Context:
Given $f(x) = x^3 - 3x + 2 + m$ with $m > 0$, in the interval $[0, 2]$ there exist three distinct real numbers $a$, $b$, and $c$, such that the triangle with sides $f(a)$, $f(b)$, and $f(c)$ is a right-angled triangle. The range of $m$ is __________.
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Context:
Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$.  Let $m=\min\{ab,bc,ca\}$.  Find the largest possible value of $m$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\{1, 2, \cdots, 60\}$, there must be three numbers that are pairwise coprime.
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Context:
\section*{Task 2 - 011012}

In the USSR, in 37 minutes today, as much gas is produced as was produced in the entire year 1913 in Tsarist Russia.

Calculate the increase in percent!
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Context:
Given that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins.
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Context:
How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum.
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Context:
what is the sum between the place values of three 7 ' s in the numeral 87953.0727
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Context:
求最小的整数 n\geq3，使得在平面内任意无三点共线的 n 个点中，必有三个点是一个非等腰三角形的顶点.
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Context:
sides of a rectangular park are in the ratio 3 : 2 and its area is 3750 sq m. Inside the park, there is a circular flower bed with a diameter equal to half the length of the shorter side of the park. The cost of fencing the park at 70 ps per meter and the flower bed at 90 ps per meter is ?
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Context:
A high school second-year class has a total of 1000 students. To investigate the vision condition of the students in this grade, if 50 samples are selected using the systematic sampling method, and all students are randomly numbered from 000, 001, 002, …, to 999, if it is determined that the second number is drawn in each group, then the student number drawn in the 6th group is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral

$$
I=\int_{0}^{\infty} \frac{x \sin a x}{x^{2}+k^{2}} \, dx \quad(a>0, k>0)
$$
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Context:
The monotonically increasing interval of the function $y=-x^{2}+4x$ is ____.
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Context:
2. Twenty numbers are arranged in a circle. It is known that the sum of any six consecutive numbers is 24. What is the number in the 12th position if the number in the 1st position is 1?
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Context:
设 5\times5 矩阵 A 的元素均取自 \{0,1,\cdots,14\}，满足 (\det A,15)=1 的矩阵 A 的个数记为 N，则 \nu_2(N)=__________.
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Context:
A pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \, \text{kg}$, slides and hits the pulley, coming to a stop. The initial distance between the block and the pulley is $1.0 \, \text{m}$. During the slide, the rope makes a constant angle of $10.0^\circ$ with the horizontal. The hanging block has a mass of $M = 2.0 \, \text{kg}$. The table has a coefficient of friction of $0.50$ with the block. The pulley is frictionless, and the block remains on the ground throughout. Let $t$ be the time in seconds for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures.
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Context:
Given that $x = 1$ is a solution to the inequality $k^2x^2 - 6kx + 8 \geq 0$ ($k \neq 0$), the range of values for $k$ is __________.
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Context:
Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding
\[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]
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Context:
Given a twentieth-degree polynomial with integer coefficients. On the plane, all points with integer coordinates whose ordinates are no less than 0 and no more than 10 are marked. What is the maximum number of marked points that can lie on the graph of this polynomial?
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Context:
Compute the sum of all positive integers $n$ for which $9 \sqrt{n}+4 \sqrt{n+2}-3 \sqrt{n+16}$ is an integer.
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Context:
In the cartesian coordinate plane $(xOy)$, the asymptote of the hyperbola $\frac{x^2}{3} - y^2 = 1$ coincides with the asymptote of the parabola $y^2 = 2px (p > 0)$. Find the value of the real number $p$.
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Context:
Given a tetrahedron $ABCD$ with each vertex on the surface of sphere $O$, $AB=AC=5$, $BC=8$, $AD\perp$ the base plane $ABC$, $G$ is the centroid of $\triangle ABC$, and the tangent of the angle formed by line $DG$ and the base plane $ABC$ is $\dfrac{1}{2}$, then the surface area of sphere $O$ is \_\_\_\_\_\_.
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Context:
In the three words "尽心尽力", "力可拔山", and "山穷水尽", each Chinese character represents a digit from 1 to 8. Identical characters represent the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each word is 19, and "尽" > "山" > "力", what is the maximum possible value of "水"?
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Context:
What is the area of the portion of the unit sphere that is enclosed within the cylinder with equation $x^2 + y^2 = \frac{1}{4}$?
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Context:
If the solution set of the inequality regarding $x$, $a \leq \frac{3}{4}x^2 - 3x + 4 \leq b$, is exactly $[a, b]$, then $a+b=$ ______.
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Context:
Find the symmetry center of the function $y=3\tan\left(\frac{1}{2}x+\frac{\pi}{3}\right)$. Express your answer as an ordered pair (x, y).
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Context:
For any 2016 complex numbers \( z_1, z_2, \ldots, z_{2016} \), it holds that
\[
\sum_{k=1}^{2016} |z_k|^2 \geq \lambda \min_{1 \leq k \leq 2016} \{ |z_{k+1} - z_k|^2 \},
\]
where \( z_{2017} = z_1 \). Find the maximum value of \( \lambda \).
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Context:
已知四面体 $A B C D$ 的外接球 $\Gamma$ 的半径为 3, 球心为 $O$, 其中 $A B$ 为球 $\Gamma$ 的直径, $C D=3, \angle A O C=\theta \in\left[\frac{\pi}{3}, \frac{5}{6} \pi\right]$, 四面体 $A B C D$的体积的最大值记为 $V(\theta)$. 则 $4\sqrt{3}V(\theta)$ 的最小值是 $\qquad$。
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Context:
Roberto is raising chickens for eggs. He buys 4 chickens of different breeds, each with a variable initial cost based on their breed and egg-laying consistency:

- Chicken 1: $25 initially, 4 eggs per week
- Chicken 2: $30 initially, 3 eggs per week
- Chicken 3: $22 initially, 5 eggs per week
- Chicken 4: $35 initially, 2 eggs per week

The weekly cost to feed each chicken is as follows:

- Chicken 1: $1.50
- Chicken 2: $1.30
- Chicken 3: $1.10
- Chicken 4: $0.90

Each chicken produces a different number of eggs per week based on their breed, and these egg production rates are expected to last forever. Roberto used to buy 1 dozen eggs a week and spent $2 per dozen. After how many weeks will the chickens be cheaper than buying his eggs at the store?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $\vec{a}=(3,2)$ and $\vec{b}=(x,1-y)$, where $\vec{a}$ is parallel to $\vec{b}$ and both $x$ and $y$ are positive numbers, find the minimum value of $\frac{3}{x}+\frac{2}{y}$. Express your answer as a single number.
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Context:
Consider the number obtained by writing the numbers $1, 2, \ldots, 1990$ one after another. In this number, every digit at an even position is omitted. In the resulting number, every digit at an odd position is omitted. Then, in the new number, every digit at an even position is omitted, and so on. What will be the last remaining digit?
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Context:
Calculate the volume of the solid enclosed by the surface $(x^2+y^2+z^2)^2=y$ using spherical coordinates.
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Context:
Given a function $f(x)$ on the interval $[a,b]$, if there exists an $x\_0∈[a,b]$ such that $f(x\_0)=∫_a^bf(x)dx$, then $x\_0$ is called an "integral point" of the function $f(x)$ on the interval $[a,b]$. Find the "integral point" of the function $f(x)=cos(2x+π/6)$ on the interval $[0,π/2]$.
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Context:
Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange, but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?
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Context:
A student rolls two dice simultaneously, with the resulting numbers being $a$ and $b$. Determine the probability that the eccentricity $e$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ satisfies $e > \frac{\sqrt{3}}{2}$. Express your answer as a simplified fraction.
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Context:
Determine the maximum number of algebraically independent homogeneous polynomials of degree $d \geq 2$ in the polynomial ring $\mathbb{Z}[x_1, \dots, x_n]$. Can this number be as large as $n^2$ or $n^3$, or is it limited to $n$?
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Context:
William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder when $M$ is divided by $1000$.
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Context:
Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Determine whether $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$.
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Context:
Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done?

