Context:
In a certain triangle, the sum of the tangents of the angles is equal to 2016. Estimate (with an accuracy of at least 1 degree) the measure of the largest angle.
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Context:
8. (5 points) In the cells of a $(2 k+1) \times(2 n+1)$ table, where $k \leqslant n$, the numbers 1, 2, and 3 are arranged such that in any $2 \times 2$ square, there are all three different numbers. What is the maximum value that the sum of the numbers in the entire table can take?
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Context:
Given the sets $A=\{|a+1|,3,5\}$ and $B=\{2a+1,a^{2}+2a,a^{2}+2a-1\}$, find $A \cup B$ when $A \cap B = \{2,3\}$.
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Context:
Given a cyclic quadrilateral \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(E\), and the following conditions: \(AC = BC\), \(AD = 5\), \(BE = 12\), \(DE = 3\), find \(\angle BCD\).
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Context:
Three people, A, B, and C, are sawing wooden sticks of the same thickness. They receive sticks of 8 meters, 10 meters, and 6 meters in length respectively, and are required to saw them into 2-meter sections. After completing the task, A, B, and C have sawn 24, 25, and 27 sections respectively. How many more times did the fastest sawyer saw compared to the slowest sawyer?
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Context:
Given a sample: 1, 3, 4, $a$, 7. Its mean is 4, then the standard deviation of this sample is
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Context:
Given the hyperbola equation x^2/16 - y^2/9 = 1 and point P(4, 4), count the number of lines that pass through P and intersect the hyperbola at only one point. Express your answer as a single integer.
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Context:
A corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?
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Context:
Using $m$ colors to paint the 6 faces of a cube $V$, with each face painted one color, find the number of distinct colorings of the cube $g(m)$.
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Context:
Find all real numbers \(a\) such that there exists a function \(f : \mathbb{R}^{+} \to \mathbb{R}^{+}\) satisfying the equation \(3(f(x))^{2} = 2f(f(x)) + ax^{4}\) for all \(x \in \mathbb{R}^{+}\).
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Context:
Calculate the volume of the body bounded by the surfaces $az = a^2 - x^2 - y^2$, $z = 0$, and $x^2 + y^2 + ax = 0$ (where $a > 0$).
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Context:
Determine if there exists a bound on the exponent $a$ such that $x^a = x^b$ holds for all $x$ in any ring $R$, independent of the ring $R$.
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Context:
Suppose $G$ is a group defined by the presentation $G=\langle a,b\mid a^2=b^3=e\rangle$. Determine the order of the group $|G|$. \(\boxed{?}\)
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Context:
Compute the Euler Characteristic of the blowing up of $\mathbb{C}\mathbb{P}^2$ at $n$ points.
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Context:
Determine the dimension of the unitary group $U(n)$, where $U(n) = \{A \in GL_n(\mathbb{C}) \mid A^\ast A = I\}$. Provide your answer as a single number.
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Context:
An arithmetic sequence $\{ b_{n}\}$ consists of positive integers. If $b_{12}-b_{1}=330$, find the limit:

$$\lim _{n\rightarrow \infty }\dfrac {n(b^{2}_{n}+1)}{\sum\limits^{n}_{k=1}\left( 2+4+6+\ldots +2k\right) }$$
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Context:
Calculate the residue of the function \( f(z) = z\exp\left(\frac{1}{1-z}\right) \) at \( z = 1 \).
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Context:
To commemorate Earth Day, the senior class of Fuxing High School visited the Earth Natural Museum. After the visit, a group of 7 students from a class took a group photo. If student A and student B stand together, and student C stands on the edge, the probability that student C is not adjacent to student A or B is ______.
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Context:
10. In a square lawn with side length $a$, there are sprinkler devices installed at each of the four corners. The sprinkler devices can rotate to spray water at a $90^{\circ}$ angle, and each sprinkler can rotate from one side of its corner to the other side, with an effective range of $a$. What is the proportion of the lawn that can be covered simultaneously by all four sprinklers to the entire lawn? $\qquad$ .
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Context:
Determine the largest possible order of any element in the matrix group ${\rm GL}_n(\mathbb{F}_q)$. Provide your answer as a single integer.
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Context:
Given the function f(x) = A sin(ωx + φ) (A > 0, ω > 0, |φ| < $\frac{\pi}{2}$) whose graph has a y-intercept of 1, its first maximum point to the right of the y-axis is (x₀, 2) and its first minimum point is (x₀ + 3π, -2). If x ∈ [0, $\frac{3\pi}{2}$], find the range of f(x).
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Context:
Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, .., 2014) until we obtain a value less than or equal to the previous roll. Let $E$ be the expected number of times we roll the die. Find the nearest integer to $100 E$.
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Context:
A trapezoid with height \(h\) is inscribed in a circle. The lateral side of the trapezoid is seen from the center of the circle at an angle of \(120^{\circ}\). Find the midsegment of the trapezoid.
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Context:
Two projective planes, each with two punctures, are joined such that one puncture from the first projective plane is connected to one puncture from the second projective plane, and the other puncture from the first projective plane is connected to the remaining puncture from the second projective plane. What is the Euler characteristic of the resulting surface?
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Context:
Let \(\Omega=\left\{(x, y, z) \in \mathbb{Z}^{3}: y+1 \geq x \geq y \geq z \geq 0\right\}\). A frog moves along the points of \(\Omega\) by jumps of length 1. For every positive integer \(n\), determine the number of paths the frog can take to reach \((n, n, n)\) starting from \((0,0,0)\) in exactly \(3 n\) jumps.
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Context:
Suppose the transformation $T$ acts on points in the plane like this:
$$T(x, y) =  \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2}\right).$$Determine the area enclosed by the set of points of the form $T(x, y)$, where $(x, y)$ is a point on the edge of a length-$2$ square centered at the origin with sides parallel to the axes.
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Context:
A man saves 15% of his monthly salary. If, due to a rise in the cost of living, he is required to increase his monthly expenses by 12%, he is only able to save Rs. 250 per month. What is his monthly salary?
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Context:
Determine the dimension of the space of Jacobi fields $J$ along a geodesic $\gamma : [0,a] \to M$ such that $J(0)=0$ and $\langle J,\gamma' \rangle(t) \equiv 0$. Assume $M$ is an $n$-dimensional manifold.
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Context:
A numerical sequence is defined by the conditions: \( a_{1}=1 \), \( a_{n+1}=a_{n}+\left \lfloor \sqrt{a_{n}} \right \rfloor \).

How many perfect squares are among the first terms of this sequence that do not exceed 1,000,000?
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Context:
Evaluate the definite integral $\int_a^{pa}\frac{ax}{\sqrt{(a-x)(x-pa)}}dx$ without using the substitution $x=asin^2\theta+pacos^2\theta$. The integral appears in the context of particle dynamics in Ganguly Saha's Applied Mathematics text.
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Context:
A building has three different staircases, all starting at the base of the building and ending at the top. One staircase has 104 steps, another has 117 steps, and the other has 156 steps. Whenever the steps of the three staircases are at the same height, there is a floor. How many floors does the building have?
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Context:
All edges of a triangular pyramid are equal to \(a\). Find the maximum area of the orthogonal projection of this pyramid onto a plane.
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Context:
A bag contains 100 unique marbles. With replacement, you pick 10 marbles at a time, at random. How many times must you pick the marbles to have a 95% probability of having seen every unique marble at least once?
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Context:
The lengths of the edges of a rectangular parallelepiped are $a$, $b$, and $c$. What is the maximum value of the area of a rectangular projection of this parallelepiped onto a plane?
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Context:
If \( b \) and \( h \) are positive integers with \( b < h \) and \( b^{2} + h^{2} = b(a + h) + a h \), find the value of \( b \).
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Context:
Let \( a^b \) be a number. If both the base and the exponent are tripled, the resulting number is denoted as \( r \). Given that \( r \) equals the product of \( a^b \) and \( x^b \), where \( x > 0 \), calculate the value of \( x \) as a function of \( a \). Express your answer in a clear and simplified algebraic form.
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Context:
The graph of the function $y=g(x)$ is given. For all $x > 5$, it is observed that $g(x) > 0.1$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, determine $A+B+C$ knowing that the vertical asymptotes occur at $x = -3$ and $x = 4$.
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Context:
A motorcyclist left point A for point B, and at the same time, a pedestrian left point B for point A. When they met, the motorcyclist took the pedestrian on his motorcycle to point A and then immediately went back to point B. As a result, the pedestrian reached point A 4 times faster than if he had walked the entire distance. How many times faster would the motorcyclist have arrived at point B if he didn't have to return?
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Context:
The hour part of a defective digital watch displays only the numbers from $1$ to $12$. After one minute from $n: 59$, although it must display $(n + 1): 00$, it displays $2n: 00$ (considering $\mod\, 12$). For example, after $7: 59$, it displays $2: 00$ instead of $8: 00$. If we set the watch to an arbitrary time, find the probability that the hour part displays $4$ after exactly one day. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
In triangle \( \triangle ABC \), \( AC = 3 \) and \( \sin C = k \sin A \) where \( k \geq 2 \). What is the maximum area of triangle \( \triangle ABC \)?
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Context:
Given a square matrix $A$ of order $2$ with a non-zero determinant ($\lvert A \rvert \neq 0$), if $\lvert A + \lvert A \rvert \adj(A) \rvert = 0$, find the value of $\lvert A - \lvert A \rvert \adj(A) \rvert$, where $\adj(A)$ denotes the adjoint (or adjugate) of $A$. Assume knowledge of determinants, adjoints, and basic matrix operations.
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Context:
Evaluate the limit: \[ \lim_{x \to 3} \left( \frac{x! - 2x}{x - 3} \right) \]
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Context:
You have 5 identical buckets, each with a maximum capacity of some integer number of liters, and a 30-liter barrel containing an integer number of liters of water. All the water from the barrel was poured into the buckets, with the first bucket being half full, the second one-third full, the third one-quarter full, the fourth one-fifth full, and the fifth one-sixth full. How many liters of water were in the barrel?
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Context:
For positive real numbers $a,$ $b,$ $c,$ and $d,$ find the minimum value of
\[\left\lfloor \frac{b + c + d}{a} \right\rfloor + \left\lfloor \frac{a + c + d}{b} \right\rfloor + \left\lfloor \frac{a + b + d}{c} \right\rfloor + \left\lfloor \frac{a + b + c}{d} \right\rfloor.\]
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\left| \frac{(n+1)^{2n^2+2n+1}}{n^{n^2} (n+2)^{(n+1)^2}}\right|$$
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Context:
A parallelogram with a perimeter of $2p$ rotates around an axis perpendicular to the diagonal of length $d$ and passing through its end. Find the surface area of the solid of revolution.
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Context:
In 1937, the German mathematician Collatz proposed a famous conjecture: for any positive integer $n$, if $n$ is even, divide it by $2$ (i.e., $\frac{n}{2}$); if $n$ is odd, multiply it by $3$ and add $1$ (i.e., $3n+1$). By repeating this operation, after a finite number of steps, you will always reach $1$. Currently, the Collatz conjecture cannot be proven or disproven. Now, please investigate: if the $8$th term after applying the rule to a positive integer $n$ (initial term) is $1$, then the number of different values of $n$ is ____.
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Context:
设S={1,2,3,4},n项的数列$a _ { 1 } , a _ { 2 } , \cdots , a _ { n } ( a _ { i } \in S )$有下列性质：对于S的任意非空子集B，在该数列中有相邻的|B|项恰好组成集合B，求n的最小值.
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Context:
Given a collection of 11,986 unique cards, determine an upper bound on the number of possible decks that can be created. Each deck must have at least 60 cards, and no more than 4 copies of any particular card can be included in a deck.
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Context:
Calculate the arc lengths of the curves given by the equations in polar coordinates.

