Context:
What is the maximum number of zeros for the Laplace transform of a function that is the difference of two sums, each containing $k$ exponential terms with positive exponents?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many structural isomers have the formula ${C_3H_6Cl_2}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the positive integers,  $m,m+1,\dots,m+n$ , only the sum of digits of  $m$  and the sum of digits of  $m+n$  are divisible by  $8$ . Find the maximum value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A criminal must pass 10 lie detector tests successfully to escape from jail. The probabilities of each lie detector detecting a lie are as follows: 0, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{4} \), \( \frac{4}{5} \), \( \frac{5}{6} \), \( \frac{6}{7} \), \( \frac{7}{8} \), \( \frac{8}{9} \), \( \frac{9}{10} \). If a detector indicates a lie, the criminal must start over from the first detector. Each test takes 1 second. Calculate the expected time in seconds for the criminal to pass all tests and escape.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ten spheres of the same radius are stacked in the form of a triangular pyramid so that each sphere touches at least three others. Find the radius of the circumscribed sphere for the pyramid of spheres, given that the radius of the sphere inscribed in the center of the pyramid of spheres, touching six identical spheres, is $\sqrt{6}-1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair die is rolled repeatedly. Let $X$ be the number of rolls needed to obtain a 5, and $Y$ the number of rolls needed to obtain a 6. Calculate the expected value $E[X \mid Y=2]$, where $Y=2$ indicates that a 6 is rolled on the second roll.
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Context:
Given the parabola y = x^2 - 2x + 1 and points A(0, y1), B(1, y2), and C(-2, y3) on the parabola, determine the relationship among y1, y2, and y3. Express your answer using inequality symbols in the format yX > yY > yZ, where X, Y, and Z are the corresponding point indices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group of 10 chickens, 9 are real and each weighs 2kg, while 1 is a false chicken that weighs either 1kg or 3kg, each with a probability of \(\frac{1}{2}\). A machine randomly selects each chicken with a probability of \(\frac{1}{2}\) for weighing, and after weighing, all chickens are returned. Three weightings are performed, resulting in sets weighing 11kg, 12kg, and 13kg, respectively. What is the probability that the false chicken weighs 3kg?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the second-order partial derivative \( \frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2 \) for the function \( z \mapsto \log|z|^2 \) defined on \( \mathbb{C} - \{0\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Walter has exactly one penny, one nickel, one dime and one quarter in his pocket.  What percent of one dollar is in his pocket?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the side  $ AB$  of a cyclic quadrilateral  $ ABCD$  there is a point  $ X$  such that diagonal  $ BD$  bisects  $ CX$  and diagonal  $ AC$  bisects  $ DX$ . What is the minimum possible value of  $ AB\over CD$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the real number set R, set A = {x|0 ≤ x ≤ 6}, and set B = {x|x > 5}, describe the resulting set from the intersection of the complement of B in R and set A. Express your answer in set notation.
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Context:
Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number $d = d(\overline{X})$ such that for any $k$-form $X$ of $\overline{X}$, the variety $X$ has a $K$-point over some finite field extension $K$ of $k$ with degree $[K:k] \le d$? Answer YES or NO.
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Context:
The sequence starts with 12, 13, 15, 17, 111, 113, 117, 119, followed by a number, then 129, and 131. What is the ninth number in the sequence?
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Context:
Let \(f(z)= \frac{z+a}{z+b}\) and \(g(z)=f(f(z))\), where \(a\) and \(b\) are complex numbers. Suppose that \(\left| a \right| = 1\) and \(g(g(z))=z\) for all \(z\) for which \(g(g(z))\) is defined. Find the difference between the largest and smallest possible values of \(\left| b \right|\). The original answer is in the form \(\sqrt{m} - n\). Please give the value of m + n.
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Context:
Determine whether the Stieltjes integral \( \int_{0} ^{4} x^{2} d([x^{2}]) \) exists. If it exists, calculate its value.
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Context:
Consider the real sequence $x_0=a$ and $x_{n+1}=\frac{1}{4}+x_n-(x_n)^2$ for $n\geq 1$. Determine the values of $a$ for which the sequence $x_n$ converges to $-\frac{1}{2}$.
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Context:
