Context:
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$.
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Context:
5. In the Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where the real numbers $x$ and $y$ satisfy $0 \leqslant x \leqslant 10$ and $0 \leqslant y \leqslant 10$, then the sum of the lengths of all the loci of points $C$ that satisfy the condition is . $\qquad$
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Context:
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $\textdollar 1$ to $\textdollar 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
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Context:
Two snails crossed the surface of a planet shaped like a torus, leaving trails behind them. One snail crawled along the outer equator, and the other along a helical line. Into how many parts did the snails' trails divide the surface of the planet? (It is sufficient to write only the answer.)
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Context:
There are 120 people that own pets. Out of these, 30 people own only dogs, 25 people own only cats, and 12 people own only snakes. Additionally, 15 people own both dogs and cats, 10 people own both cats and snakes, and 8 people own both dogs and snakes. Finally, 5 people own cats, dogs, and snakes. How many total snakes are there among these pet owners?
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Context:
Arrange the $n$ consecutive positive integers from 1 to $n$ (where $n > 1$) in a sequence such that the sum of each pair of adjacent terms is a perfect square. Find the minimum value of $n$.
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Context:
The equation \[2000x^6+100x^5+10x^3+x-2=0\]has two real roots. Compute the square of the difference between them.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:
The sum of the number of boys and girls in a school is 300. If the number of boys is a certain number, then the number of girls becomes that same number as a percentage of the total number of students. What is the number of boys?
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Context:
Cells of a  $n*n$  square are filled with positive integers in the way that in the intersection of the  $i-$ th column and  $j-$ th row, the number  $i+j$  is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to  $n$  and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the  $i-$ th column and  $j-$ row is written  $2n+2-i-j$ .
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Context:
In a board game played with dice, our piece is four spaces away from the finish line. If we roll at least a four, we reach the finish line. If we roll a three, we are guaranteed to finish in the next roll.

What is the probability that we will reach the finish line in more than two rolls?
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Context:
Let $z_1$, $z_2$, $z_3$, $\cdots$, $z_{2021}$ be the roots of the polynomial $z^{2021}+z-1$. Evaluate $$\frac{z_1^3}{z_{1}+1}+\frac{z_2^3}{z_{2}+1}+\frac{z_3^3}{z_{3}+1}+\cdots+\frac{z_{2021}^3}{z_{2021}+1}.$$
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Context:
The opposite of $\sqrt{3}$ is ____.
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Context:
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?
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Context:
Example 7.12 Find the number of second-kind circular permutations formed by selecting 6 elements with repetition from the 3-element set $A=\{a, b, c\}$.
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Context:
Diane has five identical blue disks, two identical red disks, and one yellow disk. She wants to place them on an 8-cell grid so that each cell contains exactly one disk. The two red disks must not be placed in cells that share a common edge. Calculate the total number of different-looking completed grids that Diane can produce.
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Context:
Two trains of some length and 200 meters are 630 meters apart. They are moving towards each other on parallel tracks, at speeds of 90 km/h and 72 km/h. The trains will meet after 13.998880089592832 seconds. What is the length of the first train?
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Context:
Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.
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Context:
Let the coefficient of \( x^{1992} \) in the power series \( (1 + x)^{\alpha} = 1 + \alpha x + \dots \) be \( C(\alpha) \). Find \( \int_{0}^{1} C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \, dy \).
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Context:
We call a path Valid if:

1. It only comprises the following types of steps:
   - $(x, y) \rightarrow (x + 1, y + 1)$
   - $(x, y) \rightarrow (x + 1, y - 1)$

2. It never goes below the x-axis.

Let $M(n)$ be the set of all valid paths from $(0,0)$ to $(2n,0)$, where $n$ is a natural number.

Consider a Valid path $T \in M(n)$. Denote:

$$\phi(T) = \prod_{i=1}^{2n} \mu_i$$

where $\mu_i$ is defined as:

- $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$
- $y$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y - 1)$

Now, let $f(n) = \sum_{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end of the decimal expansion of $f(2021)$.
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Context:
The minute hand of a clock moves 2 small units. How many circles does the second hand move in the same time? Express your answer as a whole number.
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Context:
Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\_\_\_\_\_\_$.
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Context:
The side of the base of a regular triangular prism $A B C A_1 B_1 C_1$ is 4, and the lateral edge is 3. On the edge $B B_1$, a point $F$ is taken, and on the edge $C C_1$, a point $G$ is taken such that $B_1 F=1, C G=\frac{2}{3}$. Points $E$ and $D$ are the midpoints of edges $A C$ and $B_1 C_1$ respectively. Find the minimum possible value of the sum $E P+P Q$, where point $P$ belongs to the segment $A_1 D$, and point $Q$ belongs to the segment $F G$.
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Context:
Let  $ABC$  be a scalene triangle whose side lengths are positive integers. It is called *stable* if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
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Context:
A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$, and find the value of $\lim _{n\to\infty} p(n)$. The original answer is in \frac{m}{n} format, please give the value of m + n.
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Context:
Define the "distance" between two sets of numbers A and B as the minimum value of $|a-b|$, where $a \in A, b \in B$. If $A=\{y|y=2x-1, x\in\mathbb{R}\}$ and $B=\{y|y=x^2+1, x\in\mathbb{R}\}$, then the "distance" between A and B is
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Context:
Let $n$ be the answer to this problem. Hexagon $ABCDEF$ is inscribed in a circle of radius 90. The area of $ABCDEF$ is $8n$, $AB=BC=DE=EF$, and $CD=FA$. Find the area of triangle $ABC$.
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Context:
If the probability of producing a Grade B product is $0.03$, and the probability of producing a Grade C product is $0.02$, calculate the probability of randomly inspecting a product and finding it to be a qualified product.
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Context:
Claudia has a total of 12 coins consisting of 5-cent and 10-cent coins, with which she can combine to make exactly 17 different values using some or all of these coins. Find the number of 10-cent coins Claudia has. Express your answer as a whole number.
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Context:
Given that $y$ is the unique solution to the IVP $xy''+y'-xy=0, y(0)=1, y'(0) = 0,$ how can we compute $\int_0^{\infty} xe^{-x^2/2}y \, dx$? The series for $y$ is given as $y = \sum\limits_{n \ge 0} \frac{(x/2)^{2n}}{(n!)^2}$.
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Context:
Find the largest integer \( A \) such that for any permutation of natural numbers from 1 to 100, there always exists a sequence of 10 consecutive numbers whose sum is greater than or equal to \( A \).
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Context:
For a positive integer $n,$ let $P_n$ be the set of sequences of $2n$ elements, each $0$ or $1,$ where there are exactly $n$ $1$’s and $n$ $0$’s. I choose a sequence uniformly at random from $P_n.$ Then, I partition this sequence into maximal blocks of consecutive $0$’s and $1$’s. Define $f(n)$ to be the expected value of the sum of squares of the block lengths of this uniformly random sequence. What is the largest integer value that $f(n)$ can take on?
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Context:
In 60 chandeliers (each with 4 shades), the shades need to be replaced. Each electrician takes 5 minutes to replace one shade. A total of 48 electricians will be working. No more than one shade can be replaced in a chandelier at the same time. What is the minimum time required to replace all the shades in all the chandeliers?
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Context:
The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression?
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Context:
In right triangle \(ABC\), \(BC = 5\), \(AC = 12\), and \(AM = x\); \(\overline{MN} \perp \overline{AC}\), \(\overline{NP} \perp \overline{BC}\); \(N\) is on \(AB\). If \(y = MN + NP\), one-half the perimeter of rectangle \(MCPN\), find the value of \(y\). The original answer is in the form \(\frac{k - mx}{n}\). Please provide the value of \(k + m + n\).
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Context:
Given the proposition $p$: the inequality $2x-x^{2} < m$ holds for all real numbers $x$; and the proposition $q$: $|m-1|\geqslant 2$. If "$\neg p$" and "$p\wedge q$" are both false propositions, find the range of values for the real number $m$.
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Context:
Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .