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


Context:
Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$.
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Context:
Count the number of functions \( f: \{1, 2, \ldots, 12\} \rightarrow \{1, 2, \ldots, 12\} \) such that if \((a, b, c)\) is a multiplication pair, then \((f(a), f(b), f(c))\) is also a multiplication pair.
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Context:
Let $P$ be an orthogonal projection onto a subspace $E$ of an inner product space $V$. If the dimensions of $V$ and $E$ are $n$ and $m$ respectively, determine the dimensions of the eigenspaces of $P$ corresponding to the eigenvalues $0$ and $1$. Provide your answer as a pair of dimensions.
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Context:
If $a > 1$, then the sum of the real solutions of
$\sqrt{a - \sqrt{a + x}} = x$
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Context:
A set of data is arranged in ascending order as follows: $9$, $10$, $12$, $15$, $x$, $17$, $y$, $22$, $26$. The median of this data set is $16$, and the $75\%$ percentile is $20$. Find the sum of $x$ and $y$. Express your answer as a single numerical value.
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Context:
8.4. In the tetrahedron $E F G H$, it is known that $E F=G H=7, E G=F H=10, E H=F G=11$. Points $K, L, M, N$ are the centers of the circles inscribed in triangles $E F G, E F H$, $E G H$ and $F G H$. Find the volume of the tetrahedron $K L M N$. Round the answer to the nearest hundredth if necessary.
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Context:
The non-terminating periodic decimal $0.124124 \ldots = 0.\overline{124}$ has period three and is abbreviated by placing a bar over the shortest repeating block.
 If only digits $0$ and $1$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2\ldots d_{12}}$ have period exactly $12$?

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Context:
Evaluate
$$
S=\sum_{k=1}^{n} k(k+1) \cdots(k+p),
$$
where \( n \) and \( p \) are positive integers.
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Context:
对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有
$f(X)=\frac{1}{|X|} \sum\limits_{a \in X} a$.
集合对 $(A, B)$ 满足
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$,
且 $1 \leqslant|A| \leqslant 98$.
任取 $p \in B$ ，令
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$.
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$ ，求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 的最大值时的$\frac{a}{b}$中的a+b.
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Context:
Let's call a natural number special if one of its digits can be changed to another such that all the digits in the resulting number are different. Numbers in which all the digits are already different are also considered special. How many special ten-digit numbers exist?
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Context:
Given the spherical metric \(ds^2 = dr^2 + r^2(d\theta^2 + \sin^2(\theta) d\phi^2)\), determine if the covariant derivative \(\nabla_\theta \theta\) is equal to 1.
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Context:
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$.  We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$.  If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$.  We continually apply this permutation to each $T_m$ to get $T_{m+1}$.  Find the smallest positive integer $n$ so that $T_n=T_0$.
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Context:
Determine whether the limit \( \lim_{x \to \pm \infty} \partial_x u(x,t) \) exists for the Burgers equation given by \( u_t + u u_x = \epsilon u_{xx} \), where \( u_t \) is the partial derivative of \( u \) with respect to \( t \), \( u_x \) is the partial derivative of \( u \) with respect to \( x \), and \( \epsilon \) is a constant.
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Context:
You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$. If\[x_{1} < x_{2} < x_{3}\quad\text{ and }\quad x_{3} - x_{2} = x_{2} - x_{1},\]which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.
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Context:
Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once.  What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\]
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Context:
Evaluate whether the following limit is true:
\[ \lim_{x\to 0}(x!\,x!!\,x!!!\,x!!!!\cdots)^{-1/x} = e \]
where \( x!! := ((x!)!) \) and \( x!!! = (((x!)!)!) \).
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Context:
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ?

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Context:
将所有非完全平方的正奇数与所有正偶数的立方从小到大排成一列（前若干项依次为 $3,5,7,8,11,13, \cdots)$ ，则该数列的第 2023 项的值为 $\qquad$。
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Context:
6. There are 900 three-digit numbers $(100,101, \cdots, 999)$. These three-digit numbers are printed on cards, with ten three-digit numbers on each card. Some cards, when flipped, still show three-digit numbers, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, which when flipped shows I $E$. Therefore, some cards can be used in two ways. Thus, at most, _ fewer cards can be printed.
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Context:
Given nine different numbers, how many different values of third-order determinants can be formed using all of them?
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Context:
Determine whether the sequence \( \frac{e^{n^2}}{n^n} \) converges or diverges as \( n \to \infty \).
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Context:
令 $I=\sqrt{\frac{2}{\pi}} \int_{-\infty}^{+\infty} e^{x-10 \cosh (2 x)} \mathrm{d} x$, 则 $[|\ln I|]=$ $\qquad$.
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Context:
8. (10 points) In a regular truncated quadrilateral pyramid $A B C D A_{1} B_{1} C_{1} D_{1}$, the section $A B C_{1} D_{1}$ is drawn. Find the ratio of the volumes of the resulting polyhedra $\frac{V_{A B C D D_{1} C_{1}}}{V_{A_{1} B_{1} C_{1} D_{1} A B}}$, if $A B: A_{1} B_{1}=3$. Round the answer to the hundredths.
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Context:
In the polar coordinate system, the minimum distance from a point on the circle $p=2$ to the line $p(\cos \theta+ \sqrt {3}\sin \theta)=6$ is $\_\_\_\_\_\_$.
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Context:
Determine the cardinality of the set of epsilon numbers (\(\epsilon\)-numbers) less than \(\omega_1\), the first uncountable ordinal. Is this set countable or uncountable?
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Context:
Find the closed form of the generating function for the series: $x^1+x^4 + x^9 +x^{16} +x^{25} + x^{36} +x^{49} +x^{64}+....$ where the exponents are the sequence of squares of positive integers.
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\boxed{\frac{1}{2} \left( \vartheta_3(0,x) - 1 \right)}


Context:
7. Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns taking 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, without violating the rules of the game. The computer plays according to the following algorithm: on each turn, it takes as many stones as needed to be in the most advantageous position. Petya always goes first. What is the probability that Petya will win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a $n \times 1$ matrix $u$ such that $u^Tu=1$, determine the matrix $A = I - 2uu^T$. Is it true that $A = -I$ for all such matrices $u$?
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Context:
A3. Determine the number of pairs of integers $(m, n)$ such that

$$
\sqrt{n+\sqrt{2016}}+\sqrt{m-\sqrt{2016}} \in \mathbb{Q}
$$

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Context:
Let O be the origin (0, 0) and C the line segment defined by \( (x, y) : x \in [1, 3], y = 1 \). Let K be the curve consisting of points \( P \) such that for some point \( Q \) on \( C \), \( P \) lies on the line segment \( OQ \) and the distance \( PQ = 0.01 \). Determine whether the length of the curve \( K \) is greater than or less than 2.
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Context:
Let the universal set U = {1,2,3,4,5,6,7}, set A = {1,3,5}, and set B = {2,3,6}. Find the result of the operation A ∪ (∁U B). Express your answer as a set of elements, separated by commas and enclosed in curly brackets.
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Context:
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$; let $G$ be the centroid of $\Delta ABC$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta ABC$.
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Context:
在 100 和 1000000 之间有多少个整数, 其数字之和等于 5 ?
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Context:
Given a recursive sequence $\left\{x_{n}\right\}$ that satisfies $x_{n+1}=x_{n}^{2}-4x_{n}$ for $n=1, 2, \cdots$, if for any initial term $x_{1} \in \mathbf{R}$ such that $x_{1} \neq 0$, there is always a term $x_{k} \geqslant b$ in the sequence, then the largest integer not exceeding $(2b-3)^{2}$ is ______.
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Context:
The Palace Museum is holding four exhibitions during the May Day holiday: "Opera Culture Exhibition," "Ming Dynasty Imperial Cellar Porcelain Exhibition," "Historical Green Landscape Painting Exhibition," and "Zhao Mengfu Calligraphy and Painting Exhibition." A student decides to visit one of them in the morning and one in the afternoon on May Day, with the condition that they must visit at least one painting exhibition. Calculate the total number of different visit plans.
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Context:
For any positive integer $n$, define
$$
S(n)=\left[\frac{n}{10^{[\lg n]}}\right]+10\left(n-10^{[\lg n]}\left[\frac{n}{10^{[\operatorname{Ig} n]}}\right]\right) \text {. }
$$

Then among the positive integers $1,2, \cdots, 5000$, the number of positive integers $n$ that satisfy $S(S(n))=n$ is $\qquad$
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Context:
Find the determinant of a matrix similar to $A_n$, where the entries of the form $(i, i+3)$ are equal to $5$.
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Context:
100. In a regular truncated quadrilateral pyramid with lateral edges $A A_{1}, B B_{1}, C C_{1}, D D_{1}$, the side of the upper base $A_{1} B_{1} C_{1} D_{1}$ is 1, and the side of the lower base is 7. A plane passing through the edge $B_{1} C_{1}$ perpendicular to the plane $A D_{1} C$ divides the pyramid into two parts of equal volume. Find the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of the real parameter $a$, such that the equation
\[ \sin 2x \sin 4x - \sin x \sin 3x = a \]
has a unique solution in the interval $[0, \pi)$.
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Context:
求满足 $\sin x _ { 1 } + \sin x _ { 2 } + \cdots + \sin x _ { n } = 0 ,$

