Context:
Given proposition $p$: The equation $\frac{x^2}{2} + \frac{y^2}{m-1} = 1$ represents an ellipse with foci on the $y$-axis, and proposition $q$: The equation in $x$, $x^2 - 2x + 4m = 0$, has no real roots. If $\neg p$ is true, and $p \lor q$ is true, find the range of the real number $m$.
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Context:
(Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \ldots$ in that order. You start at the square labeled $0$. You want to end up at the square labeled $3$. In how many ways can this be done in exactly $15$ moves?
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Context:
A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Calculate the minimum value of $a$. Express your answer as a single value.
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Context:
Given that \( a \) and \( b \) are integers, and the equation \( x^{2} + a x + 2 b^{2} = 0 \) has two roots \( \sin \alpha \) and \( \cos \alpha \) where \( 0 \leqslant \alpha < 2 \pi \), then what are all possible values of \( a + b \)?
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Context:
规定抽球试验规则如下：盒子中初始装有白球和红球各一个，每次有放回地任取一个，连续取两次，将以上过程记为一轮.如果每一轮取到的两个球都是白球，则记该轮为成功，否则记为失败.在抽取过程中，如果某一轮成功，则停止；否则，在盒子中再放入一个红球，然后接着进行下一轮抽球，如此不断继续下去，直至成功.

$附：线性回归方程\hat{y}=\hat{b}x+\hat{a}中，\hat{b}=\frac{\sum \limits^{n}_{{i=1}}x_iy_i-n\overline{x} \cdot \overline{y}}{\sum \limits^{n}_{{i=1}}x^2_i-n{\overline{x}}^2}，\hat{a}=\overline{y}-\hat{b}\overline{x}；$

$参考数据：\sum \limits^{5}_{{i=1}}x^2_i=1.46，\overline{x}=0.46，\overline{x}^2=0.212 \left(\begin{matrix}\text{其中}x_i=\frac{1}{t_i},\overline{x}=\frac{1}{5}\sum \limits^{5}_{{i=1}}x_i\end{matrix}\right).$
$某人进行该抽球试验时，最多进行三轮，即使第三轮不成功，也停止抽球，记其进行抽球试验的轮次数为随机变量X，求X的数学期望；$
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Context:
in business , a and c invested amounts in the ratio 2 : 1 , whereas the ratio between amounts invested by a and b was 2 : 3 , if rs 15,0000 was their profit , how much amount did b receive .
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Context:
A man rows at a speed of $2$ mph in still water. He sets out on a trip towards a spot $2$ miles downstream. He rows with the current until he is halfway there, then turns back and rows against the current for $15$ minutes. Then, he turns around again and rows with the current until he reaches his destination. The entire trip takes him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph, where $p$ and $q$ are relatively prime positive integers. Find $10p + q$.
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Context:
a typist uses a sheet measuring 20 cm by 30 cm lengthwise . if a margin of 2 cm is left on each side and a 3 cm margin on top and bottom , then percent of the page used for typing is
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Context:
Let $n$ be the 200th smallest positive real solution to the equation $x-\frac{\pi}{2}=\tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.
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Context:
If a non-negative integer \( m \) and the sum of its digits are both multiples of 6, then \( m \) is called a "Liuhe number." Find the number of Liuhe numbers less than 2012.
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Context:
Find the maximum value of the positive real number \( A \) such that for any real numbers \( x, y, z \), the inequality
$$
x^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \geq 0
$$
holds.
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Context:
How many ordered pairs of integers \((x, y)\) satisfy the equation:
\[
8(x^3 + x^2y + xy^2 + y^3) = 15(x^2 + y^2 + xy + 1)
\]
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Context:
Given an angle equal to $\alpha$. A point $K$ is taken on its bisector; $P$ and $M$ are the projections of $K$ onto the sides of the angle. On the segment $PM$, a point $A$ is taken such that $KA = a$. A line passing through $A$ perpendicularly to $KA$ intersects the sides of the angle at points $B$ and $C$. Find the area of the triangle $BKC$.
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\boxed{a^{2} \cot \left( \frac{\alpha}{2} \right)}


Context:
Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then
$$a_{k+1} = \frac{m + 18}{n+19}.$$
Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$.
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Context:
Evaluate the limit using L'Hopital's rule: $$\lim_{w \to 0} \frac{(1+w^2)\ln\left(\frac{1+w}{1-w}\right)- 2w}{w^2}$$
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Context:
Determine the range of values for the real number $k$ such that the line $y=k(x-2)+4$ intersects the curve $y=\sqrt{4-x^2}$ at some point(s).
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\boxed{k \in \left[\frac{3}{4}, +\infty\right)}


Context:
In the arithmetic sequence $\{a_n\}$, $a_3=0$. If $a_k$ is the geometric mean of $a_6$ and $a_{k+6}$, then $k=$ ______.
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Context:
A circular flower bed has a circumference of 314 meters. A 2-meter wide path is built around this flower bed. What is the area of this path in square meters?
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Context:
For the expression \(\left[\left(\frac{\sqrt{3}+i}{2}\right)^{8}+1\right]^{n}\), where \(n\) takes on the values from 1 to 100, calculate the number of distinct values the expression can take. Express your answer as a single integer.
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Context:
Given a cyclic quadrilateral \(ABCD\) where \(AB = BC = AD + CD\) and \(\angle BAD = \alpha\), with \(AC = d\). Find the area of \(\triangle ABC\).
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Context:
Given the hyperbola $x^2-y^2+1=0$ and the parabola $y^2=(k-1)x$, find the range of $k$ such that the two curves have at most two points in common. Express your answer as an interval, including the endpoints if they are included in the range.
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Context:
On the sides $BC$ and $CD$ of the square $ABCD$ with side length $1$, points $E$ and $F$ are chosen such that $\angle EAB = 20^\circ$ and $\angle EAF = 45^\circ$. Calculate the distance from point $A$ to the line $EF$.
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Context:
已知数列 $\left\{a_{n}\right\}:$
$a_{n}=2^{n}+3^{n}+6^{n}+1\left(n \in \mathbf{Z}_{+}\right)$.
是否存在整数 $k \geqslant 2$ ，使得 $k$ 与数列 $\left\{a_{n}\right\}$中的所有数均互素？若存在，找出最小的整数 $k$ ；若不存在，说明理由.
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Context:
How many consecutive tickets must be purchased at a bus cashier to be sure to get a lucky one? A ticket is called lucky if the sum of its first three digits equals the sum of its last three digits. The number of tickets at the cashier is unlimited.
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Context:
Zhang Hua, Li Liang, and Wang Min each sent out x, y, z New Year's cards, respectively. If it is known that the least common multiple of x, y, z is 60, the greatest common divisor of x and y is 4, and the greatest common divisor of y and z is 3, then how many New Year's cards did Zhang Hua send out?
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Context:
Let  $p(x)$  be a polynomial of degree strictly less than  $100$  and such that it does not have  $(x^3-x)$  as a factor. If   $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$   for some polynomials  $f(x)$  and  $g(x)$  then find the smallest possible degree of  $f(x)$ .
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Context:
Abbot writes the letter $A$ on the board. Every minute, he replaces every occurrence of $A$ with $A B$ and every occurrence of $B$ with $B A$, hence creating a string that is twice as long. After 10 minutes, there are $2^{10}=1024$ letters on the board. How many adjacent pairs are the same letter?
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Context:
$\left[\begin{array}{l}{[\text { Theory of algorithms (other). }} \\ {[\quad \text { Estimation + example }}\end{array}\right]$