*Proposed by Michael Ren*
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Context:
For a permutation  $P = (p_1, p_2, ... , p_n)$  of  $(1, 2, ... , n)$  define  $X(P)$  as the number of  $j$  such that  $p_i < p_j$  for every  $i < j$ . What is the expected value of  $X(P)$  if each permutation is equally likely?
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Context:
Ben wants to pick 56 sugar snap peas. At his current rate of picking, all will be picked in seven minutes. How long, in minutes, would it take Ben to pick a different number of sugar snap peas if it takes him 9 minutes to pick 72 sugar snap peas?
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Context:
(2013•Shanghai) For a function $g(x)$ defined on the interval $I$, denote $g(I) = \{y | y = g(x), x \in I\}$. Given a function $y = f(x)$ with domain $[0, 3]$ that has an inverse function $y = f^{-1}(x)$, and it is known that $f^{-1}([0, 1)) = [1, 2)$, $f^{-1}((2, 4]) = [0, 1)$. If the equation $f(x) - x = 0$ has a solution $x_0$, then $x_0 =$ .
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Context:
Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of $\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)$ (for instance, $(x+1)^{2}$ has two prime factors)?
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Context:
What is the smallest number of sides that a polygon (not necessarily convex) with an odd number of sides can have, if it can be divided into parallelograms?
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Context:
Let $A=\{x|x\geqslant 1\}$, $B=\{x|-1 \lt x \lt 2\}$. Find the intersection $A\cap B$ and express it in set notation, specifying the conditions for x.
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Context:
Given a geometric sequence ${a_n}$ with sum of the first $n$ terms $S_n$, and the equations $a_3=2S_2+5$ and $a_4=2S_3+5$, find the common ratio $q$ of this sequence. Express your answer as a single number.
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Context:
Given a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its asymptotes intersect the parabola $y^2 = 4x$ at two points A and B, distinct from the origin O. Let F be the focus of the parabola $y^2 = 4x$. If $\angle AFB = \frac{2\pi}{3}$, then the eccentricity of the hyperbola is ________.
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Context:
There are $110$ guinea pigs for each of the $110$ species, arranged in a $110 \times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs are arranged, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.
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Context:
Of 64 players on a cricket team, 37 are throwers. The rest of the team is divided so some are left-handed and the rest are right-handed. Assuming that all throwers are right-handed, there are total 55 right-handed players. What is the ratio of left-handed players to the rest of the team (excluding throwers)?
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Context:
Let $X$ be the set consisting of twenty positive integers $n, n+2, \ldots, n+38$. Determine the smallest value of $n$ such that any three numbers $a, b, c \in X$, not necessarily distinct, can form the sides of an acute-angled triangle.
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Context:
Seven cards numbered $1$ through $7$ lay stacked in a pile in ascending order from top to bottom ($1$ on top, $7$ on bottom). A shuffle involves picking a random card [i]of the six not currently on top[/i], and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after $10$ shuffles, $6$ is higher in the pile than $3$.
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Context:
If the function \( f(x) = \cos(n x) \cdot \sin\left(\frac{4}{n} x\right) \) (where \( n \in \mathbb{Z} \)) has a period of \( 3\pi \), then what is the set of possible values for \( n \)?
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Context:
Determine the number of ordered 9-tuples of positive integers \((a_{1}, a_{2}, \cdots, a_{9})\) (elements in the array can be the same) that satisfy the following condition: for any \(1 \leq i < j < k \leq 9\), there exists a different \(l (1 \leq l \leq 9)\) such that
$$
a_{i} + a_{j} + a_{k} + a_{l} = 100.
$$
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Context:
A pedestrian left point $A$ for a walk at a speed of $v$ km/h. After he had walked 6 km away from $A$, a cyclist left $A$ following him at a speed that was 9 km/h faster than the pedestrian's speed. When the cyclist caught up with the pedestrian, they turned back and returned together to $A$ at a speed of 4 km/h. At what value of $v$ will the pedestrian's total walking time be minimized?
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Context:
In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $r.$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:
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and it satisfies $S_n = (-1)^n \cdot a_n - \frac{1}{2^n}$. Define $b_n = 8a_2 \cdot 2^{n-1}$. If for any $n \in \mathbb{N}^*$, it always holds that $\lambda b_n - 1 > 0$, then the range of values for the real number $\lambda$ is __________.
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Context:
The differential equation of the family of circles whose center lies on the $x$-axis and passes through the origin is $\ldots$
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\boxed{y^2 - x^2 - 2xy\frac{dy}{dx} = 0}


Context:
In quadrilateral $ABCD$, $\angle{BAD} \cong \angle{ADC}$ and $\angle{ABD} \cong \angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in \(\frac{k}{nm}\) format, where \(m\) is given in the problem. Please find the value of \(k+n\).
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Context:
How many non-empty subsets of $\{1,2,\dots,11\}$ are there such that the product of its elements is the cube of an integer?
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Context:
Consider a linear transformation $B$ that maps a four-dimensional subspace $L$ of a five-dimensional vector space $V$ to a vector space $W$. If the intersection of the kernel of $B$ with $L$ is 0-dimensional, what are the possible dimensions of the image of $L$ under $B$?
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Context:
Expand the trinomial $(x^2+x+1)^n$. For $n=1, 2, 3, \ldots$, the expansion yields the generalized Pascal's triangle as shown below:
$$(x^2+x+1)^0 = 1$$
$$(x^2+x+1)^1 = x^2 + x + 1$$
$$(x^2+x+1)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$$
$$(x^2+x+1)^3 = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$$
$$(x^2+x+1)^4 = x^8 + 4x^7 + 10x^6 + 16x^5 + 19x^4 + 16x^3 + 10x^2 + 4x + 1$$
...
Observing the relationships between the coefficients of the polynomials, we can construct the generalized Pascal's triangle as shown. The construction rule is: the first row is 1, and each number in the following rows is the sum of the three numbers above it (if any of these three numbers do not exist, count them as 0), and the $k^{th}$ row consists of $2k+1$ numbers. If in the expansion of $(1+ax)(x^2+x+1)^5$, the coefficient of the $x^8$ term is 75, then the real number $a$ is __________.
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Context:
In an arithmetic sequence {a_n}, there are two terms a_m and a_k such that a_m = 1/k and a_k = 1/m (where m, k are positive integers and m!= k). Find the sum of the first mk terms of this sequence. Express your answer as a formula involving m and k.
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Context:
Find all values of \( m \) for which any solution of the equation

\[ 2019 \cdot \sqrt[3]{3.5x - 2.5} + 2018 \cdot \log_{2}(3x - 1) + m = 2020 \]

belongs to the interval \([1; 3]\).
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Context:
In a commercial football tournament, five teams participated. Each team was supposed to play exactly one match with each of the other teams. Due to financial difficulties, the organizers canceled some games. As a result, all teams ended up with a different number of points, and none of the teams had zero points in the points column. What is the minimum number of games that could have been played in the tournament if a win awarded three points, a draw one point, and a loss zero points?
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Context:
Find the number of primes  $p$  between  $100$  and  $200$  for which  $x^{11}+y^{16}\equiv 2013\pmod p$  has a solution in integers  $x$  and  $y$ .
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Context:
Brittany is older than Rebecca by 3 years, who is 25. Brittany goes on a 4-year vacation, during which she celebrates her birthday 3 times due to crossing the International Date Line. Additionally, there is a leap year during her vacation that affects her age calculation. How old is Brittany when she returns from her vacation?
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Context:
If the fractional equation $\frac{2k}{x-1}-\frac{3}{1-x}=1$ has a root, then $k=\_\_\_\_\_\_$.
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Context:
Once, Alice found herself in a magical forest where two amazing twin brothers, Tralalala and Trulala, lived. Tralalala always lied on Mondays, Tuesdays, and Wednesdays, but told the truth all other days of the week. Trulala always lied on Thursdays, Fridays, and Saturdays, but told the truth all other days of the week. One day, Alice met both brothers (it was impossible to distinguish who was who by appearance), and they declared:

First: "I lie on Saturdays."

Second: "I will lie tomorrow."

First: "I also lie on Sundays."