Evaluate the limit: $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$ using L'Hopital's rule.
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Context:
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers  $a$  and  $b$ . A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between  $a$  and  $b$  be for this to happen?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all pairs of positive integers \((m, n)\) such that \( \frac{m}{n} = n.m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point $P$ lies on the line $y=2x$, if there exist two points $A$ and $B$ on the circle $C(x-3)^2+y^2=4$ such that $PA \perp PB$, then the range of the abscissa $x_{0}$ of point $P$ is _______.
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Context:
Fran writes the numbers \(1,2,3, \ldots, 20\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \(n\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \(n\) that are still on the chalkboard (including \(n\) itself). What is the expected number of moves that Fran must make to erase all the numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest prime number that divides the quantity $0! + (1!) \times 1 + (2!) \times 2 + (3!) \times 3 + \cdots + (50!) \times 50$.
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Context:
Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids.
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Context:
a, b, c, and d have a total of Rs. 2000 between them. a and c together have Rs. 900, b and c together have Rs. 1100, and a and d together have Rs. 700. How much money does c have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of isomorphism classes of associative rings with identity that have 35 elements.
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Context:
One day, a beverage warehouse sold 252 cans of soda to 100 customers, and each customer bought at least one can of soda. On this day, what is the maximum possible median number of cans of soda purchased by each customer?
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Context:
A bag contains 2 white balls and 2 black balls. Each time a ball is taken from the bag, if it's white it's returned to the bag and if it's black it's replaced by a white ball. The game ends when no black balls remain in the bag. Let Y be the number of times the game is played; calculate Var(Y).
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Context:
For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$?
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Context:
Evaluate the limit \( \lim_{n\to\infty}\dfrac{1}{n^{2+\beta}}\sum_{i=1}^{n}i^{\beta} \) using the Riemann integral of suitably chosen functions.
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Context:
Elmo and Elmo\u0027s clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player\u0027s turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo\u0027s clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle \(ABC\) with side length 2011, a point \(P\) is located inside the triangle. Line segments \(DE\), \(FG\), and \(HI\) are drawn through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Given that the ratio of the lengths of these segments is \(DE : FG : HI = 8 : 7 : 10\), calculate the sum \(DE + FG + HI\).
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Context:
给定正整数 $n 、 k$, 其中, $n$ 为偶数, $k \geqslant 2$,且 $n>4 k$. 在圆周上有 $n$ 个点, 若 \frac{n}{2} 条在圆内互不相交的弦的端点恰为这 $n$ 个点，则称这些弦为一个"匹配"。求最大的整数 $m$ ，满足对任意一个匹配，均存在圆周上连续 $k$ 个点，使得这个匹配中至少有 $m$ 条弦的所有端点均在这 $k$ 个点中. 请给出当$ k = 7$ 时 $m$ 的值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the following limit is true: \( \lim_{a\to -\infty}\int_{-\infty}^a f(x)\, dx=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f: X \rightarrow S^1$ be a continuous map from a path-connected topological space $X$, and let $p: \mathbb{R} \rightarrow S^1$ be the universal covering map. Determine the condition under which there exists a continuous map $h: X \rightarrow \mathbb{R}$ such that $p \circ h = f$. Provide the condition in terms of the fundamental group of $X$ and the map $f$. \( \pi_1(X) \rightarrow \pi_1(S^1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many three-digit positive numbers \( x \) are there that are divisible by 3 and satisfy the equation \( \gcd(15, \gcd(x, 20)) = 5 \)? Find the largest of them.
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Context:
In a convex 12-sided polygon, all angles are equal. It is known that the lengths of ten of its sides are 1, and the length of one more side is 2. What could be the area of this 12-sided polygon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \oint_C \frac{\sinh(z)\cos(z)}{\cos(z)}dz \) using the Residue Theorem, where \( C \) is the contour defined by \( |z|=(n+1/2)\pi \) with \( n \in \mathbb{N}_0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school chat, a vote was held: "On which day to hold the disco: October 22 or October 29?"

