.
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Context:
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a path length of $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.) If the answer is in the form of \(\arccos\frac{k}{m}\), provide the value of k + m.
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Context:
(3) 18 football teams are participating in a round-robin tournament, which means that in each round, the 18 teams are divided into 9 groups, and each group's two teams play one match. In the next round, the teams are regrouped to play, for a total of 17 rounds, so that each team plays one match against each of the other 17 teams. After $n$ rounds of the tournament according to any feasible schedule, there always exist 4 teams that have played a total of only 1 match among themselves. Find the maximum possible value of $n$.
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Context:
【Question 25】
Among the positive integers whose digit sum is 50 and are multiples of 11, the second smallest number is $\qquad$.
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Context:
On a board, nine quadratic trinomials are written: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{2}, \ldots, b_{9}\) are arithmetic progressions. It turns out that the sum of all nine trinomials has at least one root. What is the maximum number of the original trinomials that may not have any roots?
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Context:
What is the difference between 85% of 250 raised to the power of 2, and 3/5 of 175, when they are both divided by 2.3?
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Context:
During the process of selecting test points using the 0.618 method, if the test interval is $[2, 4]$, then the second test point $x_2$ should be chosen at _____.
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Context:
The distance of a point light source from a sphere is equal to three times the radius of the sphere. How does the illuminated area of the sphere compare to the lateral surface area of the cone of light?
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Context:
Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, calculate the value of $\angle C$.
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Context:
A jewelry store is restocking its shelves. The necklace stand, which can hold 12 necklaces, currently holds 5 necklaces. The ring display, which can hold 30 rings, currently holds 18 rings. The bracelet display, which can hold 15 bracelets, currently holds 8 bracelets. The store’s supplier charges $4 per necklace, $10 per ring, and $5 per bracelet. However, the supplier offers the following volume discounts:

For necklaces:
- Buy 4 or more, get a 10% discount
- Buy 6 or more, get a 15% discount

For rings:
- Buy 10 or more, get a 5% discount
- Buy 20 or more, get a 10% discount

For bracelets:
- Buy 7 or more, get an 8% discount
- Buy 10 or more, get a 12% discount

Considering the discounts, how much, in dollars, will the store need to pay to fill the displays?
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Context:
Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.
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Context:
Evaluate the triple integral $${\int \int \int_{\Omega } xy^{2}z^{2}dxdydz}$$ where $\Omega$ is the region bounded by the surfaces $z=xy$, $y=x$, $x=1$, and $z=0$.
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Context:
A point in three-space has distances $2,6,7,8,9$ from five of the vertices of a regular octahedron. What is its distance from the sixth vertex?
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Context:
2. Find the probability that a randomly selected five-digit natural number with non-repeating digits, formed from the digits $1,2,3,4,5,6,7,8$, is divisible by 8 without a remainder. (10 points)
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Context:
How many perfect squares are between 100 and 500?
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Context:
On dividing a number by 68, we get 269 as quotient and some remainder. On dividing the same number by 67, the remainder is 1. What is the remainder in the first division?
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Context:
Figure $ABCD$ is a trapezoid with $AB||DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$, and $\angle CDA=60^\circ$. The length of $DC$ is
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Context:
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x, y)$ such that $1 \leq x, y \leq p$ and $x^3 - 3xy + y^3 + 1$ is divisible by $p$.
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Context:
In $\triangle ABC$, if angles $A$, $B$, and $C$ form an arithmetic sequence, and sides $a=2$, $c=5$, then the area of $\triangle ABC$ is $S_{\triangle ABC}=$ \_\_\_\_\_\_.
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Context:
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between \(\frac{1}{2}\) and \(\frac{2}{3}\). Armed with this information, what number should Carol choose to maximize her chance of winning? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Kanel-Belov A.Y.

The periods of two sequences are 7 and 13. What is the maximum length of the initial segment that can coincide?
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Context:
Find the maximum value of the expression for \( a, b > 0 \):

$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$
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Context:
From vertex \( B \) of triangle \( ABC \), a perpendicular \( BM \) is dropped to the bisector of angle \( C \). Let \( K \) be the point of tangency of the incircle with side \( BC \).

Find angle \( \angle MKB \) given that \( \angle BAC = \alpha \).
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Context:
In the rectangular coordinate system xOy, it is known that the line $x + \sqrt{2}y - 2\sqrt{2} = 0$ is tangent to the ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and the right focus F($c$, $0$) of the ellipse C is symmetric about the line $y = \frac{c}{b}x$ with the point E on the ellipse C. Find the area of $\triangle$OEF.
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Context:
In triangle $ABC$, $AC = 13$, $BC = 14$, and $AB=15$. Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$. Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$. Let $P$ be the point, other than $A$, of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$. Ray $AP$ meets $BC$ at $Q$. The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m-n$.
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Context:
There are $2$ boys for every $3$ girls in Ms. Johnson's math class.  If there are $30$ students in her class, what percent of them are boys?
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Context:
设集合 S=\{1,2,3,\cdots,997,998\}，集合 S 的 k 个 499 元子集 A_1, A_2, \cdots, A_k 满足：对 S 中任一二元子集 B，均存在 i\in\{1,2,\cdots,k\}，使得 B\subseteq A_i. 求 k 的最小值.
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Context:
Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.
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Context:
During one game, a certain number of people attended a baseball team's games. Forty percent and thirty-four percent of the audiences are supporters of the first and second teams, respectively. There were 3 people who attended the game and did not support either of the teams. How many people attended the game in total?
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Context:
Example 4 Given that $m, n$ are integers greater than 7, consider a rectangular array of $m \times n$ points. Color $k$ of these points red, such that the three vertices of any right-angled triangle with its two legs parallel to the sides of the rectangle are not all red points. Find the maximum value of $k$.
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Context:
Given that point $O$ is the centroid of $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2a\cdot \overrightarrow{OA}+b\cdot \overrightarrow{OB}+ \frac{2\sqrt{3}}{3}c\cdot \overrightarrow{OC}=\overrightarrow{0}$, determine the measure of angle $C$.
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Context:
Given $a, b \in \mathbb{R}$, if the inequality $x\ln x - a\ln x \geqslant x + b$ holds for all $x \gt 0$, then the range of $\frac{b}{a}$ is ____.
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Context:
The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola?
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Context:
Find the smallest natural number \( n \) for which the number \( A = n^3 + 12n^2 + 15n + 180 \) is divisible by 23.
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Context:
We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union.
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Context:
How many unordered triples $A, B, C$ of distinct lattice points in $0 \leq x, y \leq 4$ have the property that $2[ABC]$ is an integer divisible by $5$?
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Context:
The real number sequence \(\{a_n\}\) satisfies
\[1=a_0 \leqslant a_1 \leqslant a_2 \leqslant \cdots,\]