Determine which day of the week this meeting could have taken place.
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Context:
Two numbers \( x \) and \( y \) satisfy the equation \( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.
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Context:
A factory paints cubes in 6 colors (each face in its own color, with a fixed set of colors). How many different types of cubes can be produced?
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Context:
Find the value of $\displaystyle \int^{2021}_{-1}g(x)dx$ given that $g:\mathbb{R}-\{0\} \rightarrow \mathbb{R}$ satisfies the functional equation $g(x)\cdot g(y)=2g(xy)-g\bigg(\frac{2020}{x}\bigg)\cdot g\bigg(\frac{2020}{y}\bigg)\forall x,y\in \mathbb{R}-\{0\}$ and $g(2020)=1$, $g(-3)=-1$.
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Context:
During the Tree Planting Festival, students from a class are planting trees. There are three different types of saplings, labeled as A, B, and C, available in the nursery, with an ample supply of each type. They need to plant 5 saplings in a row in 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Find the total number of planting methods available. Express your answer as a single integer.
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Context:
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
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Context:
Let $A,B,C$ be angles of a triangle, where angle $B$ is obtuse, and \begin{align*}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \]where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime.  Find $p+q+r+s$.
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Context:
Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \( \frac{\text{gcd}(m,n)}{\text{lcm}(m,n)} \) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \( \frac{r}{s} \), compute \( r + s \).
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Context:
There was only enough room for some people on a raft if no one wore life jackets. There were 7 fewer people who could fit on the raft if everyone wore life jackets. If 8 people on the raft needed life jackets, 17 people could fit on the raft. How many people could fit on the raft if no one wore life jackets?
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Context:
For the quadratic function $y=x^{2}-4x+7$, when $-1 \lt x \lt 4$, the range of $y$ is ______.
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Context:
Forty children were holding hands in a circle. Among them, 22 were holding a boy's hand and 30 were holding a girl's hand. How many girls were in the circle?
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Context:
Compute the sum
\[\sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{1}{(i + j + 1)(i + j + 2)(i + j + 3)(i + j + 4)(i + j + 5)(i + j + 6)(i + j + 7)}.\]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:
Mrs. Evans has 385 crayons, 95 pieces of paper, and 210 pencils to give to her students. She wants to make sure that each student receives exactly 5 crayons, 1 piece of paper, and 2 pencils. What is the largest prime number of students she can have in her class while ensuring every student receives the exact same number of items?
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Context:
求函数$\ln ( \frac { s } { s + 9 } )$的拉普拉斯逆变换。
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Context:
There is a garden with 3 rows and 2 columns of rectangular flower beds, each measuring 6 feet long and 2 feet wide. Between the flower beds, as well as around the garden, there is a 1-foot wide path. What is the total area of the path in square feet?
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Context:
Find all integer solutions $(x, y)$ to the Diophantine equation $x^{6} = y^{2} + 53$.
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Context:
Given the probability function for selecting a digit d as log(d+1) - log(d), find the set of numbers that has a combined probability equal to twice the probability of selecting the digit 2. Express your answer as a set of numbers enclosed in curly brackets, with numbers separated by commas.
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Context:
If the sum of all binomial coefficients in the expansion of $$( \sqrt {x}- \frac {1}{2x})^{n}$$ is 512, then the constant term in this expansion is \_\_\_\_\_\_.
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Context:
Find the set of values for the real number a such that the quadratic inequality ax^2 - ax + 1 < 0 has no solution. Express your answer as an interval, including the endpoints if they are part of the solution.
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Context:
空间中点集定义如下: $A_{n}=\left\{\left.(x, y, z) \in \mathbf{R}^{3}|3| x\right|^{n}+|8 y|^{n}+|z|^{n} \leq 1\right\}, A=\bigcup_{n=1}^{\infty} A_{n}$, 求由 $A$ 中的点组成的图形的体积。
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Context:
Given points $A(-6, -1)$, $B(1, 2)$, and $C(-3, -2)$, find the coordinates of vertex $M$ of the parallelogram $ABMC$.
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Context:
Calculate the flux of the vector field $F(x,y,z) = \langle 9x, 9y, 4 \rangle$ across the part of the paraboloid $z = x^2 + y^2$ that lies between the planes $z = 0$ and $z = 4$. Assume the outward orientation for the closed surface, and take into account that the normal vector $n$ points downwards.
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Context:
Given the expression
\[
a_1 \cdot 5^1 + a_2 \cdot 5^2 + a_3 \cdot 5^3 + a_4 \cdot 5^4 + a_5 \cdot 5^5 + a_6 \cdot 5^6
\]
where \(a_1, a_2, a_3, a_4, a_5, a_6 \in \{-1, 0, 1\}\), count the number of expressions that are negative.
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Context:
After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible area of the spots becomes $S_{1}$. However, if the tablecloth is folded in half along the other diagonal, the total visible area of the spots remains $S$. What is the smallest possible value of the ratio $S_{1}: S$?
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Context:
已知 $\triangle A B C$ 为椭圆 $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ 的内接三角形，且 $A B$ 过点 $P(1,0)$ ，则 $\triangle A B C$ 的面积的最大值为 ? 请给出最大值的答案格式为 $\frac{m\sqrt{n}}{k} $的格式，请计算 $m + n+k$ 的值。
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Context:
Given that $a, b \in \mathbb{R}$, $a \neq 0$, and the curves $y = \frac{a + 2}{x}$ and $y = ax + 2b + 1$, if the two curves have at least one common point in the interval $[3, 4]$, find the minimum value of $a^2 + b^2$.
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Context:
Given a triangle \(ABC\). Points \(K, L,\) and \(M\) are placed on the plane such that triangles \(KAM, CLM,\) and \(KLB\) are all congruent to triangle \(KLM\). What inequality sign should be placed between the perimeter of triangle \(KLM\) and the semiperimeter of triangle \(ABC\)?

The vertices of the triangles are listed arbitrarily: for example, one cannot assert that in the congruent triangles \(KAM\) and \(CLM\), point \(K\) corresponds to point \(C\).
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Context:
Calculate the improper integral $\displaystyle \int ^{\infty}_{0}\frac{x^a}{(x+b)^2}\mathrm{d}x$ where $|a|<1$ and $b>0$.
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Context:
If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals
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Context:
For a Monty Hall problem with $n$ doors and $t$ cars, what is the minimum number of goats that the host must reveal to make switching choices a better strategy, assuming $n > 3$ and $t > 0$?
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Context:
已知 $a 、 b 、 c$ 是三个不全相等的实数且满足 $a=a b+c 、 b=b c+a 、 c=c a+b$. 则 $a+b+c=$ $\qquad$.
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Context:
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations

$2^x-2^y=1$
$4^x-4^y=\frac{5}{3}$

Determine $x-y$
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Context:
A book has 14 chapters. Beth read the first 4 chapters in 6 hours, and she took an additional 10 minutes break between each chapter. If she continues reading at the same pace and taking breaks, in how many hours will she be able to finish reading the remaining chapters?
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Context:
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number?
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Context:
Gina chooses what she and her sister will watch on Netflix some times as often as her sister does. If her sister watches a total of 24 shows on Netflix per week, and each show is 50 minutes long, Gina gets to choose 900 minutes of Netflix. What is the ratio of the number of times Gina chooses to the number of times her sister chooses?
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Context:
Through every three vertices of a cube, located at the ends of each trio of edges meeting at one vertex, a plane is drawn. Find the volume of the solid bounded by these planes if the edge length of the cube is $a$.
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Context:
Given \( m \) and \( n \) as natural numbers (\( m, n \in \mathbb{N} \)), find the minimum value of \( 12^m - 5^n \).
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Context:
A train of length 100 m is going with a speed of 10 m/sec. Another train of 150 m long is going with a speed of 15 m/sec. The distance between the two trains is some distance. The second train will take 60 seconds to cross the first train. What is the distance between the two trains?
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Context:
Find the limit of the following expression:

$$\lim_{n \to \infty} \frac{\sum\limits_{i = 0}^{n + 1} \binom{n + 1}{i}^3}{\sum\limits_{i = 0}^n \binom{n}{i}^3}$$
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Context:
3. We will call a pair of numbers magical if the numbers in the pair add up to a multiple of 7. What is the maximum number of magical pairs of adjacent numbers that can be obtained by writing down all the numbers from 1 to 30 in a row in some order?
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Context:
Tom's favorite number is between $100$ and $150$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Tom's favorite number?
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Context:
Call a positive integer $N$ a <i>7-10 double</i> if the digits of the base-$7$ representation of $N$ form a base-$10$ number that is twice $N$. For example, $51$ is a 7-10 double because its base-$7$ representation is $102$. What is the largest 7-10 double?
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Context:
A parallelogram with a perimeter of $2 p$ rotates around an axis perpendicular to the diagonal of length $d$ and passing through its end. Find the surface area of the solid of revolution.
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Context:
The left side of the equation $x^2-ax+9=0$ is a perfect square trinomial. Then, $a=$ .
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Context:
Given \((a + b i)^2 = 3 + 4i\), where \(a, b \in \mathbf{R}\), and \(i\) is the imaginary unit, find the value of \(a^2 + b^2\).
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Context:
Two circles with radii \( r \) and \( p \) (\( r < p \)) are externally tangent to each other, and both are internally tangent to a circle with radius \( R \). The triangle formed by the centers of these circles is isosceles, and the angle between the equal sides is greater than \( \frac{\pi}{3} \). Find the length of the base of this triangle.
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Context:
Find the common ratio of the infinite geometric series: $$\frac{-4}{7}+\frac{14}{3}+\frac{-98}{9} + \dots$$
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Context:
Team $A$ and team $B$ play a series. The first team to win three games wins the series. Before each game, each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If it turns out that team $B$ won the second game and team $A$ won the series, what is the conditional probability that team $B$ won the first game? The answer is in the form \(\frac{k}{m}\). Please provide the value of k + m.
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Context:
The positive integers $A, B, C$, and $D$ form an arithmetic and geometric sequence as follows: $A, B, C$ form an arithmetic sequence, while $B, C, D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3}$, what is the smallest possible value of $A + B + C + D$?
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Context:
In a circle, chords \( AC \) and \( BD \) intersect at point \( E \). The tangent to the circle passing through point \( A \) is parallel to \( BD \). It is known that \( CD:ED = 3:2 \) and the area of triangle \( \Delta ABE \) is 8. Find the area of triangle \( \Delta ABC \).
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Context:
Given a cubic polynomial $ 60 x^3 - 80x^2 + 24x - 2 $, which has three roots $\alpha, \beta, \gamma$, all distinct and lying in the interval $(0, 1)$. Find the sum
\[
\frac{1}{1-\alpha} + \frac{1}{1-\beta} + \frac{1}{1-\gamma}.
\]
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Context:
Let  $m = 30030$  and let  $M$  be the set of its positive divisors which have exactly  $2$  prime factors. Determine the smallest positive integer  $n$  with the following property: for any choice of  $n$  numbers from  $M$ , there exist 3 numbers  $a$ ,  $b$ ,  $c$  among them satisfying  $abc=m$ .
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Context:
Given the line y = mx is tangent to the circle x^2 + y^2 - 4x + 2 = 0, calculate the value of m. Express your answer as a single value, including the ± symbol if necessary.
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Context:
From the 12 edges and 12 face diagonals of a cube, select \( n \) line segments such that any two of the selected line segments are skew lines. What is the maximum value of \( n \)?
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Context:
Two months ago, Wilson had some money in his bank account. Then last month, he deposited $17. A few days after, he made a withdrawal. This month, he deposited $21 so he has $16 more than what is in his account two months ago. How much did Wilson have in his bank account two months ago?
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Context:
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square.  The minimum possible value of $\left\lvert d - \sqrt{n} \right\rvert$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive).  Compute $a+b+c$.

[i]Proposed by Matthew Lerner-Brecher[/i]
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Context:
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. What time does she finish the third task? Express your answer in 12-hour clock format (e.g., 3:30 PM).
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Context:
正整数 $Y=\sum_{i=0}^{2021}\left[\frac{2^{i}}{7}\right]$ 的个位数字是 $\qquad$.
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Context:
Five people are crowding into a booth against a wall at a noisy restaurant. If at most three can fit on one side, how many seating arrangements accommodate them all?
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Context:
The integer $n$, between 10000 and 99999, is represented as $abcde$ in decimal notation. The digits $a$, $b$, $c$, $d$, and $e$ are defined as follows:

- $a$ is the remainder when $n$ is divided by 2.
- $b$ is the remainder when $n$ is divided by 3.
- $c$ is the remainder when $n$ is divided by 4.
- $d$ is the remainder when $n$ is divided by 5.
- $e$ is the remainder when $n$ is divided by 6.

Find the value of $n$.
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Context:
A shop owner professes to sell his articles at a certain cost price. He uses false weights with which he cheats by 12% while buying and by some percentage while selling. His percentage profit is 40%. What is the percentage by which he cheats while selling?
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Context:
\( H O W \), \( B O W \), and \( D A H \) are equilateral triangles in a plane such that \( W O = 7 \) and \( A H = 2 \). Given that \( D \), \( A \), and \( B \) are collinear in that order, find the length of \( B A \).
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Context:
If $b$ is an even multiple of $7786$, find the greatest common divisor of $8b^2 + 85b + 200$ and $2b + 10$.
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Context:
Tim is organizing a week-long series of pranks. On Monday, he gets his friend Joe to help. On Tuesday, he can choose between two friends, either Ambie or John. For Wednesday, there are four new people willing to help. However, on Thursday, none of these previous individuals can participate, but Tim has convinced five different friends to help. On Friday, Tim decides to go solo again. Additionally, this time, whoever helps on Wednesday cannot help on Thursday. How many different combinations of people could Tim involve in his pranks during the week?
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Context:
Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism.
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Context:
Find the number of integer solutions to the equation $a^2+b^2+ab=1011$ using only modular arithmetic.
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Context:
A fruit drink is made of orange, watermelon, grape, apple, and pineapple juices. The drink contains 10 percent orange juice, 40 percent watermelon juice, 20 percent grape juice, and the remaining 30 percent consists of equal parts apple and pineapple juices. If the drink is made with 24 ounces of grape juice and 36 ounces of apple juice, how many ounces is the drink total?
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Context:
B2. Given is a quadrilateral $A B C D$

with sides $A B=16, B C=21, C D=2$ and $D A=28$.

Furthermore, $A B$ is parallel to $C D$. Two lines that are parallel to $A B$ and $C D$ divide quadrilateral $A B C D$ into three similar quadrilaterals. Calculate the perimeter of the smallest of these three quadrilaterals.
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Context:
In a plane, equilateral triangle \(ABC\), square \(BCDE\), and regular dodecagon \(DEFGHIJKLMNOP\) each have side length 1 and do not overlap. Find the area of the circumcircle of \(\triangle AFN\).
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Context:
Given the function $f(x)=x^{2}-6x+4\ln x$, find the x-coordinate of the quasi-symmetric point of the function.
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Context:
Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
You come across an ancient mathematical manuscript. It reads:

"To determine if a number is divisible by seventeen, follow this procedure:
- Take the number formed by the last two digits of the number.
- Subtract the number formed by the third- and fourth-to-last digits.
- Add the number formed by the fifth- and sixth-to-last digits, and so on.

The resulting number is divisible by seventeen if and only if the original number is divisible by seventeen."

What is the sum of the five smallest bases the ancient culture might have been using?

(Note: "Seventeen" is the number represented by $17$ in base $10$, not $17$ in the base that the ancient culture was using. Express your answer in base $10$.)
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Context:
James has barrels that he uses to store water. They store some gallons more than twice as much as a large cask. He has 4 barrels and a cask stores 20 gallons. He can store 172 gallons of water. How many gallons more do the barrels store compared to twice as much as a large cask?
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Context:
2nd Australian 1981 Problem B3 In an archery contest, the two contestants A and B start at x = 0 and walk towards the target at x = 1 together. Each chooses the distance from which he shoots. A has probability x 2 of hitting the target from x. B has probability x of hitting the target from x. If both hit the target, then the first to shoot wins. What is A's best strategy?
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Context:
设函数$f ( x ) = x ^ { 2 } - ( 4 a - 2 ) x - 6 a ^ { 2 }$在区间[0,1]上的最小值为g(a)，求g(a)的最大值.
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Context:
Find the mass percentage of B in Boric acid
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Context:
Find the maximum value of the expression \((\sqrt{8-4 \sqrt{3}} \sin x - 3 \sqrt{2(1+\cos 2x)} - 2) \cdot (3 + 2 \sqrt{11 - \sqrt{3}} \cos y - \cos 2y)\). If the answer is a non-integer, round it to the nearest whole number.
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Context:
Given a collection of stones. The mass of each stone is $≤ 2 kg$ and their total mass is equal to $100 kg$. We selected a set of stones whose total mass differs from $10 kg$ by the least possible for this set number d. What is the greatest value of d for every admissible collection of stones? The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$.
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Context:
Calculate the definite integral:

$$
\int_{0}^{\pi / 4} \frac{2 \operatorname{tg}^{2} x - 11 \operatorname{tg} x - 22}{4 - \operatorname{tg} x} \, dx
$$
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Context:
For the equation $\log_{10}x + x = 0$, define $f(x) = \log_{10}x + x$. Find an interval $[a, b]$ where $f(x)$ changes sign, indicating the existence of a real root. Express your answer in the form of an interval, e.g., [a, b].
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Context:
A sequence of one hundred natural numbers $x, x+1, x+2, \cdots, x+99$ has a sum $a$. If the sum of the digits of $a$ is 50, what is the smallest possible value of $x$?
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Context:
Using only once each of the digits  $1, 2, 3, 4, 5, 6, 7$  and  $ 8$ , write the square and the cube of a positive integer. Determine what that number can be.
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Context:
A mini train traveling at a speed of 75 kmph crosses a pole in 3 seconds. Now, there are additional four more poles placed after the first one. The distances from the first pole to the remaining poles are 500m, 800m, 1500m and 2200m respectively. What is the total time taken by the mini train to cross all the poles?
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Context:
Given square $ABCD$, points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ bisect $\angle ADC$. Calculate the ratio of the area of $\triangle DEF$ to the area of square $ABCD$.
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Context:
When "the day after tomorrow" becomes "yesterday", "today" will be as far from Sunday as "today" was from Sunday when "yesterday" was "tomorrow". What day of the week is today?
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Context:
What is the maximum number of cells that can be colored on a $6 \times 6$ board such that no four colored cells can be selected whose centers form a rectangle with sides parallel to the sides of the board?
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Context:
The equation \(\sin x = \frac{x}{2021 \pi}\) has exactly \(n\) solutions. Find \(n\).
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Context:
Given the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a > 0,b > 0)$, let $A$ and $B$ be the endpoints of its real axis. Let $P$ be a moving point on this hyperbola, and let the slopes of lines $AP$ and $BP$ be $k_1$ and $k_2$, respectively. Find the eccentricity of the hyperbola when the expression $k_1k_2-2\left( \ln |k_1|+\ln |k_2| \right)$ reaches its minimum value.
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Context:
From four different balls numbered 1, 2, 3, 4, three different balls are selected and placed into three different boxes numbered 1, 2, 3, with each box containing one ball. The total number of ways to place the balls such that ball number 1 is not in box number 1 and ball number 3 is not in box number 3 is
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Context:
A stationary stochastic process has a spectral density of $$ S_{XX}(\omega) = 1 - \frac{|\omega|}{8 \pi}. $$ What is the mean square value of the process?
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Context:
In a 7x7 geoboard, points A and B are positioned at (3,3) and (5,3) respectively. How many of the remaining 47 points will result in triangle ABC being isosceles?
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Context:
An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$.
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Context:
In a store where all items cost an integer number of rubles, there are two special offers:
1) A customer who buys at least three items simultaneously can choose one item for free, whose cost does not exceed the minimum of the prices of the paid items.
2) A customer who buys exactly one item costing at least $N$ rubles receives a 20% discount on their next purchase (regardless of the number of items).

A customer, visiting this store for the first time, wants to purchase exactly four items with a total cost of 1000 rubles, where the cheapest item costs at least 99 rubles. Determine the maximum $N$ for which the second offer is more advantageous than the first.
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Context:
An ant starts at the point $(0,0)$ in the Cartesian plane. In the first minute, the ant faces towards $(1,0)$ and walks one unit. Each subsequent minute, the ant chooses an angle $\theta$ uniformly at random in the interval $\left[-90^{\circ}, 90^{\circ}\right]$, and then turns an angle of $\theta$ clockwise (negative values of $\theta$ correspond to counterclockwise rotations). Then, the ant walks one unit. After $n$ minutes, the ant's distance from $(0,0)$ is $d_{n}$. Let the expected value of $d_{n}^{2}$ be $a_{n}$. Compute the closest integer to $$10 \lim _{n \rightarrow \infty} \frac{a_{n}}{n}$$
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Context:
693. A regular tetrahedron, octahedron, and icosahedron have equal surface areas. How do their edges relate?
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Context:
A large cube is built using 64 smaller identical cubes. Three of the faces of the large cube are painted. What is the maximum possible number of small cubes that can have exactly one face painted?
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Context:
One angle of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and a circle can be circumscribed around this trapezoid.
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Context:
Given a right circular cone $(P-ABC)$, points $P$, $A$, $B$, $C$ are all on the surface of a sphere with radius $\sqrt{3}$. If $PA$, $PB$, $PC$ are pairwise perpendicular, find the distance from the sphere's center to the cross-section $ABC$.
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Context:
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor^3$ is:
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Context:
A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the table, without slipping, and stops once the dot returns to the top. Find the length of the path followed by the dot in terms of $\pi$.
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Context:
Given that point A is on the parabola $C: x^2 = 2py$ ($p>0$), and O is the origin. If A and B are two points on the parabola $C$ that, along with point M(0, 9), form a circle with the radius equal to the length of $|OA|$, and $\triangle ABO$ is an equilateral triangle, then the value of $p$ is \_\_\_\_\_\_.
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Context:
Let P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that

$P(ux,uy, uz)=u^{2}F(y-x,z-x,z-x)$ for all real $x$, $y$, $z$, $u$,

and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let A, B, C be complex numbers with $P(A,B,C)=0$ and $|B- A|=10$. Find $|C-A|$. The original answer is in the format $(m/k)\sqrt{n}$, please give the value of m + n + k.
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Context:
Given a bag containing white and red balls with a total of $n$ balls and $m$ white balls among them. If two balls are drawn randomly, the probability that at least one ball is red is $\frac{3}{5}$. If the drawing is with replacement, and each draw consists of one ball for a total of 6 draws, then the expected number of times a white ball is drawn is 4. Now, drawing from the bag without replacement, two balls are drawn one by one. What is the probability of drawing another red ball given that the first ball drawn is red?
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Context:
Consider all polynomials of the form
\[x^7 + b_6 x^6 + b_5 x^5 + \dots + b_2 x^2 + b_1 x + b_0,\]
where \( b_i \in \{0,1\} \) for all \( 0 \le i \le 6 \). Find the number of such polynomials that have exactly two different integer roots, -1 and 0.
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Context:
Lucy started with a bag of 180 oranges. She sold $30\%$ of them to Max. From the remaining, she then sold $20\%$ to Maya. Of the oranges left, she donated 10 to a local charity. Find the number of oranges Lucy had left.
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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
\[
a_j^2 - a_j a_{j+1} + a_{j+1}^2
\]
for \(j = 1,2,3,4,5,6\) (where \(a_7\) is to be taken as \(a_1\)) are all equal to one another?
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Context:
In triangle \(A B C\), side \(B C\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \(A\) and is tangent to \(B C\), with one tangent at point \(B\) and the other at point \(C\).
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Context:
In the convex quadrilateral $\mathrm{ABCD}$, the angle bisector of $\angle \mathrm{B}$ passes through the midpoint of side $\mathrm{AD}$, and $\angle \mathrm{C} = \angle \mathrm{A} + \angle \mathrm{D}$. Find the angle $\mathrm{ACD}$.
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Context:
Given the hyperbola $ \frac{x^2}{9}-\frac{y^2}{16}=1 $, a tangent at any point $P$ on the hyperbola meets another hyperbola at $A$ and $B$. If $P$ is the midpoint of $AB$ for every choice of $P$, what is the floor of the sum of all possible values of the eccentricities of this new hyperbola?
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Context:
Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\).
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Context:
From the sequence of natural numbers $1, 2, 3, 4, \ldots$, erase every multiple of 3 and 4, but keep every multiple of 5 (for example, 15 and 20 are not erased). After removing the specified numbers, write the remaining numbers in a sequence: $A_{1}=1, A_{2}=2, A_{3}=5, A_{4}=7, \ldots$. Find the value of $A_{1988}$.
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Context:
Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.
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Context:
5. Let two regular tetrahedra $P-A B C$ and $Q-A B C$ be inscribed in the same sphere. If the dihedral angle between a lateral face and the base of the regular tetrahedron $P-A B C$ is $45^{\circ}$, then the tangent value of the dihedral angle between a lateral face and the base of the regular tetrahedron $Q-A B C$ is $\qquad$ .
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Context:
Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the smaller circle. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?
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Context:
GM Bisain's IQ is so high that he can move around in $10$-dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates.
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Context:
Given that the even function $y=f\left(x+1\right)$ is monotonically decreasing on the interval $\left[0,+\infty \right)$, the interval of monotonically increasing for the function $y=f\left(x-1\right)$ is ______.
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Context:
Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longer to jog around the outer edge of the track than around the inner edge. Calculate Michael's speed in meters per second.
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Context:
There are $360$ permutations of the letters in $MMATHS$. When ordered alphabetically, starting from $AHMMST$, $MMATHS$ is in the $n$th permutation. What is $n$?
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Context:
Cyclic quadrilateral \(ABCD\) has side lengths \(AB = 1\), \(BC = 2\), \(CD = 3\), and \(DA = 4\). Points \(P\) and \(Q\) are the midpoints of \(\overline{BC}\) and \(\overline{DA}\). Compute \(PQ^2\).
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Context:
2000 students take a paper of five multiple-choice questions each with four possible answers. Each student chooses one answer per question. What is the smallest \( n \) such that the students could have answered the questions in such a way that given any \( n \) students we can find four every two of whom answered at least two questions differently?
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Context:
Three. (50 points) Let $M=\{1,2, \cdots, 65\}, A \subseteq M$ be a subset. If $|A|=33$, and there exist $x, y \in A, x<y$, $x \mid y$, then $A$ is called a "good set". Find the largest $a \in M$ such that any 33-element subset containing $a$ is a good set.
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Context:
Find the largest integer less than 2012 all of whose divisors have at most two 1's in their binary representations.
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Context:
设$S = \{ A = ( a _ { 1 } , \cdots , a _ { 8 } ) \vert a _ { i } = 0$或中的两个元素$A = ( a _ { 1 } , \cdots , a _ { 8 } )$和$B = ( b _ { 1 } , \cdots , b _ { 8 } )$，记$d ( A , B ) = \sum _ { i = 1 } ^ { 8 } \vert a _ { i } - b _ { i } \vert$并称其为A和B之间的距离．问S中最多能取出多少元素，它们之中任何两个的距离≥5？
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Context:
A piece of music consisiting of three movements lasts for 60 minutes. No single movement is longer than the other two movements combined. The duration difference between any two movements is at least 3 minutes. What are the possible range limits for the duration of the shortest movement?
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Context:
Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of
\[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\]
where the sum is over all $12$ symmetric terms.
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Context:
The CMU Kiltie Band is attempting to crash a helicopter via grappling hook.  The helicopter starts parallel (angle  $0$  degrees) to the ground.  Each time the band members pull the hook, they tilt the helicopter forward by either  $x$  or  $x+1$  degrees, with equal probability, if the helicopter is currently at an angle  $x$  degrees with the ground.  Causing the helicopter to tilt to  $90$  degrees or beyond will crash the helicopter.  Find the expected number of times the band must pull the hook in order to crash the helicopter.
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Context:
27.4. (GDR, 78). A circle is circumscribed around a regular $2 n$-gon. A triplet of its distinct vertices is called one-sided if there exists a semicircle on which these vertices lie (the ends of the semicircle belong to it). What is the probability that a randomly chosen triplet of vertices will be one-sided?
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Context:
Find the least real number  $k$  with the following property: if the real numbers  $x$ ,  $y$ , and  $z$  are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]
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Context:
Find the smallest root of the equation