$$
\rho=2 e^{4 \varphi / 3},-\frac{\pi}{2} \leq \varphi \leq \frac{\pi}{2}
$$
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Context:
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of $a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}$.
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Context:
Candies are in the shape of $1 \times 1 \times 1$ cubes. The teacher arranged them into a $3 \times 4 \times 5$ rectangular prism and asked the children to pick candies. In the first minute, Petya took one of the corner candies. Each subsequent minute, the children took all candies that had a neighboring face with already missing candies (for example, 3 candies were taken in the second minute). How many minutes did it take for the children to take all the candies?
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Context:
Let \( A \), \( B \), and \( C \) be pairwise independent events with equal probabilities, and \( A \cap B \cap C = \varnothing \). Find the maximum possible value for the probability \( \mathrm{P}(A) \).
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Context:
Find the number of polynomials \( p(x) \) with integer coefficients satisfying \( p(x) \geq \min \left\{2 x^{4}-6 x^{2}+1,4-5 x^{2}\right\} \) and \( p(x) \leq \max \left\{2 x^{4}-6 x^{2}+1,4-5 x^{2}\right\} \) for all \( x \in \mathbb{R} \).
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Context:
In $\vartriangle ABC$, let $D$ be on $BC$ such that $\overline{AD} \perp \overline{BC}$. Suppose also that $\tan B = 4 \sin C$, $AB^2 +CD^2 = 17$, and $AC^2 + BC^2 = 21$. Find the measure of $\angle C$ in degrees between $0^o$ and $180^o$ .
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Context:
Let $a\in R$, for any real number $x$, let $f\left(x\right)=\min \{e^{x}-2$,$e^{2x}-ae^{x}+a+24\}$. If $f\left(x\right)$ has three zeros, then the range of real number $a$ is ______.
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Context:
when 242 is divided by a certain divisor the remainder obtained is 11 . when 698 is divided by the same divisor the remainder obtained is 18 . however , when the sum of the two numbers 242 and 698 is divided by the divisor , the remainder obtained is 9 . what is the value of the divisor ?
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Context:
Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\dfrac{BP}{CP} = 3$. Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$, $n$, $p$, $q$ with $\text{gcd}(m,n) = \text{gcd}(p,q) = 1$. What is $m+n+p+q$? $\phantom{}$
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Context:
Given the functions $f(x)=3\sin (ωx- \frac {π}{6})(ω > 0)$ and $g(x)=2\cos (2x+φ)+1$, their graphs have exactly the same symmetry axis. If $x∈[0, \frac {π}{2}]$, then the range of $f(x)$ is _______.
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Context:
A person has between 0 and 300,000 hairs on their head. The Marseille metropolitan area contains 2,000,000 inhabitants. How many inhabitants of Marseille have at least the same number of hairs?
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Context:
How many pairs of integers $(x, y)$ satisfy the equation \(\sqrt{x-\sqrt{x+23}}=2\sqrt{2}-y\)?
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Context:
A grocer arranges a display of cans where the top row has three cans, and each lower row has two more cans than the row above it. The total number of cans in the display is 169. How many rows does the display contain?
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Context:
In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$.
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Context:
If for any positive integer \( m \), the set
$$
\{m, m+1, m+2, \cdots, m+99\}
$$
in any \( n \)-element subset with \( n \geq 3 \), there are always three elements that are pairwise coprime, find the smallest value of \( n \).
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Context:
Let \( \triangle ABC \) be an acute triangle with circumcenter \( O \) such that \( AB = 4 \), \( AC = 5 \), and \( BC = 6 \). Let \( D \) be the foot of the altitude from \( A \) to \( BC \), and \( E \) be the intersection of \( AO \) with \( BC \). Suppose that \( X \) is on \( BC \) between \( D \) and \( E \) such that there is a point \( Y \) on \( AD \) satisfying \( XY \parallel AO \) and \( YO \perp AX \). Determine the length of \( BX \).
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Context:
Quadrilateral $ABCD$ has $\angle BCD=\angle DAB=90^{\circ}$. The perimeter of $ABCD$ is 224 and its area is 2205. One side of $ABCD$ has length 7. The remaining three sides have integer lengths. What is the integer formed by the rightmost two digits of the sum of the squares of the side lengths of $ABCD$?
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Context:
In the Cartesian plane \( xOy \), let point \( P \) be in the first quadrant, and points \( A \) and \( B \) be on the \( Ox \) and \( Oy \) axes respectively, such that \( PA = PB = 2 \). What is the maximum area of the convex quadrilateral \( PAOB \)?
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Context:
Consider a \(9 \times 9\) chessboard with a grasshopper on each square. Each grasshopper jumps once to a neighboring square diagonally. How many squares are left unoccupied at a minimum?
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Context:
A student participated in military training, which included a shooting exercise that required him to shoot 10 times. In the 6th, 7th, 8th, and 9th shots, he scored 9.0, 8.4, 8.1, and 9.3 rings, respectively. The average score of his first 9 shots was higher than the average score of his first 5 shots. If he wants the average score of his 10 shots to exceed 8.8 rings, what is the minimum score he must achieve in his 10th shot? (Each shot's score is accurate to 0.1 ring)
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Context:
For how many integers \( n \) in the inclusive range \([1, 500]\) is the sum of all positive divisors, \( \sigma(n) \), a prime number?
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Context:
A rectangular indoor football field is being designed with a length of $100 \text{m}$ and a width of $80 \text{m}$. The field must be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle with a radius equal to the height at which the spotlight is hanging. Determine the minimum possible ceiling height such that the following conditions are met: every point on the football field is illuminated by at least one spotlight; the ceiling height must be a multiple of $0.1 \text{m}$ (e.g., $19.2 \text{m}$, $28 \text{m}$, $33.9 \text{m}$, etc.).
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Context:
Evaluate the line integral \( \oint_{C}\left[\left(2x - y^{3}\right)\mathrm{d}x - xy\,\mathrm{d}y\right] \), where \( C \) is the boundary of the annular region enclosed by the circles \( x^{2} + y^{2} = 1 \) and \( x^{2} + y^{2} = 9 \).
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Context:
In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
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Context:
Highimiy and.

