Context:
Given a regular tetrahedron P-ABC, where points P, A, B, and C are all on the surface of a sphere with radius $\sqrt{3}$. If PA, PB, and PC are mutually perpendicular, then the distance from the center of the sphere to the plane ABC is ______.
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Context:
In a computer science laboratory, there are 15 workstations labeled $W_1$ through $W_{15}$ and 10 servers labeled $S_1$ through $S_{10}$. Each server can have only one direct connection with a workstation at a time. Determine the minimum number of direct connections required to ensure that any subset of up to 10 workstations can simultaneously access different servers via direct connections.
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Context:
40 persons like apple. 7 like orange and mango but dislike apple. 10 like mango and apple and dislike orange. Some people like all. There are 47 people who like apple. How many people like apple, orange, and mango?
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Context:
Let $T: V \rightarrow V$ be a linear transformation on a vector space $V$ with $\dim \operatorname{im} T = 2$. If $T^3 = bT$ for some scalar $b$, determine the possible values of $b$ for which $T + I$ is invertible.
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Context:
Determine the number of boolean functions of $n$ variables that can be expressed as a product of literals, excluding functions that always yield $0$. Provide your answer as a single integer.
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Context:
Evaluate the limit \( \lim_{x\to 0} \frac{\sqrt[3]{1+\frac{x}{3}} - \sqrt[4]{1+\frac{x}{4}}}{1-\sqrt{1-\frac{x}{2}}} \) without using L'Hospital's rule, power expansions, or derivatives.
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Context:
For $n=1,2, \ldots$ let
$$
S_{n}=-\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)+\log (\sqrt{n})
$$
where $\log$ denotes the natural logarithm. Find $\lim _{n \rightarrow \infty} S_{n}$.
 If the value can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$.
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Context:
Let $L, M, N \in \mathbb{P}^5$ be three general subspaces of codimension 3. Consider three general points $l_i$ on $L$, $m_i$ on $M$, and $n_i$ on $N$ for $i=1,2,3$. Determine the dimension of the space $I_{L \cup M \cup N}(3)$, which consists of all cubic polynomials through these subspaces with all first partial derivatives vanishing at these 9 points.
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Context:
Find the largest positive integer $N$ such that \( \sqrt{64 + 32^{403} + 4^{N+3}} \) is an integer.
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Context:
$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).
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Context:
Find all polynomials \( P \) with real coefficients such that \( P\left(X^{2}\right)=P(X) P(X-1) \).
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Context:
What is the expected value of the number of integers that will be printed from a stream of 0 to 9 (each with equal probability, and repetitions allowed) until we get the first instance of '4' followed by '5' consecutively?
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Context:
Find the smallest \( n \) such that whenever the elements of the set \(\{1, 2, \ldots, n\}\) are colored red or blue, there always exist \( x, y, z, w \) (not necessarily distinct) of the same color such that \( x + y + z = w \).
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Context:
Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that\[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\]Find $\tan^{2}\left(\measuredangle AEP\right)$.
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Context:
已知数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=1,{{a}_{2}}=3$，若$\left| {{an+1}}-{{a}_{n}} \right|={{2}^{n}}\left( n\in {{N*}} \right)$，且$\left\{ {{a2n-1}} \right\}$是递增数列、$\left\{ {{a}_{2n}} \right\}$是递减数列，则$\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a2n-1}}}{{{a}_{2n}}}=$_______.
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Context:
Given that tetrahedron P-ABC is a 'Bie'zhi' with PA⊥ plane ABC, PA=AB=2, and AC=4, and all four vertices of the tetrahedron P-ABC lie on the surface of sphere O, calculate the surface area of the sphere O.
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Context:
Solve the equation \(\ln(x) = e^{-x}\) for \(x\).
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Context:
Given the curve \( z = x^3 \), rotate this curve towards the y-axis about the z-axis to express it as \( z = f(x, y) \). Write \( z = f(x, y) = f(x^3, \theta) \). Determine the expression for \( z \) when \( \theta = 90^\circ \).
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Context:
Determine whether the functional differential equation \( D'(t) = 2 D(2t) - D(t) \) with the initial condition \( D(0) = 0 \) can be solved in a closed form.
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Context:
On an exam with $80$ problems, Roger solved $68$ of them. What percentage of the problems did he solve?
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Context:
Evaluate the integral \( \int_{C} (1 + z + z^2)(e^{\frac{1}{z}}+e^{\frac{1}{z-1}}+e^{\frac{1}{z-2}}) \, dz \), where \( C \) is the circle \(|z|=3\) in the complex plane.
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Context:
Given the matrices \(A=\begin{pmatrix} 4 & 1 & 1 \\ -1 & 1 & 0 \\ -2 & -1 & 0 \\ \end{pmatrix}\) and \(B=\begin{pmatrix} 0 & 1 & 0 \\ -3 & 4 & 0 \\ a & 2 & 1 \\ \end{pmatrix}\), determine the necessary condition on the scalar \(a\) for the existence of an invertible matrix \(P\) such that \(B=P^{-1}AP\).
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Context:
A line $l$ with a slope of $\frac{1}{3}$ passes through the left vertex $A$ of an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ and intersects the ellipse at another point $B$. If there exists a point $C$ on the $y$-axis such that $\triangle ABC$ is an isosceles right triangle with $C$ as the right-angled vertex, find the eccentricity of the ellipse.
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Context:
250. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. On the line $A A_{1}$, a point $M$ is taken, and on the line $B C$ - a point $N$ such that the line $M N$ intersects the edge $C_{1} D_{1}$. Find the minimum value of the magnitude | $M N \mid$.
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Context:
Evaluate the integral \( \int_0^{\pi / 2} \frac{\cos 2\theta}{1 + \sin \theta} \, d\theta \) using the substitution \( u = 1 + \sin \theta \).
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Context:
Find the limit \( \lim_{x \to \infty} f(x) \) for the function \( f(x) \) defined by \( f(x) = -\ln(f(x)) - x + 1 \).
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Context:
A rope of which a calf is tied is increased from 10 m to some length. The calf can graze an additional 1348.2857142857142 square meters of grassy ground. What is the new length of the rope?
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Context:
Let \( F_{1} \) and \( F_{2} \) be the two foci of an ellipse. A circle with center \( F_{2} \) is drawn, which passes through the center of the ellipse and intersects the ellipse at point \( M \). If the line \( ME_{1} \) is tangent to circle \( F_{2} \) at point \( M \), find the eccentricity \( e \) of the ellipse.
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Context:
Determine the sum of the infinite series $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$.
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Context:
Determine the limiting distribution of $Y_n = X_n/n$, where $X_n$ is a binomial random variable with parameters $n$ and $p$, as $n \to \infty$. Provide your answer in terms of a well-known distribution.
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Context:
Given a set $X$ and a finite subset $F$, determine the maximum number of ultrafilters on $X$ that can contain the subset $F$.
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Context:
Find the area of quadrilateral \(ABCD\) if \(AB = BC = 3\sqrt{3}\), \(AD = DC = \sqrt{13}\), and vertex \(D\) lies on a circle of radius 2 inscribed in the angle \(ABC\), where \(\angle ABC = 60^\circ\).
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Context:
Evaluate the integral \( \int_C \vec{F} \cdot d\vec{r} \) using Stokes' theorem, where \( \vec{F} = [-z, 5x, -y] \) and \( C \) is the ellipse defined by \( x^2 + y^2 = 4, z = x + 2 \). The direction of integration along \( C \) is clockwise as seen by a person standing at the origin.
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Context:
Let $P$ and $A$ be $3 \times 3$ real matrices such that $PAP^t = -A^t$, where $P^t$ denotes the transpose of $P$. Find $\det(P)$.
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Context:
In a scalene triangle $ABC$, one of the angles is equal to the difference of the other two angles, and one of the angles is twice as large as another angle. The angle bisectors of angles $A$, $B$, and $C$ intersect the circumcircle of the triangle at points $L$, $O$, and $M$ respectively. Find the area of triangle $LOM$ if the area of triangle $ABC$ is 8. If the answer is not an integer, round it to the nearest whole number.
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Context:
Calculate the volume of the solid bounded below by the triangle with vertices $(0,0)$, $(1,3)$, and $(3,3)$ in the $xy$-plane, and above by the surface $z = x^2y$. Provide your answer as a single numerical value.
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Context:
Two male students and three female students stand in a row. If male student A does not stand at either end, and exactly two of the three female students are adjacent, then the number of different arrangements is ______.
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Context:
Using Fourier series, we have the expression for \( x \) in the interval \( (-\pi, \pi) \):
\[ x = \sum_{n=1}^{\infty}\frac{\sin(nx)}{n}(-2\cos(n\pi)). \]
The task is to determine the limit of the following series by relating it to the given Fourier series:
\[ \lim_{N \to \infty}\sum_{n=1}^{N} \frac{\sin(n)}{n}. \]
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Context:
Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$.
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Context:
What is the total number of digits used when the first 2500 positive even integers are written?
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Context:
Rectangle $EFGH$ has area $4032.$ An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has its foci at points $F$ and $H$. Determine the perimeter of the rectangle.
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Context:
Given non-zero vectors $\overrightarrow{m}, \overrightarrow{n}$ with an angle of $\frac{\pi}{3}$ between them, and satisfying $|\overrightarrow{n}| = \lambda|\overrightarrow{m}|$ ($\lambda > 0$), consider two sets of vectors $\overrightarrow{x_{1}}, \overrightarrow{x_{2}}, \overrightarrow{x_{3}}$ composed of one $\overrightarrow{m}$ and two $\overrightarrow{n}$ vectors and $\overrightarrow{y_{1}}, \overrightarrow{y_{2}}, \overrightarrow{y_{3}}$ composed of two $\overrightarrow{m}$ and one $\overrightarrow{n}$ vectors. If the minimum possible value of $\overrightarrow{x_{1}} \cdot \overrightarrow{y_{1}} + \overrightarrow{x_{2}} \cdot \overrightarrow{y_{2}} + \overrightarrow{x_{3}} \cdot \overrightarrow{y_{3}}$ is $4|\overrightarrow{m}|^2$, find the value of $\lambda$.
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Context:
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times 30 \times 30$ cube. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
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Context:
Evaluate the improper integral: $$\int_{0}^{\infty}\frac{x^2}{e^x-1}\,dx$$
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Context:
Given two random variables $x$ and $y$, and a positive function $h(x)$, suppose the inequalities $-h(x) < f(x,y) < h(x)$ hold with probabilities $P[-h(x) < f(x,y)] = \beta$ and $P[f(x,y) < h(x)] = \beta$. Find the probability that both inequalities hold simultaneously.
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Context:
Two trains travel in opposite directions at 36 kmph and 45 kmph, and a man sitting in the slower train passes the faster train in some time. The length of the faster train is 135.0108 meters. How many seconds does it take for the man in the slower train to pass the faster train?
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Context:
Find the limit of the series \( \sum_{n=1}^\infty \frac{n}{n^2+k^2} \).
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Context:
If \(\Theta\) is a random variable uniformly distributed over \([-\pi, \pi]\), determine the probability density function \(f_\Theta(\theta)\) when \(\Theta\) is restricted to the interval \([-\pi/4, \pi/4]\).
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Context:
Flip $98$ fair coins and $1 \ HH$ coin and $1 \ TT$ coin. Given that you see an $H$, what is the probability that it was the $HH$ coin?
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Context:
In the inscribed quadrilateral $ABCD$, the diagonals intersect at point $P$, and triangle $APD$ is acute-angled. Let $E$ and $F$ be the midpoints of sides $AB$ and $CD$, respectively. A perpendicular is drawn from $E$ to $AC$, and another from $F$ to $BD$, intersecting at point $Q$. Determine the angle between lines $PQ$ and $BC$. Express your answer in degrees.
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Context:
Find a holomorphic bijection from the upper half-plane \( H = \{z \in \mathbb{C} \mid \text{Im}(z) > 0\} \) to the slit plane \( M = \mathbb{C} \setminus (-\infty,0] \).
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Context:
In $\triangle ABC$, $a=1$, $b=\sqrt{7}$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$. Find the value of $c$.
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Context:
A moving circle C is located inside the parabola $x^2=4y$ (where $x^2 \leq 4y$) and passes through the vertex of the parabola. Calculate the maximum value of the circumference of circle C. Express your answer as a multiple of pi (π).
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Context:
A group of people participate in various curriculums, with the following participation numbers: 50 practice Yoga (Y), 30 study Cooking (C), 20 study Weaving (W), 15 are involved in Pottery (P), and 10 participate in Dancing (D). The number of people participating in each pair of activities is as follows:

- 20 study both Yoga and Cooking (Y∩C)
- 13 study both Yoga and Weaving (Y∩W)
- 9 practice Yoga and Pottery (Y∩P)
- 7 are involved in Yoga and Dancing (Y∩D)
- 10 study both Cooking and Weaving (C∩W)
- 4 study Cooking and Pottery (C∩P)
- 5 practice Cooking and Dancing (C∩D)
- 3 participate in both Weaving and Pottery (W∩P)
- 2 study Weaving and Dancing (W∩D)
- 6 participate in Pottery and Dancing (P∩D)

Additionally, the following number of people are involved in each trio of activities:

- 9 study Yoga, Cooking, and Weaving (Y∩C∩W)
- 3 are involved in Yoga, Cooking, and Pottery (Y∩C∩P)
- 2 practice Yoga, Cooking, and Dancing (Y∩C∩D)
- 4 study Yoga, Weaving, and Pottery (Y∩W∩P)
- 1 practices Yoga, Weaving, and Dancing (Y∩W∩D)
- 2 participate in Cooking, Weaving, and Pottery (C∩W∩P)
- 1 studies Cooking, Weaving, and Dancing (C∩W∩D)
- 3 are involved in Cooking, Pottery, and Dancing (C∩P∩D)

Finally, 5 people participate in all five activities (Y∩C∩W∩P∩D).

How many people study both Cooking and Pottery, but not any of the other activities?
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Context:
求最小的正整数n(≥3)，使得平面内任意无三点共线的n个点中，必有3点是非等腰三角形的顶点．
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Context:
Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?
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Context:
Brandon sold some geckos, parrots, and tarantulas to a local pet shop. He sold 6 geckos for $100 each, 3 parrots for $200 each, and 10 tarantulas for $50 each. The pet store offers a bulk discount of 10% on the purchase of 5 or more animals of the same type. The pet store sells the geckos for $5 more than 3 times their purchase price, the parrots for $10 more than 2 times their purchase price, and the tarantulas for $15 more than 4 times their purchase price. What is the total profit percentage of the pet store from selling all of these animals?
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Context:
Find the derivative of the matrix expression $b^tAc$ with respect to the matrix $A$. What is the result in matrix notation?
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Context:
Solve the congruence equation: \(2357x \equiv 3^{527} \pmod{206}\).
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Context:
已知点 \(A\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) 关于 \(y=k x\) 的对称点 \(A^{\prime}\) 落在圆 \((x-2)^{2}+y^{2}=1\) 上，求 \(k\) 的值。答案的形式为\(\frac{\sqrt{a}}{b}\)，请给出a+b的值。
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Context:
Determine the minimum number of terms needed in the Taylor series expansion of \( f(x) = \ln(x) \) around \( x = e \) to ensure that the error is less than \( 0.05 \) when \( |x-e| < 1 \). Use the error term \( R_{n+1}(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-e)^{n+1} \) and assume \( x = e-1 \) for the worst-case scenario.
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Context:
There are \( n \) planets, and Luke starts his journey on planet \( A \). Each planet, except for planets \( B \) and \( C \), has two unidirectional roads leading to other planets. From at most one of these roads, Luke can find a sequence of roads that brings him back to the planet he just left. Luke's journey ends when he reaches planet \( B \) or \( C \). When leaving a planet, he chooses one of the two roads with equal probability. If the probability that he arrives at planet \( B \) is \( \frac{1}{2016} \), what is the minimum value of \( n \)?
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Context:
Given a winning probability of 50% with a return of 200% and a losing probability of 50% with a return of 50%, calculate the probability of doubling the initial money at some point using Kelly's criterion.
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Context:
Determine the number of distinct isomorphism classes of graphs with 3 labeled vertices and allowing loops.
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Context:
How many homomorphisms are there from $S_4$ to the symmetric group of $3$ symbols, $S_3$, up to conjugation?
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Context:
Determine the asymptotic distribution of the statistic \(8nB = - 8n\operatorname{ln}(\sqrt{0.2\hat{p}} + \sqrt{0.8(1-\hat{p})} )\), where \(\hat{p} = (1/n)\sum_i X_i = \bar{X}_n\) is the estimator of \(p\) from a random sample \(X_1, ..., X_n\) taken from a Bernoulli\((p)\) distribution. Assume \(H_0: p=0.2\) and \(H_1: p\neq 0.2\).
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Context:
Determine the radius of convergence, denoted as $R$, for the power series $$\sum_{n=1}^\infty \frac{\sin(n!)}{n!} x^n.$$
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Context:
Evaluate the limit \( \lim_{x \to 0}\frac{x}{\sqrt{1-e^{-x^2}}} \).
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Context:
Given the fractional equation $\frac{x}{x-2}-2=\frac{m}{x-2}$, find the value of $m$ when the equation has a root. Express your answer as a single integer.
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Context:
A certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases. If this DNA fragment is replicated twice, calculate the total number of free cytosine deoxyribonucleotide molecules required. Express your answer as a whole number.
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Context:
Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)$ in $\mathbb{R}^2$. Determine the probability that $W_t$ hits the positive x-axis before it hits the negative x-axis.
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Context:
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.