$$
\sin (\pi x)+\operatorname{tg} \chi=x+x^{3}
$$
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Context:
Given an arithmetic sequence {a_n}, the sequence {b_n} is defined by: b_1 = a_1, b_2 = a_2 + a_3, b_3 = a_4 + a_5 + a_6, etc. If the limit of b_n/n^3 as n approaches infinity is 2, find the common difference d of the sequence {a_n}. Express your answer as a single number.
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Context:
Find the sum of   $$ \frac{\sigma(n) \cdot d(n)}{ \phi (n)} $$   over all positive  $n$  that divide  $ 60$ .

Note: The function  $d(i)$  outputs the number of divisors of  $i$ ,   $\sigma (i)$  outputs the sum of the factors of  $i$ , and  $\phi  (i)$  outputs the number of positive integers less than or equal to  $i$  that are relatively prime to  $i$ .
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Context:
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes, respectively. A positive integer $n$ is called fair if $s(n) = c(n) > 1$. Find the number of fair integers less than $100$.
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Context:
A bus is traveling uphill along a road with varying slope gradients. The bus accelerates at a rate of 2 m/s^2 on a 5% slope, 3 m/s^2 on a 10% slope, and 4 m/s^2 on a 15% slope. If the bus starts from a stop (initial velocity of 0 m/s) and travels 50 meters on the 5% slope, followed by 80 meters on the 10% slope, and finally 70 meters on the 15% slope, what is the final velocity of the bus at the end of these three sections?
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Context:
Baker Ram is storing cake boxes in cartons that measure 25 inches by 42 inches by some inches. If the measurement of each cake box is 8 inches by 7 inches by 5 inches, and the maximum number of cake boxes that can be placed in each carton is 210, what is the height of the carton?
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Context:
A hexagonal prism has a height of \(165 \text{ cm}\). Its two hexagonal faces are regular hexagons with sides of length \(30 \text{ cm}\). Its other six faces are rectangles. A fly and an ant start at point \(X\) on the bottom face and travel to point \(Y\) on the top face. The fly flies directly along the shortest route through the prism. The ant crawls around the outside of the prism along a path of constant slope so that it winds around the prism exactly \(n+\frac{1}{2}\) times, for some positive integer \(n\). The distance crawled by the ant is more than 20 times the distance flown by the fly. What is the smallest possible value of \(n\)?
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Context:
Let $P(x)$ be a polynomial with positive integer coefficients and $P(0) > 1$. The product of all the coefficients is $47$ and the remainder when $P(x)$ is divided by $(x - 3)$ is $9887$. What is the degree of $P(x)$?
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Context:
In the complex plane, the vertices of the right-angled triangle $\triangle ABC$ correspond to the complex numbers $z+1$, $2z+1$, and $(z+1)^2$, with $A$ being the right-angled vertex, and $|z|=2$. Define the set $M=\left\{m \mid z^{m} \in \mathbf{R}, m \in \mathbf{N}_{+}\right\}$ and set $P=\left\{x \mid x=\frac{1}{2^{m}}, m \in M\right\}$. Find the sum of all elements in set $P$.
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Context:
In measuring the sides of a rectangle, one side is taken in excess, and the other 5% in deficit. The error percent in the area calculated from these measurements is 0.7%. What is the percentage in excess for one side?
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Context:
The base of a right prism is a right triangle with a hypotenuse of length \( c \) and an acute angle of \( 30^\circ \). A plane is drawn through the hypotenuse of the lower base and the vertex of the right angle of the upper base, forming an angle of \( 45^\circ \) with the base's plane. Determine the volume of the triangular pyramid cut off from the prism by this plane.
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Context:
Compute
\[
\log_2 \left( \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1+e^{2\pi i a b/2015}) \right)
\]
Here $i$ is the imaginary unit (that is, $i^2=-1$).
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Context:
Given the expressions x = 0.82^0.5, y = sin(1), and z = log_3(sqrt(7)), determine the correct order of magnitudes of x, y, and z. Express your answer in the format x < y < z or x > y > z, etc.
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Context:
In the game of *Winners Make Zeros*, a pair of positive integers  $(m,n)$  is written on a sheet of paper.  Then the game begins, as the players make the following legal moves:

- If  $m\geq n$ , the player choose a positive integer  $c$  such that  $m-cn\geq 0$ , and replaces  $(m,n)$  with  $(m-cn,n)$ .
- If  $m<n$ , the player choose a positive integer  $c$  such that  $n-cm\geq 0$ , and replaces  $(m,n)$  with  $(m,n-cm)$ .