$\sin x _ { 1 } + 2 \sin x _ { 2 } + \cdots + n \sin x _ { n } = 1 0 0$

的n的最小值.
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Context:
There are  $15$  people, including Petruk, Gareng, and Bagong, which will be partitioned into  $6$  groups, randomly, that consists of  $3, 3, 3, 2, 2$ , and  $2$  people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X = \cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} + \cdots + \cos\frac{2006\pi}{7} + \cos\frac{2008\pi}{7}$. Compute $\Big|\lfloor 2008 X \rfloor \Big|$.
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Context:
**Problem 2**
Determine all pairs  $(n, m)$  of positive integers satisfying the equation  $$ 5^n = 6m^2 + 1\ .  $$
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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 $60$ terms of $\{a_n\}$.
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Context:
Several people played a single-round robin tournament in table tennis. At the end of the tournament, it turned out that for any four participants, there are two who scored the same number of points in the games among these four participants. What is the maximum number of tennis players who could have participated in this tournament? In table tennis, there are no draws, one point is awarded for a win, and zero points for a loss.
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Context:
A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?
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Context:
Find the smallest possible value of

$$
x y+y z+z x+\frac{1}{x}+\frac{2}{y}+\frac{5}{z},
$$

for positive real numbers $x, y$ and $z$.
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Context:
求所有的正整数 k，使得对任意满足 k(ab+bc+ca)>5(a^2+b^2+c^2) 的正实数 a,b,c，都存在三边长分别为 a,b,c 的三角形.
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Context:
Let the sequence  $(a_{n})$  be deﬁned by  $a_{1} = t$  and  $a_{n+1} = 4a_{n}(1 - a_{n})$  for  $n \geq 1$ . How many possible values of t are there, if  $a_{1998} = 0$ ?
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Context:
Compute the number of positive integers $n \leq 1000$ such that $\text{lcm}(n, 9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.)
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Context:
Kamal's salary was decreased by 50%, then increased by 30%, and finally decreased by 20%. What is the overall percentage change in his salary?
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Context:
Determine the optimal step size \(\nabla t\) for the Forward-Backward Splitting algorithm, which minimizes the sum of two convex functions \(f(x)\) and \(g(x)\). The algorithm iteration is given by:
\[x_{t+1}=\arg\min_u\left\{\frac{1}{2}\|u-(w_t-\nabla t \partial f(w_t))\|^2+\nabla tg(u) \right\}\] where \(f(x)\) and \(g(x)\) are convex functions.
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Context:
In triangle \(ABC\), side \(AC = 42\). The angle bisector \(CL\) is divided by the point of intersection of the angle bisectors of the triangle in the ratio \(2:1\) from the vertex. Find the length of side \(AB\) if the radius of the circle inscribed in triangle \(ABC\) is 14.
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Context:
已知函数 $f(x)=|x+1|+|3 x+a|$, 若 $f(x)$ 的最小值为 1 .
若 $a>0$, 且 $m, n$ 均为正实数, 且满足 $m+n=\frac{a}{2}$, 求 $m^{2}+n^{2}$ 的最小值.
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Context:
Each cell of a 5x5 table is painted in one of several colors. Lada shuffled the rows of the table so that no row remained in its original place. Then Lera shuffled the columns so that no column remained in its original place. Surprisingly, the girls noticed that the resulting table was the same as the original one. What is the maximum number of different colors that can be used to paint this table?
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Context:
Determine whether the sum of the differences between the positive even and odd integer values of the Riemann zeta function equals \( \frac{1}{2} \).
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Context:
Assume $H$ is a Hilbert space. Given the weak convergences $u_m \rightharpoonup u$ in $L^2(0,T;H)$ and $u_m' \rightharpoonup v$ in $L^2(0,T;H^*)$, determine whether it follows that $v = u'$.
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Context:
Let $f$ be a continuous function on the interval $[0,1]$ mapping to $\mathbb{R}$. Define the linear operator $T$ by \[ T(f)(x) = \begin{cases} f(0), & \text{if } x = 0, \\ \frac{1}{x}\int_0^x f(t) \, dt, & \text{if } 0 < x \leq 1. \end{cases} \] Determine whether \[ \lim_{n \to \infty} \| T^n(f) - f(0) \|_{\infty} = 0. \]
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Context:
For what value(s) of $a$ does the limit $\lim_{x\to 0}\frac{a^{\cos x}-a^{\cot x}}{\cot x -\cos x}$ equal $-1$?
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Context:
Given the function
$$
f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\left(k \in \mathbf{Z}_{+}\right),
$$

if for any real number \(a\),
$$
\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbf{R}\},
$$

find the smallest value of \( k \).
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Context:
集合 M 为连续 n 个正整数构成的集合. 若 M 中所有元素的和恰为 2024，且 n 为偶数，则集合 M 中最小的元素为__________.
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Context:
Determine if the function \( \sum_{n=1}^{\infty} \cos \frac{n \pi x}{l} \) is periodic. If it is periodic, find its period.
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Context:
4. A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at point $(x, y)$, she takes a step based on the following rules:

(a) if $x+y$ is even she moves to either $(x+1, y)$ or $(x+1, y+1)$;

(b) if $x+y$ is odd she moves to either $(x, y+1)$ or $(x+1, y+1)$.

How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x, y)$ to $(x+1, y))$ ?

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Context:
Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}.$ Let $K$ be the area of rhombus $\mathcal{T}.$ Given that $K$ is a positive integer, find the number of possible values for $K.$

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Context:
Determine whether the series \( \sum_{n=0}^\infty \int_{[-1,1]^2} \left(1-\frac{x^2+y^2}{2}\right)^{n/2}\lambda_2(d(x,y)) \) is finite, where \( \lambda_2 \) is the 2-dimensional Lebesgue measure.
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Context:
Given the function $y=a^{x}+b (b > 0)$, its graph passes through point $P(1,3)$. Find the minimum value of $\frac{4}{a-1}+\frac{1}{b}$.
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Context:
As shown in the figure, a rectangular prism with length, width, and height of 15 cm, 5 cm, and 4 cm respectively has a smaller rectangular prism cut out of it. The smaller prism has length, width, and height of \( y \) cm, 5 cm, and \( x \) cm respectively, where \( x \) and \( y \) are integers. If the remaining volume is 120 cm\(^3\), then what is \( x + y \)?
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Context:
8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$ .
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Context:
Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$?

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Context:
Determine whether there exists a constant \( c > 0 \) such that, for any division of a pile of books between two couples \((b_1, g_1)\) and \((b_2, g_2)\), each person can receive at least \( c \) value, regardless of the values and number of books.
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Context:
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which

$a_1>a_2>a_3>a_4>a_5>a_6 \mathrm{\ and \ } a_6

An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations.

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Context:
12. (22 points) If integers $a$ and $b$ are neither coprime nor have a divisibility relationship, then $a$ and $b$ are called a "coalition" pair. Let $A$ be an $n$-element subset of the set $M=\{1,2, \cdots, 2014\}$, and any two numbers in $A$ are coalition pairs. Find the maximum value of $n$.

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Context:
Given a positive geometric sequence $\{a_n\}$, where $a_5a_{2n-5}=10^{2n}$ $(n\geqslant 3,n\in N^{*})$, find the value of the expression $\lg a_1+\lg a_2+\lg a_3+\ldots+\lg a_n$ when $n\geqslant 1$, $n\in N^{*}$.
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Context:
Call an integer  $k$  *debatable* if the number of odd factors of  $k$  is a power of two. What is the largest positive integer  $n$  such that there exists  $n$  consecutive debatable numbers? (Here, a power of two is defined to be any number of the form  $2^m$ , where  $m$  is a nonnegative integer.)
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Context:
Evaluate the following limit for $\alpha \in (0, 1)$, $a > 0$, and $\beta \in \mathbb{R} \setminus \{0\}$:

$$ \lim_{t\rightarrow \infty} \beta \, t^\alpha \sum_{k=0}^{+\infty}\dfrac{(-a\,t^\alpha)^k}{\Gamma(\alpha\,k+\alpha+1)} $$
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Context:
Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.