[i]Proposed by Huawei Zhu[/i]
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Context:
Given the bivariate normal distribution $\mathbf{X} = (X_1, X_2)' \sim N(\mu, \Lambda)$, where $\mu = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $\Lambda = \begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix}$, calculate the conditional probability $P(X_1 \geq 2 \mid X_2 + 3X_1 = 3)$. Provide your answer as a probability value.
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Context:
A builder has two identical bricks. She places them side by side in three different ways, resulting in shapes with surface areas of 72, 96, and 102. Calculate the surface area of one original brick. Express your answer as a single integer.
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Context:
How many unordered triples  $A,B,C$  of distinct lattice points in  $0\leq x,y\leq4$  have the property that  $2[ABC]$  is an integer divisible by  $5$ ?
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Context:
Determine the fundamental group of a sphere with two points removed.
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Context:
Determine the least possible value of $f(1998)$, where $f: \mathbb{N} \to \mathbb{N}$ is a function such that for all $m, n \in \mathbb{N}$,

$$f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}.$$
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Context:
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares \(\square_{1}, \square_{2}, \square_{3}\) on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set \(\{1,2,3,4,5\}\) such that no integer is used more than once. Ana's goal is to minimize the minimum value \(M\) that the polynomial \(a_{1} x^{2} + a_{2} x + a_{3}\) attains over all real \(x\), where \(a_{1}, a_{2}, a_{3}\) are the integers written in \(\square_{1}, \square_{2}, \square_{3}\) respectively. Banana aims to maximize \(M\). Assuming both play optimally, compute the final value of \(100 a_{1} + 10 a_{2} + a_{3}\).
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Context:
Let $f (x) = 2x -1$. Find the value of $x$ that minimizes $| f ( f ( f ( f ( f (x)))))-2020|$.
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Context:
Let $X$ be a random variable following a geometric distribution with parameter $p = \frac{2}{3}$, i.e., $X \sim \text{Geo}(\frac{2}{3})$. Define $Y = \left\lfloor \frac{X}{2} \right\rfloor + 1$. Find the expected value $\mathbb{E}[Y]$. \( \text{Express your answer as a single number.} \)
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Context:
Given a positive integer \( n \) and an integer \( k \) where \( 2 \leq k \leq 9 \), determine the \( n \)-th number that either contains the digit \( k \) or is divisible by \( k \). For example, if \( n = 15 \) and \( k = 3 \), the answer is \( 33 \).
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Context:
Calculate \( \int \int_X f(x, y) \, dx \, dy \) where \( f(x, y) = x \) and the region \( X \) is defined by the inequalities \( 2rx \leq x^2 + y^2 \leq R^2 \) with \( 0 < 2r < R \).
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Context:
Provide an example of an open set in the domain of the function \( f(x) = x^2 \) whose image under \( f \) is not open in the codomain.
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Context:
Determine the number of non-isomorphic graphs with 30 vertices and 432 edges.
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Context:
In Papa Carlo's room, there is a clock on each wall, and they all show incorrect times: the first clock is off by 2 minutes, the second by 3 minutes, the third by 4 minutes, and the fourth by 5 minutes. One day, Papa Carlo decided to find out the exact time before leaving the house, and he saw the following times on the clocks: 14:54, 14:57, 15:02, and 15:03. Help Papa Carlo determine the exact time.
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Context:
What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23$?
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Context:
Find a function \( f(x) \) such that the sequence \( f(n) = a_n \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.
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Context:
A matrix $H$ over $\Bbb F_2$ has as its first seven rows the identity matrix $I_7$, and the other rows are all vectors of $\{0,1\}^7$ with exactly three $1$'s per vector (weight-3 vectors). This matrix $H$ serves as a parity-check matrix for a linear code $C$. Determine the distance $d(C)$ of this linear code.
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Context:
Let \( b_n \) be the sequence of all positive integers such that the decimal expression for \( \frac{1}{b_n} \) terminates in an odd digit: 1, 2, 4, 8, 10, \ldots (For instance, 3 is not included because \( \frac{1}{3} = 0.33\ldots \) does not terminate, 4 is included because \( \frac{1}{4} = 0.25 \) terminates in 5, which is odd; 5 is not included because \( \frac{1}{5} = 0.2 \) terminates in 2, which is even.) Find the sum to infinity of \( \frac{1}{b_n} \).
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Context:
Let $f(x) \geq 0$ be a Riemann integrable function such that \( \int_0^1 f(x) \, \mathrm{d}x = 1 \) and \( \int_0^1 x f(x) \, \mathrm{d}x = \frac{1}{6} \). Find the minimum value of \( \int_0^1 f^2(x) \, \mathrm{d}x \).
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Context:
The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees?
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Context:
A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. Find the number of checkerboard squares which are completely covered by the disc.
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Context:
Let \(ABCD\) be a regular tetrahedron and let \(E\) be a point inside the face \(ABC\). Denote by \(s\) the sum of the distances from \(E\) to the faces \(DAB\), \(DBC\), \(DCA\), and by \(S\) the sum of the distances from \(E\) to the edges \(AB\), \(BC\), \(CA\). The original answer is in \(\frac{k\sqrt{m}}{n}\) format. Please give the value of \(k + m + n\).
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Context:
Evaluate the integral \( \iint_D \left(\sqrt{x^2+y^2}-\sqrt{a^2-x^2-y^2}~\right)dxdy \) where \(D_{xy}\) is the disk \(x^2+y^2\le a^2\), using a change of variables different from polar coordinates.
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Context:
At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
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Context:
Determine the set of values of $m\in\mathbb{C}$ such that the equation $(1+iz)^n=m(1-iz)^n$ has only real roots, where $n\in\mathbb{N}$ and $z$ is a complex variable.
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Context:
Find the square root of the real number 4. Express your answer using the ± symbol to denote both positive and negative roots.
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Context:
Evaluate the limit: $$\lim_{x \to 0} \left(\ln\left(\frac{\cos x}{1+\sin x}\right)\right)^{\frac{1}{x}}$$
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Context:
Given the function $f\left(x\right)=2\sin x+\sin 2x$, the minimum value of $f\left(x\right)$ is ______.
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Context:
Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all $a + p$ for the solutions.
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Context:
James is standing at the point $(0,1)$ on the coordinate plane and wants to eat a hamburger. For each integer $n \geq 0$, the point $(n, 0)$ has a hamburger with $n$ patties. There is also a wall at $y=2.1$ which James cannot cross. In each move, James can go either up, right, or down 1 unit as long as he does not cross the wall or visit a point he has already visited. Every second, James chooses a valid move uniformly at random, until he reaches a point with a hamburger. Then he eats the hamburger and stops moving. Find the expected number of patties that James eats on his burger.
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Context:
A diagonal of a regular 2006-gon \( P \) is called good if its endpoints divide the boundary of \( P \) into two parts, each containing an odd number of sides. The sides of \( P \) are also called good. Let \( P \) be divided into triangles by 2003 diagonals, none of which have common interior points. What is the greatest number of isosceles triangles, each of which has two good sides, that such a division can have?
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Context:
Let \( S \) be the set of points \(\left(x_{1}, x_{2}, \ldots, x_{2012}\right)\) in 2012-dimensional space such that \(\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{2012}\right| \leq 1\). Let \( T \) be the set of points in 2012-dimensional space such that \(\max _{i=1}^{2012}\left|x_{i}\right|=2\). Let \( p \) be a randomly chosen point on \( T \). What is the probability that the closest point in \( S \) to \( p \) is a vertex of \( S \)?
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Context:
The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \( O \) of one rectangle and the midpoint \( M \) of one edge of the other. What is the distance \( OM \)? Express your answer as a whole number.
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Context:
Evaluate the integral \( \int_0^n \frac{(1-\frac{x}{n})^n}{ne^{-x}}dx \) as \( n \to \infty \).
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Context:
Determine the Krull dimension of the trivial ring.
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Context:
Given the probability density function \( f(x, \theta) = \frac{\theta}{x^{2}} \) for \( 0 < \theta \le x < \infty \), and an i.i.d. sample \( (X_i)_{i=1}^{n} \), find the sufficient statistic for \( \theta \).
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Context:
Use Taylor's inequality to estimate the accuracy of the approximation \( f(x) \approx T_3(x) \) for \( f(x) = e^{-x^2} \cos(x) \) at \( a = 0 \), where \( T_3(x) = 1 - \frac{3x^2}{2} \). Estimate the error for \( |x| \leq 1 \).
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Context:
In triangle \(ABC\), where \(AB = AC\), the distances from vertex \(C\) to the trisection points of side \(AB\) are 17 cm and 20 cm, respectively. Determine the distances from vertex \(A\) to the trisection points of side \(BC\).
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Context:
Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?
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Context:
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all  integers $n \ge 1$ are defined:
$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
$O_n$ is the center of $\odot (PA_nB_n)$
$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$