When  $m$  or  $n$  becomes  $0$ , the game ends, and the last player to have moved is declared the winner.  If  $m$  and  $n$  are originally  $2007777$  and  $2007$ , find the largest choice the first player can make for  $c$  (on his first move) such that the first player has a winning strategy after that first move.
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Context:
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?
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Context:
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is selected. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 12 and \( \cos \alpha = \frac{1}{4} \).
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Context:
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. Find the number of distinct lines that pass through at least two members of $S$.
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Context:
In a party with $1982$ people, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
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Context:
Find all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < d_{4} < n$ and $1 + n = 5(d_{1} + d_{2} + d_{3} + d_{4})$.
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Context:
Mary is baking a cake. The recipe calls for 11 cups of sugar, 6 cups of flour, and 9 cups of salt. She already put in some cups of flour. She needs to add 2 more cups of sugar than cups of salt now. How many cups of flour has she already put in?
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Context:
The six-digit number $20210A$ is prime for only one digit $A$. What is the value of $A$?
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Context:
A TV show costs $100,000 per episode for the first season and twice that much for every other season. The first season had a certain number of episodes, and every season after that had 50% more episodes except the last season, which had 24 episodes. The show had 5 seasons. It cost $16,800,000 to produce all the episodes. How many episodes were in the first season?
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Context:
Aaron and his brother Carson each saved up $150 for a weekend outing. They first go to a fancy restaurant where the bill amounts to 3/4 of their total money inclusive of a 15% service charge. Later, they decide to treat themselves with some expensive gourmet ice cream. Each scoop of ice cream costs $4. If they both get the same number of ice cream scoops and they are both left with $4 at the end of the day, how many scoops of ice cream did they each buy?
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Context:
8. Consider each permutation of the nine digits $1,2, \cdots, 9$ as a nine-digit number. Among them, the number of those that are divisible by 11 is $\qquad$.
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Context:
Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$.
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Context:
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
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Context:
It is given that the number $4^{11} + 1$ is divisible by some prime greater than $1000$. Determine this prime.
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Context:
A rigid board with a mass \( m \) and a length \( l = 20 \) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \( 2m \) was placed at its very edge. How far from the stone can a person with a mass of \( m / 2 \) walk on the board? Neglect the sizes of the stone and the person compared to the size of the board.
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Context:
A $12 \times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number $N$ such that if $N$ rooks are placed on the unit squares, one rook per square, we can always find $7$ rooks such that no two are attacking each other.
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Context:
给定正整数 $n$, 求最大的正整数 $k$, 使得如下命题成立: 对每个 $1 \leq i \leq 2 n$,设 $A_{i}$ 是若干个相邻的整数构成的集合（即每个 $A_{i}$ 都形如 $\{a+1, a+2, \cdots, a+r\}$ ，其中 $a$ 是整数, $r$ 是正整数），如果对任何 $1 \leq i \leq n, n+1 \leq j \leq 2 n$ 都有 $A_{i} \cap A_{j} \neq \emptyset$ ，则存在整数 $x$ ，使得集合 $\left\{1 \leq i \leq 2 n \mid x \in A_{i}\right\}$ 包含至少 $k$ 个不同的元素. 请给出当n=5时的k值.
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Context:
Suppose that global mean temperature is normally distributed $N(\mu, \sigma)$ and $N$ samples have been drawn from this distribution: $T_1, \cdots, T_N$. Let $T_{maxN}$ be the maximum of those first $N$ samples. What is the probability that the next three samples drawn are successively greater than the maximum of the first $N$ samples, i.e., $$P(T_{N+3} > T_{N+2} > T_{N+1} > T_{maxN})$$
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Context:
A point is randomly thrown on the segment \([6, 11]\) and let \( k \) be the resulting value. Find the probability that the roots of the equation \(\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
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Context:
If there exists an $x \in [-1, 2]$ such that $a \cdot 3^x \geq x - 1$, find the minimum value of the real number $a$.
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Context:
Three positive integers are each greater than $1$, have a product of $1728$, and are pairwise relatively prime. What is their sum?
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Context:
In the interval [0, 2π], how many solutions does the equation tan(2x) = cos(x/2) have? Express your answer as a whole number.
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Context:
Given the function \( f(x) = \frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1} \), calculate the sum \( \sum_{i=1}^{2009} f\left( \frac{i}{2009} \right) \). Express your answer as a single integer.
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Context:
Given a real number \( a \) such that there is only one real number \( x \) satisfying the inequality \( \left|x^{2}+2 a x+3 a\right| \leqslant 2 \), what is the number of real numbers \( a \) that meet this condition?
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Context:
Given \( a, b \in \mathbf{N}^{*} \) and satisfying \( \frac{1}{a} - \frac{1}{b} = \frac{1}{2021} \), determine the number of positive integer pairs \( (a, b) \).
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Context:
. Let \( n \geqslant 3 \) be an integer. A frog is to jump along the real axis, starting at the point 0 and making \( n \) jumps: one of length 1, one of length 2, \(\ldots\), one of length \( n \). It may perform these \( n \) jumps in any order. If at some point the frog is sitting on a number \( a \leqslant 0 \), its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number \( a > 0 \), its next jump must be to the left (towards the negative numbers). Find the largest positive integer \( k \) for which the frog can perform its jumps in such an order that it never lands on any of the numbers \( 1, 2, \ldots, k \).
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Context:
a typist uses a sheet measuring 20 cm by 30 cm lenghtwise . if a margin of 2 cm is left on each side and a 3 cm margin on the top and bottom , then what is the percentage of page used by the typist ?
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Context:
Let  $ m, n \geq 1$  be two coprime integers and let also  $ s$  an arbitrary integer. Determine the number of subsets  $ A$  of   $ \{1, 2, ..., m \plus{} n \minus{} 1\}$  such that  $ |A| \equal{} m$  and  $ \sum_{x \in A} x \equiv s \pmod{n}$ .
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Context:
Three truck drivers went into a roadside cafe. One driver bought four sandwiches, a cup of coffee, and ten donuts for a total of 1 dollar and 69 cents. The second driver bought three sandwiches, a cup of coffee, and seven donuts for 1 dollar and 26 cents. How much did the third driver pay for a sandwich, a cup of coffee, and a donut?
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Context:
Byan is playing a game called "raven, raven, falcon" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either "raven" or "falcon," each with a probability of $\frac{1}{2}$. The game ends when Byan has said "falcon" twice. The probability that one of his friends will be called a "falcon" twice can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
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Context:
Let $x_i \in \mathbb{R}$ such that $|x_i|<1$ for $i=1, 2,..., n$ and $x_1^2+x_2^2+...+x_n^2 = (x_1+x_2+...+x_n)^2 + 2561$. Find the least value of $n$ so that the equation is true.
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Context:
A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\frac{1}{n^{2}}$ for $n=1,2, \ldots, 2016$. Find \lfloor 2017 P(2017)\rfloor.
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Context:
A man spends 1/5 of his salary on food, 1/10 of his salary on house rent, 3/5 of his salary on clothes, and saves 1/20 of his salary. After accounting for taxes, which take away 1/8 of his initial salary, he still has $18,000 left. Find his original salary before taxes.
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Context:
In a regular quadrangular pyramid, the plane angle at the apex is equal to the angle between the lateral edge and the base plane. Determine the dihedral angles between adjacent lateral faces of this pyramid.
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Context:
a cone of height 9 cm with diameter of its base 18 cm is carved out from a wooden solid sphere of radius 9 cm . the percentage of the wood wasted is :
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Context:
Given the function \( f(x) = x^2 + x + \sqrt{3} \), if for all positive numbers \( a, b, c \), the inequality \( f\left(\frac{a+b+c}{3} - \sqrt[3]{abc}\right) \geq f\left(\lambda \left(\frac{a+b}{2} - \sqrt{ab}\right)\right) \) always holds, find the maximum value of the positive number \( \lambda \).
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Context:
Given the sets $A={1,4,x}$ and $B={1,2x,x^{2}}$, if $A \cap B={4,1}$, find the value of $x$.
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Context:
Given the function f(x) = lg [sin (πx)⋅sin (2πx)⋅sin (3πx)⋅sin (4πx)], find the number of open intervals in the domain of f(x) that intersect with the interval [0,1]. Express your answer as an integer.
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Context:
Given the set $S = \{ xy(x + y) \mid x, y \in \mathbb{N} \}$. Let $a$ and $n$ be natural numbers such that $a + 2^k \in S$ for all $k = 1, 2, 3, \ldots, n$. Find the greatest value of $n$.
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Context:
In the country of Limonia, the currency consists of coins worth $2^{n}, 2^{n-1} \cdot 3,2^{n-2} \cdot 3^{2}$, $2^{n-3} \cdot 3^{3}, \ldots, 2 \cdot 3^{n-1}, 3^{n}$ piastres, where $n$ is a natural number. A resident of the country went to the bank without any cash. What is the largest amount that the bank cannot provide?
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Context:
Distinct points $A, B, C, D$ are given such that triangles $A B C$ and $A B D$ are equilateral and both are of side length 10 . Point $E$ lies inside triangle $A B C$ such that $E A=8$ and $E B=3$, and point $F$ lies inside triangle $A B D$ such that $F D=8$ and $F B=3$. What is the area of quadrilateral $A E F D$ ?
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Context:
In Anchuria, a checkers championship takes place over several rounds. The days and cities for hosting the rounds are determined by a draw. According to the championship rules, no two rounds can be held in the same city and no two rounds can be held on the same day. A lottery is held among the fans: the grand prize goes to the person who correctly guesses the cities and days for all the rounds before the championship starts. If no one guesses correctly, the grand prize is awarded to the championship organizing committee. There are eight cities in Anchuria, and the championship is scheduled to take place over eight days. How many rounds should there be in the championship to maximize the probability that the organizing committee wins the grand prize?
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Context:
How many connected components are there in the complement in $\mathbb{R}^2$ of the set $ \{(x,1/x)|x \in \mathbb{R} , x \neq 0 \}$?
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Context:
In each cell of a chessboard with $2$ rows and $2019$ columns, a real number is written such that:
1. There are no two numbers written in the first row that are equal to each other.
2. The numbers written in the second row coincide with (in some other order) the numbers written in the first row.
3. The two numbers written in each column are different, and they add up to a rational number.