In Anchuria, a day can be either clear, when the sun shines all day, or rainy, when it rains all day. And if today is not like yesterday, the Anchurians say that the weather has changed today. Once, Anchurian scientists established that January 1 is always clear, and each subsequent day in January will be clear only if the weather changed exactly a year ago on that day. In 2015, January in Anchuria was quite varied: sunny one day, rainy the next. In which year will the weather in January first change in exactly the same way as it did in January 2015?
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Context:
If the graph of the function $y=\cos 2x$ has exactly two extreme points on the interval $(-\frac{π}{4},m)$, then one possible value of the real number $m$ is ______.
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Context:
In a school's mentoring program, several first-grade students can befriend one sixth-grade student, while one sixth-grade student cannot befriend multiple first-grade students. It is known that $\frac{1}{3}$ of the sixth-grade students and $\frac{2}{5}$ of the first-grade students have become friends. What fraction of the total number of students in the first and sixth grades are these friends?
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Context:
Two strips of width 1 intersect in a parallelogram whose area is 2. Find the angle between the strips. Express your answer in degrees.
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Context:
Determine the number of isomorphism classes of $\mathbb{Z}[i]$-modules that have exactly 5 elements.
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Context:
Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Given that $AB = 8, AC = 10$, and $\angle BAC = 60^\circ$, find the area of $BCHG$.
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Context:
Let $n$ be a given positive integer. Ana and Bob play the following game: Ana chooses a polynomial $p$ of degree $n$ with integer coefficients. In each round, Bob can choose a finite set $S$ of positive integers, and Ana responds with a list containing the values of the polynomial $p$ evaluated at the elements of $S$ with multiplicity (sorted in increasing order). Determine, in terms of $n$, the smallest positive integer $k$ such that Bob can always determine the polynomial $p$ in at most $k$ rounds.
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Context:
Determine the number of boolean functions of $n$ variables that can be expressed as a single product of literals.
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Context:
In triangle \(ABC\), points \(B_1\) and \(C_1\) are chosen on sides \(AB\) and \(AC\) respectively, such that \(AB_1: AB = 1:3\) and \(AC_1: AC = 1:2\). A circle is drawn through the points \(A\), \(B_1\), and \(C_1\). A line through point \(B_1\) intersects the segment \(AC_1\) at point \(D\) and the circle at point \(E\). Find the area of triangle \(B_1C_1E\) given that \(AC_1=4\), \(AD=1\), \(DE=2\), and the area of triangle \(ABC\) is 12.
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Context:
Five socks, colored blue, brown, black, red, and purple are in a drawer. In how many different ways can we choose three socks from the drawer if the order of the socks does not matter?
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Context:
Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \).
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Context:
Calculate the Lie bracket \([X_1, [X_1, X_2]]\) for the vector fields \(X_1 = \frac{\partial}{\partial x_1} + x_2\frac{\partial}{\partial x_3} + x_2^2\frac{\partial}{\partial x_4}\) and \(X_2 = \frac{\partial}{\partial x_2}\).
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Context:
On Friday, Remy paid Sally's Woodworking LLC a total of $20,700 for a bulk purchase of 150 pieces of furniture, consisting of chairs, tables, and cabinets. For every 3 chairs, Remy bought 5 tables and 2 cabinets. The base prices for chairs, tables, and cabinets are $120, $220, and $340 respectively. The company offered tiered discounts based on the quantity of each item purchased:

- Chairs: 10% discount if purchasing 15 or more, and an additional 5% discount if purchasing 30 or more.
- Tables: 5% discount if purchasing 25 or more, and an additional 3% discount if purchasing 40 or more.
- Cabinets: 2% discount if purchasing 10 or more, and an additional 5% discount if purchasing 20 or more.

Additionally, there was a 7% sales tax applied to the total payment after discounts, and a $250 delivery charge along with a $350 assembly charge. Later that evening, the company's accountant discovered that a new intern in the sales department had overcharged Remy by not applying the tiered discounts and additional fees correctly. How much money will Sally's Woodworking LLC reimburse Remy?
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Context:
Find the length of the normal chord of the parabola \(y^2 = 4x\) that subtends a right angle at the vertex.
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Context:
Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, but Carlson requires 5 minutes to do the same. In total, they had 10 jars of jam and 10 pots of honey. For how many minutes did they eat everything, given that they started and finished eating at the same time? (Each eats a pot of honey or a jar of jam completely.)
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Context:
Determine the expected profit from a game where a fair die with $n$ sides is rolled at each step, and you can choose to stop and keep the last roll's value as your profit or pay $1$ to continue playing. Find an asymptotic formula for the expectation as $n$ approaches infinity.
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Context:
Given that $α$ is in the interval $(0,π)$ and $\cos(α - \frac{π}{6}) = \frac{3}{5}.$ Find $\cos α$.
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Context:
Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10(that is, $10, 100, 1000, 10 000$ , ...). If $p>q$, the last digit of $p - q$ cannot be
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Context:
A square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. Calculate the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle. Express your answer as a simplified fraction.
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Context:
Find the minimal polynomial of the linear operator \( T \) on the vector space \( V = M_{n \times n}(R) \), where \( T(A) = A^t \) for each matrix \( A \).
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Context:
Let $x_i \in [-11, 5]$ for $i = 1, 2, \ldots, 2014$, and suppose $x_1 + x_2 + \cdots + x_{2014} = 0$. Determine the maximum value of $x_1^2 + x_2^2 + \cdots + x_{2014}^2$. \(\text{Express your answer as a single number.}\)
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Context:
After exploring the mountain country, she went to the famous beach on the continent. There, she collected 820 seashells in various colors including red, green, blue, yellow, and purple. The number of shells of each color follows the Fibonacci sequence. If she has 144 red shells and 89 green shells, and knowing that blue shells are the least numerous, how many blue, yellow, and purple shells does she have in total, given that the number of each color is a unique term in the Fibonacci sequence?
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Context:
Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[ \sum_{i=1}^7 n_i^6 = 96957. \]
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Context:
Find the Gini index for the probability density function f(x) = 1 on [0, 1] and f(x) = 0 elsewhere.
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Context:
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in  $10$ ,  $9$ ,  $11$ , and  $12$  days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
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Context:
Find a tight bound for the recurrence relation: \[ T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + T\left(\left\lfloor \frac{n}{4} \right\rfloor \right) + T\left(\left\lfloor \frac{n}{8} \right\rfloor \right) + n \]
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Context:
Suppose  $28$  objects are placed along a circle at equal distances. In how many ways can  $3$  objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
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Context:
Given the sets $A=\{x\mid -2 < x < 4\}$ and $B=\{x\mid x < a\}$, and their union $A\cup B=\{x\mid x < 4\}$, determine the range of values for the real number $a$.
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Context:
8.5 On an island, there live 25 people: knights, liars, and tricksters. Knights always tell the truth, liars always lie, and tricksters answer the questions posed to them in turn, alternating between truth and lies. All the islanders were asked three questions: "Are you a knight?", "Are you a trickster?", "Are you a liar?" (the questions were asked in the exact order specified). To the first question, 21 people answered affirmatively, to the second - 17 people, to the third - 6 people. How many knights live on this island?
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Context:
There are three video game systems: the Paystation, the WHAT, and the ZBoz2 \pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2 \pi, and Dirac owns a ZBoz2 \pi and a Paystation. A store sells 4 different games for the Paystation, 6 different games for the WHAT, and 10 different games for the ZBoz2 \pi. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys 3 random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?
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Context:
Identify the smallest group that is not isomorphic to a cyclic group, a direct product of cyclic groups, or a semi-direct product of cyclic groups.
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Context:
Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset \mathcal{O}_p(C)$ is its maximal ideal, find $\text{dim}_\mathbb{C} \mathfrak{m}^k/\mathfrak{m}^{k+1}$.
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Context:
Find the probability assigned by the measure $\mu$ to the union of the top and left edges of the unit square $[0,1]^2$. Express your answer as a single value.
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Context:
Let $k \ll n$ denote that $k < n$ and $k \mid n$. Let $f: \{1, 2, \ldots, 2013\} \rightarrow \{1, 2, \ldots, M\}$ be such that, if $n \leq 2013$ and $k \ll n$, then $f(k) \ll f(n)$. What is the least possible value of $M$?
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Context:
Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$.
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Context:
Given the identity $(x^3+px^2+2x-5)^{19}(x^2+qx-41)^{8}(x^4-x^3+x-7)^6 = x^{97} + 391x^{96} + a_{95}x^{95} + \cdots + a_1x + a_0$, where $p, q, a_{95}, \cdots, a_0$ are integers, find the smallest possible value of $p$.
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Context:
Given an isosceles right triangle \(ABC\) with hypotenuse \(AB\). Point \(M\) is the midpoint of side \(BC\). A point \(K\) is chosen on the smaller arc \(AC\) of the circumcircle of triangle \(ABC\). Point \(H\) is the foot of the perpendicular dropped from \(K\) to line \(AB\). Find the angle \(\angle CAK\), given that \(KH = BM\) and lines \(MH\) and \(CK\) are parallel.
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Context:
Given that $\binom{21}{13}=20349$, $\binom{21}{14}=11628$, and $\binom{23}{15}=490314$, find $\binom{22}{15}$.
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Context:
10. $1991^{2000}$ divided by $10^{6}$, the remainder is
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Context:
Given the line $l: \lambda x-y-\lambda +1=0$ and the circle $C: x^{2}+y^{2}-4y=0$, calculate the minimum value of $|AB|$.
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Context:
A man walks at a rate of 10 mph. After every ten miles, he rests for a certain amount of time. He takes 332 minutes to walk 50 miles. How many minutes does he rest after every ten miles?
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Context:
A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is $3$ feet deep. How deep, in feet, would the oil have been if the tank had been standing upright on one of its bases? Express your answer as a decimal to the nearest tenth.
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Context:
Determine the exact value of the series
\[
\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.
\]
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Context:
China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq  \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I}  }x_i\le \sum_{i\in \mathbb{I}  }a_i$ , find the maximum value of $x_1x_2\cdots  x_n.$
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Context:
Determine the Galois group of the polynomial \( f(x) = x^6 - 9 \) over \( \mathbb{Q} \), given that the splitting fields of its factors \( x^3 - 3 \) and \( x^3 + 3 \) are identical.
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Context:
What angle is formed by the hour and minute hands at 4:12?
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Context:
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime, while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
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Context:
The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the cross-section passing through this lateral edge and the height of the base is equal to \( Q \). Determine the volume of the prism.
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Context:
In a convex hexagon $ABCDEF$, let $P$, $Q$, $R$, $S$, $T$, and $U$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The areas of triangles $ABR$, $BCS$, $CDT$, $DEU$, $EFP$, and $FAQ$ are given as $12$, $34$, $56$, $12$, $34$, and $56$, respectively. Find the area of hexagon $ABCDEF$. \( \text{Express your answer as a single number.} \)
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Context:
# Problem 10. (2 points)

The sum of a geometric progression consisting of four natural numbers is 1417. Find the second largest of these numbers.