The graph shows how the votes were distributed an hour after the start of the vote.

Then, 80 more people participated in the vote, voting only for October 22. After that, the voting finished. The final distribution of votes is also shown on the graph.

How many people participated in the vote?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis.
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Context:
The hyperbola $C$: $\dfrac{x^2}{4} - y^2 = 1$ has its left and right foci as $F_1$ and $F_2$, respectively. A line passing through $F_1$ intersects the left branch of the hyperbola at points $A$ and $B$. The minimum perimeter of $\Delta F_2AB$ is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the constant $k$ so that the graphs of the parabolas $y = x^2 + k$ and $x = y^2 + k$ are tangent to each other.
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Context:
When taking the Fast Fourier Transform (FFT) of a 2D Gaussian Dispersal Kernel given by the function \( G(x,y) = \frac{1}{4\pi \mu} e^{-\frac{x^2 + y^2}{4\mu}} \), where the spatial domain is a 7x7 grid with 500m cell length, and using a grid of 8 intervals (\(N = 8\)) for efficiency, the kernel is scaled by a factor of 4 during the calculation \(fK2D = dx \cdot dy \cdot \text{fft2}(K2D)\). Determine the normalization factor required for the kernel to ensure correct population values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The focus of the parabola $y^{2}=-8x$ coincides with the left focus of the hyperbola $\frac {x^{2}}{a^{2}}-y^{2}=1$. Find the angle between the two asymptotes of the hyperbola.
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Context:
The competition with $n (n \geqslant 2)$ contestants lasts for $k$ days. Each day, the scores of the contestants are exactly $1, 2, \cdots, n$ (with no two contestants having the same score). After $k$ days, the total score of each contestant is 26. Find all possible values of $(n, k)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the vertices of a graph $G$ be the integers $1,2,\dots,99$. The numbers $i \neq j$ are connected if their greatest common divisor is at least $3$. Find the chromatic number of $G$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We divide the height of a cone into three equal parts, and through the division points, we lay planes parallel to the base. How do the volumes of the resulting solids compare to each other?
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Context:
Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. Note that numbers like $012$ and $007$ are not considered $3$-digit numbers.
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Context:
The last digit in the natural number is 2016 times less than the number itself. Find all such numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of triples $0 \le k,m,n \le 100$ of integers such that \[ 2^mn - 2^nm = 2^k. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
例5 若自然数n满足这样的条件：存在由n个实数组成的数列，使得任何连续17个项之和为正，而任何连续10个项之和为负，求n的最大值．（改编题）
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum $\sum_{n=-\infty}^\infty \frac{\sin nx}{1+n^2}$ for $x\in[-\pi,\pi]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can we place two bishops on a chessboard in different rows, columns, and squares of different colors?
Please reason step by step, and put your final answer within \boxed{}.