[i]Proposed by Isabella Grabski[/i]
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Context:
There are 4 numbers written on the board. Vasya multiplied the first of these numbers by \(\sin \alpha\), the second by \(\cos \alpha\), the third by \(\operatorname{tg} \alpha\), and the fourth by \(\operatorname{ctg} \alpha\) (for some angle \(\alpha\)) and obtained a set of the same 4 numbers (possibly written in a different order). What is the maximum number of distinct numbers that could be written on the board?
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Context:
What is the units digit of the expression 7 ^ y + 6, where y is a positive integer, and the units digit of the expression is 9?
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Context:
In the convex quadrilateral \(ABCD\), \(\angle ABC=60^\circ\), \(\angle BAD=\angle BCD=90^\circ\), \(AB=2\), \(CD=1\), and the diagonals \(AC\) and \(BD\) intersect at point \(O\). Find \(\sin \angle AOB\).
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Context:
Given the following two propositions:
Proposition $p$: The quadratic equation with real coefficients $x^{2}+mx+2=0$ has imaginary roots;
Proposition $q$: For the equation $2x^{2}-4(m-1)x+m^{2}+7=0$ ($m\in\mathbb{R}$), the sum of the moduli of its two imaginary roots does not exceed $4\sqrt{2}$,
If both propositions $p$ and $q$ are true, find the range of the real number $m$.
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Context:
 How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
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Context:
设函数 $f(x)=a x^{2}+b x+c \quad(a \neq 0)$ 满足
$|f(0)| \leq 2, \quad|f(2)| \leq 2, \quad|f(-2)| \leq 2$,
求当 $x \in[-2,2]$ 时 $y=|f(x)|$ 的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$
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Context:
Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.
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Context:
The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\mu = 1$ and the variance $\sigma^2 = 4$. Given that $P(X \geq 2) = 0.2$, calculate the probability $P(0 \leq X \leq 1)$.
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Context:
Compute the integral: $$\int_{0}^{1}\frac{x^2}{x+\sqrt{1-x^2}}dx$$
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Context:
设四面体四个面的面积分别为 $S_{1}, S_{2}, S_{3}, S_{4}$，它们的最大值为 $S$，记 $\lambda=\left(\sum_{i=1}^{4} S_{i}\right) / S$。找出 $\lambda$ 一定满足的范围。答案的形式为 $a < \lambda \leqslant b$，请给出 $a + b$ 的值。
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Context:
Determine the number of integers $D$ such that whenever $a$ and $b$ are both real numbers with $-1 / 4<a, b<1 / 4$, then $\left|a^{2}-D b^{2}\right|<1$.
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Context:
Let $ABC$ be a triangle such that $AB=\sqrt{10}$, $BC=4$, and $CA=3\sqrt{2}$. Circle $\omega$ has diameter $BC$, with center at $O$. Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P'$, where $AP < AP'$. Suppose line $P'O$ intersects $AC$ at $X$. Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p}$, where $n$ and $p$ are squarefree, find $m+n+p$.
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Context:
At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If $13$ married couples attended, how many handshakes were there among these $26$ people?
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Context:
In ten boxes, there are balls in the following quantities: 1, 3, 5, ..., 19 balls. Two friends, Peter and Vasily, take turns taking one ball from any box, with Peter going first. The player who loses is the one after whose turn the number of balls in any two boxes becomes the same (including possibly zero). Determine which player can guarantee a win regardless of the opponent's moves.
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Context:
In the following equation, the same Chinese characters represent the same digit, different Chinese characters represent different digits, and no Chinese character represents 7. "迎", "春", and "杯" are not equal to 1. What is the sum of the three digits represented by "迎", "春", and "杯"?
学数学 $\times 7 \times$ 迎 $\times$ 春 $\times$ 杯 $=$ 加油加油 $\times$ 吧.
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Context:
Determine if the congruence $n \equiv 1 \pmod{\lambda^5}$, where $\lambda = 1 - \zeta_5$ and $\zeta_5$ is a 5th root of unity, can be simplified to a congruence modulo an integer in $\mathbb{Z}$.
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Context:
Two trains are moving on parallel tracks in the same direction: Train A is 400 meters long and has a speed of 63 km/hr, while Train B is 300 meters long and has a speed of 54 km/hr. At the same time a man is walking in the direction of the moving trains with a speed of 3 km/hr. How long will it take for the faster Train A to completely overtake Train B and the man walking, considering that Train A starts from behind the man and has to completely cross both the man and Train B?
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Context:
Given a positive integer N, count the number of pairwise coprime integer triples (a, b, c) such that 1 ≤ a, b, c ≤ N. For example, when N = 3, what is the count of valid triples?
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Context:
Consider a Poisson process $N_t$ with $\\langle N \\rangle_t = N_t$, and a continuous process of finite variation $X_t$ with $\\langle X \\rangle_t = 0$. Determine if the quadratic covariation $\\langle X, N \\rangle_t = 0$ is true.
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Context:
Last summer \( {30}\% \) of the birds living on Town Lake were geese, \( {25}\% \) were swans, \( {10}\% \) were herons, and \( {35}\% \) were ducks.
 What percent of the birds that were not swans were geese?

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Context:
方程$( x ^ { 2 } - 6 x + c _ { 1 } ) ( x ^ { 2 } - 6 x + c _ { 2 } ) ( x ^ { 2 } - 6 x + c _ { 3 } ) = 0$的自然数解集为$\{ x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$,$x _ { 5 } , x _ { 6 } \}$，那么$c _ { 1 } - c _ { 2 }$的最大值为
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Context:
Given the quadratic equation $2x^2-2x+3m-1=0$ with two real roots $x_1$ and $x_2$, and the inequality $x_1 \cdot x_2 > x_1 + x_2 - 4$, determine the range of the real number $m$ that satisfies both conditions. Express your answer as an inequality in terms of $m$.
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Context:
In a regular tetrahedron \( ABCD \) with edge length \( a \), points \( A_1 \), \( B_1 \), \( C_1 \), and \( D_1 \) are chosen in the planes \( BCD \), \( CDA \), \( DAB \), and \( ABC \), respectively, such that the line \( A_1B_1 \) is perpendicular to the plane \( BCD \), the line \( B_1C_1 \) is perpendicular to the plane \( CDA \), the line \( C_1D_1 \) is perpendicular to the plane \( DAB \), and the line \( D_1A_1 \) is perpendicular to the plane \( ABC \). Find the volume of the tetrahedron \( A_1B_1C_1D_1 \).
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Context:
A train 180 meters long running at the rate of 54 kmph takes 55.99552035837134 seconds to cross a bridge of certain length. How long is the bridge?
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Context:
给定正整数 $n=3$. 求最大的实数 $\lambda$, 使得不等式 $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ 对任何满足 $a_{1}<a_{2}<\cdots<a_{n}$ 的正整数 $a_{1}, a_{2}, \cdots, a_{n}$ 均成立.
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Context:
Miyuki texted a six-digit integer to Greer. Two of the digits of the six-digit integer were 3s. Unfortunately, the two 3s that Miyuki texted did not appear and Greer instead received the four-digit integer 2022. What is the number of possible six-digit integers that Miyuki could have texted?
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Context:
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?
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Context:
Calculate the measure \( \mu(A) \), where \( \mu = L^1 \otimes \sum_{k = 1}^{\infty}k\delta_k \) and \( A = \{(x, y) \in \mathbb{R}^2: 3 - |x| > y > 2|x|\} \). Use Fubini's theorem to evaluate the integral \( \mu(A) = \int_{A} d\mu \).
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Context:
\( S \) is the sum of the first 10 terms of an increasing arithmetic progression \( a_{1}, a_{2}, a_{3}, \ldots \) consisting of integers. It is known that \( a_{6} a_{12} > S + 1 \) and \( a_{7} a_{11} < S + 17 \). Indicate all possible values of \( a_{1} \).
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Context:
Determine whether the statement "$$\forall\varepsilon\gt0,\exists N\in\mathbb{Z^+},\text{s.t.}\forall n\gt N,\left| a_n-a_N\right|\lt\varepsilon$$" is equivalent to the convergence of the sequence $\{a_n\}$. Answer 'yes' if they are equivalent, otherwise answer 'no'.
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Context:
The right focus of the hyperbola $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ is $F$, and its left and right vertices are $A_{1}$ and $A_{2}$, respectively. A perpendicular line to $A_{1}A_{2}$ passing through $F$ intersects the hyperbola at points $B$ and $C$. If $A_{1}B \perp A_{2}C$, then the slope of the asymptotes of this hyperbola is ______.
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Context:
Suppose \( \{a_1,a_2,\ldots,a_7\} \) is a set of pair-wisely different positive integers. If \( a_1, 2a_2, \ldots, 7a_7 \) can form an arithmetic series (in this order), find the smallest positive value of \(|a_7-a_1|\).
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Context:
For what values of $ \alpha $, does this integral converge? $$ \int_0^\infty x^\alpha \left( 1- e^{-\frac{1}{\sqrt{x}}} \right) \, \mathrm d x $$
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Context:
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that $$\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}$$whenever $\tan 2023x$ is defined. What is $a_{2023}?$