If there are multiple possible answers, list them in ascending order separated by commas or semicolons.
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Context:
Let $g(x)$ be a continuous function for all real values of $x$ that satisfies the following equation:

$$\int^x_1 g(t)dt =\int^0_{x} t^3g(t)dt +\frac{x^{18}}{9}+\frac{x^8}{4}+b$$

Find the value of $b$.
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Context:
Translate the function $f(x)=1-2\sqrt{3}\cos^2{x}-(\sin{x}-\cos{x})^2$ by shifting its graph to the left by $\dfrac{\pi}{3}$ units to obtain the graph of the function $y=g(x)$. If $x\in\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$, then the interval of monotonic increase for the function $g(x)$ is ______.
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Context:
Determine the dimension of the ideal \( I = \langle x+1 \rangle / \langle x^9+1 \rangle \) in the ring \( R = \mathbb{Z}_2[x]/\langle x^9+1 \rangle \).
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Context:
10. Given $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in \mathbf{N}$ and satisfy $a_{1}+a_{2}+a_{3}=b_{1}+b_{2}+b_{3}$
$$
\begin{array}{c}
a_{1} a_{2} a_{3}=b_{1} b_{2} b_{3} \\
a_{1} a_{2}+a_{1} a_{3}+a_{2} a_{3} \neq b_{1} b_{2}+b_{1} b_{3}+b_{2} b_{3}
\end{array}
$$

then the minimum value of the maximum number among $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ is $\qquad$
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Context:
求最大的实数 $k$, 使得对于任意的正数 $a 、 b$ ,均有
$$
(a+b)(a b+1)(b+1) \geqslant k a b^{2}.
$$  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:
In the triangle \( \triangle ABC \), \( AB = AC \), \( \angle ABC = 40^\circ \) and the point \( D \) is on \( AC \) such that \( BD \) is the angle bisector of \( \angle ABC \). If \( BD \) is extended to the point \( E \) such that \( DE = AD \), find the measure of \( \angle ECA \) in degrees.
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Context:
Given the parabola $y^{2}=4x$ and the circle $E:(x-4)^{2}+y^{2}=12$. Let $O$ be the origin. The line passing through the center of circle $E$ intersects the circle at points $A$ and $B$. The lines $OA$ and $OB$ intersect the parabola at points $P$ and $Q$ (points $P$ and $Q$ are not coincident with point $O$). Let $S_{1}$ be the area of triangle $OAB$ and $S_{2}$ be the area of triangle $OPQ$. Find the maximum value of $\frac{S_{1}}{S_{2}}$.
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Context:
Sergei and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the greatest possible total weight of all the chocolate bars?
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Context:
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $n$ between 1 and 25. Betül draws $n$ balls from the bag one by one such that no ball is put back after it is drawn. If exactly 2 of the $n$ balls are blue and the second blue ball is drawn at the $n^{th}$ order, Ahmet wins; otherwise, Betül wins. To increase the possibility of winning, what integer should Ahmet choose?
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Context:
A regular tetrahedron has two spheres, one inscribed within it and the other circumscribed around it. Between each face of the tetrahedron and the circumscribed sphere, there are four smaller spheres centered on each face. Given a point \( P \) within the circumscribed sphere, the probability that \( P \) lies within one of the five smaller spheres is closest to:
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Context:
Find the smallest positive integer $n$ such that the last 30 digits of the decimal representation of $n!(n+1)!(2n+1)! - 1$ are all 9s.
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Context:
若 $x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4}=799$ 有正整数解，则正整数 $n$ 的最小值为多少？
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Context:
Suppose $X$ and $Y$ are independent and identically distributed (i.i.d.) random variables, each following a normal distribution $N(0,1)$. Calculate the conditional probability $P(X > 0 \mid X + Y > 0)$. Express your answer as a single probability value.
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Context:
Myrtle’s 3 hens lay 3 eggs a day. She was gone for a certain number of days and told her neighbor to take as many as they would like. The neighbor took 12 eggs. Once home, Myrtle collected the remaining eggs, dropping 5 on the way into her house. Myrtle has 46 eggs. How many days was Myrtle gone?
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Context:
Evaluate the integral \[ I = \iint_{[0,1]^2} \frac{dx \, dy}{(1+x^2+y^2)^{3/2}}. \]
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Context:
Estimate the upper bound of the following integral using the ML lemma, without evaluating it:

$$\left|\int_{Γ}\frac{z^{\frac{3}{2}}}{z^2 + 4}dz\right|$$

where $Γ$ is the semicircular arc {$z ∈ C : |z| = 2,\quad 0 ≤ Arg(z) ≤ π$} described in the clockwise direction. Consider the principal branch of $z^{3/2}$.
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Context:
Compute the limit \( \displaystyle\lim_{x \to +\infty} f(x) \), where \( f(x) = \frac{1}{\Gamma(x)}\int_x^{+\infty}t^{x-1}e^{-t}dt \) for all \( x \in \Bbb R_+^* \).
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Context:
Given a closed, smooth, oriented Riemannian manifold $([M]^n, g)$, with two closed, smooth, oriented submanifolds $[S]^k$ and $[T]^{n-k}$ intersecting transversely, let $\omega$ be a $k$-form on $M$ such that $\oint_{[S]} \omega = r \ne 0$. If $\eta$ is the Hodge star of $\omega$, determine $\oint_{[T]} \eta$ given that the $\mathbb{Z}\pi_1(M)$-intersection number of $S$ and $T$ is $\pm 1$, where $\pi = \pi_1(M)$. Provide your answer as a single value or expression.
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Context:
In $\triangle ABC$, given $C= \sqrt {2}$, $\angle B= \frac {\pi}{4}$, and $b=2$, find $\angle A=$ ______.
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Context:
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is selected. On one side of \(AO\), points \(B\) and \(C\) are chosen on the circle such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\), if the radius of the circle is 10 and \(\cos \alpha = \frac{2}{5}\).
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Context:
There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different
rooms will have people in them?
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Context:
Calculate the area bounded by the inverse of the function \( f(x) = x^3 + 3x + 2 \) between \( x = -2 \) and \( x = 6 \).
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Context:
Find the value of the limit: \[ \lim_{n\to \infty }\left(\sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}\right). \]
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Context:
The function $f(x)=k\ln x-x^{2}+3x (k\neq 0)$ has extreme values on the interval $(0,\frac{3}{2})$. The range of real number $k$ is ______.
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Context:
As shown in Figure 2, let $E$ and $F$ be the midpoints of sides $AD$ and $BC$ respectively of rectangle $ABCD$. Points $M$ and $N$ are on segments $AB$ and $CF$ respectively. Line segment $MN$ intersects $AF$ at point $P$, with $MN = 12$, $AD = 2AB$, and $PE \perp MN$. Find the length of $PE$.
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Context:
The numbers 1, 2, 3, and 4 are assigned, in some order, to the variables p, q, r, and s. What is the maximum possible value of p^q + r^s? Express your answer as a single numerical value.
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Context:
Given that $2^{2004}$ is a $604$-[[digit]] number whose first digit is $1$, how many [[element]]s of the [[set]] $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$?
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Context:
Find the coefficient of \(x^{25}\) in the expansion of \((1 + x^2 + x^4 + x^7)^{10}\).
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Context:
Two cyclists set off simultaneously towards each other from points $A$ and $B$ and met 70 km from $A$. Continuing with the same speeds, they reached points $A$ and $B$ and turned back. The second time they met 90 km from $B$. Find the distance from $A$ to $B$.
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Context:
Evaluate the surface integral \(\iint_{S} xz \, dy \, dz\) where \(S = \{(x,y,z) \mid x \geq 0, y \geq 0, z \geq 0, 2x + 2y + z = 2\}\).
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Context:
Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is  $social$  if it knows at least  $20$  other persons and at least  $2$  of those  $20$  know each other. For a person, we say that it is  $shy$  if it doesn't know at least  $20$  other persons and at least  $2$  of those  $20$  don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any  $social$  nor  $shy$  persons.
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Context:
Given a random variable $Y=\frac{c(X_1-X_2)}{\sqrt{(X_3)^2+(X_4)^2}}$, where $X_1$, $X_2$, $X_3$ and $X_4$ are independent standard normally distributed variables, find the value of $c$ such that $Y$ follows a t-distribution.
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Context:
Given circle $C: x^2 + y^2 = 1$, and point $P(x_0, y_0)$ is on the line $l: 3x + 2y - 4 = 0$. If there always exist two different points $A, B$ on circle $C$ such that $\vec{OA} + \vec{OB} = \vec{OP}$, then the range of values for $x_0$ is ______.
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Context:
Solve the system of equations for \((x, y, z)\), where \(\{x\}\) denotes the fractional part of \(x\):