The sequence \(\{b_n\}\) satisfies
\[b_n = \sum_{k=1}^{n}\left(1-\frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}}, \quad (n=1,2, \cdots).\]

Find the largest real number \(C\) such that for any real number \(x \in [0, C)\), there exists a sequence \(\{a_n\}\) satisfying the condition (1), such that the corresponding sequence \(\{b_n\}\) has infinitely many terms greater than \(x\).
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Context:
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone.
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Context:
## Task Condition

Find the derivative.

$y=\arccos \frac{x^{2}-4}{\sqrt{x^{4}+16}}$
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Context:
13.

A box of hawthorns has more than one hundred grains. When counted in groups of 3, there is 1 extra; when counted in groups of 4, there are 2 extra; when counted in groups of 5, there are 3 extra; and when counted in groups of 6, there are 4 extra. The box of hawthorns has at most ( ) grains.
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Context:
A line passing through the focus of the parabola $y^2=2px$ ($p > 0$) intersects the parabola at points $P(x_1, y_1)$ and $Q(x_2, y_2)$. If $x_1 + x_2 = 3p$, then the length of $PQ$ is $\boxed{\_\_\_\_\_\_\_\_}$.
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Context:
$已知函数f(x)=x-a(1+\ln x).$
$若函数f(x)有两个零点x_1，x_2，求a的取值范围.$
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Context:
Given an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity as $e$. If there exists a point $P$ on the ellipse such that $\dfrac{|PF_1|}{|PF_2|}=e$, then the range of the eccentricity $e$ is.
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Context:
Given the line $4x+3y+2m=0$ intersects with the circle $C:\left(x+3\right)^{2}+\left(y-1\right)^{2}=1$, one possible value of the integer $m$ is ______.
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Context:
Given the circle $\left(x-a\right)^{2}+y^{2}=1\left(a\geqslant 0\right)$ and the circle $x^{2}+\left(y-2\right)^{2}=25$, find the range of the real number $a$ such that the two circles have common points. Express your answer in interval notation.
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Context:
How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)?
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Context:
Find the arithmetic mean of the reciprocals of the first four prime numbers, including the number 7 instead of 5.
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Context:
Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials:
\[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \]
and
\[ Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \]
Let $S$ be the set of all complex numbers which are a root of either $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$
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Context:
Given an obtuse triangle $ABC$ with an obtuse angle at $C$. Points $P$ and $Q$ are marked on its sides $AB$ and $BC$ respectively such that $\angle ACP = CPQ = 90^\circ$. Find the length of the segment $PQ$, given that $AC = 25$, $CP = 20$, and $\angle APC = \angle A + \angle B$.
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Context:
Find the derivative.

\[ y = 2x - \ln \left(1 + \sqrt{1 - e^{4x}}\right) - e^{-2x} \cdot \arcsin \left(e^{2x}\right) \]
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Context:
In relation to Smiths Falls, Ontario, the local time in St. John's, Newfoundland, is 90 minutes ahead, and the local time in Whitehorse, Yukon, is 3 hours behind. When the local time in St. John's is 5:36 p.m., what is the local time in Whitehorse? Express your answer in 12-hour format (HH:MM a.m./p.m.).
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Context:
Find the cosine of the angle between adjacent lateral faces of a regular quadrilateral pyramid, where the lateral edge is equal to the side of the base.
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Context:
A certain company plans to build a container (neglecting thickness, with length unit in meters). The bottom of the container is cylindrical with a height of 1 and a bottom radius of r. The top is a hemisphere with a radius of r. According to the design requirements, the volume of the container is 28/3π cubic meters. Assuming that the construction cost of the container is only related to its surface area, it is known that the construction cost per square meter for the cylindrical part is 30,000 yuan and for the hemispherical part is 40,000 yuan. Calculate the value of the radius r that minimizes the construction cost of the container.
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Context:
A total of 2015 tickets, numbered 1, 2, 3, 4, ..., 2014, 2015, are placed in an empty bag. Alfie removes ticket \( a \) from the bag. Bernice then removes ticket \( b \) from the bag. Finally, Charlie removes ticket \( c \) from the bag. They notice that \( a < b < c \) and \( a + b + c = 2018 \). In how many ways could this happen?
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Context:
During the flower exhibition in Zhengzhou Green Expo Garden, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that areas A and B each have one person, and the remaining two areas each have two people. Among them, Little Li and Little Wang cannot be together. Determine the total number of different arrangements possible.
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Context:
An $8 \times 8 \times 8$ cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
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Context:
In the isosceles triangle \(ABC\) with equal sides \(AB\) and \(CB\), the sides \(AB\) and \(CB\) are extended beyond point \(B\) to points \(D\) and \(E\) respectively. The segments \(AE\), \(ED\), and \(DC\) are equal to each other, and \(\angle BED \neq \angle BDE\). Find the angle \(ABE\).
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Context:
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
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Context:
Evaluate the limit:

$$\lim_{k \to \infty} \frac{\int_{0}^{1} x^{2k+1} e^x \, dx}{\int_{0}^{1} x^{2k} e^x \, dx}$$
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Context:
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property. Provide your answer as an integer.
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Context:
Given the parabola C: y²=4x, its focus is F, and its directrix is l. Point A is on l, and line segment AF intersects parabola C at point B. If $$\overrightarrow {FA}=3 \overrightarrow {FB}$$, find the length of $$\overrightarrow {AF}$$.
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Context:
A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z^2 \), where \( a \) and \( b \) are real numbers. The function has the property that for each complex number \( z \), \( f(z) \) is equidistant from both \( z \) and the origin. Given that \( |a+bi| = 5 \), find \( b^2 \).
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Context:
In the right triangle \(ABC\) with \(\angle B = 90^\circ\), \(P\) is a point on the angle bisector of \(\angle A\) inside \(\triangle ABC\). Point \(M\) (distinct from \(A\) and \(B\)) lies on the side \(AB\). The lines \(AP\), \(CP\), and \(MP\) intersect sides \(BC\), \(AB\), and \(AC\) at points \(D\), \(E\), and \(N\) respectively. Given that \(\angle MPB = \angle PCN\) and \(\angle NPC = \angle MBP\), find \(\frac{S_{\triangle APC}}{S_{ACDE}}\).
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Context:
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
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Context:
Given $a^{2}+b^{2}=6ab$, find $\frac{a+b}{a-b}$.
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Context:
Suppose \( n \) is a natural number. For any real numbers \( x, y, z \), the inequality \(\left(x^{2} + y^{2} + z^{2}\right) \leqslant n \left(x^{4} + y^{4} + z^{4}\right)\) always holds. Then the minimum value of \( n \) is \(\qquad\)
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Context:
Determine the largest constant  $K\geq 0$  such that   $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$   holds for all positive real numbers  $a,b,c$  such that  $ab+bc+ca=abc$ .

*Proposed by Orif Ibrogimov (Czech Technical University of Prague).*
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Context:
Given the inequality ax^2 + bx + c < 0 with solution set (-∞, α) ∪ (β, +∞) and α < β < 0, find the solution set for the inequality cx^2 - bx + a > 0. Express your answer in interval notation.
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Context:
19 The sequence of positive integers $\left\{a_{n}\right\}$ satisfies: for any positive integers $m, n$, if $m \mid n, m<n$, then $a_{m} \mid a_{n}$, and $a_{m}<a_{n}$. Find the minimum possible value of $a_{2000}$.
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Context:
Find the sum of the reciprocals of the distances from the centroid of the triangle formed by the intersection of the plane $lx+my+nz=p$ (where $l^2+m^2+n^2=1$) with the coordinate axes to the points where the perpendicular from the centroid to the plane meets the coordinate planes.
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Context:
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
 ( $1$ ) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$ );
 ( $2$ ) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break. Find the number of possible values of $n$ with $1 < n < 1000.$
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Context:
[Pairing and Grouping; Bijections] $[$ Evenness and Oddness $]$

Given a 29-digit number $X=\overline{a_{1}} \overline{-} \overline{-} \overline{29}\left(0 \leq a_{k} \leq 9, a_{1} \neq 0\right)$. It is known that for every $k$ the digit $a_{k}$ appears in the representation of this number $a_{30-k}$ times (for example, if $a_{10}=7$, then the digit $a_{20}$ appears seven times). Find the sum of the digits of the number $X$.
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Context:
已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\frac{2 \sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于原始答案的形式为 $\frac{m \sqrt{k}}{n} \pi$，请给出 m + n +k的值。
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Context:
In triangle \( \triangle ABC \), given that \( 2 \sin A + 3 \cos B = 4 \) and \( 3 \sin B + 2 \cos A = \sqrt{3} \), find the measure of angle \( \angle C \) in degrees.
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Context:
It is known that in a right triangle \(c^{2}=a^{2}+b^{2}\), where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. Find the acute angles of the triangle if \(\left(\frac{c}{a+b}\right)^{2}=0.5\).
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Context:
Among all fractions of the form \(\frac{m}{n}\), where \(m\) and \(n\) are four-digit numbers with the same sum of digits, the largest one was chosen. Find it and write your answer as an improper fraction (do not round or simplify it!).
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Context:
Person A and Person B are standing 30 meters apart, facing each other. They play "Rock, Paper, Scissors." The winner moves forward 3 meters, the loser moves back 2 meters, and in the case of a tie, both move forward 1 meter. After 15 rounds, Person A is 17 meters from the starting point, and Person B is 2 meters from the starting point. How many times did Person A win?
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Context:
If \( a, b, c \) are integers (where \( 0 < c < 90 \)), and \( \sqrt{9 - 8 \sin 50^{\circ}} = a + b \sin c^{\circ} \), find the value of \( \frac{a+b}{c} \).
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Context:
7. Let the set $T=\left\{(a, b, c) \mid a, b, c \in \mathbf{N}^{*}\right.$, and $a, b, c$ can form the side lengths of some triangle $\}$. Then the value of $\sum_{(a, b, c) \in T} \frac{2^{a}}{3^{b} \cdot 5^{c}}$ is $\qquad$.
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Context:
Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
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Context:
Given column vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if for all distinct $i, j \in \{1, 2, 3\}$, the minimum eigenvalue of the matrix $v_iv_i^T + v_jv_j^T$ satisfies $\lambda_{\min}(v_iv_i^T+v_jv_j^T) \leq 1$, what is an upper bound for $\lambda_{\min}(v_1v_1^T+v_2v_2^T+v_3v_3^T)$?
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Context:
Given the following premises:
1. The diagonals of a parallelogram are equal.
2. A square is a parallelogram.