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Context:
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.
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Context:
Consider two totally finite measure spaces $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$. Let $A\subseteq \Omega_1\times \Omega_2$ satisfy the given measurability conditions. If \(\int_{\Omega_1} \mu_2(A_x) \, d\mu_1 = \int_{\Omega_2} \mu_1(A^y) \, d\mu_2 = 0\), determine whether we can conclude that \(\mu_1\times\mu_2(A) = 0\).
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Context:
The 61st question: Find the maximum value of the positive integer $n$, such that for any simple graph of order $n$ with vertices $v_{1}, v_{2}, \ldots, v_{n}$, there always exist $n$ subsets $A_{1}, A_{2}, \ldots, A_{n}$ of the set $\{1,2, \ldots, 2020\}$, satisfying: $A_{i} \cap A_{j} \neq \varnothing$ if and only if $v_{i}$ is adjacent to $v_{j}$.
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Context:
A regular tetrahedron has two spheres that are inscribed in and circumscribed around it. Additionally, in each face of the regular tetrahedron and between it and the circumscribed sphere, there are four spheres touching the center of the face. Given that point P is inside the circumscribed sphere, calculate the probability that P lies inside one of the five small spheres. Express your answer as a decimal value between 0 and 1, rounded to two decimal places.
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Context:
Bin $A$ has one white ball and four black balls.  Bin $B$ has three balls labeled $\$1$ and one ball labeled $\$7$.  Bin $W$ has five balls labeled $\$8$ and one ball labeled $\$500$.  A game is played as follows: a ball is randomly selected from bin $A$.  If it is black, then a ball is randomly selected from bin $B$; otherwise, if the original ball is white, then a ball is randomly selected from bin $W$.  You win the amount printed on the second ball selected.  What is your expected win?
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Context:
Suppose $f_n$ is a sequence of bounded measurable functions on a probability space, and $f_n$ does not converge to $0$ almost everywhere. Determine whether there exists a set of positive measure $E$ such that $\limsup\limits_{n\to\infty}|f_n(x)|>2\epsilon$ for all $x\in E$. Answer "yes" or "no".
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Context:
4.1. Two identical cylindrical vessels are connected at the bottom by a small cross-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg \(/ \mathrm{m}^{3}\), and the density of oil is 700 kg \(/ \mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Provide the answer in centimeters.
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Context:
Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.
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Context:
Consider an analytic function $f$ defined on the open unit disk $\mathbb{D}$ with the power series representation $$f(z) = \sum_{n = 0}^\infty a_n z^n.$$ Define the sequence of functions $f_n(z)$ as $$f_n(z) = a_0 + a_1z + \ldots + a_nz^n.$$ If this sequence converges uniformly to $f$ on the closed unit disk, determine whether the radius of convergence of the power series is strictly greater than 1 or can be exactly 1.
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Context:
The diagram shows an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Find a positive real number $R$ such that for the complex function $p(z) = z^{10} + 100z + 9$, the inequality $|p(z)| > 1$ holds whenever $|z| > R$. Provide a rationale for your choice of $R$ without using complex number manipulations.
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Context:
Regular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find the minimum possible value of $(AM + MN + NA)^2$.
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Context:
Find the limit of the following expression as $n$ approaches infinity: $$\lim_{n \to +\infty} \frac{1}{\sqrt{(n^2)}} + \frac{1}{\sqrt{(n^2+n)}} + \frac{1}{\sqrt{(n^2+2n)}}......+ \frac{1}{\sqrt{(n^2+2n^2)}}$$
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Context:
A circle of radius 2 has center at (2,0). A circle of radius 1 has center at (5,0). A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line?