If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$

[hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]
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Context:
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\frac{2}{3}$ probability of voting for Alice and a $\frac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?
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Context:
The radius of the circle circumscribed around a triangle is equal to half of one of the sides of the triangle. What can be said about the angle opposite this side?
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Context:
Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.
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Context:
Given $f\left(x\right)=ax\cdot e^{x}(a \gt 0)$, if the range of $y=f\left(f\left(x\right)\right)$ is the same as the range of $y=f\left(x\right)$, then the range of the real number $a$ is ______.
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Context:
For which natural numbers $n$ does the inequality $2^{n} \geq n^{3}$ hold?
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Context:
Lolita can drink 3 boxes of milk during weekdays. During Saturdays, she can drink twice the number of boxes of milk than on the weekdays, and during Sundays, she can drink some boxes of milk. She drinks 30 boxes of milk per week. What is the ratio of the number of boxes of milk she drinks on Sundays to the number of boxes she drinks on weekdays?
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Context:
Calculate: ${(\frac{81}{16})}^{-\frac{1}{4}}+\frac{1}{4}\log_{\sqrt{2}}3\times\log_{3}4\times{[(-\frac{1}{3})}^{2]}^{\frac{1}{2}}+{7}^{\log_{7}\frac{1}{2}}$.
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Context:
Let $\Omega = D(0,2)/\{\frac{1}{2}\}$, where $D(0,2)$ is a disc, $f$ holomorphic in $\Omega$. $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate $$ lim_{n \to \infty} \frac{f^{n}(0)}{2^{n}n!}$$
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Context:
At a football game, the total initial attendance was 1300 people comprising boys, girls, and adults. At the start of the game, there were 350 boys, 450 girls, and the rest were adults. After the first quarter, one-fourth of the boys, one-eighth of the girls, and one-fifth of the adults left the stadium. During halftime, 5% of the remaining boys, 7% of the remaining girls, and 50 more adults entered the stadium. At the beginning of the third quarter, 3% of the remaining boys and 4% of the remaining girls left the stadium, and 2% of the remaining adults entered the stadium. Lastly, 15 boys and 25 girls entered the stadium before the start of the fourth quarter and 1% of the remaining adults left the stadium. Find the total number of spectators in the stadium at the end of these changes.
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Context:
Consider the ring $R=\mathbb{Z}_p[x]/(x^p-x)$. Find the number of elements in $R$ that satisfy $r^3=r$.
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Context:
Tom has a list of nine integers: 1, 2, 3, 4, 5, 6, 7, 8, and 9. He creates a second list by adding 2 to some of the integers in the first list and by adding 5 to all of the other integers in the first list. What is the smallest number of different integers he can obtain in the second list? Express your answer as a single integer.
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Context:
Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1.
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Context:
What is the difference between a number and the local value of 3 in 28943712, if the local value of 7 in the difference is 7000?
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Context:
Given that the area of the parallelogram $ABCD$ is $240$, $E$ and $H$ are the midpoints of sides $AD$ and $AB$ respectively. Point $G$ is on side $BC$ such that $BG = 2GC$, and point $F$ is on side $CD$ such that $DF = 3FC$. Point $K$ is on side $AC$ such that the area of $\triangle EKF$ is 33. Find the area of $\triangle HKG$.
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Context:
On a map, the distance between two mountains is 312 inches. The actual distance between the mountains is 136 km. Ram is camped at a location that on the map is 34 inches from the base of the mountain. How many km is he from the base of the mountain?
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Context:
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, and it is given that $S_n=(-1)^n a_n-\frac{1}{2^n}$. Find the value of $S_1+S_3+S_5+\cdots +S_{2017}=\_\_\_\_\_\_\_\_\_$.
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Context:
A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent cuts, forming a tray with taped edges. Determine the height of this tray, which is the perpendicular distance from the base of the tray to the plane formed by the upper edges.
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Context:
Determine the minimum number of shifts required for a group of 6 people on a ship, such that for any two distinct persons $A$ and $B$, there exists a shift where $A$ is awake while $B$ is asleep.
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Context:
Given \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in \mathbf{N} \) such that \( a_{1} + a_{2} + a_{3} = b_{1} + b_{2} + b_{3} \),
\[
a_{1}a_{2}a_{3} = b_{1}b_{2}b_{3}
\]
\[
a_{1}a_{2} + a_{1}a_{3} + a_{2}a_{3} \neq b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3}
\]

Then find the minimal value of the largest number among \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \).
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Context:
In how many different ways can the digits $0,1,2,3,4,5,6$ form a seven-digit number that is divisible by 4? (The number cannot start with 0.)
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Context:
The coefficients of the polynomial \(P(x)\) are nonnegative integers, each less than 100. Given that \(P(10)=331633\) and \(P(-10)=273373\), compute \(P(1)\).
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Context:
Calculate the following integral:

$$I_p = \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr$$

where $p$ is a real number. Determine the values of $p$ for which the integral converges.
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Context:
A circle \( O \) with radius \( R \) is given. There is a line \( l \) outside the circle. Let the projection of \( O \) on \( l \) be \( H \). The distance \( OH = d \). \( OH \) intersects the circle \( O \) at points \( C \) and \( D \) with \( CH > DH \). Let \( P \) and \( Q \) be points on \( l \), on the same side of \( H \), such that \( PH = a \) and \( QH = b \) with \( a > b \). There are 2008 chords \( A_i B_i \) \((i=1,2,\ldots, 2008)\) in the circle parallel to \( l \), and these 2008 chords divide \( CD \) into equal segments. Find the value of \(\frac{1}{2008} \sum_{i=1}^{2008} (PA_i^2 + PB_i^2 + QA_i^2 + QB_i^2)\) in terms of \( a, b, d, \) and \( R \).
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Context:
What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?
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Context:
$\mathrm{Find}\lim\limits_{a\to\infty}\frac{1}{a^2}\int_0^a\log(1+e^x)dx.$
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Context:
In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). If the area of triangle \( BQC \) is 1, find the area of triangle \( ABC \).
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Context:
Evaluate

\[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]
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Context:
Given the function \( f(x) = \frac{1+10x}{10-100x} \), and the composition \( f^n = \underbrace{f\circ f\circ f \circ\cdots \circ f}_{n} \), find the sum \( f(1/2) + f^2(1/2) + f^3(1/2) + \cdots + f^{6000}(1/2) \).
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Context:
Calculate the expected exit time of a random walk on the interval $(-a, b)$ starting at position $x$ using a recurrence relation.
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Context:
集合$\{ x \vert - 1 \leq \log _ { \frac { 1 } { x } } 1 0 < - \frac { 1 } { 2 } , x \in N _ { + } \}$的真子集个数是 .
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Context:
Compute the value of the infinite series \[
\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n \cdot \left(n^4+4\right)}
\]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:
Determine the rate at which \( \mathrm{E} \min \left(\sigma_{2n}, 2n\right) \) approaches infinity as \( n \to \infty \), where \( \sigma_{2n} = \min \{ 1 \leq k \leq 2n : S_k = 0 \} \) and \( \sigma_{2n} = 2n \) if \( S_k \neq 0 \) for all \( 1 \leq k \leq 2n \).
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Context:
In the arithmetic sequence $\{a_{n}\}$, if $\frac{{a}_{9}}{{a}_{8}}＜-1$, and its sum of the first $n$ terms $S_{n}$ has a minimum value, determine the minimum value of $n$ for which $S_{n} \gt 0$.
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Context:
Find the smallest integer $n \geq 5$ for which there exists a set of $n$ distinct pairs $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ of positive integers with $1 \leq x_{i}, y_{i} \leq 4$ for $i=1,2, \ldots, n$, such that for any indices $r, s \in\{1,2, \ldots, n\}$ (not necessarily distinct), there exists an index $t \in\{1,2, \ldots, n\}$ such that 4 divides $x_{r}+x_{s}-x_{t}$ and $y_{r}+y_{s}-y_{t}$.
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Context:
Find the sum of $453_6$, $436_6$, and $42_6$ in base 6.
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Context:
Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center?
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Context:
Sally, Mark, Sarah, Steve, Jim, Laura, and Matthew are inheriting the money from their grandmother. She left them $874,532.13 to split equally between all of them. If the money is split equally, how much money will each person receive?
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Context:
In the acute triangle $ABC$, the distance between the feet of the altitudes to sides $AB$ and $BC$ is $24$. Given that $\angle B = 37^\circ$, calculate the circumradius of triangle $ABC$. \( \boxed{?} \)
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Context:
Determine the maximum order of persistence of excitation for the periodic input signal $u(t)$ given by:
\[
u(t) = \begin{cases} 0, & t = 0,3 \\ 1, & t = 1,4 \\ -1, & t = 2,5 \end{cases}\]
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Context:
Let \( n \) be a positive integer. Calculate
\[
P = \prod_{k=0}^{n-1} \sin \left(\theta + \frac{k \pi}{n} \right).
\]
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Context:
How many ways are there to arrange the numbers $1$ through $8$ into a $2 \times 4$ grid such that:
1. The sum of the numbers in each of the two rows is a multiple of $6$.
2. The sum of the numbers in each of the four columns is a multiple of $3$?
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Context:
Determine the smallest positive integer $n$ for which $\sin\left(\frac{\pi}{n}\right)$ cannot be expressed in terms of radicals of integers.
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Context:
Given that the function \( f(x) = \sin(2x) - 8(a+1)\sin x + (4a^2 + 8a - 14)x \) is strictly increasing for all \( x \in \mathbb{R} \) and has no critical points for all \( x \in \mathbb{R} \), and the set of all values of the parameter \( a \) is \((-
fty, m - \sqrt{n}) \cup (\sqrt{n}, \infty)\), where \( m \) and \( n \) are prime numbers, find the value of \( m^2 + n^2 \).
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Context:
Henry scored 50 points on his Geography test, 70 on his Math test, 66 on his English test, 84 on his Science test, and 75 on his French test. If his History score is a weighted average of these 5 scores, where Geography has a weight of 25%, Math and English have a weight of 20% each, Science has a weight of 15%, and French has a weight of 10%, what was his total score across all 6 subjects?
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Context:
For any positive integer $n$, we define the integer $P(n)$ by :

$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.

Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.
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Context:
Let $\min (x, y)$ denote the smaller of the numbers $x$ and $y$. For what $\lambda$'s does the equation
$$\int_{0}^{1} \min (x, y) f(y) d y=\lambda f(x)$$
have continuous solutions which do not vanish identically in ( 0,1$)$? The largest $\lambda$ can be expressed in the form of $\frac{m}{n\pi^{2}}$. Find the value of $m + n$.
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Context:
Let \((X, d)\) be a metric space and \(Y \subseteq X\). Suppose there exists an \(x \in X\) and \(r > 0\) such that \(Y \subseteq B(x, r)\). Determine the smallest \(r''\) such that \(conv(Y) \subseteq B(x, r'')\), where \(conv(Y)\) denotes the convex hull of \(Y\).
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Context:
Let \(ABCD\) be an isosceles trapezoid with parallel bases \(\overline{AB}\) and \(\overline{CD}\), where \(AB > CD\). Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles with areas \(2\), \(3\), \(4\), and \(5\), starting with the triangle with base \(\overline{CD}\) and moving clockwise. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k + \sqrt{m}\). Please calculate the value of \(k + m\).
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Context:
Let \( p(x) \) be a polynomial of degree strictly less than 100. If \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^2+1} =\frac{f(x)}{g(x)} \) for some polynomials \( f(x) \) and \( g(x) \), what is the smallest possible degree of \( f(x) \) when \( p(x) \) does not have \( x^2+1 \) as a factor?
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Context:
Find the greatest number which leaves the same remainder when it divides 25, 57, and a number. The greatest number is 16. What is the third number?
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Context:
Evaluate the following limit without using L'Hopital's rule:
$$\lim_{x\to\infty}\left[\frac{2}{\pi}\left((x+1)\cos^{-1}\left(\frac{1}{x}\right)\right)-x\right]$$
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Context:
Among the eight vertices of the cube ABCD-A1B1C1D1, a plane passes through four of them, and the remaining four vertices are all at equal distances from the plane. Calculate the number of such planes.
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Context:
The function $y=(m^2-m-1)x^{m^2-2m-1}$ is a power function, and it is a decreasing function on $x\in(0, +\infty)$. Then, the real number $m=$ ______.
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Context:
If the inequality \(x^{2}-\log _{m} x<0\) always holds in the interval \(\left(0, \frac{1}{2}\right)\), find the range of values for the real number \(m\).
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Context:
A 12% stock yielding 10% is quoted at, with a 3% tax rate on dividend income and 2% inflation rate. Calculate the adjusted yield taking into account tax and inflation implications.
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Context:
Find the inclination angle of the line $\sqrt {2}x+ \sqrt {6}y+1=0$.
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Context:
Find the number of solutions to
\[\sec \theta + \csc \theta = \sqrt{15}\]where $0 \le \theta \le 2 \pi.$
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Context:
In the 19th century, Danish mathematician Kierkegaard made outstanding contributions to mathematical analysis, especially in the concavity and convexity of functions and inequalities. It is defined that the derivative of a function $f(x)$ on the interval $(a, b)$ is denoted as $f'(x)$, and the derivative of $f'(x)$ on the interval $(a, b)$ is denoted as $f''(x)$. If $f''(x) < 0$ holds for all $x$ in $(a, b)$, then the function $f(x)$ on $(a, b)$ is called a "strictly convex function." If the function $f(x) = e^x - x\ln x - \frac{m}{2}x^2$ is a "strictly convex function" on $(1, 4)$, then the range of $m$ is ____.
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Context:
Calculate the expected difference between the second and third highest numbers when selecting 5 numbers (with repetition allowed) from the set \( \{1, 2, \ldots, 100\} \) and arranging them in descending order.
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Context:
In the class, there are 30 students: excellent students, C-grade students, and D-grade students. Excellent students always answer questions correctly, D-grade students always make mistakes, and C-grade students answer the questions given to them strictly in turn, alternating between correct and incorrect answers. All students were asked three questions: "Are you an excellent student?", "Are you a C-grade student?", and "Are you a D-grade student?". 19 students answered "Yes" to the first question, 12 to the second, and 9 to the third. How many C-grade students are there in this class?
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Context:
Given a right prism $ABC-A'B'C'$ with volume $V$, points $P$ and $Q$ are on edges $AA'$ and $OC'$ respectively, and $AP = C'Q$. Calculate the volume of the tetrahedron $B-APQC$ in terms of the original volume $V$. Express your answer as a fraction of $V$.
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Context:
Steve writes a letter every 3rd day to his friend. He spends 20 minutes writing the letters. It takes 10 minutes to write 1 page. He also writes a long letter at the end of the month which he puts more thought into. It takes a certain ratio of time per page compared to regular letters, but he spends 80 minutes writing. He writes 24 pages a month. What is the ratio of time spent per page for the long letter compared to a regular letter?
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Context:
Find the angle between the tangents to the curve \(3x^2+4xy+5y^2-4=0\) at the points where the lines \(y=(-3/2)x\) and \(y=(-2/5)x\) intersect the curve.
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Context:
From a sphere of what is the smallest radius can a regular quadrilateral pyramid with a base edge of 14 and an apothem of 12 be cut?
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Context:
Evaluate the limit: $$ \lim_{x\to\infty}\frac{\ln(1+e^{ax})}{\ln(1+e^{bx})} $$ as \( x \) approaches infinity.
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Context:
Solve the equation \(2^x = -2x + 11\) for \(x\).
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Context:
Let $N$ denote the number of permutations of the $18$-character string $AAAAABBBBBBCCCCCC$ such that:
1. None of the first five letters is an $A$.
2. None of the next six letters is a $B$.
3. None of the last seven letters is a $C$.
Find the remainder when $N$ is divided by $500$.
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Context:
Let \( T \) be the set of ordered triples \((x, y, z)\), where \( x, y, z \) are integers with \( 0 \leq x, y, z \leq 9 \). Players \( A \) and \( B \) play the following guessing game: Player \( A \) chooses a triple \((x, y, z)\) in \( T \), and Player \( B \) has to discover \( A \)'s triple in as few moves as possible. A move consists of the following: \( B \) gives \( A \) a triple \((a, b, c)\) in \( T \), and \( A \) replies by giving \( B \) the number \( |x+y-a+b|+|y+z-b+c|+|z+x-c+a| \). Find the minimum number of moves that \( B \) needs to be sure of determining \( A \)'s triple.
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Context:
在梯形 $A B C D$ 中, $A D / / B C, M$ 在边 $C D$ 上, 有 $\angle A B M=\angle C B D=\angle B C D$, 则 $\frac{A M}{B M}$ 的取值范围是一个区间，该区间的有限端点形如\frac{m}{n}，请给出m+n的值
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Context:
Let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with the following probability mass function (pmf): \( P(X = k_1) = \frac{1 - \theta}{2} \), \( P(X = k_2) = \frac{1}{2} \), and \( P(X = k_3) = \frac{\theta}{2} \), where \( 0 < \theta < 1 \). Find a non-trivial sufficient statistic for \( \theta \).
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Context:
Given that the random vector $(X, Y, Z)$ follows a multivariate normal distribution with expected values $E[X] = 0$, $E[Y] = 1$, $E[Z] = 1$, and covariance matrix:
\[
\begin{bmatrix}
1 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{bmatrix}
\]
calculate the variance $\operatorname{Var}(X(Y+Z))$. Express your answer as a single numerical value.
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Context:
Let $g: W_m \to W_m$ be an endomorphism such that $g^2 = 0$ and the rank of $g$ is $k$. Calculate the dimension of the intersection of the kernel and image of $g$. Express your answer as a single integer.
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Context:
Find the area enclosed by the ellipse given by the equation \(5x^2 + 6xy + 2y^2 + 7x + 6y + 6 = 0\).
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Context:
Calculate the expected value of the random variable \( S = \sum_{i=1}^{N} \frac{X_i}{2^i} \), where \( X_1, X_2, \ldots \) are i.i.d. random variables uniformly distributed on \([0,1]\), and \( N \) is the smallest positive integer \( k \) such that \( X_k < X_{k+1} \). If no such \( k \) exists, then \( N = \infty \). Given that \( \Pr(N = n) = \frac{n}{(n + 1)!} \), find \( \mathbb{E}[S] \).
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Context:
Use the power series expansion of \( \frac{z}{e^z-1} \) to find the value of \( B_5 \).
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Context:
已知函数 $y=\left(a \cos ^{2} x-3\right) \sin x$ 的最小值为 -3 ，则实数 $a$ 的取值范围是。请提供该区间的两个端点之和的整数部分。
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Context:
146. Find the particular solution of the equation $y^{\prime \prime}-2 y^{\prime}+y=0$, if $y(0)=4, y^{\prime}(0)=2$.
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Context:
把边长为$1$的正方形$ABCD$沿对角线$AC$折起，当以$A,B,C,D$四点为顶点的三棱锥体积最大时，此三棱锥的外接球的表面积的大小等于__________．
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Context:
What is the maximum value that the area of the projection of a regular tetrahedron with an edge length of 1 can take?
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Context:
An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
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Context:
In a classroom, there are ten students. Aline writes ten consecutive integers on the board. Each student chooses one of the ten integers written on the board, ensuring that no two students choose the same integer. Each student then calculates the sum of the nine integers selected by the other nine students. Each student whose result is a perfect square receives a gift.

What is the maximum number of students who will receive a gift?