Context:
3.3. Two identical cylindrical vessels are connected at the bottom by a small-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \, \text{cm} \). At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg \(/ \text{m}^3\), and the density of oil is 700 kg \(/ \text{m}^3\). Neglect the volume of the connecting pipe. Provide the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product of three prime numbers equals five times their sum. Find these three numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the chart, each number below the top row is the positive difference of the two numbers to the right and left in the row immediately above it. What is the value of x? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________.
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Context:
In the diagram, triangle \(ABC\) lies between two parallel lines as shown. If segment \(AC\) has length \(5 \ \text{cm}\), what is the length (in \(\text{cm}\)) of segment \(AB\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the given grid, all circles share the same center. What is the ratio of the area of the highlighted gray region to the area of the largest circle? Express your answer as a fraction or ratio.
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Context:
Rectangle \(PQRS\) is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. Calculate the length of \(QS\), rounded to 2 decimal places.
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Context:
In the given graph of the function \( y = kx + b \), compare \( |k| \) and \( |b| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
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Context:
For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which
\[r(n, 1000) > r(n, 1001).\]
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Context:
Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done?
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Context:
Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple?
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Context:
In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. Calculate the area of $\triangle AEF$ in square centimeters (cm^2).
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Context:
Two trains of length 120 m and 300 m are running towards each other on parallel lines at 42 kmph and a certain speed respectively. They will be clear of each other from the moment they meet in 20.99832013438925 seconds. What is the speed of the second train?
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Context:
Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)?
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Context:
Find the x-coordinate of the point of intersection of the functions \( y = 2^x \) and \( y = \frac{1}{x} \).
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Context:
In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter.
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Context:
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$
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Context:
Find the mass percentage of N in NH4Br
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Context:
What is the molecular weight of a compound having 2 H, 3 C, 1 N, 1 Cl, and 3 O?
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Context:
Determine the values of the parameter \(a\) for which the function \(f(x) = 8ax - a\sin 6x - 7x - \sin 5x\) is increasing and has no critical points for all real \(x\).
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Context:
Vasya solved problems for 10 days, solving at least one problem each day. Each day (except for the first day), if the weather was cloudy, he would solve one more problem than on the previous day, and if it was sunny, he would solve one less problem. In the first 9 days, Vasya solved a total of 13 problems. What was the weather on the tenth day?
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Context:
Arrange the numbers 1 through 9 in a 3x3 grid such that consecutive numbers are adjacent (share an edge), and the sum of the numbers in the four corners is 18. What is the number in the center of the grid?
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Context:
For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of
\[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]
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Context:
Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas?
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Context:
Jones is waiting for one of two telephone booths occupied by A and B. Let the duration of their calls be an exponential random variable with $\lambda=1/8$. What is the probability that of the three of them, Jones is not the last to finish their call?
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Context:
In an isosceles triangle \(ABC\) with \(AB = BC = 25\) cm and \(AC = 14\) cm, find the radius of a circle that touches \(BC\) at point \(D\), which is the foot of the altitude \(AD\), and passes through the midpoint of \(AC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. By how much is the sum that Mena calculated greater than the sum that Emily calculated?
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Context:
Bill picked 50 apples from the orchard with his wife and two children. He sends each of his kids to school with 3 apples for their two favorite teachers. His wife Jill bakes two apple pies, using a certain number of apples per pie. Bill has 24 apples left. How many apples does Jill use per pie?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.

The first person said: "Exactly 1 percent of the people present in this room are liars."

The second person said: "Exactly 2 percent of the people present in this room are liars."

and so on.

The person with number $n$ said: "Exactly $n$ percent of the people present in this room are liars."