\[\begin{cases} x + \lfloor y \rfloor + \{z\} = 12.5\\ \{x\} + y + \lfloor z \rfloor = 16.2\\ \lfloor x \rfloor + \{y\} + z = 15.7 \end{cases}\]
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Context:
Bob, a bomb-defuser, encounters a bomb with two wires: red and green. Initially, he decides to cut the green wire. However, every second from time \( t = 15 \) seconds down to \( t = 1 \) second, there is a \(\frac{1}{2t^2}\) chance that Bob, due to panic, will switch his wirecutters to the other wire. At \( t = 1 \) second, Bob will cut the wire his wirecutters are currently on without switching. What is the probability that Bob cuts the green wire?
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Context:
If $A$ and $B$ are two positive definite matrices such that $A < B$, determine the relationship between $A^{-1}$ and $B^{-1}$.
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Context:
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.
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Context:
A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y), (x + 2, y), (x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.
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Context:
Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that
\[ (P(X))^2+(Q(X))^2=X^{2n}+1\]
and $ \text{deg}P<\text{deg}{Q}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the function \( f(x) = 12x^2\int_0^1 yf(y) \, dy + 20x\int_0^1 y^2 f(y) \, dy + 4x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \geq 3 \) be an odd integer. Define \( [\minus{}n,n] \) as the set of all integers from \( -n \) to \( n \), inclusive. Player \( A \) selects a positive integer \( k \), and player \( B \) chooses a subset \( S \) of \( k \) distinct elements from \( [\minus{}n,n] \). Player \( A \) wins if every number in \( [\minus{}n,n] \) can be expressed as the sum of exactly \( n \) distinct elements from \( S \). Otherwise, player \( B \) wins. Determine the smallest value of \( k \) such that player \( A \) can always win.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the last two digits of the number \( A = (2016^{2015^{2014}} + 2014^{2015^{2016}} + 2017)^{2017} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose there are ten genes that determine tallness, each represented by two bits (either 0 or 1). Each gene can be 00, 01, 10, or 11, and a person's height is determined by the number of '11' genes they possess. If both parents have a height of 7, meaning they each have seven '11' genes, what is the expected height of their offspring, assuming the bits of the three remaining non-'11' genes are uniformly distributed? Inheritance is such that for each gene, one bit is randomly chosen from the male and the other from the female.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be filled?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle $ABC$, three lines are drawn perpendicular to each side, intersecting within the triangle. Blue points are placed such that any line connecting two blue points is parallel to one of the sides of the triangle. Each perpendicular line divides the parallel lines it intersects into three segments, with each segment containing at most one blue point. What is the maximum number of blue points that can be placed within the triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc = m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether there exists a field of characteristic 0 that is not algebraically closed and cannot be ordered to form an ordered field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_{S} \vec{F} \cdot \text{d}S \) where \( \vec{F} = 3x\hat{i} + 2y\hat{j} - 5z\hat{k} \), and \( S \) is the portion of the surface \( y = x^2 + z^2 \) that lies in the region \( y \leq 1 \), oriented with the normal vector pointing in the positive \( y \)-direction. Use Gauss' theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest constant \( k \) such that
 \( \frac{x}{\sqrt{x + y}} + \frac{y}{\sqrt{y + z}} + \frac{z}{\sqrt{z + x}} \leq  k\sqrt{x + y + z} \)
 for all positive \( x,y,z \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n\to \infty}\left(\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+\cdots+\dfrac{1}{\sqrt{n^2+2n}}\right) \) using the Squeeze Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## 265. Math Puzzle $6 / 87$

For a motorcycle, the optimal rotational speed of the engine shaft is $6000 \mathrm{U} / \mathrm{min}$.

What is the gear ratio of the rear axle to the engine shaft produced by the transmission if the optimal rotational speed results in a speed of $60 \mathrm{~km} / \mathrm{h}$?

The diameter of the rear wheel is $53 \mathrm{~cm}$ (circumference 1.67 m).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \geq 3 \) be an integer. \( n \) balls are arranged in a circle and labeled with numbers \( 1, 2, \ldots, n \) in order. There are 10 colors, and each ball is colored with one of them such that two adjacent balls have different colors. The sum of the labeled numbers of balls of each color is the same for all colors. Find the minimum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-equivalent formulas that can be constructed using the propositions \( p_1, p_2, \ldots, p_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Aerith and Bob take turns picking a nonnegative integer, each time changing exactly one digit from the other's last number. The first person to pick a number that (s)he picked before loses. If Aerith goes first, and both play optimally, who wins?

(Note: There are no leading zeroes, except in the number 0 itself. For instance, if one person picks 2020, the other could respond by picking 0020 = 20, however the reverse does not hold.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red or blue. Each time he sees a marble, he can either keep it or pass, but cannot return to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for keeping a blue marble, and gains 0 points for passing. All distributions of colors are equally likely, and Merble can only see the color of his current marble. If his goal is to end with exactly one point and he plays optimally, what is the probability that he fails?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a vector field $F(x,y)=\left(-\frac{y}{x^2 + y^2} , \frac{x}{x^2+y^2}\right)$ in $\mathbb{R}^2$, find the value of the line integral $\int_{\vert x \vert + \vert y \vert = 5 } -\frac{y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2}dy $.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{k=0}^{2013} 2^k a_k \) for the sequence \( a_0, a_1, a_2, \ldots, a_n \) defined by \( a_0 = 2013 \) and \( a_n = -\frac{2013}{n} \sum_{k=0}^{n-1} a_k \) for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find
$$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \(M\) is a linear operator on \(\mathbb{R}^3\) with unique and real eigenvalues \(\lambda_1 < \lambda_2 < \lambda_3\), and there exists a vector \(x \in \mathbb{R}^3 \setminus \{0\}\) such that \(\lim_{n \to \infty} ||M^n x|| = 0\), what are the possible values of \(\lambda_1\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse of the element \(1010\) in the field \(GF(2^4)\) with multiplication modulo \(x^4 + x^3 + 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parallelogram \( ABCD \), \( \angle B = 111^\circ \) and \( BC = BD \). Point \( H \) is on segment \( BC \) such that \( \angle BHD = 90^\circ \). If \( M \) is the midpoint of side \( AB \), find the measure of angle \( AMH \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the series \(\sum_{k=0}^{n}(-1)^{k-1}k\binom{n}{k}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval of convergence of the series \(\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{4^n \ln(n)}.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral:

$$\int_0^{2\pi}\frac{\cos^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\cos^2(x/2)} dx$$
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Context:
Consider the interval $[0,1]$. At each stage, remove the closed middle third interval from each remaining segment. For example, from $[0,1]$, remove $[rac{1}{3}, rac{2}{3}]$ to get $[0, rac{1}{3}) \cup (\frac{2}{3}, 1]$. Continue this process indefinitely. How many elements are left in the intersection of all these sets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $\sqrt{a-5}+\left(b-3\right)^{2}=\sqrt{c-4}+\sqrt{4-c}$, then the simplified result of $\frac{\sqrt{c}}{\sqrt{a}-\sqrt{b}}$ is ____.
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Context:
A player starts with $X and plays a betting game where they have a 49.8% chance to double their bet and a 50.2% chance to lose it. If the player's total wealth increases to $X + Y, what is the expected total amount of money they must wager to return to $X? For example, if a player starts with $100 and their wealth increases to $150, calculate the expected total amount they must wager to return to $100.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the letters $F$,$L$,$Y$,$B$,$U$,$G$ represent distinct digits. Suppose $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}$ is the greatest number that satisfies the equation \[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\] What is the value of $\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ordered quadruples \((a, b, c, d)\) of distinct positive integers such that \(\binom{\binom{a}{b}}{\binom{c}{d}} = 21\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A criminal must undergo a series of 10 lie detector tests to secure his release from jail. The probabilities that each detector correctly identifies a lie are as follows: 0, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{4} \), \( \frac{4}{5} \), \( \frac{5}{6} \), \( \frac{6}{7} \), \( \frac{7}{8} \), \( \frac{8}{9} \), \( \frac{9}{10} \). If a detector fails, the test restarts from the first detector. The criminal is released only if all tests are successfully passed in a row. Given that the criminal always lies and each test takes 1 second, determine the expected time (in seconds) for the criminal to be released from jail.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \sum_{d\mid N}\Lambda(d) \) for a given positive integer \( N \), where \( \Lambda(n) = \log p \) if \( n = p^r \) for some prime \( p \) and integer \( r \), and \( \Lambda(n) = 0 \) otherwise.
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Context:
If $\sin \theta= \frac {3}{5}$ and $\frac {5\pi}{2} < \theta < 3\pi$, then $\sin \frac {\theta}{2}=$ ______.
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Context:
Given that \( \mathbb{E}[X] = 10 \), find a lower bound for the expression \( 0.8\mathbb{E}[X^2] + 0.2\mathbb{E}[X^4] \) in terms of \( \mathbb{E}[X] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements of order 7 in a given group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the Riemann zeta function at zero, \( \zeta(0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set $\mathbf{Z}_n = \{0, 1, \ldots, n - 1\}$ with modular addition. A subset $S_n$ of $\mathbf{Z}_n$ is such that the intersection of $(S_n + k)$ and $S_n$ is nonempty for all $k \in \mathbf{Z}_n$. Let $f(n)$ denote the minimum number of elements in such a subset. Determine the limit $$\lim_{n \to \infty} \frac{\ln f(n)}{\ln n}.$$
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Context:
Let \( X \) be a set with 56 elements. Determine the smallest positive integer \( n \) such that for any 15 subsets of \( X \), if the union of any 7 of these subsets contains at least \( n \) elements, then there must exist 3 of these 15 subsets whose intersection is non-empty.
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Context:
Determine the greatest common divisor (GCD) of all possible values of the product $a \cdot b \cdot c \cdot d \cdot e \cdot f$, where $a, b, c, d, e, f$ are natural numbers satisfying the equation $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_{\mathbb{R}} \left(\int_{\mathbb{R}} f(x,y)\,dx \right)dy \) where the function \( f: \mathbb{R}^2 \rightarrow \mathbb{R} \) is defined as follows:

\[
f(x,y) = \begin{cases}
1, & \text{for } x \ge 0, \ x \le y < x+1 \\
-1, & \text{for } x \ge 0, \ x+1 \le y < x+2 \\
0, & \text{elsewhere}
\end{cases}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many known classifications and types of indifferent fixed-points are there in complex dynamics?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知△ABC中，$AC=2\sqrt{3}$，D是BC边上的一点，且△ABD为等边三角形，则△ACD面积S的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
对于四面体 $A B C D, A B \perp B C, C D \perp$ $B C, B C=2$ ，且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\circ}$. 若四面体 $A B C D$ 的外接球半径为 $\sqrt{5}$,则四面体 $A B C D$ 的体积的最大值为 $\qquad$ .原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P$ be the transition matrix of a random walk on an $n$-cycle, where $n$ is odd. Determine the smallest integer $t$ such that $P^t(x, y) > 0$ for all states $x$ and $y$.
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Context:
Given a random variable $X$ that follows a Student's $t$ distribution with $k$ degrees of freedom, determine the distribution of $Y = X^2$. The probability density function of $X$ is given by:
$$f_X(x;k) = \frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{k\pi} \ \Gamma \left(\frac{k}{2}\right)}{\left(1+\frac{x^2}{k}\right)}^{-\frac{k+1}{2}}$$
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Context:
Let \( \omega = ze^{xy} dx \wedge dy \) be a 2-form on \( \mathbb{R}^3 \). Consider the map \( \varphi: \mathbb{R}^2_{\ne 0} \to \mathbb{R}^3 \) defined by \( \varphi(u,v) = \left( \frac{u}{v}, \frac{v}{u}, uv \right) \). Compute the pullback \( \varphi^*\omega \).
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Context:
A 24-ton mixture of crushed rock is needed in a construction job; it will cost $800. The mixture is composed of rock costing a certain amount per ton and $40 per ton. 8 tons of each should be added. How much does the cheaper rock cost per ton?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: \[ \lim_{x\to\infty}\frac{\gamma(x,-c\sqrt{x}+x)}{\Gamma(x,c\sqrt{x}+x)} \] where \( \gamma(a,b) \) and \( \Gamma(a,b) \) are the lower and upper incomplete gamma functions, respectively.
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Context:
Determine the values of \( n \) for which the improper integral converges.
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Context:
已知 $T=\{1,2, \cdots, 8\}$ 。对于 $A \subseteq T, A \neq \varnothing$ ，定义 $S(A)$ 为 $A$ 中所有元素之和。问： $T$ 有多少个非空子集 $A$ ，使得 $S(A)$ 是 3 的倍数，但不是 5 的倍数？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the disk of convergence for the power series \( \sum_{n=1}^\infty n!\,(z-i)^{n!} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the harmonic function $u(x, y)$ at the point $(1, 0)$ using Poisson's representation formula, given that the boundary condition is $u(x, y) = 2+3y^2$ on the circle $x^2 + y^2 = 9$. Provide your answer as a single number.
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Context:
In the figure, $AB = 200$ is a diameter of the circle. Points $A$ and $B$ are assigned the numbers $1$ and $1$. The two semi-circles are bisected at points $C$ and $D$, which are assigned the numbers $2$ and $2$. Each quarter circle is then bisected and assigned the number $3$. This process continues, with each arc being bisected and the new point being assigned the sum of the numbers at the endpoints of the arc. This process is repeated until the sum of all numbers on the circle is at least equal to the product of $4$ times the area of the circle and $rac{1}{3}$ of the radius. Find the least number of points on the circle. (Take $\pi$ to be $3.14$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Choose a random subset of $\{2^1,2^2,⋯,2^{10}\}$ by selecting each of the 10 elements independently with probability $\frac{1}{2}$. Find the expected value of the smallest element in the subset.
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Context:
In a convex quadrilateral \(ABCD\), the midpoints of the sides \(AB, BC, CD,\) and \(DA\) are marked as \(M, N, K,\) and \(L\) respectively. Given that \(|AC| = |BD| = 2a\) and \(|MK| + |NL| = 2b\), find the area of quadrilateral \(MNKL\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\rightarrow\infty}\Delta^x_{k=1}\sin \frac{k\pi}{2x}$$ Does this limit exist, and if so, what is its value?
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Context:
Determine the value of \( \alpha \) that makes the metric \( g = (1-x_1^2-x_2^2-\cdots-x_n^2)^{-\alpha}(dx_1\otimes dx_1+dx_2\otimes dx_2+\cdots+dx_n\otimes dx_n) \) geodesically complete on the set \( \{\vec{x}\in \mathbb{R}^n : x_1^2+x_2^2+\cdots+x_n^2<1\} \) in \( \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parallelogram ABCD, let K and L be the midpoints of sides BC and CD, respectively. Given that ∠BAD = π/3 and the points A, B, K, and L are concyclic, determine the measure of ∠ADB.
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Context:
Evaluate the limit \( \lim_{n\to\infty}\frac{\displaystyle\sum_{n=1}^{N}a_{n}}{\ln(n)} \), given that \( a_1 + a_2 + a_3 + \ldots + a_n = \int_{1}^{e}\ln^{k}(x)dx \) for \( n \ge 2 \) where \( a_n = \int_{0}^{1}x^{n}e^{x}dx \).
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Context:
Determine the order of the Galois group of the polynomial \( f(x) = x^2 + 2ix + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine which function is asymptotically greater as $n$ approaches infinity: $f(n) = 3^{2^n}$ or $g(n) = n! \times n^3$.
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Context:
Find the constant term in the antiderivative of the function \( f(x) = \log(\log x) + \log^{-2}x \) given that the antiderivative passes through the point \((e, 1998-e)\).
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Context:
Evaluate the integral of the 2-form \( x_2 \, dx_2 \wedge dx_3 + x_1 x_3 \, dx_1 \wedge dx_3 \) over the image set \( Y \) of the mapping \( \alpha: (0,1)^2 \to \mathbb{R}^3 \) given by \( \alpha(u,v) = (u,v,u^2+v^2+1) \).
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Context:
In a right-angled triangle $\triangle ABC$ with $\angle B = 90^\circ$ and $\angle C = 15^\circ$, given that $|AC| = 7$, a point $D$ is chosen on $AC$. Perpendicular lines $DE$ and $DF$ are drawn to $AB$ and $AC$, respectively. Calculate the probability that $DE \cdot DF > 3$. Provide your answer as a decimal rounded to two decimal places.
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Context:
Determine the values of \( p \) for which the series \( \sum_{n=N}^\infty \frac{1}{n(\log(n))(\log\log(n))\dotsm(\log_{j}(n))(\log_{j+1}(n))^p} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value \( E(\ln(X)) \) where \( X \sim \text{Beta}(1/y, 1) \).
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Context:
Suppose that a sequence \( {a}_{0},{a}_{1},\ldots \) of real numbers is defined by \( {a}_{0} = 1 \) and
 \( {a}_{n} = \left\{  \begin{array}{ll} {a}_{n - 1}{a}_{0} + {a}_{n - 3}{a}_{2} + \cdots  + {a}_{0}{a}_{n - 1} & \text{ if }n\text{ odd } \\  {a}_{n - 1}{a}_{1} + {a}_{n - 3}{a}_{3} + \cdots  + {a}_{1}{a}_{n - 1} & \text{ if }n\text{ even } \end{array}\right. \)
 for \( n \geq  1 \) . There is a positive real number \( r \) such that
 \( {a}_{0} + {a}_{1}r + {a}_{2}{r}^{2} + {a}_{3}{r}^{3} + \cdots  = \frac{5}{4}. \)
 If \( r \) can be written in the form \( \frac{a\sqrt{b} - c}{d} \) for positive integers \( a,b,c,d \) such that \( b \) is not divisible by the square of any prime and
 \( \gcd \left( {a,c,d}\right)  = 1 \) ,then compute \( a + b + c + d \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
An acute triangle $ABC$ is inscribed in a unit circle. Points $P$ and $Q$ are interior points on sides $AB$ and $AC$ respectively, such that $\frac{AP}{PB} = 2$ and $\frac{AQ}{QC} = 1$, and the points $P, O, Q$ are collinear. Determine the range of possible lengths for $AB$.
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Context:
In a group of 43 children, each child throws exactly one snowball at another child. The throwing pattern is as follows:

- The first child throws a snowball at the child who threw a snowball at the second child.
- The second child throws a snowball at the child who threw a snowball at the third child.
- The forty-third child throws a snowball at the child who threw a snowball at the first child.