A perfect square is an integer of the form \( n^{2} \), where \( n \) is a natural number.
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Context:
Let $T=[-1,0)\cup [1,2]$ and $g:T\to R$ be a strictly decreasing function such that $g(T)$ is connected. How many discontinuities does $g$ have?
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Context:
Determine the number of lines that are both normal and tangent to the curve \(y^2 = x^3\), intersecting the curve at two distinct points.
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Context:
The equation $ax^3 + bx^2 + cx + d = 0$ has three distinct solutions. How many distinct solutions does the following equation have:

$4(ax^3 + bx^2 + cx + d)(3ax + b) = (3ax^2 + 2bx + c)^2$?
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Context:
Compute the area of the region on the cylinder $x^2+y^2=R^2$ bounded by the planes $z=mx$ and $z=nx$, where $z \ge 0$, $m > n > 0$. Express your answer in terms of $R$, $m$, and $n$.
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Context:
在平面直角坐标系中，点 A,B,C 在双曲线 xy=1 上，满足 \triangle ABC 为等腰直角三角形. 求 \triangle ABC 的面积的最小值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimeter of $\triangle APR$.
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Context:
Maria gave a third of her money to her friend Isha, half of what she gave to Florence. Florence received three times as much money as Maria's cousin Rene. Maria gave her three friends $1650. How much money did Rene receive?
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Context:
The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is:
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Context:
A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \(1 \mathrm{~cm}\) to a cylinder of radius \(3 \mathrm{~cm}\). When the bottle is right side up, the height of the water inside is \(20 \mathrm{~cm}\). When the bottle is upside down, the height of the liquid is \(28 \mathrm{~cm}\). What is the total height, in cm, of the bottle?
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Context:
Compute the integral \( \iint_D (3x^3 + 3x^2y + x + y) \, dA \) over the region \( D \) bounded by the curves \( y = x^3 - 4 \), \( y = x^3 + 4 \), \( x + y = 6 \), and \( x + y = 0 \), using the change of variables \( u = x^3 \) and \( v = x + y \).
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Context:
The graph of the function $y=\sin 2x - \sqrt{3} \cos 2x$ is translated to the left by $\frac{\pi}{6}$ units, resulting in the graph of the function $y=g(x)$. Find the value of $g(\frac{5\pi}{6})$.
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Context:
Determine the smallest integer $N$ such that among the $\binom{N}{3}$ circles formed by every triple of $N$ points in general position in 3-D space, there exists at least one pair of linked circles. Note that circles sharing one of the $N$ points do not count as linked.
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Context:
Given that \(T_1\) is a lower \(\alpha \times 100\%\) confidence limit and \(T_2\) is an upper \(\alpha \times 100\%\) confidence limit for the parameter \(\theta\), such that \(P(T_1 < T_2) = 1\), determine the \((2\alpha - 1) \times 100\%\) confidence interval for \(\theta\). Assume \(T_1\) and \(T_2\) are not independent.
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Context:
由曲线 $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ 围成的图形绕 $y$ 轴旋转一周所得旋转体的体积为 $V_{1}$ ；满足 $x^{2}+y^{2} \leqslant 16, x^{2}+(y-2)^{2} \geqslant 4, x^{2}+(y+2)^{2} \geqslant 4$ 的点组成的图形绕 $y$ 轴旋转一周所得旋转体的体积为 $V_{2}$ ，求 $V_{1}$ 和 $V_{2}$ 的比值。
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Context:
For each integer $m\ge 2$, find the value of $f(2^m)$.
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Context:
Bob chooses a $4$-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob's string is $0000$ and the stream of bits begins $101000001\ldots$, then $N = 7$.
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Context:
Evaluate the contour integral \( \int_\gamma \frac{z}{z^2-i} \, dz \), where \( \gamma \) is a counterclockwise unit square in the complex plane.
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Context:
Find the values of \( \lambda \) where the functions \( f(\lambda) = e^{\frac{\ln(\frac{-\lambda\theta^k}{3\lambda-2T})}{k}} \) and \( g(\lambda) = \frac{\ln(\frac{-\lambda}{3\lambda-2T})}{k}+\theta \) intersect, given that \( \theta, k, \) and \( T \) are positive real numbers.
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Context:
Determine the minimum number of spheres required to completely shield a point source of light from all directions.
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Context:
How can we express the sum $\sum_{n=1}^\infty \frac{\Lambda(n)}{n^3 \ln^2 n}$ in terms of the prime zeta function?
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Context:
Given that $a$, $b$, and $c$ are the roots of the cubic equation $x^3 - 4x + 1 = 0$, find the value of the expression \[ \frac{a^2bc}{a^3+1} + \frac{ab^2c}{b^3+1} + \frac{abc^2}{c^3+1}. \]
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Context:
What is the minimum number of vertices required in a bipartite graph with a minimum degree of $d$ to ensure the existence of a cycle of length at least $2d+1$?
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Context:
Find the third derivative \( y'''(x) \) of the function implicitly defined by the equation \( x^2 + xy + y^3 = 1 \) at the point where \( x = 1 \).
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Context:
The angles \( \alpha ,\beta ,\gamma \) of a triangle are in arithmetic progression. If \( \sin {20\alpha },\sin {20\beta } \) , and \( \sin {20\gamma } \) are in arithmetic progression,
 how many different values can \( \alpha \) take?
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Context:
In the trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the diagonals \(AC\) and \(BD\) intersect at point \(E\). A circle is circumscribed around triangle \(ECB\), and the tangent to this circle at point \(E\) intersects line \(AD\) at point \(F\) such that points \(A, D,\) and \(F\) lie consecutively on this line. Given that \(AF = a\) and \(AD = b\), find \(EF\).
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Context:
Determine the set of all possible values of \(\varepsilon\) such that the set \(\mathcal{O}\), defined by the condition \(\exists \varepsilon > 0 : \forall x \in \mathcal{O} \; (x-\varepsilon,x + \varepsilon ) \subseteq \mathcal{O}\), is equal to the set of all real numbers.
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Context:
Around a circle, an isosceles trapezoid \(ABCD\) is described. The side \(AB\) touches the circle at point \(M\), and the base \(AD\) touches the circle at point \(N\). The segments \(MN\) and \(AC\) intersect at point \(P\), such that \(NP: PM=2\). Find the ratio \(AD: BC\).
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Context:
Determine the minimum number of different table arrangements needed for a dinner party with 20 guests, seated in groups of 4, so that every guest meets every other guest at least once.
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Context:
Let $PA$, $PB$, and $PC$ be three non-coplanar rays originating from point $P$, with each pair of rays forming a $60^\circ$ angle. A sphere with a radius of 1 is tangent to each of these three rays. Find the distance from the center of the sphere $O$ to point $P$.
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Context:
Calculate the flux \( \iint \vec{F} \cdot \hat{n} \, ds \) of the vector field \( \vec{F} = 4x \hat{i} - 2y^2 \hat{j} + z^2 \hat{k} \) through the surface \( S = \{(x,y,z) : x^2 + y^2 = 4, z = 0, z = 3\} \).
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Context:
Determine the minimum number of questions Player B needs to ask to guess the secret numbers selected by Player A, given that Player A chooses a group of single-digit integers \( x_{1}, x_{2}, \cdots, x_{n} \) which can be positive or negative, and Player B can ask about the sum \( a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \) with any set of numbers \( a_{1}, a_{2}, \cdots, a_{n} \).
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Context:
Evaluate the limit: $$\lim_{n\rightarrow \infty} \dfrac{P_{n-2}\cdot P_{n+2}}{P_{n-1}\cdot P_{n+1}}$$ where $P_{n}$ is the product of the numbers in the nth row of Pascal's Triangle.
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{(\sin(x))^{n}}{1-\sin{(x)}}\,\d x.$$
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Context:
Calculate the probability of having exactly 40 Type A arrivals before either a single Type B arrival occurs or the time reaches \( t = 3 \). Type A customers arrive at a rate given by \( \lambda_a = 0.7\frac{-t^2+8t+84}{10} \), and Type B customers arrive at a constant rate of \( \lambda_b = 6 \). Use the provided information and the concept of super-positioning to determine this probability.