How many people could be in the room, given that it is known that at least one of them is a knight?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(a\) such that \(\lim_{x\to\infty} \frac{(ax)!}{x^x} = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The radii of five concentric circles \(\omega_{0}, \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\) form a geometric progression with common ratio \(q\). What is the largest value of \(q\) for which it is possible to draw an open broken line \(A_{0} A_{1} A_{2} A_{3} A_{4}\), consisting of four segments of equal length, where \(A_{i}\) lies on \(\omega_{i}\) for all \(i = 0, 1, 2, 3, 4\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A taxi driver passes through six traffic checkpoints on the way from the restaurant to the train station. Assuming that the events of encountering a red light at each checkpoint are independent of each other and the probability is $\frac{1}{3}$ at each checkpoint, calculate the probability that the driver has passed two checkpoints before encountering a red light.
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Context:
Find the expectation \( E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}] \) where \( X_t \) is a Brownian bridge.
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Context:
Suppose $15\%$ of $x$ equals $20\%$ of $y$. What percentage of $x$ is $y$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two small weights are attached to the ends of a vertically oriented uniform spring. The weight with mass $m_{1}$ is on the top, and the weight with mass $m_{2}$ is at the bottom. A person grasps the middle of the spring and holds it vertically in the air. As a result, the upper half of the spring is deformed by $x_{1} = 8 \, \text{cm}$, and the lower half by $x_{2} = 15 \, \text{cm}$. Then, without inverting it, the person places the spring on a horizontal surface and releases it. Determine the deformation of the spring in this case. Neglect the size of the person's hand compared to the length of the spring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an  ${8}$ × ${8}$  squares chart , we dig out  $n$  squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .
Then find the mininum value of  $n$  .
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Context:
In a 150 member association consisting of men and women, exactly 10% of men and exactly 20% of women are homeowners. What is the least number of members who are homeowners?
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Context:
In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Furthermore, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
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Context:
Cagney can frost a cupcake in 15 seconds and Lacey can frost a cupcake in 25 seconds. After a 1-minute warm-up, their frosting speeds increase by 20%. Calculate the total number of cupcakes they can frost together in 10 minutes.
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Context:
If $V = gt + V_0$ and $S = \frac{1}{2}gt^2 + V_0t$, then $t$ equals:
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Context:
Given a positive sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, for $n \geq 2$, $(a_n - S_{n-1})^2 = S_nS_{n-1}$, and $a_1 = 1$. Let $b_n = \log_2 \frac{a_{n+1}}{6}$. Derive an expression for $b_n$ in terms of $n$.
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Context:
A cylindrical glass weighs 200 pounds and has a center of gravity 4.8 cm high. The base area of the glass is 20 cm². Determine the height of the water at which the center of gravity of the glass and water system is the lowest.
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Context:
2. (10 points) In each cell of a $50 \times 50$ square, a number is written that is equal to the number of $1 \times 16$ rectangles (both vertical and horizontal) in which this cell is an end cell. In how many cells are numbers greater than or equal to 3 written?
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Context:
Two swimmers, A and B, start swimming towards each other from opposite ends of a 100-meter pool at the same time. Swimmer A takes 72 seconds to swim 100 meters, and Swimmer B takes 60 seconds to swim 100 meters. Ignoring the time it takes to turn around, calculate how many times they meet in 12 minutes. Express your answer as a whole number.
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Context:
A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements? Express your answer in the form of \frac{k}{m}, where k and m are coprime integers. Please give the value of k + m.
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Context:
You have a  $3 \times 2021$  chessboard from which one corner square has been removed. You also have a set of  $3031$  identical dominoes, each of which can cover two adjacent chessboard squares. Let  $m$  be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the remainder when  $m$  is divided by  $19$ ?
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Context:
Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then find $\tan B.$
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Context:
Given that $| \overrightarrow{a}|=12$, $| \overrightarrow{b}|=9$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-54 \sqrt {2}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
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Context:
The circle C is tangent to the y-axis and the line l: y = (√3/3)x, and the circle C passes through the point P(2, √3). Determine the diameter of circle C.
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Context:
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations

$ax+by=1\,$
$x^2+y^2=50\,$
has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers.  How many such ordered pairs $(a,b)\,$ are there?
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Context:
In a group of people, 40% of the total number of people plus 60 more people constitutes the number of people wearing a specific type of hat. If 25% of the total number of people are wearing shoes of a specific brand, find the difference between the number of people wearing that specific type of hat and those wearing the specific brand of shoes, given that the number of people wearing both the hat and the shoes is 20% of the total population size. What is the total number of people in the group?
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Context:
What is the molecular weight of a compound having 1 Aluminium, 3 Bromine, 2 Oxygen, and 1 Chlorine in its molecular structure?
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Context:
Given three independent, identically distributed continuous random variables $X_1$, $X_2$, and $X_3$, each following an Exponential Distribution, calculate the probability that the median of these variables is closer to the smallest value $X_1$ than it is to the largest value $X_3$. Provide your answer as a probability value.
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Context:
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_0^{2\pi} x \frac{\cos x}{2-\cos^2 x} \, dx \]
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Context:
A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the separated parts was then alloyed with the remainder of the other piece, resulting in two new alloys with the same percentage of copper. What is the mass of each part separated from the original alloy pieces?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the molecular weight of a compound having 2 Al, 4 O, 6 H, 3 N, and 1 P?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y = g(x) is symmetric about the line x = 2, g(2) = 4, determine the value of \sum _{k=1}^{22}f(k).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
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Context:
设$T$为$2004^{100}$的所有正约数的集合，$T$的子集$S$满足$S$中的元素不是$S$中其他任意元素的整数倍.求$S$的元素个数的最大可能值.
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Context:
Given the function \( f(x) = \left(\frac{1}{a^{x}-1} + \frac{1}{2}\right) x^{2} + b x + 6 \) (where \(a\) and \(b\) are constants and \(a > 1\)), and \( f(\log \log_{8} 1000) = 8 \), find the value of \( f(\log \log 2) \). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the two smallest consecutive natural numbers, each of which has a digit sum divisible by 7.
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Context:
Principal Skinner is thinking of two integers  $m$  and  $n$  and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information.  Chalmers asks Skinner the numerical value of  $mn+13m+13n-m^2-n^2$ .  From the value of this expression alone, he miraculously determines both  $m$  and  $n$ .  What is the value of the above expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S = \{1, 2, 3, \ldots, 280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.
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Context:
Right triangle \(ABC\) has side lengths \(BC=6\), \(AC=8\), and \(AB=10\). A circle centered at \(O\) is tangent to line \(BC\) at \(B\) and passes through \(A\). A circle centered at \(P\) is tangent to line \(AC\) at \(A\) and passes through \(B\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
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Context:
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $2b\cos B = a\cos C + c\cos A$, then $B = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\). Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit of the sequence $\left(\frac{N_{2r}}{N_r}\right)_{r\in\mathbb{N}}$, where $N_r$ is the number of unit squares with integral rectangular coordinates that intersect a circle of radius $r$ centered at the origin.
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Context:
The decimal representation of \(\dfrac{1}{25^{10}}\) consists of a string of zeros after the decimal point, followed by non-zero digits. Find the number of zeros in that initial string of zeros after the decimal point.
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Context:
Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?
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Context:
Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For all real numbers $x$ such that $x \neq \pm1, \pm2, \dots, \pm10$, find the constant $k$ that satisfies the equation:

$$\sum_{j=1}^{10} \frac{4j}{x^2 - j^2} = k \sum_{j=1}^{10} \frac{1}{(x - j)(x + 11 - j)}$$