Determine the number of the child who threw a snowball at the third child.
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Context:
已知一个圆柱形水杯的质量为 $a \mathrm{~g}$, 其重心在圆柱轴的中点处 (杯底厚度及质量均忽略不计，且水杯直立放置）。质量为 $b \mathrm{~g}$ 的水恰好装满水杯，装满水后的水杯的重心还在圆柱轴的中点处. 若 $b=3 a$, 求装入半杯水的水杯的重心到水杯底面的距离与水杯高的比值，化为最简分数的形式后求分子加分母的值是多少。
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the local value of a digit in the product of the face value of 7 and the local value of 6 in 7098060, if the local value of 6 in the product is 6000?
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Context:
Given the force field $\bar{F} (x, y, z) = (x − yz)\bar{i} + (y − xz)\bar{j} +(x(1−y) +z^2)\bar{k}$, calculate the work done in moving a particle along the closed path formed by the line segments connecting the points $(0, 0, 0), (1, 1, 1), (1, 1, 0), (0, 0, 1)$ in that order.
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Context:
What is the maximum possible dimension of a subspace $W$ in $\mathbb{R}^5$ that does not include the standard basis vectors $e_2$, $e_4$, and $e_5$? Given that $W$ includes $e_1$ and $e_3$, which are linearly independent.
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Context:
Let $P$ be a polyhedron where every face is a regular polygon, and every edge has length 1. Each vertex of $P$ is incident to two regular hexagons and one square. Choose a vertex $V$ of the polyhedron. Find the volume of the set of all points contained in $P$ that are closer to $V$ than to any other vertex.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \rightarrow \infty} \int_{\gamma} \frac{1}{(z+a)^2} \cot(\pi z) \, dz \), where \( \gamma \) is the path defined by \( \gamma=[n+\frac{1}{2}+ni,-n-\frac{1}{2}+ni,-n-\frac{1}{2} -ni, n+\frac{1}{2}-ni,n+\frac{1}{2}+ni] \).
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Context:
What is the minimum number of comparisons required to determine if a given number $x$ is present in a matrix $B$, where the matrix satisfies the condition $B_{ij} \le B_{kl}$ for all $i \ge k$ and $j \le l$?
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Context:
In square \(ABCD\), point \(M\) divides the diagonal \(AC\) in the ratio \(MC : AM = 1 : 4\). A line through \(M\) divides the square into two regions with areas in the ratio \(1 : 11\). Determine the ratio in which this line divides the perimeter of the square.
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Context:
Determine the minimum number of button presses required to try all possible 5-digit passwords on a 10-key keyboard without pressing enter.
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Context:
If the mass of the gun is 2 kg, the mass of the bullet is 20 gm, and the velocity of the bullet after firing is 200 m/s, then what is the velocity of the gun after firing?
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Context:
Reflect a square in sequence across each of its four sides. How many different transformations result from the sequence of four reflections?
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Context:
Let \( M \) be a real vector space of \( 2 \times 3 \) matrices with real entries. Define a linear transformation \( T: M \rightarrow M \) as follows:
\[ T\left( \begin{pmatrix} x_{1} & x_{2} & x_{3} \\ x_{4} & x_{5} & x_{6} \end{pmatrix} \right) = \begin{pmatrix} -x_{6} & x_{4} & x_{1} \\ x_{3} & x_{5} & x_{2} \end{pmatrix} \]
Determine the determinant of the linear transformation \( T \).
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Context:
Calculate the limit of the sum of binomial coefficients:
$$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
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Context:
Given the complex numbers \(a = \cos(q) + i\sin(q)\), \(b = \cos(r) + i\sin(r)\), and \(c = \cos(e) + i\sin(e)\), if \(\frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1\), find the value of \(\cos(r - e) \cos(e - q) + \cos(q - r)\).
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Context:
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could turn out to be rational numbers?
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Context:
There are eight seats at a round table. Six adults $A_1, \ldots, A_6$ and two children sit around the table. The two children are not allowed to six next to each other. All the seating configurations where the children are not seated next to each other are equally likely. What is the probability that the adults $A_1$ and $A_2$ end up sitting next to each other?
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Context:
Opal won $100.00 betting on a horse race. She put some portion of her winnings into savings and bet the other portion of her winnings. This time, she made a 60% profit and again, put half of her earnings into savings. She put $90 into her savings. What is the ratio of the amount she put into savings to the amount she bet after winning the horse race?
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Context:
In a vast oak tree, numerous species of birds are perched on its branches. There are twice as many cardinals as bluebirds, and the number of goldfinches is equal to the product of bluebirds and swallows. The number of sparrows is half the sum of cardinals and goldfinches, while the number of robins is 2 less than the quotient of bluebirds divided by swallows. If there are 12 swallows, which is half as many as the number of bluebirds, what is the total number of birds perched on the oak tree?
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Context:
Find the constant \(a\) such that the primitive function \(F(x)\) of \(f(x) = x^3 - 4a^2x\) has a minimum value of 0 and satisfies \(F(2) = 4\).
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Context:
Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.
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Context:
For how many sets of natural numbers \((x, y, z)\) with \(1 < x < y < z\) is \((x-1)(y-1)(z-1)\) a factor of \(xyz-1\)?
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Context:
Prince Accelerated invited Princess Slow on a journey to his castle. After waiting for a long time, he set out to meet her. After two days of traveling, he met her at one-fifth of her journey. Together, they continued traveling twice as fast as the princess traveled alone. They arrived at the prince's castle on the second Saturday after they met. On which day did they meet, given that the princess started her journey from her castle on a Friday?
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Context:
Suppose that starting at $t=0$, we introduce bulbs to light a room one by one, immediately replacing any failed bulb. Each new bulb is chosen randomly, with equal probability of being either type A or type B. The lifetime of a type-A bulb follows an exponential distribution with a rate of 1 ($f_{X_A}(x) = e^{-x}, x \geq 0$), while a type-B bulb has an exponential distribution with a rate of 3 ($f_{X_B}(x) = 3e^{-3x}, x \geq 0$). What is the probability that the combined lifetime of the first two type-B bulbs exceeds the lifetime of the first type-A bulb?
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Context:
Let \( V \) be a vector space of functions from \([0,1]\) to \( \mathbb{R} \). Determine the maximal possible dimension of a subspace \( U \subseteq V \) consisting of monotone functions.
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Context:
设双曲线 $C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$, 若 $C_{2}$ 的短轴长与 $C_{1}$ 的实轴长的比值等于 $C_{2}$ 的离心率，则求 $C_{1}$ 在 $C_{2}$ 的一条准线上截得线段的长。
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Context:
将正整数 1,2,\cdots,100 填入 10\times10 方格表中，每个小方格恰好填 1 个数，要求每行从左到右 10 个数依次递减，对 1\leq i\leq 10，记第 i 行的 10 个数之和为 S_i. 设 n\in\{1,2,\cdots,10\} 满足：存在一种填法，使得 S_1, S_2, \cdots, S_{10} 均大于第 n 列上的 10 个数之和，求 n 的最小值.
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Context:
A street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $15$ feet and each stripe is $50$ feet long. Find the distance, in feet, between the stripes.
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Context:
In a sports club with 150 members, 65 play badminton, 80 play tennis, 60 play basketball, 20 play both badminton and tennis, 15 play both badminton and basketball, 25 play both tennis and basketball, and 10 play all three sports. Furthermore, 12 members don't participate in any of these sports. How many members play only one of badminton, tennis or basketball?
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Context:
In a 3D cubic grid of size $8 \times 8 \times 8$, how many straight lines can be formed that contain exactly 8 symbols in a row?
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Context:
Determine the condition on the real numbers \(a\) and \(b\) such that the series \[ S = \frac{a}{b} + \frac{a(a+1)}{b(b+1)} + \frac{a(a+1)(a+2)}{b(b+1)(b+2)} + \ldots \] converges. The sequence of terms is given by \[ u_n = \frac{\prod_{i=0}^{i=n-1} (a+i)}{\prod_{j=0}^{j=n-1} (b+j)}. \]
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Context:
Find the analytical solution of the equation \( x(\ln(\ln(9)) - \ln(\ln(x))) = \ln(8) \).
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Context:
Compare the values of $a = 211!$ and $b = 106^{211}$. Determine which is greater.
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Context:
In an inscribed quadrilateral $ABCD$, given $|AB|=a$ and $|AD|=b$, where $a > b$. Find the side $|BC|$, given that $BC$, $CD$, and $AD$ are tangent to a circle whose center is on $AB$.
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Context:
Let \( {ABC} \) be a triangle with circumcenter \( O \) ,incenter \( I \) ,and circumcircle \( \Gamma \) . It is known that \( {AB} = 7,{BC} = 8,{CA} = 9 \) . Let
 \( M \) denote the midpoint of major arc \( \overset{⏜}{BAC} \) of \( \Gamma \) ,and let \( D \) denote the intersection of \( \Gamma \) with the circumcircle of \( \bigtriangleup {IMO} \) (other
 than \( M \) ). Let \( E \) denote the reflection of \( D \) over line \( {IO} \) . Find the integer closest to \( {1000} \cdot  \frac{BE}{CE} \) .
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Context:
Suppose lines $p$ and $q$ in the first quadrant are defined by the equations $y = -2x + 8$ and $y = -3x + 9$, respectively. What is the probability that a point randomly selected in the 1st quadrant and below $p$ will fall between $p$ and $q$? Express your answer as a decimal to the nearest hundredth.
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Context:
Determine the maximum number of squares a chess piece, which can move 8 or 9 squares horizontally or vertically without revisiting any square, can cover on a $15 \times 15$ board. The piece may start from any square.
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Context:
In an acute-angled, non-isosceles triangle $ABC$, the altitudes $AA'$ and $BB'$ intersect at point $H$, and the medians of triangle $AHB$ intersect at point $M$. The line $CM$ bisects segment $A'B'$ into two equal parts. Find the angle $C$.
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Context:
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1\times1$ squares of area $2010\times2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$ where $\frac{a}{b}$ is a reduced fraction, find $a+b$
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Context:
Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order?
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Context:
Jimmy Butler, a shooting guard, makes 30% of his three-point shot attempts. In any game, the number of three-point shots he attempts follows a Poisson distribution with a mean of 5. What is the probability that he makes exactly 2 three-point shots in a random game?
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Context:
Evaluate the integral \( \int_0^\infty \frac{1 - \cos(x\sqrt{e-1})}{xe^x} \, dx \).
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Context:
Let \( f:[0,\infty)\to\mathbb{R} \) be a continuously differentiable function such that \( \lim _{x\to \infty} f'(x)=\ell \) for a real number \( \ell \). Find \( \lim _{x \to \infty} \frac{f(6x)}{x} \).
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Context:
Find the minimal polynomial of $\beta^k$ over the field $L$, where $\beta$ is a root of the irreducible polynomial $x^p-b$ and $k \in \mathbb{N}$ divides $p$. Provide your answer in terms of $b$, $p$, and $k$.
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Context:
Evaluate the integral \( \int_{0}^{\pi/2}\cos(x)\ln(\tan(x))\,dx \).
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Context:
Given that the ABC stock pays no dividend and is currently priced at $S_0 = \$10$, with the stock price at expiry $T$ going up to $uS_0$ with probability $0 < p < 1$ and down to $dS_0$ with probability $1 - p$, where $d < 1 < u$. Assume no arbitrage and a zero interest rate. A European put option with a strike price of $\$9$ is priced at $\$14/9$, and a European put option with a strike price of $\$8$ is priced at $\$8/9$. What is the fair value of a European call option with a strike price of $\$7$?
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Context:
Calculate the limit \( \lim_{n\to\infty}P\left(\sum^n_{i=1}X_i\ge\sum^n_{i=1}Y_i\right) \), where \( X_1, X_2, \ldots \) are independent and identically distributed with a uniform distribution over \((0,1)\), and \( Y_1, Y_2, \ldots \) are independent and identically distributed with density \( e^{-x}\cdot\mathbb{I}\{x\ge0\}\).
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Context:
Approximate the integral \( \int_{0}^{1}\frac{\sin(x)}{x}dx \) with an error less than \( 10^{-5} \).
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Context:
Find the $x^6$ term in the numerator of the limit expression \(\lim_{x\to 0^+} \frac{-\tan^{-1}(2x^3)+\frac{(x+2)x^5}{x^2+3}+2x\sin^2(x)}{x e^{-1/x}-2x^2+\ln(\frac{x^2+1}{1-x^2})}\) by expanding \(\sin x\) up to order 3 and taking the square.
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Context:
Determine the range of values for \(c \in \mathbb{R}^+\) such that the Banach fixed-point theorem's condition is satisfied for the function \(g(x_1, x_2) = \left(\frac{1}{6}(x_1x_2 + cx_2 - 1), \frac{1}{6}(x_1^2 - x_2 + 1)\right)\) on the set \(M = [-1,1]\times [-1,1]\).
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Context:
Find a nonzero function $f(x)$ in the span of \( \{x^2, x^3, x^4\} \) that satisfies the conditions \( \langle f, x^2 \rangle = 0 \) and \( \langle f, x^3 \rangle = 0 \), where the inner product is defined as \( \langle f, g \rangle = \int_{-1}^1 f(x)g(x) \, dx \).
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Context:
In the diagram, $ABCDEFGH$ is a cube with an edge length of $12 \text{ cm}$. An ant sits on the cube at vertex $A$. The ant can only walk along the edges of the cube, and cannot walk along any edge more than once. What is the greatest distance that the ant can walk before it cannot continue? Express your answer in centimeters as a whole number.
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Context:
Find the positive integer that is coprime with every term of the sequence \( \{a_{n}\} \) where the general term is given by \( a_{n} = 2^{n} + 3^{n} + 6^{n} - 1 \) for \( n \in \mathbf{Z}_{+} \).
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Context:
In triangle $ABC$, the median $CM$ is perpendicular to the angle bisector $AL$, and their lengths are in the ratio $\sqrt{2} : 1$. Find $\cos A$.
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Context:
A scientist has a package of 10 pills for forgetfulness. Each time he takes a pill, he notes the remaining count. When only one pill is left, he orders a new package with instant delivery. If a package is empty, he discards it. What is the probability that at 10:00 AM on December 31, the scientist will have exactly two packages of pills in his cabinet?
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Context:
Given the function $f\left(x+ \frac {1}{2}\right)= \frac {2x^{4}+x^{2}\sin x+4}{x^{4}+2}$, calculate the value of $f\left( \frac {1}{2017}\right)+f\left( \frac {2}{2017}\right)+\ldots+f\left( \frac {2016}{2017}\right)$. Express your answer as a single integer.
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Context:
In a right triangle, the sine of the smaller angle is \(\frac{1}{3}\). A line is drawn perpendicular to the hypotenuse, dividing the triangle into two regions of equal area. Determine the ratio in which this line divides the hypotenuse.
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Context:
The extension of the altitude $BH$ of triangle $ABC$ intersects its circumcircle at point $D$ (points $B$ and $D$ are on opposite sides of line $AC$). The degree measures of arcs $AD$ and $CD$ that do not contain point $B$ are $120^{\circ}$ and $90^{\circ}$, respectively. Determine the ratio in which segment $BD$ is divided by side $AC$.
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Context:
Determine the number of integer solutions for the equation \( \sum_{i=1}^n i a_i = n \), where \( a_i \in \{0, \dotsc, n\} \).
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Context:
Let $\|\|_1$ and $\|\|_2$ be norms on the space $C[0,1]$. Define a linear operator $S$ from $(C[0,1], \|\|_2)$ to $(C[0,1], \|\|_1)$ by:

$$S(f)(t)=\frac{d}{dt}f(t)$$

Determine the norm of the operator $S$. \( \|S\| \)
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Context:
$\text{How many ordered integer pairs }(x,y)(0\leq x,y<31)\text{are there satisfying }(x^{2}-18)^{2}\equiv y^{2}({\mathrm{mod}}31)?$
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Context:
Express $7.\overline{123}$ as a common fraction in lowest terms.
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Context:
The polynomial $$P(x)=(1+x+x^2+\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\left[\cos(2\pi\alpha_k)
+i\sin(2\pi\alpha_k)\right]$, $k=1,2,3,\ldots,34$, with $0<\alpha_1\le\alpha_2\le\alpha_3\le\dots\le\alpha_{34}<1$ and $r_k>0$. Find $\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Find the inverse of the polynomial \(2x^2 + x + 1\) in the field \(\mathbb{Z}_3[x]/(x^3 + x^2 + x + 2)\).
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Context:
Provide an example of a function $f(x)$ such that the integral \( \int_{1}^{\infty} f(x^2) \, dx \) converges, but the integral \( \int_{1}^{\infty} f(x) \, dx \) diverges.
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Context:
During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got the same number of points. Determine the maximum number of students that could participate in the Olympics.
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Context:
Compute the remainder when $1^{2547} + 2^{2547} + \ldots + 2547^{2547}$ is divided by $25$.
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Context:
Consider two functions, \( f \) and \( g \), defined on a real variable such that \( g(x) = \int_0^x f(y)(y-x)dy \) for all \( x \). If \( g \) is three times continuously differentiable, find the smallest integer \( n \) such that \( f \) must be \( n \) times continuously differentiable.
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