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Context:
Evaluate the following sum, where $a, b, c$ are natural numbers and $a \leq c$:
$$\sum_{i=0}^{b}\frac{a\binom{b}{i}\binom{c}{a}}{(i+a)\binom{b+c}{i+a}}$$
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Context:
The volume of a cube \( ABCD-A_1B_1C_1D_1 \) is 512. \( M \) and \( N \) are the midpoints of edges \( CB \) and \( BB_1 \), respectively. A plane passing through points \( A \), \( M \), and \( N \) cuts a triangular prism \( B-AMN \). Find the surface area of the triangular prism \( B-AMN \).
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Context:
Determine the linear minimum mean squared error (MMSE) estimator for \( y \) given \( x \) for a 2D random point \((x,y)\) uniformly distributed on a shaded region. Express the estimator in the form \( \hat{y} = wx \).
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Context:
Let $K = \mathbb{C}(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$, where $G \cong \mathbb{Z}$. The field of $G$-invariants, $K^G$, is an extension of $\mathbb{C}$ with transcendence degree $d$. For a generic $n$-tuple $x \in \mathbb{C}^n$, consider the Zariski closure of the $G$-orbit $\{ g(x) \}_{g \in G} \subset \mathbb{C}^n$. Can you conclude that the dimension of this Zariski closure is generically $n-d$?
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Context:
In how many ways can each square of a $2 \times 9$ board be colored red, blue, or green so that no two squares that share an edge are the same color?
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Context:
If $A$ and $B$ are subsets of $\mathbb{N}$ with $\delta(A) = \delta(B) = 0$, does the natural density $\delta(A \cup B)$ exist?
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Context:
Determine the group to which $Gal(\mathbb{Q}(i+\sqrt{2}):\mathbb{Q})$ is isomorphic. The order of this group is 4. Is it isomorphic to the Klein four-group or the cyclic group of order 4?
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Context:
设$S={1,2,⋯,2005}$，若$S$中任意$n$个两两互素的数组成的集合中都至少有一个素数，试求$n$的最小值.
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Context:
Find all values of $n$ such that there exists a rectangle with integer side lengths, perimeter $n$, and area $2n$.
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Context:
Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers?
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Context:
7点多少分的时候，分针落后时针100度．
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Context:
Given three unit vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ in a plane that are not collinear and satisfy $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = \mathbf{0}$. If $0 \leq t \leq 1$, what is the range of values of $|-2 \boldsymbol{a} + t \boldsymbol{b} + (1-t) \boldsymbol{c}|$?
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Context:
Given $D \in \mathbb{R}^{N\times M}$, $\psi\in \mathbb{R}^M$, and $s^0 \in \mathbb{R}^N$, if for all $h\in\mathbb{R}^N$ the relationship $h^T s^0 = h^T D\psi$ holds, does it imply that $s^0 = D\psi$?
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Context:
Calculate the value of $\displaystyle\frac{245^2-225^2}{20}$.
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Context:
A collection of $8$ cubes consists of one cube with edge - length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by $1000$?
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Context:
In a ring $R$, if there exists an element $z \in R$ such that $za = az = z$ for all $a \in R$, does it necessarily follow that $z = 0$?
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Context:
In isosceles $\triangle A B C, A B=A C$ and $P$ is a point on side $B C$. If $\angle B A P=2 \angle C A P, B P=\sqrt{3}$, and $C P=1$, compute $A P$.
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Context:
Given that $\log_{10}2 \approx 0.30103$, ﬁnd the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.
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Context:
A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters?
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Context:
If the inequality $\cos \alpha_{1} \cos \alpha_{2} \cdots \cos \alpha_{n} + \sin \alpha_{1} \sin \alpha_{2} \cdots \sin \alpha_{n} \leqslant M$ always holds, then what is the minimum value of $M$?
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Context:
An eight-sided polygon ABCDEFGH is divided into a rectangle and a square. The side lengths are integers, the area of the square is greater than the area of the rectangle, and the product of the two areas is 98. Calculate the possible perimeter of the polygon ABCDEFGH. Express your answer as a whole number.
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Context:
10.4. Let's call a snake in a convex $n$-gon an open, non-self-intersecting broken line with $n-1$ segments, the set of vertices of which coincides with the set of all vertices of the $n$-gon. Find the number of different snakes in an $n$-gon. (Snakes are equal if they coincide as geometric loci of points. For example, the number of snakes in a triangle is 3)
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Context:
Given three distinct non-zero numbers, Petya and Vasya create quadratic equations by using these numbers as coefficients in various orders. If a quadratic equation has at least one root, Petya gets a candy wrapper. If it has no roots, Vasya gets a candy wrapper. The first three candy wrappers went to Petya, and the next two to Vasya. Determine who will get the last, sixth candy wrapper.
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Context:
Find the greatest positive real number  $M$  such that for all positive real sequence  $(a_n)$  and for all real number  $m < M$ , it is possible to find some index  $n \ge 1$  that satisfies the inequality  $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$ .
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Context:
The cricket team of 11 members is 25 years old and the wicket keeper is older by a certain number of years. If the ages of these 2 are excluded, the average age of the remaining players is 1 year less than the average age of the whole team. The average age of the team is 22 years. How much older is the wicket keeper?
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Context:
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
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Context:
A lock code is made up of four digits that satisfy the following rules: - At least one digit is a 4, but neither the second digit nor the fourth digit is a 4. - Exactly one digit is a 2, but the first digit is not 2. - Exactly one digit is a 7. - The code includes a 1, or the code includes a 6, or the code includes two 4s. How many codes are possible?
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Context:
Find the minimum value of the expression \(\left\lfloor \frac{8(a+b)}{c} \right\rfloor + \left\lfloor \frac{8(a+c)}{b} \right\rfloor + \left\lfloor \frac{8(b+c)}{a} \right\rfloor\), where \(a\), \(b\), and \(c\) are arbitrary natural numbers.
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Context:
King Qi and Tian Ji are having a horse race. Tian Ji's top horse is better than King Qi's middle horse but worse than King Qi's top horse. Tian Ji's middle horse is better than King Qi's lower horse but worse than King Qi's middle horse. Tian Ji's lower horse is worse than King Qi's lower horse. Now, if a horse is randomly selected from both sides for a race, what is the probability of Tian Ji winning? Express your answer as a simplified fraction.
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Context:
A dishonest dealer professes to sell his goods at cost price, but uses varying weights for different items. For potatoes, he uses a weight of 900 grams instead of a kilogram. For onions, he uses a weight of 850 grams instead of a kilogram, and for carrots, he uses a weight of 950 grams for each kilogram. In one day, he sells 10 kilograms of potatoes, 15 kilograms of onions, and 25 kilograms of carrots at their respective cost prices. Calculate his overall gain percentage for that day.
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Context:
Let $f$ be a function taking the positive integers to the positive integers, such that
\[f(mf(n)) = nf(m)\]for all positive integers $m$ and $n.$  Find the smallest possible value of $f(2007).$
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Context:
For a positive integer n, 5^n is a factor of a certain factorial, but 5^(n+1) is not a factor of the same factorial. The value of n is 4. What is the factorial?
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Context:
Given a bounded sequence $\{x_n\}$ with limit superior $x^*$ and limit inferior $x_*$, determine whether for every $c \in [x_*,x^*]$, there exists a subsequence of $\{x_n\}$ that converges to $c$. Answer "yes" or "no".
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Context:
Square \(ABCD\) has sides of length \(4\), and \(M\) is the midpoint of \(\overline{CD}\). A circle with radius \(2\) and center \(M\) intersects a circle with radius \(4\) and center \(A\) at points \(P\) and \(D\). Find the distance from \(P\) to \(\overline{AD}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
设复数 $x, y, z$ 满足: $|x+2 y+3 z|=1$ 。
求 $|x|^{2}+|y|^{2}+|z|^{2}+\left|x^{2}+y^{2}+z^{2}\right|$ 的最小值.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
4. How many sets of three consecutive three-digit natural numbers have a product that is divisible by 120?
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Context:
In a 100 x 100 grid, Petya and Vasya play a game where Petya starts by coloring one cell black. On each subsequent turn, a player can color a vertical or horizontal white strip of size $1 \times n$ black, where $n$ is either equal to or one more than the number of cells colored by the previous player. The player who cannot make a move loses. Determine the winner if both play optimally.
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Context:
In a commutative ring $R$ with unity, consider two distinct maximal ideals $I$ and $J$. Determine if the ideal generated by $I^2$ and $J$, denoted by $(I^2, J)$, is equal to the entire ring $R$.
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Context:
Let $g:[a,b]\to\mathbb{R}$ be a Riemann-integrable function. If there exists a dense subset $A\subseteq[a,b]$ such that $g(x)=0$ for all $x\in A$, is it always true that $\int_{a}^{b} g = 0$?
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Context:
Determine whether the expression \( \frac{1^\alpha + 2^\alpha + \cdots + n^\alpha}{n^{\alpha+1}} \) diverges to infinity as \( n \to \infty \) for \( \alpha \in (-\infty, -1) \).
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Context:
What is the sum of the six positive integer factors of 30, each multiplied by 2?
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Context:
Given an arithmetic sequence $\{a_n\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for which $S_n > 0$ holds is ______.
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Context:
What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle?
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Context:
Calculate the number of different ways to arrange 2014 points into either a triangular or a trapezoidal point lattice, where the difference in the number of points between two adjacent rows is always 1, and there are at least two layers. Express your answer as a single integer.
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Context:
In the first 3 seconds, Kevin can dribble a basketball 13 times while moving straight ahead. As he continues, he encounters a zigzag-shaped path with obstacles: during the next 5 seconds, he only dribbles 18 times while navigating the path. If Kevin continues to dribble for the remainder of the 27 seconds, slowing down at an arithmetic decreasing rate of 3 fewer dribbles for each additional 5 seconds as he navigates through three more increasingly complex obstacles, how many times will he dribble the basketball in total?
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Context:
The military kitchen needs 1000 jin of rice and 200 jin of millet for dinner. Upon arriving at the rice store, the quartermaster finds a promotion: "Rice is 1 yuan per jin, with 1 jin of millet given for every 10 jin purchased (fractions of 10 jins do not count); Millet is 2 yuan per jin, with 2 jins of rice given for every 5 jin purchased (fractions of 5 jins do not count)." How much money does the quartermaster need to spend to buy enough rice and millet for dinner?
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Context:
Two swimmers start at opposite sides of a 90-foot swimming pool. One swims at 3 feet per second, while the other swims at 2 feet per second. They swim back and forth for 12 minutes without taking any time to turn around. Calculate the number of times they meet. Express your answer as a whole number.
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Context:
Problem 11-2. Mr. $A$ stood at the point with coordinates $(0,0)$ for an hour. During the same hour, moving uniformly and in a straight line, Mr. $B$ traveled from the point $(22,0)$ to the point $(2,20)$. During this same hour, Miss $C$, also moving uniformly and in a straight line, traveled from the point $(30,4)$ to the point $(0,24)$. How many times during the observation period did the area of triangle $A B C$ take on integer values? The initial and final moments are included.
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Context:
Consider the set of $30$ parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
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Context:
设 S 是一个 35 元集合，\mathcal{F} 是一个由一些 S 到自身的映射组成的集合. 对正整数 k，称 \mathcal{F} 具有性质 P(k)，如果对任意 x,y\in S，都存在 \mathcal{F} 中的 k 个映射f_1, f_2, \cdots, f_k（可以相同），使得f_k(\cdots(f_2(f_1(x)))\cdots)=f_k(\cdots(f_2(f_1(y)))\cdots).求最小的正整数 m，使得所有具有性质 P(2019) 的集合 \mathcal{F} 都具有性质 P(m).
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Context:
The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
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Context:
A middle school holds a sports meeting, with participation by grade. There are three events: high jump, long jump, and 100m sprint, with the top three in each event receiving points: 5 points for first place, 3 points for second place, and 1 point for third place. It is known that the total scores of the seventh and eighth grades are equal, and they are tied for first place. Moreover, the number of students from the eighth grade who placed in the top three is twice that of the seventh grade. What is the total score of the ninth grade?
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Context:
9. Cars A and B depart from locations $A$ and $B$ respectively at the same time, and travel back and forth between $A$ and $B$ at a constant speed. If after the first meeting, Car A continues to drive for 4 hours to reach $B$, while Car B only drives for 1 hour to reach $A$, then when the two cars meet for the 15th time (excluding meetings at $A$ and $B$), they have driven $\qquad$ hours.
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Context:
If $q^k n^2$ is an odd perfect number with special prime $q$, is $\sigma(q^k)$ coprime to $\sigma(n^2)$? Assume $q \equiv k \equiv 1 \pmod{4}$ and $\gcd(q,n)=1$. The function $\sigma(x)$ denotes the sum of divisors of $x$. Determine if the conjecture $\gcd(\sigma(q^k),\sigma(n^2))>1$ holds true.
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\boxed{\gcd(\sigma(q^k), \sigma(n^2)) > 1}


Context:
Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$
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Context:
Xiao Ming places several chess pieces into a $3 \times 3$ grid of square cells. Each cell can have no pieces, one piece, or more than one piece. After calculating the total number of pieces in each row and each column, we obtain 6 different numbers. What is the minimum number of chess pieces required to achieve this?
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Context:
Let  $a_1 = \sqrt 7$  and  $b_i = \lfloor a_i \rfloor$ ,  $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$  for each  $i\geq i$ . What is the smallest integer  $n$  greater than  $2004$  such that  $b_n$  is divisible by  $4$ ? ( $\lfloor x \rfloor$  denotes the largest integer less than or equal to  $x$ )
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Context:
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\).
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Context:
Evaluate the following double integral:

$$\int_0^{\pi / 2} \int_y^{2 y} \frac{\sin x}{x} d x d y+\int_{\pi / 2}^\pi \int_y^\pi \frac{\sin x}{x} d x d y$$
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Context:
Evaluate the series: $$\sum\limits_{n = 1}^\infty {(2n+1)\frac{{{{(\alpha )}^{2n}}}}{{(2n)!}}}$$
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\boxed{\(\alpha \sinh \alpha + \cosh \alpha - 1\)}


Context:
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is

$
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Context:
Minimize \(x^2 - y^2 - z^2\) subject to the constraint \(x^4 + y^4 + z^4 \leq 1\) using the Karush-Kuhn-Tucker (KKT) conditions. Determine the KKT points and find the minimum value of the objective function.
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Context:
A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule these 6 projects.
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Context:
Find the time period for which a certain sum of money was invested at 5% per annum if the amount being Rs. 896 and the principal is Rs. 799.9999999999999?
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Context:
Let $ABCDEFG$ be a regular heptagon with center $O$.  Let $M$ be the centroid of triangle $ABD$.  Find $\cos^2 \angle GOM$.
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Context:
How many pairs of positive integers(a,b)are there such that the roots of polynomial\( {x}^{2} - {ax} - b \) are not greater than 5 ?