Calculator usage is not allowed.
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Context:
$在\triangle ABC中，a=3，b-c=2，\cos  B=-\frac{1}{2}.$
$求\sin (B-C)的值.$
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Context:
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$.
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Context:
Will the number $\frac{1}{1996}$ decrease or increase, and by how many times, if in the decimal notation of this number, the first non-zero digit after the decimal point is crossed out?
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Context:
Calculate the surface integral \(\iint_S F \cdot dS\) for the vector field \(F(x, y, z) = \langle x^3 + y^2, 2yz + e^z, y^2 - z^2 \rangle\), where \(S\) is the boundary of the solid \(E\) enclosed by the paraboloids \(z = x^2 + y^2\) and \(z = 12 - 2x^2 - 2y^2\), with outward-pointing normal. Use the Divergence Theorem to find and evaluate the integral.
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Context:
Company C sells a line of 50 products with an average retail price of $2,500. If none of these products sells for less than $800, and exactly 20 of the products sell for less than $2,000, what is the greatest possible selling price of the most expensive product?
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Context:
Two workers were assigned to produce a batch of identical parts; after the first worked for \(a\) hours and the second for \(0.6a\) hours, it turned out that they had completed \(\frac{5}{n}\) of the entire job. After working together for another \(0.6a\) hours, they found that they still had \(\frac{1}{n}\) of the batch left to produce. How many hours will it take for each of them, working separately, to complete the whole job? The number \(n\) is a natural number; find it.
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Context:
A weight of 200 grams rests on a table. It was flipped and placed on the table on its other side, the area of which is 15 sq. cm smaller. As a result, the pressure on the table increased by 1200 Pa. Find the area of the side on which the weight originally rested. Give your answer in sq. cm, rounded to two decimal places if necessary.
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Context:
Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the total number of triples of integers  $(x,y,n)$  satisfying the equation  $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$ , where  $n$  is either  $2012$  or  $2013$ .
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Context:
Let \( a = \lg z + \lg \left[ \frac{x}{yz} + 1 \right] \), \( b = \lg \frac{1}{x} + \lg (xyz + 1) \), and \( c = \lg y + \lg \left[ \frac{1}{xyz} + 1 \right] \). Denote the maximum of \( a \), \( b \), and \( c \) as \( M \). What is the minimum value of \( M \)?
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Context:
Let $a=256$. Find the unique real number $x>a^{2}$ such that $$\log _{a} \log _{a} \log _{a} x=\log _{a^{2}} \log _{a^{2}} \log _{a^{2}} x$$
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Context:
Calculate the volume of the solid that lies below the surface \( z = 3x^2 + y^2 \) and above the region in the xy-plane bounded by the curves \( y = x \) and \( y^2 - y = x \).
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Context:
A regular triangular prism is circumscribed around a sphere, and another sphere is circumscribed around the prism. Find the ratio of the surface areas of these spheres.
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Context:
Evaluate the integral: \[ \int_{-\infty}^{+\infty} \mathrm{sinc}(8t)\mathrm{sinc^2(t)}\cos(8\pi t) \, \mathrm{d}t \] where the sinc function is defined as \( \mathrm{sinc}(t) = \frac{\sin(\pi t)}{\pi t} \) for \( t \neq 0 \) and \( \mathrm{sinc}(0) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sequence of numbers, a term is called golden if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of \(1, 2, 3, \ldots, 2021\)?
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Context:
Find four numbers such that all their pairwise sums are consecutive natural numbers, the smallest of which is 2023.
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Context:
Let $a, b, c$ be positive real numbers, and let $\Gamma$ denote the Gamma function. Evaluate the sum:
\[ \sum_{n=0}^\infty \frac{a^n}{\Gamma(b+nc)} \]
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Context:
Given real numbers $x$ and $y$ that satisfy: $(x-3)^3+2014(x-3)=1$ and $(2y-3)^3+2014(2y-3)=-1$, find the minimum value of the expression $x^2+4y^2+4x$. Express your answer as a single number.
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Context:
The cubic polynomial $q(x)$ satisfies $q(3) = 2,$ $q(8) = 20,$ $q(16) = 12,$ and $q(21) = 30.$  Find
\[q(2) + q(3) + q(4) + \dots + q(22).\]
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Context:
A group of $25$ friends were discussing a large positive integer. The first friend said, "It can be divided by $1$." The second friend said, "It can be divided by $2$." The third friend said, "And by $3$." The fourth friend added, "And by $4$." This pattern continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the partial differential equation \( \frac{\partial z}{\partial x}\cdot \frac{\partial z}{\partial y}=1 \) with the initial condition \( z(0,0)=0 \). Find \( z(0,1) \).
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Context:
In a right triangular pyramid P-ABC, where PA, PB, and PC are mutually perpendicular and PA=1, the center of the circumscribed sphere is O. Find the distance from O to plane ABC.
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Context:
John decides to purchase dinner plates, dessert plates, forks, knives, spoons, and glassware for his home. The forks cost $25, the knives cost $30, the spoons cost $20, and the glassware set costs $35. The dinner plates cost 50% as much as the overall cost of silverware, while dessert plates cost 75% as much as the glassware set. John has a 10% discount on silverware items and a 15% discount on the dinner and dessert plates. How much did John pay for everything after applying the respective discounts?
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Context:
Is the following number rational or irrational?