There are 2004 boxes on the table, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee identifying a box that contains a white ball?
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Context:
Find the number of positive integers $n$ between $1$ and $1000$, inclusive, satisfying
\[ \lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor\]
where $\lfloor n \rfloor$ denotes the greatest integer not exceeding $n$.
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Context:
What is the largest power of 1991 which divides \( 1990m + 1992n \), where \( m = 1991^{1992} \) and \( n = 1991^{1990} \)?
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Context:
After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The original answer is in the format k:m, where k and m are integers. Please find the value of k + m.
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Context:
Xiaoqiang and Xiaohua, two classmates, agreed to meet beside the fountain at Daliang Clock Tower Park in the afternoon. They agreed that whoever arrives first must wait for 10 minutes, and if the other person has not arrived by then, they can leave. If Xiaoqiang arrives between 1:20 and 2:00, and assuming Xiaohua arrives sometime between 1:00 and 2:00, with the arrival time within this hour being equally likely, then the probability that they will meet is ______.
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Context:
You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled $1,2, \ldots$. Each ball has probability $1 / 2^{n}$ of being put into box $n$. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls?
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Context:
设关于$\textit{x}$的不等式$a{{x}^{2}}+8(a+1)x+7a+16\ge 0,(a\in Z)$，只有有限个整数解，且0是其中一个解，则全部不等式的整数解的和为____________
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Context:
The formula for converting a Fahrenheit temperature  $F$  to the corresponding Celsius temperature  $C$  is  $C=\frac{5}{9}(F-32)$ .  An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer.  For how many integer Fahrenheit temperatures  $T$  with  $32 \leq T \leq 1000$  does the original temperature equal the final temperature?
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Context:
A book library charges fifty cents per day on any book borrowed by its members to read outside the library's premises. At the start of May, Celine borrowed three books from the library to read at home for her upcoming examinations. She returned one book 20 days after borrowing it, and the other two stayed at her house for a certain period. She paid a total of $41 for lending the three books. When did she return the other two books?
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Context:
Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$.
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Context:
A bag contains 8 yellow marbles, 7 red marbles, and 5 black marbles. Without looking in the bag, Igor removes N marbles all at once. What is the maximum possible value of N such that, no matter which choice of N marbles he removes, there are at least 4 marbles of one color and at least 3 marbles of another color left in the bag? Express your answer as a single integer.
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Context:
Find the largest $A$ for which there exists a polynomial
$$
P(x)=A x^{4}+B x^{3}+C x^{2}+D x+E
$$
with real coefficients, which satisfies
$$
0<P(x)<1 \text { for }-1<x<1
$$
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Context:
Dana Point beach has four times the number of sharks as Newport Beach, and Huntington Beach has half the number of sharks as Dana Point beach. If Newport Beach has 22 sharks and the total shark population across the three beaches was initially 65, what is the percentage increase or decrease in shark populations across the three beaches in comparison to the initial population?
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Context:
Find the mass percentage of H in Dihydrogen monoxide
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Context:
On the board, there are two-digit numbers. Each number is composite, but any two numbers are coprime. What is the maximum number of such numbers that can be written?
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Context:
How many unique pairs of parallel edges does a regular octahedron have?
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Context:
We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to purchase gifts from all three stores and still have money left?
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Context:
Yvon has 4 different notebooks and 5 different pens. Determine the number of different possible combinations of notebooks and pens he could bring.
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Context:
When a cone is divided by two planes parallel to its base such that its height is cut into three equal parts, what is the ratio of the volumes of the three parts of the cone?
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Context:
Michel starts with the string $H M M T$. An operation consists of either replacing an occurrence of $H$ with $H M$, replacing an occurrence of $M M$ with $M O M$, or replacing an occurrence of $T$ with $M T$. For example, the two strings that can be reached after one operation are $H M M M T$ and $H M O M T$. Compute the number of distinct strings Michel can obtain after exactly 10 operations.
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Context:
对任意正实数 $a$ ，记函数 $f(x)=|\lg x|$ 在 $[a,+\infty)$ 上的最小值为 $m_{a}$, 函数 $g(x)=\sin \frac{\pi x}{2}$ 在 $[0, a]$ 上的最大值为 $M_{a}$. 若 $M_{a}-m_{a}=\frac{1}{2}$,  $a$ 的值可能为$\frac{a}{b}$ 或 $\sqrt{c}$，请给出 $a + b + c$ 的值。
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Context:
Given two sets $A$ and $B$ that satisfy the conditions $A\cap B \neq \emptyset$ and $A\cup B = \{1, 2, 3, 4, 5\}$. When $A \neq B$, the pair $(A, B)$ and $(B, A)$ are considered as two different pairs of sets. The total number of such set pairs $(A, B)$ that meet the conditions is __________.
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Context:
In $\triangle ABC, D$ and $E$ are the midpoints of $BC$ and $CA$, respectively. $AD$ and $BE$ intersect at $G$. Given that $GEC$D is cyclic, $AB=41$, and $AC=31$, compute $BC$.
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Context:
Let \( S = \{1, 2, \ldots, n\} \). Define \( A \) as an arithmetic sequence with at least two terms and a positive common difference, where all terms are in \( S \). Also, adding any other element of \( S \) to \( A \) does not result in an arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \) (a sequence of only two terms is also considered an arithmetic sequence).
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Context:
As shown in Figure 1, in the triangular prism $P-ABC$, $PA \perp$ plane $ABC$, $\angle ABC = 120^\circ$, and $PA = 4$. If the radius of the circumscribed sphere of the triangular prism $P-ABC$ is $2\sqrt{2}$, find the tangent value of the angle between the line $PC$ and the plane $ABC$.
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Context:
Four pieces of lumber are placed in parallel positions, perpendicular to line M:

- Piece W is 5 m long
- Piece X is 3 m long and its left end is 3 m from line M
- Piece Y is 5 m long and is 2 m from line M
- Piece Z is 4 m long and is 1.5 m from line M

A single cut, perpendicular to the pieces of lumber, is made along the dotted line L. The total length of lumber on each side of L is the same. What is the length, in metres, of the part of piece W to the left of the cut?
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Context:
After counting all her money in her local currency, Michelle decided to travel and exchange her savings into whole $100 US Dollar bills. She went to a currency exchange and noticed that the exchange rate was 1 US Dollar equals 0.85 units of her local currency. Additionally, there was a 3% exchange fee on the total amount that she would need to pay to the currency exchange. After completing the exchange, Michelle acquired 8 pieces of $100 US Dollar bills. How much was the total amount of Michelle's savings in her local currency?
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Context:
Given points $A(4, y_{1})$, $B(\sqrt{2}, y_{2})$, $C(-2, y_{3})$ are all on the graph of the quadratic function $y=(x-2)^{2}-1$, then the relationship between $y_{1}$, $y_{2}$, $y_{3}$ is ____.
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Context:
A chord \( AB \) of fixed length slides its ends along a circle of radius \( R \). A point \( C \) on this chord, which is at distances \( a \) and \( b \) from the ends \( A \) and \( B \) of the chord respectively, traces another circle as the chord makes a full rotation. Compute the area of the annulus enclosed between the given circle and the circle traced by point \( C \).
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Context:
The function $g(x)$ satisfies
\[g(x + g(x)) = 5g(x)\] for all \(x\), and \(g(1) = 5\). Find \(g(26)\).
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Context:
Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). Determine the number of possible DNA sequences on Rthea.
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Context:
The diameter of a circle with radius $r$ is the base of an equilateral triangle. Find the part of the area of the triangle that lies outside the circle.