Using syllogistic reasoning, deduce a logical conclusion about the diagonals of a square. Express your answer as a clear and concise statement.
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\boxed{The diagonals of a square are equal.}


Context:
If a three-digit number $m$ satisfies that the digits in each place are all different and not zero, then this three-digit number is called a "mutually exclusive number." Remove the unit digit of the "mutually exclusive number" $m$ to get a two-digit number ${m'}$, and let the difference between ${m'}$ and the unit digit of $m$ be denoted as $F(m)$, and the difference between the tens digit and the unit digit of $m$ be denoted as $G(m)$. It is known that a three-digit positive integer $m=20(5x+1)+2y$ (where $x$ and $y$ are integers, and $1\leqslant x\leqslant 9, 1\leqslant y\leqslant 9$) is a "mutually exclusive number," $\frac{{F(m)}}{{G(m)}}$ is an integer and can be divided by $13$. Then the maximum value of the "mutually exclusive number" $m$ that satisfies the conditions is ______.
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Context:
4.94. Solve the Cauchy problem: $y^{\prime \prime}-8 y^{3}=0 ; y(0)=-1 ; y^{\prime}(0)=2$.
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Context:
Dorothy is 15 years old and wants to go to a museum with her family. Her family consists of her, her younger brother, her parents, and her grandfather. The regular ticket cost is $10. People of a certain age or younger have a discount. Dorothy had $70 before the trip and had $26 left after the trip. What is the discount percentage for people 18 years old or younger?
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Context:
Find the value of the ratio $$\frac{\int_0^1 x^n \sqrt{1-x^2} dx}{\int_0^1 x^{n-2} \sqrt{1-x^2} dx}$$ using appropriate techniques.
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Context:
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.
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Context:
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is:
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Context:
Two circles intersect at points \( A \) and \( B \). Chords \( A C \) and \( A D \) are drawn in each of these circles such that each chord in one circle is tangent to the other circle. Find \( A B \) if \( C B = a \) and \( D B = b \).
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Context:
In year $N$, the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?
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Context:
A rhombus has a side length \( a \) and an acute angle of \( 60^\circ \). A circle is inscribed in the rhombus. Determine the area of the quadrilateral whose vertices are the points of tangency of the circle with the sides of the rhombus.
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Context:
For any positive integer \( n \), let \( D_n \) be the set of all positive divisors of \( n \), and define \( f_i(n) \) as the number of elements in the set
\[ F_i(n) = \{ a \in D_n \mid a \equiv i \pmod{4} \}. \]
Find the smallest positive integer \( m \) such that
\[ 2 f_1(m) - f_2(m) = 2017. \]
(The 14th China Southeastern Mathematical Olympiad)
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Context:
If 5 + 3 + 2 = 151022, 9 + 2 + 4 = 183652, then 7 + 2 + some number = 143547. What is the third number in the third equation?
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Context:
Circle $O$ has chord $A B$. A circle is tangent to $O$ at $T$ and tangent to $A B$ at $X$ such that $A X=2 X B$. What is \frac{A T}{B T}?
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Context:
Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2, \ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that \(\sum_{1 \leq i \leq k} t_i=4000.\) Find the value of \(\sum_{1 \leq i \leq k} a^2_i.\)
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Context:
Teams A and B each have 7 members who appear in a predetermined order to compete in a Go game relay. The first player of each team competes first; the loser is eliminated, and the winner advances to compete against the next player of the losing team. This process continues until all the players of one team are eliminated. The other team wins, forming a competition sequence. What is the total number of possible competition sequences? $\qquad$ .
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Context:
In triangle $ABC$, $AB = 7$, $BC = 24$, and the area of triangle $ABC$ is 84 square units. Given that the length of median $AM$ from $A$ to $BC$ is 12.5, find $AC$.
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Context:
Grandma Olga has 6 daughters and 5 sons. If all her daughters each have 10 sons, 9 daughters, and 4 step-children (2 sons and 2 daughters), and each of her sons has 8 daughters, 7 sons, and 3 in-laws (who have an additional 2 children each, 1 son and 1 daughter), how many grandchildren and extended family members does she have in total?
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Context:
Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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Context:
Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$?
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Context:
5 men are equal to as many women as are equal to some boys. All of them earn Rs. 150 only. Men's wages are Rs. 10. How many boys are there?
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Context:
We color an equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the resulting set.
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Context:
$\left.\frac{\text { Auxiliary similar triangles }}{[\quad \text { Law of Cosines }}\right]$