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Context:
In a game involving two players, A and B, two standard six-sided dice are rolled repeatedly. Player A wins if a total of 2 is rolled first, while player B wins if two consecutive totals of 7 are rolled first. What is the probability that player A wins the game?
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Context:
Find an asymptotic bound for the sum \( \sum_{k = 0}^{\lg n} 2^k \lg(\lg(n) - k) \).
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Context:
Determine the number of solutions \(x(t)\) to the differential equation \(x^{\prime}(t) = x(t) - e^{-t^2}\) that satisfy \(\lim_{t \to \pm \infty}x(t) = 0\).
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Context:
The first floor where the seventh-grade boys live has $x$ rooms. If each room accommodates 6 people, exactly one room remains empty; if each room houses 5 people, 4 people will have nowhere to stay. How many rooms are there in total on the first floor? The equation related to $x$ can be stated as _______.
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Context:
Max attended 40 college courses with varying credit hours in 2 years. On average, he attended 5 courses per semester, with each course worth 3 credit hours, but in the last semester, he took two additional courses, each worth 4 credit hours. Sid attended four times as many college courses as Max in the same period, and he took twice as many 4-credit-hour courses as Max in the last semester. What's the total number of credit hours they both completed?
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Context:
Let $u=\{a_1, \ldots, a_5\}$, where all $a_i \in \{0,1\}$. Let $D(u)$ be the set of all subsequences of $u$ with length 4. Let $A$ be a set of sequences $u_1, \ldots, u_{|A|}$ with the property that $|D(u_i) \cap D(u_j)| = 0$ for all $i \neq j$. What is the maximum value of $|A|$?
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Context:
Paul has a collection of 137 baseball cards that he wants to place in his binder. The binder has a variety of pages, with some pages accommodating 4 cards, some accommodating 6 cards, and others accommodating 8 cards. He distributes the cards equally among the page types, starting with the 4-card pages, followed by the 6-card pages, and finally the 8-card pages. How many cards will he have on the last partially-filled page?
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Context:
What is the sum of the ten terms in the arithmetic sequence $-3, 4, \dots, 40$?
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Context:
Find the area of the region that lies inside both curves $r = 1 - \cos \theta$ and $r = \frac{1}{2}$. Express the area in the form $\frac{a \pi}{b} - \frac{c \sqrt{d}}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Calculate $a + b + c + d + e$. Provide your answer as a single integer.
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Context:
The sequence $\{a_{n}\}$ satisfies $a_{{n+1}}+(-1)^{n} a_{n} =2n-1$. Find the sum of the first 80 terms of the sequence.
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Context:
Wrap a line segment of 20 cm in length into a hexagon. What is the range of the longest side of the hexagon?
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Context:
Let $V$ be an infinite-dimensional vector space over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. If the dimension of $V$ is $\alpha$, a cardinal number, determine the dimension of $V^*$, the algebraic dual of $V$. It is known that $\dim V < \dim V^*$ when $V$ is infinite-dimensional.
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Context:
Find the integer $n$ such that $8001 < n < 8200$ and $2^n - 1$ divides $2^{k(n-1)! + k^n} - 1$ for all integers $k > n$. What is the value of $n$?
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Context:
Find the dimension of the quotient ring \( \mathbb{C}[x, y] / \langle xy^3, x^2y^2, x^3y, \text{all monomials of degree 5} \rangle \).
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Context:
Given the sets
$$
A=\{x \mid |x-a|<2\}, \quad B=\left\{x \mid \frac{2x-1}{x+2}<1\right\},
$$
if $A \cap B = A$, find the range of values for the real number $a$.
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Context:
Given the sequence \(\{a_{n}\}\) defined by
\[ a_{n} = \left\lfloor (2+\sqrt{5})^{n} + \frac{1}{2^{n}} \right\rfloor \quad \text{for} \quad n \in \mathbb{Z}_{+}, \]
where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\). Let \(C\) be a real number such that for any positive integer \(n\),
\[ \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leq C. \]
Find the minimum value of \(C\).
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Context:
Let points \( C \) and \( D \) be the trisection points of \( AB \). At 8:00, person \( A \) starts walking from \( A \) to \( B \) at a constant speed. At 8:12, person \( B \) starts walking from \( B \) to \( A \) at a constant speed. After a few more minutes, person \( C \) starts walking from \( B \) to \( A \) at a constant speed. When \( A \) and \( B \) meet at point \( C \), person \( C \) reaches point \( D \). When \( A \) and \( C \) meet at 8:30, person \( B \) reaches \( A \) exactly at that time. At what time did person \( C \) start walking?
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Context:
Calculate the arc length of the curve given by the equation \(e^{2x+2y}=x-y\) from the point \((0,0)\) to the point \((1,1)\).
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Context:
设 $\boldsymbol{u}=\left(u_{1}, u_{2}, u_{3}\right)$ 和 $\boldsymbol{v}=$ $\left(v_{1}, v_{2}, v_{3}\right)$ 是空间向量，满足 $u_{i} 、 v_{i}(i=1,2,3 )$ 均为整数，且
$0.9999<\cos \langle\boldsymbol{u}, \boldsymbol{v}\rangle<1$.
记 $S=\left|u_{1}\right|+\left|u_{2}\right|+\left|u_{3}\right|+\left|v_{1}\right|+\left|v_{2}\right|+\left|v_{3}\right|$.求 $[\sqrt{S}]$ 的最小可能值，其中， $[x]$ 表示不超过实数 $x$ 的最大整数。
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Context:
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.
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Context:
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen. On one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\), given that the radius of the circle is 16 and \(\sin \alpha = \frac{\sqrt{39}}{8}\).
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Context:
Given that $\sin \alpha \cos \alpha = \frac{1}{8}$, and $\alpha$ is an angle in the third quadrant. Find $\frac{1 - \cos^2 \alpha}{\cos(\frac{3\pi}{2} - \alpha) + \cos \alpha} + \frac{\sin(\alpha - \frac{7\pi}{2}) + \sin(2017\pi - \alpha)}{\tan^2 \alpha - 1}$.
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Context:
Evaluate the integral \( \int_R z\,dA \) where \( R \) is the region enclosed by the intersection of the cone \( z^2=x^2+y^2 \) and the cylinder \( x^2+y^2=4 \).
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Context:
If the price of a bag of cherries is $5, the price of a bag of olives is $7, and a box of grapes is priced at $11, how much would Jordyn pay for buying 50 bags of cherries with a 12% discount, 75 bags of olives with an 8% discount, and 25 boxes of grapes with a 15% discount, given a 5% sales tax and a 2% service charge on the final amount?
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Context:
Determine the number of nine-digit odd numbers divisible by 375, where the digits from the second to the second-last are in non-decreasing order. If the number is represented as \( n = \overline{a_1a_2a_3a_4\ldots a_9} \), then \( a_2 \le a_3 \le a_4 \le a_5 \le \ldots \le a_8 \).
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Context:
Determine for which values of \( n \) the integral \( \int_0^1 \frac{x^n}{\sqrt{1-x^4}} \, dx \) is convergent.
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Context:
Calculate the surface integral \( \oint_A F \cdot v \, dS(x) \) for the vector field \( F(x,y,z) = (-y+e^{x+z}, 0, e^{x+z}) \) over the set \( A := \delta B_1 \cap(\{x>0, z=0\}\cup\{x=0, z>0\}) \), where \( B_1 \) is the unit ball in \( \mathbb{R}^3 \) and \( v \) is a tangent vector on \( A \).
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Context:
Find \( \epsilon > 0 \) such that for all partitions \( P \) of \([1,2]\) with mesh \( \lambda(P) < \epsilon \), the inequality \(|U_{f,P} - L_{f,P}| < 0.01\) holds, where \( U_{f,P} \) and \( L_{f,P} \) are the upper and lower Darboux sums of the function \( f(x) = \frac{1}{x} \) on \([1,2]\).
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Context:
From point $C$ a tangent is drawn to a circle with radius $2 \sqrt{5}$ and center at point $O$. Point $A$ is the point of tangency. The segment $CO$ intersects the circle at point $B$. From point $B$, a perpendicular is drawn to the line $BC$ and intersects the line $AC$ at point $F$. Find the radius of the circumcircle of triangle $ABC$ if $BF = 2$.
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Context:
Find the following limit:

$$\lim_{x\rightarrow -\infty}{e^{\frac {1}{2-x}}\cdot\frac{x^2+2x-1}{x-2}}-x$$
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Context:
Find the smallest natural number $n$ such that the group $\mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45}$ is isomorphic to the direct product of $n$ cyclic groups.
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Context:
Given the function expression y = x^2 + 1 and the range {1,3}, how many homologous functions can be defined? Express your answer as a single integer.
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Context:
Evaluate the definite integral: \[ \int_0^4\sqrt{x^2+4}\,dx + \int_2^{2\sqrt{5}}\sqrt{x^2-4}\,dx \]
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Context:
Two employees M and N are paid a total of $616 per week by their employer. M is paid a certain percentage more than N, and N is paid $280 per week. What is the percentage of M's salary compared to N's salary?
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Context:
For which values of the natural number \( n > 2 \) is the graph \( G_n = (V, E) \), defined by \( V = \{A \subset \{1, 2, \ldots, n\} : |A| = 2\} \) and \( \{A, B\} \in E \iff A \cap B = \emptyset \), connected?
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Context:
Find the shortest distance between the lines $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}$
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Context:
Let \( A = \left( a_{ij} \right) \) be the \( n \times n \) matrix, where \( a_{ij} \) is the remainder of the division of \( i^j + j^i \) by 3 for \( i, j = 1, 2, \ldots, n \). Find the greatest \( n \) for which \(\operatorname{det} A \ne 0\).
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Context:
Determine the isomorphic form of the quotient group \(\mathbb{C}^\times / S_1\), where \(S_1\) is the unit circle subgroup of the multiplicative group of non-zero complex numbers, defined as \(\{e^{i\theta} \mid \theta \in [0,2\pi)\}\).
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Context:
Find the least positive integer \( k \) such that \( \cos^k \left(\frac{\pi}{2k}\right) \geq \frac{99}{100} \).
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Context:
There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone in these 20 rooms has had a turn, how many rooms will have their lights off?
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Context:
7.5. At each vertex of a cube lives a number, not necessarily positive. All eight numbers are distinct. If a number is equal to the sum of the three numbers living in the adjacent vertices, then it is happy. What is the maximum number of happy numbers that can live at the vertices of the cube?
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Context:
In an octagon, let the number of obtuse angles be $x$. What is the minimum value of $x$?
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Context:
a placed 3 sheets with two carbons to get two extra copies of the original . then he decided to get more carbon copies and folded the paper in such a way that the upper half of the sheets were on top of the lower half . then he typed . how many carbon copies did he get ?
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Context:
We define a positive integer as "special" if it is divisible by all of its non-zero digits. Determine the maximum number of consecutive special numbers that can exist.
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Context:
What is the greatest positive integer $m$ such that $ n^2(1+n^2-n^4)\equiv 1\pmod{2^m} $ for all odd integers $n$?
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Context:
Given the functions \( f(x) = e^{x+1} - 1 \) and \( g(x) = 100|f(x)| - \sum_{k=1}^n|f(x^k)| \), where \( n \in \mathbb{N} \), determine the sum of all possible integer values of \( n \) such that \( g(x) \) is differentiable over the entire set of real numbers.
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Context:
Let the set \( M = \{1, 2, \cdots, 2020\} \). The subset \( A \subseteq M \) is such that for any element \( x \) in \( A \), \( 4x \) is not in \( A \). What is the maximum number of elements in the set \( A \)?
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Context:
In the sequence $\{a_n\}$, $a_1= \sqrt{2}$, $a_n= \sqrt{a_{n-1}^2 + 2}\ (n\geqslant 2,\ n\in\mathbb{N}^*)$, let $b_n= \frac{n+1}{a_n^4(n+2)^2}$, and let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. The value of $16S_n+ \frac{1}{(n+1)^2}+ \frac{1}{(n+2)^2}$ is ______.
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Context:
From the digits 0, 1, 2, 3, 4, 5, select 3 digits to form a three-digit number without repeating digits. How many of such three-digit numbers are divisible by 5? (Answer with a number)
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Context:
10.228. One end of the diameter of a semicircle coincides with the vertex of the angle at the base of an isosceles triangle, and the other lies on this base. Find the radius of the semicircle if it touches one lateral side and divides the other into segments of lengths 5 and 4 cm, measured from the base.
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Context:
A bicycle wheel has a certain diameter. It makes 393.1744908390343 complete revolutions in 1 km. What is the diameter of the wheel?
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Context:
On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the digits is equal to 13. What is the distance from Yolkino to Palkino?
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Context:
Calculate the expected hitting time \( t_3 \) for a simple random walk on a 3-dimensional hypercube \( C_3 \), where \( t_3 \) is the expected time to hit a vertex \( B \) starting from a vertex \( A \) with Hamming distance 3. Assume the random walk has a \( 1/n \) chance of moving to each of the \( n \) adjacent vertices.
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Context:
Given the equations \( x^2 - y^2 - u^3 + v^2 + 4 = 0 \) and \( 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \), find \( \frac{\partial u}{\partial x} \) at \((x,y) = (2,-1)\). Assume that these equations determine functions \( u(x,y) \) and \( v(x,y) \) near the point \((x,y,u,v) = (2, -1, 2, 1)\).
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Context:
Determine the order of the element \((1,2)+\langle(0,4)\rangle\) in the quotient group \(\Bbb Z_{4}\oplus \Bbb Z_{16}/\langle(0,4)\rangle\).
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Context:
How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer?
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Context:
What is the average case complexity of the stable marriage problem?
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Context:
Determine the number of continuous functions \( f(x) \) defined on \([0, 1]\) that satisfy the following conditions: \( \int_0^1 f(x)\,dx = 2 \), \( \int_0^1 (x-0.5)f(x)\,dx = 0 \), and \( \int_0^1 (x-0.5)^2f(x)\,dx = 0.5 \).
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Context:
In a plane, determine the minimum number of non-parallel straight lines required such that at least one angle formed between any two lines is less than 26 degrees.
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Context:
6. How many ways are there to choose integers $a, b$ and $c$ with $a<b<c$ from the list $1,5,8,21,22,27,30,33,37,39,46,50$ so that the product abc is a multiple of 12 ?
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Context:
In a guessing game, if the number of possible options, $N$, is even and the target number of options, $M$, is $\frac{N}{2}$, what is the minimum number of guesses required for Bob to ensure that Alice confirms with a YES?
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Context:
On an infinite chessboard, a bishop and a knight are placed on squares in the same row. A meteor storm places a meteor on each square independently with probability \( p \). Neither the bishop nor the knight is hit, but their movements may be obstructed. Find the value of \( p \) such that the expected number of valid squares the bishop can move to equals the expected number of valid squares the knight can move to. Express \( p \) as \( \frac{a}{b} \) for relatively prime positive integers \( a \) and \( b \), and compute \( 100a + b \).
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Context:
Let $X$ and $Y$ be positive integers such that the first few digits of the fractions \( \frac{29}{70}, \frac{70}{169}, \frac{169}{408}, \frac{408}{X}, \frac{X}{Y} \) are all equal to 0.4142. What is \( X + Y \)? Note: The fractions are not equal.
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Context:
Suppose you have 100 distinct video games and 50 distinct friends. You must offer each friend 3 video games from your collection of 100, and each friend will select one of the three games offered to them. Assuming all offers are made simultaneously, you may offer the same game to multiple friends, and each set of 3 games and each game chosen has an equal probability of selection. How many distinct games will be chosen more than once?
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Context:
Let $  l  $ be a tangent line at the point $  T\left(t, \frac{t^2}{2}\right) \ (t > 0)  $ on the curve $  C : y = \frac{x^2}{2}  $. A circle $  S  $ touches the curve $  C  $ and the $  x  $ axis.
Denote by $  (x(t), y(t))  $ the center of the circle $  S  $. Find $ \lim\limits_{r \to +0} \int_r^1 x(t)y(t) \, dt. $
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Context:
求出最小正整数 $n$, 使其恰有 144 个正约数, 并且其中 10 个是连续整数.
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Context:
Find the limit of the complex logarithm function in the following case: $$\lim_{y\to 0} [\log(-a+iy)-\log(-a-iy)].$$ where $a>0$ and $y$ approaches 0 from above and below.
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Context:
Determine the asymptotic behavior of the expected number of flips required for the number of heads to exceed the number of tails in a sequence of fair coin flips.
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Context:
Vasya has \( n \) candies of several types, where \( n \geq 145 \). It is known that if any group of at least 145 candies is chosen from these \( n \) candies (in particular, one can choose the group of all \( n \) candies), there exists a type of candy such that the chosen group contains exactly 10 candies of that type. Find the largest possible value of \( n \).
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Context:
Find the ratio of the number of natural numbers from 1 to 1,000,000 that can be expressed as the sum of a perfect square and a perfect cube to the number of numbers that cannot be expressed in this way.
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Context:
A team of five people is laying a 30-meter pipe into the ground. To prepare the trench, they decided to dig in turns with the condition that each person digs for the amount of time it takes the other four together to dig one-third of the trench. The 30-meter trench was completed when the last team member had worked their allotted time. How many meters of the trench could they have dug in the same time if they all worked together?
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Context:
Calculate the expectation of the stopped local time $L_t^y$ for a Brownian motion $B_t$ starting at $x$, where $x > y$ and $T_a=\inf\{ s \geq 0 \mid B_s=a\}$ with $a > x$. Express your answer in terms of $x$, $y$, and $a$. \( \boxed{\text{Your answer here}} \)
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Context:
Let \( F(x) \) be a real-valued function defined for all real numbers except \( x=0 \) and \( x=1 \). Given that \( F(x) + F\left(\frac{x-1}{x}\right) = 1 + x \), find \( F(x) \).
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Context:
Determine the condition on the complex-valued matrix $V$ such that the determinant \( \det(I+[aA,bB]VV^*[aA,bB]^*) \) is increasing in both real positive parameters \(a\) and \(b\), where \(A\) and \(B\) are complex-valued matrices, and \(^*\) denotes the conjugate transpose.
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Context:
Determine the values of $p$ and $q$ for which the series \( \sum_{n=1}^\infty \frac{x}{n^p+n^qx^2} \) converges uniformly on \( \mathbb{R} \).
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Context:
If the area of a circle decreases by 42%, then the radius of the circle decreases by a certain percentage. What is the percentage decrease in the radius of the circle?
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Context:
The altitudes of an acute-angled triangle \(ABC\) from points \(B\) and \(C\) are extended to intersect the circumcircle at points \(B_1\) and \(C_1\). It turns out that the segment \(B_1C_1\) passes through the center of the circumcircle. Find the angle \(BAC\).
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Context:
Given $f(x)=2x+1$, $g(x)=x|x-2|$, if for any $x_{1}$, $x_{2}\in [0,t]$, when $x_{1}\neq x_{2}$, $\frac{{g(x_{1})-g(x_{2})}}{{f(x_{1})-f(x_{2})}}<2$ holds, then the maximum value of $t$ is ______.
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Context:
In an isosceles triangle with a base of \(a\) and an angle \(\alpha\) at the base, an inscribed circle is drawn. Find the radius of a circle that is tangent to both the inscribed circle and the legs of the triangle.
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Context:
Given that \( T = \sin 50^{\circ} \times \left(S + \sqrt{3} \times \tan 10^{\circ}\right) \), find the value of \( T \).
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Context:
A $10 \times 10$ square is divided into unit squares. How many triangles are formed after drawing one diagonal?
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Context:
The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \mathrm{km} / h$, the passenger train would travel from $A$ to $B$ 2 hours and 24 minutes faster than the freight train. Determine the distance from $A$ to $B$.
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Context:
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth?
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Context:
Square $PQRS$ lies in the first quadrant. Points $(3,0)$, $(5,0)$, $(7,0)$, and $(13,0)$ lie on lines $SP$, $RQ$, $PQ$, and $SR$, respectively. If the sum of the coordinates of the center of the square $PQRS$ is $x$, what is the value of $10x$?
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Context:
2. At what moment between 13:00 and 14:00 do the hour and minute hands of a clock form a straight angle $\left(180^{\circ}\right)$?
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Context:
Find the coefficient of $x^{10}$ in the expansion of $(x+2)^{10}(x^2-1)$. Express your answer as a single integer.
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Context:
Determine the order type of the set \( \omega \times n \) with the lexicographic order, considering the first position first.
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