#
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Context:
Compute the number of ways to completely fill a $3 \times 15$ rectangle with non-overlapping $1 \times 3$ rectangles.
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Context:
The clock hands show exactly one o'clock. A mosquito and a fly are sitting at the same distance from the center on the hour hand and the minute hand respectively. When the hands overlap, the insects swap places. By how many times is the distance traveled by the mosquito in half a day greater than the distance covered by the fly over the same period?
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Context:
Given the function f(x) = e^x + ax, find the range of values for a such that the function has a local maximum point greater than zero. Express your answer as an inequality using standard mathematical notation.
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Context:
Given the equation $\log_{\frac{1}{3}}(a-3^x) = x-2$, find the minimum value of the real number $a$ for the equation to have a solution. Express your answer as a single real number.
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Context:
In Kaya's teacher's desk, there are 18 pink highlighters, 14 yellow highlighters, 11 blue highlighters, 8 green highlighters, and 7 purple highlighters. However, 3 pink highlighters, 2 yellow highlighters, and 1 blue highlighter are broken, and Kaya has borrowed 1 green and 2 purple highlighters from her teacher as well. After one week, the teacher plans to increase the total number of remaining highlighters by 25%. How many highlighters will be left after the teacher's increase next week?
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Context:
A seven-digit number has the following properties: the hundreds digit is twice the ten millions digit, the tens digit is twice the hundred thousands digit, the units digit is twice the ten thousands digit, the thousands digit is 0, and it must be divisible by a five-digit number \( a \). What is \( a \)?
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Context:
Given positive numbers $a$ and $b$ satisfying $2a+b=ab$, find the minimum value of $a+2b$.
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Context:
Given that \( m \) is a constant, and the function
\[
f(x) = \sin^{4} x + m \sin x \cdot \cos x + \cos^{4} x
\]
has the range \(\left[0, \frac{9}{8}\right]\). Find the value of \( m \).
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Context:
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$.  For how many values of $n$ is $n+S(n)+S(S(n))=2007$?
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Context:
用 $[x]$ 表示不超过实数 x 的最大整数, 若数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{3})^{2^{n}}\right](n \in$
$\left.\mathbf{N}^{*}\right)$ ，则 $a_{2020}$ 的末尾两位数字是 $\qquad$.
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Context:
In a room, there are 2019 aligned switches, each connected to a light bulb. Initially, all the light bulbs are switched on. Then, 2019 people enter the room one by one and perform the following operation:

- The first person toggles every switch.
- The second person toggles every second switch.
- The third person toggles every third switch.
- This continues until the 2019th person toggles every 2019th switch.

Determine how many light bulbs remain switched on after all 2019 people have entered the room and completed their operations.
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Context:
Let $S$ be the set of ordered pairs $(x, y)$ such that $0 < x \le 1, 0<y\le 1,$ and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The notation $[z]$ denotes the greatest integer that is less than or equal to $z.$
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Context:
In an acute triangle \( ABC \), \(\angle A = 30^\circ\). Taking \( BC \) as the diameter, a circle is drawn which intersects \( AB \) and \( AC \) at points \( D \) and \( E \) respectively. Connecting \( DE \) divides triangle \( ABC \) into triangle \( ADE \) and quadrilateral \( BDEC \). Suppose their areas are \( S_1 \) and \( S_2 \) respectively. Find the ratio \( S_1 : S_2 \).
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Context:
35. In triangle $ABC$, points $M, N$, and $P$ are taken: $M$ and $N$ are on sides $AC$ and $BC$, and $P$ is on segment $MN$, such that

$$
\frac{|AM|}{|MC|}=\frac{|CN|}{|NB|}=\frac{|MP|}{|PN|}
$$

Find the area of triangle $ABC$, if the areas of triangles $AMP$ and $BNP$ are $T$ and $Q$.
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Context:
Problem 9.5. On an island, there live knights who always tell the truth, and liars who always lie.

One day, 80 residents of the island gathered, each wearing a T-shirt with a number from 1 to 80 (different residents had different numbers). Each of them said one of two phrases:

- "Among those gathered, at least 5 liars have a T-shirt number greater than mine."
- "Among those gathered, at least 5 liars have a T-shirt number less than mine."

What is the smallest number of knights that could have been among these 80 residents?
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Context:
Given differentiable functions $f(x)$ and $g(x)$ on the interval $[0,2]$, where $f''(x) - g''(x) = 0$, $f'(A) = 2$, $g'(A) = 4$, and $f(B) = 3$, $g(B) = 9$, find the value of $f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right)$.
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Context:
A function \( f \) satisfies the following conditions for all nonnegative integers \( x \) and \( y \):

- \( f(0, x) = f(x, 0) = x \)
- If \( x \geq y \geq 0 \), \( f(x, y) = f(x - y, y) + 1 \)
- If \( y \geq x \geq 0 \), \( f(x, y) = f(x, y - x) + 1 \)

Find the maximum value of \( f \) over \( 0 \leq x, y \leq 100 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sequence {a_n} satisfying: a_1 = 2, a_{n+1} = 1 - 1/a_n. Let T denote the product of the first n terms of the sequence {a_n}. Calculate T_{2017}.
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Context:
设函数$f ( x ) = 4 \sin x \cdot \sin ^ { 2 } ( \frac { \pi } { 4 } + \frac { x } { 2 } ) + \cos 2 x .$.若|f(x)-m|<2成立的充分条件是$\frac { \pi } { 6 } \leq x \leq \frac { 2 \pi } { 3 }$，则实数m的取值范围是 .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is the value of $f(2023)$?
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Context:
Let $A=\{V, W, X, Y, Z, v, w, x, y, z\}$. Find the number of subsets of the 2-configuration \( \{\{V, W\}, \{W, X\}, \{X, Y\}, \{Y, Z\}, \{Z, V\}, \{v, x\}, \{v, y\}, \{w, y\}, \{w, z\}, \{x, z\}, \{V, v\}, \{W, w\}, \{X, x\}, \{Y, y\}, \{Z, z\}\} \) that are consistent of order 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the coefficients of all even degree terms in x in the expansion of $(x + \sqrt{x^3 - 1})^6 +( x - \sqrt{x^3 -1})^6 $ , ($x>1$)
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Context:
Five standard six-sided dice are rolled.  We are told there is no three-of-a-kind, but there is a pair of dice that show the same number.  These two dice are set aside, and the other three dice are re-rolled.  What is the probability that after re-rolling these three dice, at least three of the five dice show the same value?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:
A train 120 m long crosses a platform with a certain speed in 15 sec. The length of the platform is 130.02 m. What is the speed of the train in kmph?
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Context:
Given the second-order linear differential equation $$y''+2y'+2y=0$$ with initial conditions $y(0)=0$ and $y'(0)=5$, find the maximum value of the function $y(x)$ over the interval $0 \leq x < \infty$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the following expression:
\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum:
\[ T = r_{100} + r_{101} + r_{102} + \cdots + r_{200} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Count the matches. A friend writes that he bought a small box of short matches, each one inch long. He found that he could arrange them in a triangle whose area in square inches was equal to the number of matches. Then he used 6 matches, and it turned out that from the remaining matches, he could form a new triangle with an area that contained as many square inches as there were matches left. After using another 6 matches, he was able to do the same thing again.

How many matches did he originally have in the box? This number is less than 40.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have?
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Context:
$已知f(x)=|x-a|x+|x-2|(x-a).$
$当a=1时，求不等式f(x)<0的解集；$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya invented a new chess piece called the "super-bishop." One "super-bishop" (denote it as $A$) attacks another (denote it as $B$) if they are on the same diagonal, there are no pieces between them, and the next cell on the diagonal beyond "super-bishop" $B$ is empty. For example, in the figure, piece $a$ attacks piece $b$, but does not attack any of the pieces $c, d, e, f,$ or $g$.

What is the maximum number of "super-bishops" that can be placed on a chessboard so that each one is attacking at least one other super-bishop?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The shortest path from Atown to Cetown runs through Betown. Two signposts on this path are shown, but one of them is broken and a number is missing. Given that the distance from Atown to Betown is 4 km and the distance from Atown to the second signpost is 6 km, calculate the distance that was written on the broken sign. Express your answer in kilometers (km) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangles $\triangle ABC$ and $\triangle PQR$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,10)$, $C(20,0)$, $P(30,0)$, $Q(30,20)$, $R(50,0)$. A rotation of $n$ degrees counterclockwise around the point $(x,y)$ where $0<n<180$, will transform $\triangle ABC$ to $\triangle PQR$. Find $n+x+y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum difference between two positive integer factors of 1998. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that out of 12 bottles of beverages, 2 have expired, calculate the probability of selecting exactly 1 expired bottle from these 12 bottles. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle A's base is 20% greater than the base of Triangle B, and A's height is some percent less than the height of Triangle B. The area of Triangle A is 0.25% less than the area of Triangle B. What is the percentage by which Triangle A's height is less than Triangle B's height?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A boy named Vasya wrote down the nonzero coefficients of a polynomial \( P(x) \) of degree seven in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, continuing this process until he arrived at a constant, which he also wrote down.