In triangle $ABC$, a point $D$ is taken on side $AC$, such that $AD=3$, $\cos \angle BDC=13/20$, and $\angle B+\angle ADB=180^{\circ}$. Find the perimeter of triangle $ABC$ if $BC=2$.
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Context:
The distance between towns A and B is 999 km. At every kilometer of the road that connects A and B a sign shows the distances to A and B as follows:
$$\boxed{0|999}\quad\boxed{1|998}\quad\boxed{2|997}\quad\ldots\quad\boxed{998|1}\quad\boxed{999|0}$$
How many signs are there, with both distances written with the help of only two distinct digits?
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Context:
已知 \( \sin \alpha+2 \cos \beta=2 \). 则\( y=\sin \left(\alpha+\frac{\pi}{4}\right)+2 \sin \left(\beta+\frac{\pi}{4}\right) \)的值域为 $\left[\sqrt{a}-\frac{\sqrt{b}}{c}, \sqrt{d}+\frac{\sqrt{e}}{f}\right]$.请提供a+b+c+d+e+f的值
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Context:
In triangle \( KLM \), a bisector \( MN \) is drawn. A circle passes through vertex \( M \), touches side \( KL \) at point \( N \), intersects side \( KM \) at point \( P \), and side \( LM \) at point \( Q \). Segments \( KP \), \( QM \), and \( LQ \) are equal to \( k \), \( m \), and \( q \) respectively. Find \( MN \).
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Context:
Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) lie on lines \( AB \) and \( BC \), respectively, such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) given that \( \angle BAC = 17^\circ \).
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Context:
In triangle \(ABC\), \(\angle ABC=50^\circ\) and \(\angle ACB=70^\circ\). Let \(D\) be the midpoint of side \(BC\). A circle is tangent to \(BC\) at \(B\) and is also tangent to segment \(AD\); this circle intersects \(AB\) again at \(P\). Another circle is tangent to \(BC\) at \(C\) and is also tangent to segment \(AD\); this circle intersects \(AC\) again at \(Q\). Find \(\angle APQ\) (in degrees).
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Context:
Color the vertices of a quadrilateral pyramid so that the endpoints of each edge are different colors. If there are only 5 colors available, what is the total number of distinct coloring methods?
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Context:
Find the smallest natural number that can be expressed as the sum of 2002 natural addends with the same sum of digits and as the sum of 2003 natural addends with the same sum of digits.
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Context:
John needs to get a new seeing-eye dog. The adoption fee cost $150 for an untrained dog, but there is a 10% discount available. It then takes 12 weeks of training which costs $250 a week.  However, a local non-profit organization offers a 15% subsidy for half of the training costs. After the training, she needs certification which costs $3000 but her insurance covers 90% of that. Meanwhile, she also needs to purchase additional supplies for the dog, such as a harness, leash, and food, costing $200. What is her out-of-pocket cost?
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Context:
Simplify the expression and then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a=-1$ and $b=1$.
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Context:
Find the sum, 'P', in terms of the unknown interest rate 'r', when the difference between compound interest and simple interest on a sum for 2 years is Rs. 65, with interest rates of 10% p.a. for the first year and 'r' % p.a. for the second year, without knowing the actual value of 'r'.
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Context:
A rectangle measuring 19 cm by 65 cm is divided by lines parallel to its sides into small squares with a side length of 1 cm. Into how many parts will this rectangle be divided if a diagonal is also drawn within it?
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Context:
If the internal angles of $\Delta ABC$ satisfy $\sin A+ \sqrt{2}\sin B=2\sin C$, then the minimum value of $\cos C$ is ______.
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Context:
Given a parallelogram \(A B C D\) with \(\angle B = 111^\circ\) and \(B C = B D\). On the segment \(B C\), there is a point \(H\) such that \(\angle B H D = 90^\circ\). Point \(M\) is the midpoint of side \(A B\). Find the angle \(A M H\). Provide the answer in degrees.
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Context:
Sang-jun has 10 more books on economics than a quarter of his total books, 5 fewer books on social studies than 3/5 of the rest, and 12 books on science. If Sang-jun has 13 books that are not economics, social studies, or science, how many books does Sang-jun have in total?
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Context:
Calculate the smallest root \(x_0\) of the equation

$$
x^{2}-\sqrt{\lg x +100}=0
$$

(with a relative error of no more than \(10^{-390} \%\)).
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Context:
Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose $$(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.$$ Compute the product of all possible values of $a$.
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Context:
There is a circle with a circumference of 1 meter and a point marked on it. Two cockroaches start running in the same direction from the marked point at different speeds.

- Whenever the faster cockroach catches up with the slower one, the slower one instantly turns around and starts running in the opposite direction with the same speed.
- Whenever they meet face-to-face, the faster one instantly turns around and starts running in the opposite direction with the same speed.