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Context:
For every positive integer  $n$ , determine the biggest positive integer  $k$  so that  $2^k |\ 3^n+1$
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Context:
Let \(E\) be a locally compact second-countable metric space, and let \(E^\ast = E \cup \{\infty\}\) denote the Alexandroff one-point extension. Consider a sequence \((x_n)_{n \in \mathbb{N}} \subseteq E^\ast\) and a point \(x \in E^\ast\) such that \(x_n \xrightarrow{n \to \infty} x\). Let \(B_n \subseteq E^\ast\) be open sets with \(x_n \in B_n\) for each \(n \in \mathbb{N}\) and \(\bigcap_{n \in \mathbb{N}} B_n = \{\infty\}\). Assume that the sequence \((B_n)_{n \in \mathbb{N}}\) is nonincreasing. Determine if it follows that \(x_0 = \infty\).
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Context:
Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 12th list differed from the 11th list. How many draws were there?
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Context:
Compute
\[\frac{5}{3^2 \cdot 7^2} + \frac{9}{7^2 \cdot 11^2} + \frac{13}{11^2 \cdot 15^2} + \dotsb.\]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:
How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have?
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Context:
Determine whether the limit \( \lim_{x\to a^+}\frac{\cos{x}\ln(x-a)}{\ln(e^x-e^a)} \) exists.
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Context:
How many integer numbers are no greater than $2004$, with remainder $1$ when divided by $2$, with remainder $2$ when divided by $3$, with remainder $3$ when divided by $4$, and with remainder $4$ when divided by $5$?

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Context:
Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$.
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Context:
If the circle $(x-a)^2+(x+4)^2=9$ is tangent to the y-axis, then $a=$ ___.
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Context:
A large tanker can be filled by two pipes A and B in 60 minutes and some minutes respectively. It takes 29.999999999999993 minutes to fill the tanker from an empty state if B is used for half the time and A and B fill it together for the other half. How many minutes does it take for pipe B to fill the tanker alone?
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Context:
Sixty $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table. Each cube has a total of 6 faces. When a single cube is placed on the table, the bottom face is not exposed, leaving 5 exposed faces. However, when the cubes are joined in a row, some faces between them are not exposed. Calculate the total number of exposed $1 \times 1$ faces for the 60 joined cubes. Express your answer as a whole number.
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Context:
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
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Context:
Let $p(x)$ be defined on $2 \le x \le 10$ such that $$p(x) = \begin{cases} x + 1 &\quad \lfloor x \rfloor\text{ is prime} \\ p(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ where $y$ is the greatest prime factor of $\lfloor x\rfloor.$ Express the range of $p$ in interval notation.
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Context:
A dealer purchases 15 articles of type A for Rs. 25 each and 20 articles of type B for Rs. 35 each. If he gets a discount of 5% on bulk purchase of type A articles and a 10% discount on type B articles, he then sells 12 articles of type A and 18 articles of type B. On the day of sale, due to the fluctuating market, he has to increase the selling price of type A articles by 10% which is Rs. 33 each but had to decrease the selling price of type B articles by 15% which becomes Rs. 45 each. Find the overall profit or loss percentage the dealer made from these transactions?
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Context:
Given two permutations of the set \( \{1, 2, \ldots, N\} \) where \( 3 \leq N \leq 1000 \), find the number of permutations \( X_1, X_2, \ldots, X_N \) such that when arranged in a 3xN matrix with the two given permutations, each column contains unique elements.

For example, for \( N = 4 \):

First permutation: \( \begin{pmatrix}3 & 1 & 2 & 4\end{pmatrix} \).
Second permutation: \( \begin{pmatrix}2 & 4 & 1 & 3\end{pmatrix} \).

Determine the number of possible permutations for the third row \( \begin{pmatrix}X_1 & X_2 & X_3 & X_4\end{pmatrix} \) such that:
\( X_1 \neq 3, 2 \),
\( X_2 \neq 1, 4 \),
\( X_3 \neq 2, 1 \),
\( X_4 \neq 4, 3 \).

The answer for this example is 2, with possible permutations \( \begin{pmatrix}1 & 3 & 4 & 2\end{pmatrix} \) and \( \begin{pmatrix}4 & 2 & 3 & 1\end{pmatrix} \).
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Context:
Five people stand in a line, each wearing a different hat numbered $1, 2, 3, 4, 5$. Each person can only see the hats of the people in front of them. Xiao Wang cannot see any hats; Xiao Zha can only see hat $4$; Xiao Tian does not see hat $3$, but can see hat $1$; Xiao Yan can see three hats, but not hat $3$; Xiao Wei can see hat $3$ and hat $2$. What number hat is Xiao Wei wearing?
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Context:
If the graph of the function $f(x)$ passes through the point $(0, 1)$, then the graph of the inverse function of $f(x+3)$ must pass through the point ______.
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Context:
Find a condition on the real numbers \(a, b, c\) such that the following equality holds:

\[ \alpha = \beta \]

where:

\[ \alpha = \sqrt{a} + \sqrt{2b + 2\sqrt{a}} \]
\[ \beta = \sqrt{b + 2\sqrt{c}} + \sqrt{a + b - 2\sqrt{c} + 2\sqrt{ab - 2a\sqrt{c}}} \]

and \(a = b^2 - 4c\). Assume all square roots are well-defined.
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Context:
Find the greatest integer \( n \) ,such that there are \( n + 4 \) points \( A,B,C,D,{X}_{1},\ldots ,{X}_{n} \) in the plane with \( {AB} \neq  {CD} \) that
 satisfy the following condition: for each \( i = 1,2,\ldots ,n \) triangles \( {AB}{X}_{i} \) and \( {CD}{X}_{i} \) are equal
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\boxed{
The greatest integer \( n \) is 4.

}


Context:
复平面上，非零复数 $z_{1}, z_{2}$ 在以 $i$ 为圆心， 1 为半径的圆上， $\overline{z_{1}} z_{2}$ 的实部为零， $z_{1}$ 的辐角主值为 $\frac{\pi}{6}$ ，则 $z_{2}=$。请注意，$z_{2}$ 的原始答案为$-\frac{\sqrt{3}}{2}+\frac{3}{2}i$，请给出虚部分子和分母之和。
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Context:
For all \( n \in \mathbb{N}^{*} \), let \( p_{n} \) be the number of sequences of \( A \) and \( B \) of length \( n \) containing neither \( AAAA \) nor \( BBB \). Calculate

$$
\frac{p_{2004}-p_{2002}-p_{1999}}{p_{2000}+p_{2001}}
$$
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Context:
Determine whether \( \frac{\partial}{\partial x} f(x,y(x)) \) is equal to \( x \) or \( 2x \) given that \( f(x,y) = xy^2 \) and \( y^2 = x \).
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Context:
Given \(\frac{ab}{a+b}=3\), \(\frac{bc}{b+c}=6\), and \(\frac{ac}{a+c}=9\), determine the value of \(\frac{c}{ab}\).
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Context:
Three. (50 points) Given that the difference between any two adjacent interior angles of a convex $n$-sided polygon is $20^{\circ}$. Find the maximum value of $n$.