What is the smallest number of distinct numbers he might have obtained?

The coefficients are recorded with their signs; if there are free terms, they are also written down. If there's a monomial of the form \( \pm x^{n} \), \( \pm 1 \) is written.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$?
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Context:
During the unloading of a barge, initially, four cranes with equal power operated for 2 hours. Then, two additional cranes with lesser, but equal power were brought into action. After this, it took another 3 hours to complete the unloading. If all these cranes had started working simultaneously, the unloading would have been completed in 4.5 hours. If one crane of greater power and one crane of lesser power worked together, how long would it take to unload the barge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The light glows for some seconds. It glowed 331.27 times between 1:57:58 and 3:20:47 am. What is the frequency of the light glow in seconds?
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Context:
Find the number of positive divisors $d$ of $15! = 15 \cdot 14 \cdot \cdots \cdot 2 \cdot 1$ such that $\gcd(d, 60) = 5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f=y^2-x^3$ and $g=y^3-x^7$. Calculate the intersection multiplicity of $f$ and $g$ at $(0,0)$.
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Context:
On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle?
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Context:
For each positive integer \( n \), define \( s(n) \) to be the sum of the digits of \( n \). For example, \( s(2014) = 2+0+1+4 = 7 \). Determine all positive integers \( n \) with \( 1000 \leq n \leq 9999 \) for which \( \frac{n}{s(n)} = 112 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a geometric progression \( b_1, b_2, \ldots, b_{3000} \) with all positive terms and a total sum \( S \). It is known that if every term with an index that is a multiple of 3 (i.e., \( b_3, b_6, \ldots, b_{3000} \)) is increased by 50 times, the sum \( S \) increases by 10 times. How will \( S \) change if every term in an even position (i.e., \( b_2, b_4, \ldots, b_{3000} \)) is increased by 2 times?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB = 5$, $BC = 9$, $CD = 10$, and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:

1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 22 games.

How many games did the Warriors win? The win numbers given in the bar chart are 23, 28, 33, 38, and 43 games respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( g \) is a twice differentiable function over the positive real numbers such that
\[ g(x) + 2x^3 g'(x) + x^4 g''(x) = 0 \quad \text{for all positive reals } x \]
and
\[ \lim_{x \to \infty} x g(x) = 1, \]

find the real number \(\alpha > 1\) such that \( g(\alpha) = \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \ldots + a_{12}$.
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Context:
There was an equation of the form \(x^{2} + px + q = 0\) written on the board, with non-zero integer coefficients \(p\) and \(q\).

From time to time, different students would come to the board, erase the equation, and then create and write an equation of the same form, the roots of which were the coefficients of the erased equation. At some point, the newly created equation matched what was originally written on the board. What equation was originally written on the board?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let set $A=\{(x,y)|y=a|x|\}$, and set $B=\{(x,y)|y=x+a\}$. If $A\cap B$ is a singleton set, the range of the real number $a$ is ____.
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Context:
410 deputies sat around a round table, each of whom was either a knight who always tells the truth or a liar who always lies. Each of the deputies said: "Among my twenty neighbors to the left and twenty neighbors to the right, there are exactly 20 liars in total." It is known that at least half of the people at the table are liars. How many knights are at the table?
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Context:
在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\mathbf{0}$. 设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$的面积，则 $\frac{t}{s}=$ $\qquad$.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:
If $w,w^2$ are non-real cube roots of unity and $a,b,c \in R$ such that $$\frac{1}{a+w}+\frac{1}{b+w}+\frac{1}{c+w}=2 w^2$$ and $$\frac{1}{a+w^2}+\frac{1}{b+w^2}+\frac{1}{c+w^2}=2 w$$ then find $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Over all real numbers $x$ and $y$ such that $$x^{3}=3 x+y \quad \text { and } \quad y^{3}=3 y+x$$ compute the sum of all possible values of $x^{2}+y^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x, y, z$ be real numbers satisfying $$\frac{1}{x}+y+z=x+\frac{1}{y}+z=x+y+\frac{1}{z}=3$$ The sum of all possible values of $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify \[\frac{1}{\dfrac{2}{\sqrt{5}+2} + \dfrac{3}{\sqrt{7}-2}}.\]
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Context:
A is half as efficient as B, and B is twice as efficient as C. Together, A, B, and C can finish a job in 26 days. When they work in pairs, A and B can finish the job in 13 days, while A and C can finish it in 39 days. How many days would it take for B and C to finish the job together, assuming they all work at a constant rate?
Please reason step by step, and put your final answer within \boxed{}.


Context:
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数, 其中两个根的和为 $2+i$, 另两根的积为 $5+6 i$,则 $b$ 等于多少？
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Context:
Two candles of the same thickness and material have lengths in the ratio \(21:16\). They start burning at the same time. After 18 minutes, the length ratio of the longer candle to the shorter one is \(15:11\). How many more minutes can the longer candle continue to burn?
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Context:
What is the sum of the digits of the smallest possible value of n for which there are more than 1,000,000 ways to arrange n identical black socks and 2n identical gold socks in a row so that there are at least 2 gold socks between any 2 black socks?
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Context:
In how many ways can the letters in $BEEKEEPER$ be rearranged so that two or more $E$s do not appear together? Provide the number of such arrangements.
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Context:
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
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Context:
The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{225} - \frac{y^2}{144} = \frac{1}{36}\]
coincide. Find \(b^2\).
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Context:
In a field arranged in 10 rows side by side, choose 2 rows to plant two different crops, A and B, with one row for each crop. To facilitate the growth of the crops, it is required that the interval between crops A and B is not less than 6 rows. How many different planting methods are there in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four red candies and three green candies can be combined to make many different flavors. Flavors are different if the percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green; and likewise 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is to be made using some or all of the seven candies, how many different flavors are possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知实数 $x, y$ 满足 $x|x|+\frac{y|y|}{3}=1$, 则 $|\sqrt{3} x+y-4|$ 的取值范围为 [p, q)。请给出 p + q 的整数部分。
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Context:
The base radius of a cone is 6 cm, and the central angle of the sector in the unfolded lateral surface of the cone is 240°. Find the slant height of the cone. Express your answer in centimeters as a whole number.
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Context:
Given $N=2^n$ (where $n\in\mathbb{N}^*, n\geq 2$), place $N$ numbers $x_1, x_2, \ldots, x_N$ sequentially into $N$ positions labeled from 1 to $N$, resulting in the permutation $P_0=x_1x_2\ldots x_N$. Extract the numbers located in the odd and even positions of this permutation and place them in order into the corresponding first $\frac{N}{2}$ and last $\frac{N}{2}$ positions, forming permutation $P_1=x_1x_3\ldots x_{N-1}x_2x_4\ldots x_N$. Let's call this operation a C transformation. Divide $P_1$ into two segments, each containing $\frac{N}{2}$ numbers, and perform a C transformation on each segment to obtain $P_2$. For $2\leq i\leq n-2$, divide $P_i$ into $2^i$ segments, each with $\frac{N}{2^i}$ numbers, and apply C transformations to each segment to get $P_{i+1}$. For example, when $N=8$, $P_2=x_1x_5x_3x_7x_2x_6x_4x_8$, and $x_7$ is in the fourth position in $P_2$. When $N=32$, find the position of $x_{21}$ in $P_3$.
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Context:
A train travelling at 80 kmph passes through a tunnel of 70 km at 5:12 am. The train leaves the tunnel at a certain time. The length of the train is 1 km. At what time did the train leave the tunnel?
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Context:
A class of 30 students wrote a history test. Of these students, 25 achieved an average of 75%. The other 5 students achieved an average of 40%. Calculate the overall class average percentage, rounded to the nearest whole number. Express your answer as a percentage.
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Context:
Rachel starts with the number $1000$ in her hands. Each minute, she flips a fair coin. If it lands heads, she adds 1 to the number by placing it in her left pocket, transforming it to $x+1$. If it lands tails, she inverts the number by placing it in her right pocket, transforming it to $x^{-1}$. After eight minutes, the expected value of the number in Rachel's hands is $E$. Compute $\left\lfloor\frac{E}{10}\right\rfloor$.
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Context:
Emily wants to know how much it rained last week. She sees that it rained 2 inches on Monday morning and 1 more inch later that day. It rained some amount on Tuesday compared to Monday. It did not rain on Wednesday but on Thursday it rained 1 inch. The biggest storm was on Friday when the total was equal to Monday through Thursday combined. The daily average rain total for the week was 4 inches. What is the ratio of the rain on Tuesday to the total rain on Monday?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=m$, $b_1-a_1=1$, $b_2-a_2=2$, $b_3-a_3=3$. If the general term formula of the sequence $\{a_n\}$ is unique, then the value of the positive real number $m$ is.
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Context:
A $39$-tuple of real numbers $(x_1,x_2,\ldots, x_{39})$ satisfies the following conditions:
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots, x_{39})$ satisfying the given conditions. This ratio is expressed as $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find the sum $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 60 three-digit numbers between 100 and 500, where the sum of the first two digits is equal to a certain number. What is that number?
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Context:
For \(0 \leq x \leq 1\) and positive integer \(n\), let \(f_0(x) = |1 - 2x|\) and \(f_n(x) = f_0(f_{n-1}(x))\). How many solutions are there to the equation \(f_{10}(x) = x\) in the range \(0 \leq x \leq 1\)?
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Context:
Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$.
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Context:
Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
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Context:
若函数$f ( x ) = \lg [ ( k ^ { 2 } - 1 ) x ^ { 2 } - ( k + 1 ) x + 1 ]$的值域是R，则实数k的取值范围是
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compare the sizes: $-\frac{3}{5}$ ____ $-\frac{1}{5}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let point $P$ be on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$. A line through $P$ intersects the asymptotes at $P_{1}$ and $P_{2}$, and $\overrightarrow{P_{1} P} \overrightarrow{P P_{2}}$ $=3$. Let $O$ be the origin. Find the area of $\triangle O P_{1} P_{2}$.
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Context:
How many distinct four-digit positive integers are there such that the product of their digits equals 18?
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Context:
Let $a > 1$, and $m = \log_a(a^2+1)$, $n = \log_a(a-1)$, $p = \log_a(2a)$. Determine the relationship among $m$, $n$, and $p$.
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Context:
a volunteer organization is recruiting new members . in the fall they manage to increase their number by 9 % . by the spring however membership falls by 19 % . what is the total change in percentage from fall to spring ?
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Context:
Determine the coefficient of the fourth term in the expansion of $( 3x - \frac{1}{2 \cdot 3x} )^{8}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task 4 - 261244