Determine the maximum quantity of irrational numbers that can be in the chessboard.
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Context:
On triangle $ABC$, let $D$ be the point on $AB$ such that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ such that $AE$ bisects angle $BAC$. Let $G$ be the intersection of $AE$ and $CD$, and let point $F$ be the intersection of side $AC$ and the ray $BG$. If $AB$ has length $28$, $AC$ has length $14$, and $CD$ has length $10$, then the length of $CF$ can be written as $\frac{k-m\sqrt{p}}{n}$ where $k$, $m$, $n$, and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $k - m + n + p$.
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Context:
In $\triangle PQR$, points $X$ and $Y$ lie on $\overline{QR}$ and $\overline{PR}$, respectively. If $\overline{PX}$ and $\overline{QY}$ intersect at $Z$ such that $PZ/ZX = 2$ and $QZ/ZY = 5$, what is $RX/RY$?
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Context:
设 V 为关于 x 的次数小于或等于 10 的实多项式构成的集合，且 a 为实数. 定义 V 到 V 的线性映射 F 为 F(f)=(x^2f)''+f'-af. 记 I 为使得 F 不是满射的 a 的所有可能值之和，则 [I]=__________.
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Context:
Sam has 1 Among Us task left. He and his task are located at two randomly chosen distinct vertices of a 2021-dimensional unit hypercube. Let $E$ denote the expected distance he has to walk to get to his task, given that he is only allowed to walk along edges of the hypercube. Compute $\lceil 10E \rceil$.
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Context:
After eating a hearty meal they went to see the Buckingham palace. There, Rachel learned that 45 visitors came to the Buckingham palace on the previous day. If 829 people visited the Buckingham palace within the past 85 days. How many visitors visited the Buckingham palace on that day?
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Context:
Find a bound for $\left|\int_x^\infty \frac{e^{-t}}{t^{n+1}}\mathrm dt\right|$ when $|x| \gg 0$.
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Context:
If Greg rolls six fair six-sided dice, what is the probability that he rolls more 2's than 5's?
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Context:
Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. Compute the sum of $a+b$ across all expressions that Rishabh writes.
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Context:
A train 70 meters long takes 14.998800095992321 seconds to cross a bridge of a certain length if the speed of the train is 36 kmph. How long is the bridge?
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Context:
We know that the total weight of several crates is 10 tons, and no single crate weighs more than 1 ton. The crates need to be transported using trucks that each have a capacity of 3 tons. What is the minimum number of trucks that is always sufficient to transport the crates?
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Context:
Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 3$. Simplify:
\[
\frac{1}{b^2 + c^2 - 3a^2} + \frac{1}{a^2 + c^2 - 3b^2} + \frac{1}{a^2 + b^2 - 3c^2}.
\]
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Context:
Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have?
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{5 / 2}^{10 / 3} \frac{\sqrt{x+2}+\sqrt{x-2}}{(\sqrt{x+2}-\sqrt{x-2})(x-2)^{2}} d x
$$
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Context:
The quotient of dividing a certain three-digit number by 11 is equal to the sum of the squares of the digits of the dividend. Find this number.
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Context:
In a one-dimensional random walk with 10 steps, each of size 1, the process $S_n$ is defined as the sum of the first $n$ steps, where $S_n = \sum_{i=1}^{n} X_i$ and $X_i$ can be +1 or -1. The condition $S_i \geq 0$ must hold for all $i$ from 1 to 10. Given that $S_{10} = 4$, how many distinct paths are there that satisfy this constraint?
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Context:
How many solutions does the equation $\tan x = \tan (\tan x + \frac{\pi}{4})$ have in the interval $0 \leq x \leq \tan^{-1} 1884$?
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Context:
Alex and Katy play a game on an \(8 \times 8\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the number of Ks on the grid at the end of the game. What is the highest score Katy can be sure to get if she plays well, no matter what Alex does?
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Context:
Calculate $f_5\left(\frac{8\pi}{3}\right)$, where $f_n(x) = \cos x \cos \frac{x}{2} \cos \frac{x}{4} \cdots \cos \frac{x}{2^{n-1}}$. Express your answer as a fraction with a square root.
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Context:
Example 3 Add three digits after 764 to make the resulting six-digit number divisible by 8, 9, and 11.
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Context:
设S={1,2,3,4}，n项的数列：$q _ { 1 } , q _ { 2 } , \cdots , q _ { n }$有下列性质：对S的任意非空子集B，该数列中恰有相邻的｜B｜项组成集合B，则n的最小值为.
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Context:
A train of a certain length running at the speed of 72 km/hr takes 14.098872090232781 seconds to cross a bridge 132 m in length. How long is the train?
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Context:
Two circles are given. Their common internal tangents are mutually perpendicular. The chords connecting the points of tangency are 3 and 5. Find the distance between the centers of the circles.
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Context:
How many normal $4\times4$ magic squares have the property that the sum of every row, column, and $2\times2$ subsquare is equal to a constant?
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Context:
求最大正整数 $m$ ，使得可以在 $m$ 行 $8$列的方格表的每个方格中填入 $C、G、M、O$ 这四个字母之一，并且具有如下性质：对于方格表的任意不同两行，至多存在一列，使得这两行在此列处的字母相同。
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Context:
An old car can drive 8 miles in one hour. After 5 hours of constant driving, the car needs to get cooled down which takes some time. This car can drive 88 miles in 13 hours. How long does it take to cool down the car?
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Context:
10. The base of the right prism $A B C A_{1} B_{1} C_{1}$ is a triangle $A B C$ with angle $B$ equal to $90^{\circ}$ and angle $C$ equal to $30^{\circ}$. Find the area of the section of the prism by a plane passing through the center of the lateral face $A A_{1} C_{1} C$ and vertex $B$ and parallel to the diagonal of the lateral face $A B_{1}$, if the distance from vertex $C$ to the cutting plane is 2, and the hypotenuse of the base of the prism is 4?
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Context:
nine (not necessarily distinct) nine-digit numbers have been formed; each digit has been used in each number exactly once. What is the maximum number of zeros that the sum of these nine numbers can end with

#
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Context:
From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:

$0|999,1|998, \ldots, 999|0$.

How many of these markers have only two different digits?
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Context:
直角ΔABC的斜边BC在半平面α上，A在$\alpha$上的射影为D，若$\angle A B D = 3 0 ^ { \circ }$.$\angle A C D = 4 5 ^ { \circ }$，则ΔABC所在半平面与$\alpha$所成二面角的平面角为弧度.
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Context:
有三根绳子，第一根(7/8)米，比第二根长(1/4)米，第三根比第二根长(2/5)米，第三根绳子有多长？
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Context:
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. Given that $S_{25} > 0$ and $S_{26} < 0$, find which term is the largest in the sequence $\left\{ \frac{S_1}{a_1}, \frac{S_2}{a_2}, \ldots, \frac{S_{25}}{a_{25}} \right\}$.
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Context:
Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three 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:
Sixteen 6-inch wide square posts are evenly spaced with 4 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence?
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Context:
Each page number of a 488-page book is printed one time in the book.  The first page is page 1 and the last page is page 488. When printing all of the page numbers, how many more 4's are printed than 8's?
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Context:
Compute $\gcd(6^210^2,15^4)$.
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Context:
Given that point $A(x_{0},y_{0})(x_{0}{\neq }0)$ is the intersection point of one asymptote of the parabola $C_{1}$: $y^{2}{=}2px(p{ > }0)$ and the hyperbola $C_{2}$: ${\dfrac{x^{2}}{a^{2}}{-}}{\dfrac{y^{2}}{b^{2}}}{=}1(a{ > }0{,}b{ > }0)$, if the distance from point $A$ to the axis of symmetry of the parabola $C_{1}$ is $p^{2}$, and the eccentricity of the hyperbola is equal to ${\sqrt{5}}$, find the value of $p$.
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Context:
If the acute angles \( A, B, C \) satisfy \( \sin^2 A + \sin^2 B + \sin^2 C = 2 \), find the minimum value of \( \frac{1}{\sin^2 A \cos^4 B} + \frac{1}{\sin^2 B \cos^4 C} + \frac{1}{\sin^2 C \cos^4 A} \).
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Context:
设$A = \{ 1 , 2 , \cdots , 2 0 0 2 \}$,$M = \{ 1 0 0 1 , 2 0 0 3 , 3 0 0 5 \}$.对$A$的任一非空子集$B$，当$B$中任意两数之和不属于$M$时，称$B$为$M-$自由集.如果$A = A _ { 1 } \cup A _ { 2 } , A _ { 1 }$U$A _ { 2 } = \varnothing$，且$A _ { 1 } 、 A _ { 2 }$均为$M-$自由集，那么，称有序对$( A _ { 1 } , A _ { 2 } )$为A的一个$M-$划分.试求$A$的所有$M-$划分的个数.
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Context:
Sara used $\frac{5}{8}$ of a roll of wrapping paper to wrap four presents. She used an additional $\frac{1}{24}$ of a roll on one of the presents for decorative purposes. How much wrapping paper did she use on each of the other three presents?
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