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Context:
A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.
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Context:
Solve the following equation for $x , y , z \in \mathbb{N}$ :
\[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]
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Context:
Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number.
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Context:
Find the number of ways to assign 15 students to 5 classes such that each class has at least 2 students. Ensure that any overcounting errors are corrected in your solution.
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Context:
Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median.
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Context:
The altitudes of triangle \(ABC\) drawn from vertices \(A\) and \(C\) intersect at point \(M\). Find \(\angle AMC\) if \(\angle A = 70^\circ\) and \(\angle C = 80^\circ\).
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Context:
Find the value of ( 85 + 32 / 113 ) × 113. What is the result?
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Context:
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,

$$
a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}
$$

Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{2011}$. What is the sum of all integers $k$, $1\le k \le 2011$, such that $a_k=b_k?$

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Context:
Determine whether the series \(\sum_{n=1}^\infty n^r \exp\left(-k\sum_{m=1}^n \frac{1}{m}\right)\) converges when \(r-k < -1\) or \(r<k\), using the approximation \(\frac{u_{n+1}}{u_n} \approx 1 + \frac{r-k}{n}\).
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Context:
A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\{a, a+b, a+b+c, a+b+c+d\}$ is equal to 4?
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Context:
In a game involving a 5x9 rectangular grid, Petya starts by cutting the grid into two parts along the border lines of the cells, forming an "L" shape with two perpendicular segments. Vasya then cuts one of the resulting figures in the same manner, followed by Petya with one of the three resulting figures, and so on. The player who cannot make a move loses. Determine the winner of the game, Petya or Vasya, regardless of the moves made by the other player.
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Context:
Determine the number of ordered quintuples $(a,b,c,d,e)$ of integers with $0\leq a<$ $b<$ $c<$ $d<$ $e\leq 30$ for which there exist polynomials $Q(x)$ and $R(x)$ with integer coefficients such that\[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\]
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Context:
The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a \(0.5\) probability of winning any game), find the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\).
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Context:
In the diagram, there is a map consisting of 8 cities connected by 12 roads. How many ways are there to close 5 roads simultaneously for repairs such that it is still possible to travel from any city to any other city?
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Context:
Given the circle \( M: 2x^{2} + 2y^{2} - 8x - 8y - 1 = 0 \), and the line \( l: x + y - 9 = 0 \), let \( A \) be a point on \( l \). Construct \(\triangle ABC\) such that \(\angle BAC = \frac{\pi}{4}\), with side \( AB \) passing through the center of the circle \( M \) and points \( B \) and \( C \) lying on the circle \( M \). Determine the range of the y-coordinate of point \( A \).
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Context:
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $
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Context:
Let $a, b, c$ be orthonormal vectors in $\mathbb{R}^n$. Determine whether the sum $\sum_{k=1}^n a_{k}b_{k}c_{k}$ is equal to zero.
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Context:
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
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Context:
Let  $ABCD$  be a unit square. For any interior points  $M,N$  such that the line  $MN$  does not contain a vertex of the square, we denote by  $s(M,N)$  the least area of the triangles having their vertices in the set of points  $\{ A,B,C,D,M,N\}$ . Find the least number  $k$  such that  $s(M,N)\le k$ , for all points  $M,N$ .

*Dinu Șerbănescu*
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Context:
Triangle $\vartriangle ABC$ has sidelengths $AB = 10$, $AC = 14$, and, $BC = 16$. Circle $\omega_1$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $B$. Circle $\omega_2$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $C$. Let $\omega_1$, $\omega_2$ intersect at points $X, Y$ . The square of the perimeter of triangle $\vartriangle AXY$ is equal to $\frac{a+b\sqrt{c}}{d}$ , where $a, b, c$, and, $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
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Context:
Given the group \(G=\mathbb{Z}_6 \times \mathbb{Z}_3\), the element \(a=(4,1)\) of \(G\), the subgroup \(H=\langle a\rangle\) of \(G\), and the group \(G_1=G/H\), find the order \(k\) of the class of \((3,1)\) in \(G_1\) and the order \(\ell\) of the class of \((2,2)\) in \(G_1\). What is \((k+\ell) \mod{5}\)?
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Context:
If sequences $x_n$ and $y_n$ converge almost surely to constants $a$ and $b$ respectively, does the sequence $x_n + y_n$ converge almost surely to $a + b$?
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Context:
A light ray falls at an angle $\alpha=60^{\circ}$ on the front surface of a plane-parallel glass plate. The refractive index of the glass is $n=1.6$. By what angle from the direction of the incident ray is the ray, reflected from the back surface of the plate and exiting back through the front surface, deviated?
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Context:
At a table tennis team meeting, each member sits on a stool. There are two types of stools: square stools (with four legs) or round stools (with three legs). A child walks into the meeting room and counts a total of 33 legs (not including the child's own legs). How many team members are attending the meeting?
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Context:
At a race track, a maximum of 6 horses can race together at a time. There is a total of 30 horses. Races not only depend on speed, but also rely on track conditions and unpredictable horse behavior during each race. Track conditions vary with each race. What is the minimum number w of races we need to conduct to confidently determine the top 3 fastest horses under different conditions?
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Context:
A centaur, a chess piece that can move one square up, left, or diagonally up-right, stands on the square marked with a star on a $1000 \times 1000$ board. Four rectangles $2 \times 994$ have been removed from the board (as shown in the diagram). Two players take turns moving the centaur. The player who cannot make a move loses. Who wins with optimal play?
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Context:
Problem 6.3. A country has the shape of a square and is divided into 25 identical square counties. In each county, either a knight-count, who always tells the truth, or a liar-count, who always lies, rules. One day, each count said: "Among my neighbors, there are an equal number of knights and liars." What is the maximum number of knights that could have been? (Counts are neighbors if their counties share a common side.)
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Context:
A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $A B$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 5 more than the other.
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Context:
A list of six positive integers \( p, q, r, s, t, u \) satisfies \( p < q < r < s < t < u \). The sums of the 15 pairs of numbers that can be formed by choosing two different numbers from this list are:

\[ 25, 30, 38, 41, 49, 52, 54, 63, 68, 76, 79, 90, 95, 103, 117 \]

Calculate the sum \( r+s \) using the given information. Express your answer as a single integer.
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Context:
圆周上 \(A_{1}, A_{2}, \cdots, A_{7}\) 七个点两两相连。求任选两条线段，这两条线段无公共点的概率。答案的形式为\(\frac{k}{m}\)，请计算k+m的值。
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Context:
Compare the following numbers: $2\sqrt{2}-3\_\_\_\_\_\_0.$
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Context:
What is the maximum number of rooks that can be placed on an 8x8 chessboard such that each rook attacks at most one other? A rook attacks all squares on its file and rank.
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Context:
Viola has been practising the long jump. At one point, the average distance she had jumped was $3.80 \mathrm{~m}$. Her next jump was $3.99 \mathrm{~m}$, and that increased her average to $3.81 \mathrm{~m}$. After the following jump, her average had become $3.82 \mathrm{~m}$. Calculate the distance of her final jump. Express your answer in meters, rounded to two decimal places.
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Context:
The number $K$ is divisible by 7 and its binary representation is given by:

$$
K = 10110101010101xyz110
$$

Determine the digits $x$, $y$, and $z$.
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Context:
To make nine circular cardboard discs for Carnival, square sheets with sides of 10 cm are used, as indicated in the figure. Calculate the area (in cm²) of the unused paper after cutting out the circular discs. Express your answer as a decimal number, rounded to one decimal place.
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Context:
Determine all real numbers $a > 0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region \[ R = \{ (x,y) ; 0 \le x \le a, 0 \le y \le f(x) \} \] has perimeter $k$ units and area $k$ square units for some real number $k$.
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Context:
If $a+b=1$, find the supremum of $$- \frac {1}{2a}- \frac {2}{b}.$$
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Context:
设复数 $z$ 满足 $|3 z-7 \mathrm{i}|=3$, 求 $\left|\frac{z^{2}-2 z+2}{z-1+\mathrm{i}}\right|$ 的最大值。答案的形式为 $\frac{k}{m}$，请给出 $k + m$ 的值。
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Context:
Given the inequality $x^{3}-3x+3- \frac{x}{e^{x}}-a\leqslant 0$ with respect to $x$, where $x\geqslant -2$, find the minimum value of the real number $a$. Express your answer as a single value.
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Context:
Suppose that $A$ is an abelian group with order $p^4$ where $p$ is a prime number. And there exists a subgroup $N$ such that $A/N\simeq \mathbb{Z}/p^3\mathbb{Z}$. How many abelian groups $A$ satisfy the condition above?
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Context:
Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?
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Context:
Find the numbers \( x \) between 0 and 30 for which the sine of \( x \) degrees equals the sine of \( x \) radians. How many such numbers exist between 30 and 90?
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Context:
Tom, Dick, and Harry started out on a $100$-mile journey. Tom and Harry traveled by automobile at a speed of $25$ mph, while Dick walked at a speed of $5$ mph. After a certain distance, Harry got off and continued walking at $5$ mph, while Tom went back for Dick and brought him to the destination at the same time that Harry arrived. Find the number of hours required for the trip.
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Context:
Find the last 5 digits of the number $M = 5^{2010}$.
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