Determine the smallest positive integer $a$ for which $(a+1)^{5}-a^{5}-1$ is divisible by 18305.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a grid of size $m \times n$ (with $m, n > 1$). The centers of all the cells are marked. What is the maximum number of marked centers that can be chosen such that no three of them form the vertices of a right triangle?
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Context:
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
a, b, and c can do a work in 4 hours, 12 hours, and 6 hours respectively. a starts the work at 6 am, and they work alternately for one hour each in the order of a, b, and c. When will the work be completed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer $n > 1$ such that $n^2(n - 1)$ is divisible by 2009.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A child lines up $2020^2$ pieces of bricks in a row and removes bricks whose positions are square numbers (i.e., the 1st, 4th, 9th, 16th, ... bricks). Then, he lines up the remaining bricks again and removes those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( y = (x - 16)(x - 14)(x + 14)(x + 16) \), find the minimum value of \( y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$  $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdots,20\}$ for $i=1,\cdots,5$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x) = \ln x \) and \( g(x) = a x + \frac{b}{x} \). These functions have a common tangent at a common point on the x-axis. When \( x > 1 \), what is the relationship between \( f(x) \) and \( g(x) \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least positive integer $n$ greater than $1$ such that $n^3 - n^2$ is divisible by $7^2 \times 11$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marie has 40 notebooks, each containing 150 stamps, 12 binders, each filled with 315 stamps, and 25 folders, each holding 80 stamps. If she decides to keep only 28.5% of her stamps and give away the rest, how many stamps can she give away?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $\Gamma_1$  be a circle with radius  $\frac{5}{2}$ .  $A$ ,  $B$ , and  $C$  are points on  $\Gamma_1$  such that  $\overline{AB} = 3$  and  $\overline{AC} = 5$ . Let  $\Gamma_2$  be a circle such that  $\Gamma_2$  is tangent to  $AB$  and  $BC$  at  $Q$  and  $R$ , and  $\Gamma_2$  is also internally tangent to  $\Gamma_1$  at  $P$ .  $\Gamma_2$  intersects  $AC$  at  $X$  and  $Y$ .  $[PXY]$  can be expressed as  $\frac{a\sqrt{b}}{c}$ . Find  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=2\sin(\omega x+\varphi)$, where $(\omega > 0, |\varphi| < \frac{\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\left(\frac{\pi}{18}, \frac{\pi}{3}\right)$. Additionally, the graph of $f(x)$ coincides with its original graph after being shifted to the left by $\pi$ units. If $x_{1}, x_{2} \in \left(-\frac{17\pi}{12}, -\frac{2\pi}{3}\right)$ and $x_{1} \neq x_{2}$, and $f(x_{1}) = f(x_{2})$, calculate $f(x_{1}+x_{2})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
G.H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: "I came here in taxi-cab number 1729. That number seems dull to me, which I hope isn't a bad omen." "Nonsense," said Ramanujan. "The number isn't dull at all. It's quite interesting. It's the smallest number that can be expressed as the sum of two cubes in two different ways." Ramanujan had immediately seen that $1729 = 12^{3} + 1^{3} = 10^{3} + 9^{3}$. What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Li Ming and Li Hua both arrive at the bus station to wait for either bus number 1 or bus number 2 to go home. Bus number 1 arrives every 8 minutes, while bus number 2 arrives every 10 minutes. What is the probability that Li Ming boards bus number 1 before Li Hua boards bus number 2?
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Context:
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. Find the number of two-digit positive integers $n$ such that $R(n) = R(n+1)$.
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Context:
How many solutions does the equation $x = [P_1(x)] + 2[P_2(x)] + 3[P_3(x)] + 4[P_4(x)]$ have, where $P_1(x): x \le 5$, $P_2(x): x \ge 3$, $P_3(x): x$ is odd, and $P_4(x): x \ge 6$ for all integers $x$?
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Context:
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, what number could be obtained?
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Context:
On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of the car ahead to the front of the car in the same lane is one car length for every 10 kilometers per hour of speed or fraction thereof. Suppose each car is 5 meters long, and a photoelectric eye at the side of the road detects the number of cars that pass by in one hour. Determine the whole number of cars passing the eye in one hour if the speed in the right lane is 50 kilometers per hour. Calculate $M$, the maximum result, and find the quotient when $M$ is divided by 10.
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Context:
A plane contains points \(A\) and \(B\) with \(AB = 1\). Let \(S\) be the union of all disks of radius \(1\) in the plane that cover \(\overline{AB}\). The original answer is in the form \(k\pi - \frac{\sqrt{m}}{n}\). Please find the value of \(k + m + n\).
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Context:
A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.
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 $a + 2b + 3c = d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a class of 50 students, a sample of size 4 is drawn using the systematic sampling method. It is known that students with identification numbers 6, 30, and 42 are included in the sample. Determine the identification number of the fourth student in the sample.
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Context:
The ratio of width to length of a book is the golden ratio. Given that the length of the book is 20cm, calculate the width of the book. Express your answer in centimeters (cm), rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find
$$
\{\ln (1+e)\}+\left\{\ln \left(1+e^{2}\right)\right\}+\left\{\ln \left(1+e^{4}\right)\right\}+\left\{\ln \left(1+e^{8}\right)\right\}+\cdots,
$$
where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a chess festival held at a school with $2017$ students, each pair of students played at most one match against each other. By the end of the event, it was observed that for any pair of students who played a match against each other, at least one of them had played at most $22$ matches. What is the maximum possible number of matches in this event?
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Context:
10.3. Find the greatest solution of the inequality