Determine how far from the marked point their 100th meeting could occur.
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Context:
How many natural numbers exist such that their square results in a thirty-digit number with the following property: When divided from left to right into three groups of ten digits, the numbers formed by the middle group and the right group are both four times the number formed by the left group?
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Context:
Calculate the minimum distance of the cyclic code of length 9 generated by the polynomial \( g(x) = x^7 + x^6 + x^4 + x^3 + x + 1 \) over GF(2).
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Context:
Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$. If $N = 100a + b$, where $a, b$ are positive integers less than $100$, find $a + b$.
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Context:
The sum of the integer parts of all positive real solutions \( x \) that satisfy \( x^{4}-x^{3}-2 \sqrt{5} x^{2}-7 x^{2}+\sqrt{5} x+3 x+7 \sqrt{5}+17=0 \) is
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Context:
In a right-angled triangle $ABC$ with the right angle at $C$, the angle bisector $BD$ and the altitude $CH$ are drawn. A perpendicular $CK$ is dropped from vertex $C$ to the angle bisector $BD$. Find the angle $HCK$ if $BK: KD = 3:1$.
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Context:
A duty mechanic descended on a downward-moving metro escalator. His entire journey from the upper platform to the lower one lasted $24 \mathrm{s}$. Then he ascended and descended again at the same pace, but this time on a stationary escalator. It is known that the descent took $42 \mathrm{s}$. How many seconds would it take for a person to descend on the downward-moving escalator while standing on a step?
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Context:
For which positive integers \( n \) does the determinant \( \mathcal{D}_n \) of an \( n \times n \) matrix, filled row-by-row with the first \( n^2 \) elements of the Thue-Morse sequence, not equal zero?
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Context:
Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand \((m - a)(n - b)\), where \(a, b, m, n\) are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for \(m\) and \(n\), he realizes that the value of ChatSFFT’s result is the same as the real result in terms of \(a\) and \(b\). How many such pairs are there?
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Context:
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. What is the total number of triangles of any size in the figure? Express your answer as a whole number.
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Context:
Let $E \subset F$ be a field extension with degree $n$. Determine the degree of the extension $E(x) \subset F(x)$, where $E(x)$ and $F(x)$ are the rational function fields over $E$ and $F$, respectively.
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Context:
Given proposition $p$: The equation $\frac{x^2}{2} + \frac{y^2}{m} = 1$ represents an ellipse with foci on the $y$-axis; proposition $q$: $\forall x \in \mathbb{R}$, $4x^2 - 4mx + 4m - 3 \geqslant 0$. If $(\neg p) \land q$ is true, find the range of values for $m$.
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Context:
Jane started babysitting when she was a certain age. Whenever she babysat for a child, that child was no more than half her age at the time. Jane is currently 34 years old, and she stopped babysitting 12 years ago. The current age of the oldest person for whom Jane could have babysat is 25. At what age did Jane start babysitting?
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Context:
Angelina drove at an average rate of $80$ kmh and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ kmh. Altogether she drove $250$ km in a total trip time of $3$ hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
$
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Context:
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. What is the minimum number of moves required to reach the destination point from the starting point? Express your answer as a whole number.
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Context:
Calculate the probability of selecting two subsets from a set of 100 integers (50 odd, 50 even) such that both subsets have the same number of odd and even integers, given that the subsets are of different sizes.
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Context:
There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this?
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Context:
Calculate the rank of the symmetric bilinear form \( f(A,B) = \operatorname{tr}A\operatorname{tr}B + n\operatorname{tr}AB \).
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Context:
A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals.
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Context:
There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible?
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Context:
In the range of $0^\circ$ to $360^\circ$, the angle(s) with the same terminal side as $-35^\circ$ is/are $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a$  and  $b$  be positive integers satisfying
	\[ \frac a{a-2} = \frac{b+2021}{b+2008} \]
Find the maximum value  $\dfrac ab$  can attain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the degrees of irreducible divisors of the polynomial $x^4-3$ in $F_{31}[x]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
ii. $n \geqslant 5$ is a positive integer, $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \geqslant 0$. It is known that
$$
\sum_{i=1}^{n} a_{i}^{2}=1, \sum_{i=1}^{n} b_{i}=1
$$

Find
$$
\sum_{i=1}^{n} a_{i}^{1+b_{i}}
$$

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


Context:
Find the constant $c$ such that the limit \( \lim_{n \to \infty} P\left(\sqrt{\left(\sum_{i=1}^{n}X_i\right)^2+\left(\sum_{i=1}^{n}Y_i\right)^2}>c\sqrt{n}\right) = 0.05 \) holds, where $(X_n, Y_n)$ are independent two-dimensional random variables with each $X_n$ and $Y_n$ having a Laplace distribution with density $f(x) = \frac{1}{2}e^{-|x|}$, and the density function of $\sum_{i=1}^{n}X_i$ and $\sum_{i=1}^{n}Y_i$ is $g(x) = \frac{1}{2^n}e^{-n|x|}$. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the pyramid is a square $ABCD$ with a side length of 1.5, the lateral edge $SC$ is perpendicular to the base plane and equals 1.75. Points $S, B$, and $D$ lie on the lateral surface of a cone with the vertex at point $A$, and point $C$ lies in the base plane of this cone. Find the lateral surface area of the cone.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following alternating series using the Leibniz alternative test for convergence:
$$
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}
$$
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Context:
At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 10 minutes, and when she came back, she found Ivan and Foma in front of the door, with Foma having apparently arrived after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the shop to reopen.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ be a real number such that the equation
$$
x^{4}+3 a x^{3}+a\left(1-5 a^{2}\right) x-3 a^{4}+a^{2}+1=0
$$
has a real root, and there are exactly 2 distinct real roots. Determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} u(uu')'\,dx \) given that \( u = u(x) \), \( u' = \frac{du}{dx} \), and \( u(x) \rightarrow 0 \) and \( u'(x) \rightarrow 0 \) as \( x \rightarrow \pm \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each edge of a regular tetrahedron is divided into three equal parts. Through each resulting division point, two planes are drawn, parallel to the two faces of the tetrahedron that do not pass through that point. Into how many parts do the constructed planes divide the tetrahedron?
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Context:
Given a floor plan composed of \( n \) unit squares, Albert and Berta want to cover this floor with tiles. Each tile is either a \( 1 \times 2 \) domino or a T-tetromino. Albert has tiles of one color, while Berta has dominoes of two colors and tetrominoes of four colors. Albert can cover the floor plan in \( a \) ways, and Berta in \( b \) ways. Assuming \( a \neq 0 \), find the ratio \( \frac{b}{a} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For an agricultural experiment, seeds were planted in five different plots as follows:

Plot 1: 300 seeds with 25% germination rate and 90% transplant success rate
Plot 2: 200 seeds with 35% germination rate and 85% transplant success rate
Plot 3: 400 seeds with 45% germination rate and 80% transplant success rate
Plot 4: 350 seeds with 15% germination rate and 95% transplant success rate
Plot 5: 150 seeds with 50% germination rate and 70% transplant success rate

After germination, the sprouted seeds were transplanted to a separate growth area. What percent of the total number of seeds planted across all five plots successfully germinated and were transplanted to the growth area?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $C$ be the set of all 100-digit numbers consisting of only the digits $1$ and $2$. Given a number in $C$, we may transform the number by considering any $10$ consecutive digits $x_0x_1x_2 \dots x_9$ and transform it into $x_5x_6\dots x_9x_0x_1\dots x_4$. We say that two numbers in $C$ are similar if one of them can be reached from the other by performing finitely many such transformations. Let $D$ be a subset of $C$ such that any two numbers in $D$ are not similar. Determine the maximum possible size of $D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two fleas are jumping on a large sheet of paper. The first flea jumps in a sequence: right, up, left, down, with each jump increasing by 1 cm. The second flea jumps in a sequence: left, up, right, down, also increasing each jump by 1 cm. After 100 jumps, the fleas are 3 meters apart. What was the initial distance between the fleas?
Please reason step by step, and put your final answer within \boxed{}.


Context:
One integer is selected at random from the following list of 15 integers: $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5$. The probability that the selected integer is equal to $n$ is $\frac{1}{3}$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two particles are positioned at the vertices of a cube. If at a given time $t$ the particles are on the same edge, they remain in the same position up to time $t + 1$. Otherwise, one of them is chosen at random with equal probability, and at time $t + 1$ it is moved to one of the 3 neighboring vertices with equal probability. If the particles start at opposite vertices of the same face, what is the expected value of the least time they are on the same edge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Anderson is bored in physics class. His favorite numbers are $1$, $7$, and $33$. He randomly writes $0$., and then writes down a long string consisting of those numbers juxtaposed against one another (e.g., $0.1337173377133$...) on a sheet of paper. Since physics class is infinitely long, he writes an infinitely long decimal number. If the expected value of the number he wrote down is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the parabola $y^{2} = 4cx$, where $c = \sqrt{a^{2} + b^{2}}$, find the eccentricity of the hyperbola given that $|AB| = 4c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
用代数式表示 $a$ 的倒数的 70% 与 $b$ 的 2 倍的倒数的和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the multiplicity of the point \( Q = (0:1:0) \) for the curve given by the equation \( G = X^4Z + X^4Y + Y^4Z \) using the method of partial derivatives.
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Context:
Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of $n$, required to eliminate all the soldiers?

[i]David Yang.[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the point of maximum for the function \( f_n(x) = x^n (\ln x)^2 \) on the interval \((0,1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hexagon has each angle measuring $120^{\circ}$, and its sides alternately measure 1 cm and $\sqrt{3} \mathrm{~cm}$. What is the area of the hexagon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Santa Claus has 36 identical gifts divided into 8 bags. The number of gifts in each of the 8 bags is at least 1 and all are different. You need to select some of these bags to evenly distribute all their gifts to 8 children, such that all gifts are distributed completely (each child receives at least one gift). How many different selections are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of terms required in the Taylor series expansion of \( \ln(1+x) \) about 1 to approximate \( \ln(4) \) with an error less than \( 2^{-10} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a financial market with a bond $A$, a stock $S$, and call options $f$ on the stock $S$, consider the following values: $A(0) = 100$, $A(T) = 120$, $S(0) = 50$, $S^u(T) = 70$, and $S^d(T) = 30$. The call options have a strike price $K = 50$ and expire at $T$. Construct a portfolio $\\phi$ consisting of shares of $S$ and $f$ such that its value $V(T)$ is $120$, regardless of whether $S$ goes up or down. Calculate the price of the call option at time $0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Stella is filling up a truck with a capacity of 6000 stone blocks. In the first 2 hours, she can put 250 blocks per hour, while Twinkle can put 200 blocks per hour. After 2 hours, their rates change: Stella can put 220 blocks per hour, and Twinkle can put 230 blocks per hour. They work for four hours and are then joined by 6 other people. For the first hour, three of these people have a work rate of 300 blocks per hour, and the other three have a work rate of 180 blocks per hour. After one hour, their work rates change: the first three now have a rate of 280 blocks per hour, and the other three have a rate of 190 blocks per hour. How many hours did filling the truck take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$: $$x_{n+1} = \sum_{k=1}^{n} \sqrt[n]{x_k}.$$ Find the value of $$\lim_{n\to\infty} \frac{x_n - n}{\ln n},$$ assuming it exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside a convex pentagon, a point is marked, and it is connected to all the vertices. What is the maximum number of the ten segments (five sides and five segments connecting the marked point with the vertices of the pentagon) that can have a length of 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rabbit runs in a perfect circle of radius $r$ with a constant speed $v$. A fox chases the rabbit, starting from the center of the circle and also moves with a constant speed $v$, always positioning itself between the center of the circle and the rabbit. Determine the time it will take for the fox to catch the rabbit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Write the four numbers \(1, 9, 8, 8\) in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, then write the difference between these two numbers in between them. This completes one operation. Then, perform the same operation on the new series of 7 numbers. Continue doing this a total of 100 times. Find the sum of the numbers in the final row.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Hunter was observing the wildlife in the pond and noticed an intriguing mix of amphibians. He saw 4 different species of frogs, including:

- 3 x 2 = 6 green frogs sitting on lily pads,
- 5 tree frogs, which made up 1/3 of the total tree frogs in the area,
- 2 bullfrogs,
- 8 exotic tree frogs, where half of them were blue and half were red.

Apart from these, he saw:

- 3 salamanders swimming near the surface,
- 2/5 of the water surface filled with 50 tadpoles moving in a group,
- 10 baby frogs hopping onto a big rock jutting out from the pond,
- An additional group of 20 tadpoles that were 20% smaller in number than the first group,
- A newt, two toads, and a caecilian hiding under the vegetation at the water's edge.

How many total amphibians (including different species of frogs, salamanders, tadpoles, toads, newts, and caecilians) did Hunter observe in the pond?
Please reason step by step, and put your final answer within \boxed{}.


Context:
正整数$a _ { 1 } ,a _ { 2 } ,\cdots,a_{2006}$（可以有相同的）使得$\frac { a _ { 1 } } { a _ { 2 } },frac { a _ { 2 } } { a _ { 3 } },\cdots, \frac { a _ { 2 0 0 5 } } { a _ { 2 0 0 6 } }$两两不相等.问：$a _ { 1 } , a _ { 2 },\cdots,a_{2006}$中最少有多少个不同的数？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jimmy and Irene go shopping for clothes on a Tuesday, where senior citizens get a 10% discount on their purchases. There is also a "buy 2, get 1 free" promotion on all summer clothes, and there is a 5% sales tax applied on the final sale amount. Jimmy picks out 3 shorts from the $15 rack, and Irene grabs 5 shirts from the $17 rack. Taking into account the senior citizens' discount, sales tax, and the "buy 2, get 1 free" promotion, how much money do they give to the cashier?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Omi is twice as old as Kimiko. Arlette is 3/4 times as old as Kimiko. Xander is 5 years younger than the square of Kimiko's age, and Yolanda's age is the cube root of Xander's age. If Kimiko is 28 years old, calculate the average age of the five individuals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Natural numbers \(a\), \(b\), and \(c\) greater than 2022 have the following properties:
- \(a + b\) is divisible by \(c - 2022\)
- \(a + c\) is divisible by \(b - 2022\)
- \(b + c\) is divisible by \(a - 2022\)

What is the greatest possible value of \(a + b + c\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system, the coordinates of the point $\left(2,-1\right)$ with respect to the $x$-axis are ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
利用24小时制计时，可得到某一时刻的四个数字之和比时刻$20:21$的四个数字之和小的概率为_______。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
已知圆锥面 $x^{2}+y^{2}=\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\sqrt{3}, 3,6)$ 到 $Q(\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 $I$, 则 $[10 I]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of $6(7+1)(7^2+1)(7^4+1)(7^8+1)+1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the recurrence relation $$T(n) = nT(\sqrt{n}) + n^2$$ with the initial condition $$T(2) = 1$$ and considering $n = 2^{2^k}$, find a closed-form solution for $T(n)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A teacher teaches one class for each of three grades in a day, with a total of 8 periods in a day, 5 in the morning and 3 in the afternoon. The teacher cannot teach three consecutive classes (periods 5 and 6 do not count as consecutive). Calculate the total number of possible schedules for the teacher in a day. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find any four distinct natural numbers that have the following property: if you add the product of any two of them to the product of the remaining two numbers, you will get a prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On weekdays, a Scatterbrained Scientist commutes to work on the circular line of the Moscow metro from the "Taganskaya" station to the "Kievskaya" station, and in the evening, he returns back. Upon entering the station, the Scientist boards the first train that arrives. It is known that trains run at approximately equal intervals in both directions. The northern route (through "Belorusskaya") takes 17 minutes from "Kievskaya" to "Taganskaya" or vice versa, while the southern route (through "Paveletskaya") takes 11 minutes. The Scientist, by habit, calculates everything. One day, he calculated based on many years of observations:
- A train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after a train traveling clockwise arrives there. The same is true for "Taganskaya".
- The commute from home to work takes, on average, 1 minute less than the commute from work to home.

Find the expected interval between trains traveling in one direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a standard Brownian motion $\{B(t); t\ge 0\}$ (zero mean and $\sigma^{2} = 1 $) at times $0 < u < u+v < u+v+w < u+v+w+x $, where $u, v, w, x > 0$. Evaluate the product moment: $$E[B(u)B(u+v)B(u+v+w)B(u+v+w+x)] $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the bisection method to find an approximate value of a positive root of the equation $x^2 - 2 = 0$, accurate to $0.1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An isosceles trapezoid $ABCD$ is circumscribed around a circle. The lateral side $AB$ touches the circle at point $M$, and the base $AD$ at point $N$. Segments $MN$ and $AC$ intersect at point $P$, and $NP: PM=2$. Find

the ratio $AD: BC$.
Please reason step by step, and put your final answer within \boxed{}.