$$
\frac{-\log _{3}(100+2 x \sqrt{2 x+25})^{3}+\left|\log _{3} \frac{100+2 x \sqrt{2 x+25}}{\left(x^{2}+2 x+4\right)^{4}}\right|}{3 \log _{6}(50+2 x \sqrt{2 x+25})-2 \log _{3}(100+2 x \sqrt{2 x+25})} \geqslant 0
$$
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Context:
A chemical factory plans to improve the technology for a chemical product and has decided to optimize the processing temperature, with the experimental range set between 60°C and 81°C, requiring an accuracy of ±1°C. If the technician prepares to use the method of fractions to optimize, what is the maximum number of experiments needed to find the optimal temperature?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$  Find the maximum value of
\[7x + 2y + 8z.\]
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Context:
In a campus singing competition, a contestant received scores of $92$, $93$, $88$, $99$, $89$, and $95$ from $6$ judges. The $75\%$ percentile of this data set is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer.  Compute the remainder when $3^n$ is divided by $1000$.
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Context:
Let k and m be non-negative integers, n be a negative integer, and 15^k, 20^m, and 10^n be divisors of 1,031,525. If x = 3^k - k^3 and y = 5^m - m^5, find the value of z = (x + y) - 2^(n+2).
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Context:
在凸四边形 $A B C D$ 的外部分别作正 $\triangle A B Q 、 \triangle B C R 、 \triangle C D S 、 \triangle D A P$, 记四边形 $A B C D$ 的对角线的和为 $x$, 四边形 $P Q R S$ 的对边中点连线的和为 $y$. 求 $\frac{y}{x}$ 的最大值。原始答案为 $\frac{a+\sqrt{b}}{c}$ 的格式，请给出 a + b + c的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence $\left\{a_{n}\right\}$ that satisfies
$$
a_{n+1} = -\frac{1}{2} a_{n} + \frac{1}{3^{n}}\quad (n \in \mathbf{Z}_{+}),
$$
find all values of $a_{1}$ such that the sequence $\left\{a_{n}\right\}$ is monotonic, i.e., $\left\{a_{n}\right\}$ is either increasing or decreasing.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles triangle $ABC$ with an angle $\angle A = 20^\circ$ and base $BC = 12$, a point $E$ on the side $AC$ is chosen such that $\angle ABE = 30^\circ$. Additionally, a point $F$ is on the side $AB$ such that $EF = FC$. Find the length of $FC$.
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Context:
Given that \( f(x) = x^2 + ax + b \) is the common factor of \( x^3 + 4x^2 + 5x + 6 \) and \( 2x^3 + 7x^2 + 9x + 10 \), find the value of \( Q \) if \( f(P) = Q \).
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Context:
If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any other student?
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Context:
One hour after Adrienne started walking the 60 miles from point X to point Y, Harold started walking from point X to point Y as well. Adrienne walks at a speed of 3 miles per hour, and Harold walks 1 mile per hour faster than Adrienne. At the same time when Harold starts walking, Christine also begins walking from point X to point Y, and her walking speed is 0.5 miles per hour slower than Harold's. What will be the location for all three of them to meet (the distance from point X) along the line between X and Y?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $81^4 = 27^x$, what is the value of $3^{-x}$? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two bikers, Bill and Sal, simultaneously set off from one end of a straight road. Neither biker moves at a constant rate, but each continues biking until he reaches one end of the road, at which he instantaneously turns around. When they meet at the opposite end from where they started, Bill has traveled the length of the road eleven times and Sal seven times. Find the number of times the bikers passed each other moving in opposite directions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangle has an inscribed circle with a radius of 8 cm. Calculate the length of $\overline{AB}$ if the triangle is a right triangle with one angle measuring $30^\circ$. Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\overline{AB} = 16(\sqrt{3} + 1)}


Context:
Given an arithmetic progression $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}=\left(x+2y+1\right)3^{n}+\left(x-y-4\right)$ (where $x \gt 0, y \gt 0$). The minimum value of $\frac{1}{x}+\frac{2}{y}$ is ______.
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Context:
设数列 $\left\{x_{n}\right\}$ 满足
$x_{1}=1, x_{n+1}=4 x_{n}+\left[\sqrt{11} x_{n}\right]$ 。
求 $x_{2012}$ 的个位数字.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Professor Oscar has to evaluate his students on the final exams, but this semester, he included a mix of question types and different class sizes.

Class A has 30 students, and their exam consists of 12 multiple-choice questions, 3 essay questions, and 1 presentation.

Class B has 25 students, and their exam consists of 15 multiple-choice questions, 5 short-answer questions, and 2 essay questions.

Class C has 35 students, and their exam consists of 10 multiple-choice questions, 3 essay questions, and 1 group project presentation per group of 5 students.

Class D has 40 students, and their exam consists of 11 multiple-choice questions, 4 short-answer questions, and 3 essay questions.

Class E has 20 students, and their exam consists of 14 multiple-choice questions, 5 short-answer questions, and 2 essay questions.

Given the varying number of questions, students, and components for each of the 5 classes, how many evaluations must Professor Oscar review in total? Consider the evaluations for each component separately (multiple-choice, short-answer, essay, and presentations).
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Context:
A cuckoo clock chimes "cuckoo" on the hour, with the number of "cuckoo" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger until he moved the clock forward by 7 hours. How many times did the clock chime "cuckoo" during this period?
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Context:
A technician has to visit a total of 5 service centers (A, B, C, D, and E) along the same route, making a round-trip by returning back to center A. The technician completes the drive to center B, then finishes 40% of the journey from center B to center C, and after that, completes 60% of the drive from center C to center D. What percent of the entire round-trip, including the return trip to center A, has the technician completed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative.

\[ y = -\frac{1}{3 \sin^{3} x} - \frac{1}{\sin x} + \frac{1}{2} \ln \frac{1+\sin x}{1-\sin x} \]
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Context:
How can I solve the following limit problem involving factorials and powers of x: $$\lim_{x\to 1} \frac{(1-x)(1-x^2)\cdots(1-x^{2n})}{[(1-x)(1-x^2)\cdots(1-x^n)]^2}?$$
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Context:
For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$.

[i]Proposed by Mehtaab Sawhney[/i]
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Context:
The regular tetrahedron, octahedron, and icosahedron have equal surface areas. How are their edges related?
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Context:
[ Geometry (miscellaneous).]

A sphere of radius $\sqrt{5}$ with center at point $O$ touches all sides of triangle $ABC$. The point of tangency $N$ bisects side $AB$. The point of tangency $M$ divides side $AC$ such that $AM=\frac{1}{2} MC$. Find the volume of the pyramid $OABC$, given that $AN=NB=1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $r$  be a positive integer. Find the smallest positive integer  $m$  satisfying the condition: For all sets  $A_1, A_2, \dots, A_r$  with  $A_i \cap A_j = \emptyset$ , for all  $i \neq j$ , and  $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$ , there exists  $a, b \in A_k$  for some  $k$  such that  $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\cos(2x- \frac{\pi}{6})$, find the horizontal shift required to transform the graph of $y=\sin 2x$ into the graph of $y=\cos(2x- \frac{\pi}{6})$.
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Context:
If $a$, $b$, and $c$ are positive numbers such that $ab = 24\sqrt{3}$, $ac = 30\sqrt{3}$, and $bc = 40\sqrt{3}$, find the value of $abc$.
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Context:
Sherry is waiting for a train. Every minute, there is a 75% chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a 75% chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?
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Context:
Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?
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Context:
A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?
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Context:
Piazzanos Pizzeria has a special offer on triple cheese pizzas, and a buy-2-get-1-free special on meat lovers pizzas. If the standard price for a pizza is $5, it costs $55 to purchase some triple cheese and 9 meat lovers pizzas using the special pricing. What is the ratio of the number of triple cheese pizzas you need to buy to get one free?
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Context:
Two trains of length 100 m and 280 m are running towards each other on parallel lines at a certain speed and 30 kmph respectively. From the moment they meet, they will be clear of each other in 18.998480121590273 seconds. What is the speed of the first train in kmph?
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Context:
Consider the collection of all 5-digit numbers whose sum of the digits is 43. One of these numbers is chosen at random. What is the probability that it is a multiple of 11?
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Context:
How many odd, 5-digit palindromic numbers are there with all unique digits?
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Context:
Let
\[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ]. \]

Find the remainder when $N$ is divided by 1000.
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Context:
Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
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Context:
若锐角 A,B,C 满足 \sin^2A+\sin^2B+\sin^2C=2，则 \df{1}{\sin^2A\cos^4B}+\df{1}{\sin^2B\cos^4C}+\df{1}{\sin^2C\cos^4A} 的最小值是__________.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:
Reduce the fraction \(\frac{2x^{6} + 5x^{4} - 3x^{3} + 2x^{2} - 12x - 14}{4x^{6} - 4x^{4} - 6x^{3} - 3x^{2} + 25x - 28}\). As a result of the reduction, the degrees of the polynomials in the numerator and the denominator should decrease.
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Context:
Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\]
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Context:
On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red points?
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Context:
Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value of $n$.
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Context:
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?
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Context:
For an annuity with $3n$ equal payments and a periodic interest rate of $i$, it is observed that the present value of the first $n$ payments, one period before the first payment, equals the present value of the last $2n$ payments. Determine the value of $v^n$ where $v = \left(\frac{1}{1+i}\right)$.
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Context:
Given \( n \) sticks. From any three, it is possible to form an obtuse triangle. What is the maximum possible value of \( n \)?
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Context:
Let f(x) be a function defined on R with a period of 2, and for any real number x, it always holds that f(x) - f(-x) = 0. When x is in [0, 1], f(x) = -sqrt(1-x^2). Find the number of zeros of the function g(x) = f(x) - e^x + 1 in the interval [-2017, 2017].
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Context:
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other.
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Context:
Replace the symbols $\$ last\$ with the same numbers so that the equation becomes true: $\$ \$ |\frac{20}{x} - \frac{x}{15} = \frac{20}{15} \$ \$$
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Context:
Maya loves to read books. Last week she read 5 books. Each book had 300 pages of text. This week she read a certain amount compared to last week. Maya read a total of 4500 pages. What is the ratio of the number of books she read this week to the number of books she read last week?
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Context:
There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues until someone cannot say a new number, and the person who cannot say a new number loses the game. Who will be the last person in the game?
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Context:
Plane $S$ makes an angle of $30^{\circ}$ with the horizontal. Plane $S$ is then rotated by $120^{\circ}$ around a vertical line as an axis. What is the angle of inclination between the resulting plane and plane $S$?
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Context:
4. What is the greatest length that a closed, non-self-intersecting broken line can have, which runs along the grid lines of a square grid of size $8 \times 8$?

#
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Context:
On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links?
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Context:
On the bisector of angle \( B A C \) of triangle \( A B C \), there is a point \( M \), and on the extension of side \( A B \) beyond point \( A \), there is a point \( N \) such that \( A C = A M = 1 \) and \( \angle A N M = \angle C N M \). Find the radius of the circumcircle of triangle \( C N M \).
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Context:
Calculate the value of N such that the number of odd integers between 4 and N is equal to the number of even integers between 1 and 103. Express your answer as a whole number.
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Context:
In a middle school math competition, there were three problems labeled A, B, and C. Among the 25 students who participated, each student solved at least one problem. Among the students who did not solve problem A, the number who solved problem B was twice the number of those who solved problem C. The number of students who solved only problem A was one more than the number of students who solved A along with any other problem. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B?
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Context:
11.2. On the sides $A B$ and $A C$ of an equilateral triangle $A B C$ with side length 10, points $P$ and $Q$ are taken such that the segment $P Q$ is tangent to the inscribed circle of the triangle and its length is 4. Find the area of triangle $A P Q$.
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Context:
There are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from $1$ to $64$. At the center of the table, there are $1996$ light bulbs, all initially turned off.

Every minute, the chips move simultaneously in a circular pattern as follows: chip $1$ moves one booth forward, chip $2$ moves two booths forward, and so on, such that multiple chips can occupy the same booth at the same time.

A bulb is lit whenever chip $1$ shares a booth with any other chip.

Determine the position of chip $1$ on the first minute when all $1996$ bulbs are lit.
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Context:
Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\left\lfloor\sqrt{P(x)}\right\rfloor = \sqrt{P(\left\lfloor x \right\rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + e$.
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Context:
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\varphi$.
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Context:
已知欧拉常数 $\gamma=\lim\limits _{n \rightarrow+\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\right)$ 约等于 0.5772156649 。记 $I=\frac{100}{\int_{0}^{1}\left(\frac{1}{x}-\left[\frac{1}{x}\right]\right) \mathrm{d} x}$, 则 $[I]=$ $\qquad$
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Context:
Given that $a \in A$, and $a-1 \notin A$ and $a+1 \notin A$, $a$ is called an isolated element of set $A$. How many four-element subsets of the set $M=\{1,2, \cdots, 9\}$ have no isolated elements?
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Context:
Given that 800 students were surveyed, and their pasta and pizza preferences were lasagna (150 students), manicotti (120 students), ravioli (180 students), spaghetti (200 students), and pizza (150 students), calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred pizza.
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Context:
In the rhombus \(ABCD\), the heights \(BP\) and \(BQ\) intersect the diagonal \(AC\) at points \(M\) and \(N\) (point \(M\) lies between \(A\) and \(N\)), \(AM = p\) and \(MN = q\). Find \(PQ\).
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Context:
There are two concentric circles, with six non-overlapping points on the circumference of the outer circle and three non-overlapping points on the circumference of the inner circle. Determine the minimum number of straight lines that can be formed using these nine points. Express your answer as a whole number.
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Context:
A train 280 meters long travels at a speed of 18 meters per second across a bridge. It takes 20 seconds from the moment the tail of the train gets on the bridge until the head of the train leaves the bridge. Therefore, the length of the bridge is $\qquad$ meters.
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Context:
If $\operatorname{tg} \alpha$ and $\operatorname{tg} \beta$ are the two roots of the quadratic equation $x^{2} + p x + q = 0$, then find the value of $\frac{\sin (\alpha + \beta)}{\cos (\alpha - \beta)} =$.
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Context:
The rules of a card arrangement game are as follows: Arrange nine cards marked with $1,2, \cdots, 9$ randomly in a row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is called an operation. The game stops when no operation can be performed (i.e., the number on the first card is "1"). If an arrangement cannot be operated on and can be obtained by performing one operation on a unique other arrangement, it is called a "second stop arrangement". Find the probability of a second stop arrangement occurring among all possible arrangements.
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Context:
There are 6 people in the elevator. Their average weight is 156 lbs, not including the weight of their clothing. A group of 3 individuals, A, B, and C, enter the elevator one by one, each carrying a backpack and wearing various clothing items. The total weight of A's clothing and backpack is 18 lbs, B's is 20 lbs, and C's is 22 lbs. After each person enters, the average weight, without considering clothing and backpacks, gets updated.

After A enters, the average weight becomes 159 lbs. When B enters, the average weight increases to 162 lbs. After C enters, the new average weight is reduced to 161 lbs. What is the true weight (excluding clothing and backpack) of the third person, C, who entered the elevator?
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Context:
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+2)a_n$, and $a_nb_n=\frac{1}{2}$. If $\lambda > T_n$ holds true for any $n\in\mathbb{N}^*$, find the minimum value of the real number $\lambda$.
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Context:
Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room.
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Context:
An arithmetic sequence alternates between even and odd terms. The sums of the odd terms and the even terms are 24 and 30, respectively. The last term exceeds the first term by 10.5. Find the total number of terms in the arithmetic sequence. Express your answer as a whole number.
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Context:
10. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ with edges $A B=4, A D=2$, $A A_{1}=3 \sqrt{2}$, a plane is drawn through the diagonal $B D_{1}$, intersecting the edge $A A_{1}$ such that the section of the parallelepiped by this plane has the smallest perimeter. Find the area of this section.
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Context:
The angle bisectors of triangle \( A B C \) intersect at point \( I \), and the external angle bisectors of angles \( B \) and \( C \) intersect at point \( J \). The circle \( \omega_{b} \) with center at point \( O_{b} \) passes through point \( B \) and is tangent to line \( C I \) at point \( I \). The circle \( \omega_{c} \) with center at point \( O_{c} \) passes through point \( C \) and is tangent to line \( B I \) at point \( I \). The segments \( O_{b} O_{c} \) and \( I J \) intersect at point \( K \). Find the ratio \( I K / K J \).
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Context:
8. Let $A, B, C, D$ be four non-coplanar points in space, and connect an edge between each pair of points with a probability of $\frac{1}{2}$. Whether any two pairs of points are connected is independent of each other. Then the probability that $A$ and $B$ can be connected by (a single edge or a sequence of edges forming a) spatial polyline is $\qquad$ .
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Context:
Let \( M \) be a subset of the set \(\{1, 2, 3, \cdots, 15\}\), and suppose that the product of any three different elements in \( M \) is not a perfect square. Let \( |M| \) denote the number of elements in the set \( M \). Find the maximum value of \( |M| \).
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Context:
From two pieces of alloys (with different lead contents) weighing 6 kg and 12 kg, respectively, equal-mass pieces were cut off. Each of the cut pieces was fused with the remainder of the other piece, after which the percentage of lead in both resulting alloys became identical. What are the masses of each of the cut pieces?
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Context:
Given the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$.
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Context:
Let the set \( M = \{1, 2, \cdots, 19\} \) and \( A = \{a_1, a_2, \cdots, a_k\} \subseteq M \). Determine the smallest \( k \), such that for any \( b \in M \), there exist \( a_i, a_j \in A \), satisfying \( a_i = b \) or \( a_i \pm a_j = b\) (where \(a_i\) and \(a_j\) can be the same).
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Context:
Let $f(x)$ be a polynomial of degree 3 with real coefficients, satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.
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Context:
Let $P(x) \in \mathbb{Z}[x]$ be a polynomial with degree $t-1$, and suppose it satisfies the equation $$xP(x) = x^t + P(x-1)$$ for all real $x$. Determine $t \equiv ? \pmod{3}$.
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Context:
A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}ax^2 + bx$ has an extreme value.
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Context:
Sally has a gold credit card with a certain spending limit, and a platinum card with twice the spending limit of the gold card. She has a balance on her gold card that is a certain fraction of the spending limit on that card, and she has a balance on her platinum card that is 1/9 of the spending limit on that card. If Sally transfers the entire balance on her gold card to her platinum card, 0.7222222222222222 portion of her limit on the platinum card will remain unspent. What fraction of the spending limit on the gold card is her current balance?
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Context:
A person crosses a 1080 m long street in some time. His speed is 5.4 km per hour. How much time did he take to cross the street?
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Context:
Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\frac{[H_PH_QH_RH_S]}{[PQRS]}$.
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Context:
The coordinates of the fixed point that the function $f(x)=\log_{x}(3x-2)+2$ (where $a > 0$ and $a \neq 1$) always passes through are ______.
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Context:
A railway freight station decided to organize and dispatch 8 coal freight trains into two groups, each containing 4 trains, with the conditions that trains A and B cannot be in the same group, train A departs first, and train B departs last. Find the total number of different departure sequences for these 8 trains. Express your answer as a single integer.
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Context:
Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0, 1, 2, \ldots, 12$ such that the value of the resulting expression is divisible by 5.
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Context:
There are 10 different electronic products, among which 2 products are unstable. Technicians test them one by one until both unstable products are identified, and the testing ends. Calculate the number of methods to exactly finish the testing in 3 tries.
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Context:
A positive integer $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \overline{aa}$, which is a multiple of $a^2$. Find all possible values of $\frac{b}{a^2}$.
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Context:
Find the sum of the alternating series: $$\sum_{n=-\infty}^{\infty} (-1)^n \frac{2 n-1}{(2 n-1)^2+y^2}$$
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Context:
Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed.
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Context:
Select 5 different letters from the word "equation" to arrange in a row, ensuring that "qu" are together and in the same order. Calculate the total number of different arrangements. Express your answer as a single integer.
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Context:
A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base?  Give your response in base 2.
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Context:
A 360 m long train is running at a speed of 55 km/hr. It crossed a platform of certain length in 57.59539236861051 seconds. What is the length of the platform?
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Context:
The length of a cuboid is increased by 40%, its width is halved, and its height is increased by 25%. What is the percentage change in the volume of the cuboid?
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Context:
In the context of group theory, determine the largest prime number $p$ for which the multiplicative group $(U(p^2), \cdot \mod p)$, where $U(p^2)$ represents the group of units modulo $p^2$, is isomorphic to a subgroup of the symmetric group $S_{50}$.
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Context:
某班有 47 个学生，所用教室有 6 排，每排有 8 个座位，用 (i,j) 表示位于第 i 排第 j 列的座位. 新学期准备调整座位，设某学生原来的座位为 (i,j)，如果调整后的座位为 (m,n)，则称该生作了移动 [a,b]=[i-m,j-n]，并称 a+b 为该生的位置数. 所有学生的位置数之和记为 S，求 S 的最大值与最小值之差.
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Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{de} f$ 是偶数的概率为 $\qquad$.
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:
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing each watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on a piece of paper the weights of the scales he used. What is the smallest number of different weights that could appear in his records if the mass of each weight is an integer kilogram?
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Context:
Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \).
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Context:
Given the function $f(x)={2^x}+\frac{x}{3}$, find the interval in which the zero point $x_0$ of the function lies.
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Context:
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$, and let $x^{*2}$ be defined as $x^{*2} = x*x$. Then, the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$ is ____.
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Context:
Let \( n \) be the 200th smallest positive real solution to the equation \( x - \frac{\pi}{2} = \tan x \). Find the greatest integer that does not exceed \( \frac{n}{2} \).
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Context:
Suppose a Markov chain is given as follows. $p_{ii}=1-3a$ and $p_{ij}=a\ \forall i\ne j$ where $P=(p_{ij}), 1\le i,j\le 4$. Find $P_{1,1}^n$.
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Context:
An isosceles triangle \( ABC \) \((AB = BC)\) is inscribed in a circle. The diameter \( AD \) intersects the side \( BC \) at point \( E \), with \( DE : EA = k \). Find the ratio \( \frac{CE}{BC} \).
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Context:
The words "PUNK CD FOR SALE" are painted on a clear window. Count how many of the letters in the sign look the same from both sides of the window due to being symmetrical either vertically or horizontally. Provide your answer as a whole number.
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Context:
Using Qin Jiushao's algorithm, calculate the value of v_3 when evaluating the polynomial f(x) = x^5 + 3x^4 - x^3 + 2x - 1 at x = 2. Express your answer as a single integer.
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Context:
Find the smallest positive integer $x$ such that $10^x\equiv 1 \pmod{1997}$.
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Context:
If John makes a contribution to a charity fund at school, the average contribution size will increase by 75% reaching $100 per person. If there were 10 other contributions made before John's, what is the size of his donation?
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Context:
If the function $f(x)=\frac{1}{3}x^3-ax^2+x-5$ has no extreme points, then the range of real number $a$ is ____.
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Context:
There were 3 candidates in an election and they received some votes. The winning candidate got 65.21739130434783% of the total votes in that election. If the other two candidates received 5000 and 15000 votes respectively, how many votes did the first candidate receive?
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\sin \frac{A}{2},\cos \frac{A}{2})$ and $n=(\cos \frac{A}{2},-\cos \frac{A}{2})$, and $2m\cdot n+|m|=\frac{ \sqrt{2}}{2}$, find $\angle A=$____.
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{2 \operatorname{arctan} 2}^{2 \operatorname{arctan} 3} \frac{d x}{\cos x(1-\cos x)}
$$
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