$$
\sqrt[3]{2016^{2} + 2016 \cdot 2017 + 2017^{2} + 2016^{3}} ?
$$
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Context:
在数列 \{x_n\}_{n=1}^{+\infty} 中，x_1=1，且对任意正整数 n，x_{n+1}=x_n+[\sqrt{x_n}]，这里 [x] 表示不超过 x 的最大整数. 则满足 x_n\geq2019 的最小正整数 n 为__________.
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Context:
Determine the number of zeroes of the function \( f(z) = \sin(z) + \frac{1}{z+i} \) within the region \( D = \{ z : |\operatorname{Im}(z)| \leq 1, \pi(100+\frac{1}{2}) \leq \operatorname{Re}(z) \leq \pi(200+\frac{1}{2}) \} \).
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Context:
James has 20 pairs of red socks and half as many black socks.  He has twice as many white socks as red and black combined.  How many total socks does he have combined?
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Context:
For triangle $ABC$, with sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, if $\frac{a-b+c}{c} = \frac{b}{a+b-c}$ and $a=2$, then the maximum area of $\triangle ABC$ is ______.
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Context:
On  $5\times 5$  squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
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Context:
There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result?
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Context:
In the following diagram, \(\angle ACB = 90^\circ\), \(DE \perp BC\), \(BE = AC\), \(BD = \frac{1}{2} \mathrm{~cm}\), and \(DE + BC = 1 \mathrm{~cm}\). Suppose \(\angle ABC = x^\circ\). Find the value of \(x\).
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Context:
For which natural numbers \( n \), for any numbers \( \alpha, \beta, \gamma \) that are the angles of an acute-angled triangle, the inequality

$$
\sin n \alpha + \sin n \beta + \sin n \gamma < 0
$$

holds true?
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Context:
Let \[ p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1, \] and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4 + x^3 + 2x^2 + x + 1$. Find the remainder when $|r(2008)|$ is divided by $1000$.
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Context:
Given 2006 positive integers \( a_{1}, a_{2}, \ldots, a_{2006} \), where the ratios \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \ldots, \frac{a_{2005}}{a_{2006}} \) are all distinct, determine the minimum number of distinct integers among \( a_{1}, a_{2}, \ldots, a_{2006} \).
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Context:
$(MON 1)$  Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
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Context:
A grocery shop has a sale of Rs. 6535, Rs. 6927, Rs. 6855 in the first three months, some amount in the fourth month, and Rs. 6562 in the fifth month. The shop must have a sale of Rs. 4891 in the sixth month to get an average sale of Rs. 6500. How much sale did the shop have in the fourth month?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 0-1 binary sequences of ten 0's and ten 1's which do not contain three 0's together.
Please reason step by step, and put your final answer within \boxed{}.


Context:
30 people are arranged in six rows of five people each. Each of them is either a knight, who always tells the truth, or a liar, who always lies, and they all know who among them is a knight or a liar. A journalist asked each of them: "Is it true that there are at least 4 rows in which more than half are liars?" What is the maximum number of "yes" answers he could hear?
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Context:
What is the largest number of solid $2 \times 2 \times 1$ blocks that can fit in a $3 \times 2 \times 3$ box?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of $p$ such that the equation
\[\frac{2x + 3}{px - 2} = x\]
has exactly one solution.
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Context:
A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知集合 $A=\{x \mid 5 x-a \leqslant 0\}, B=\{x \mid 6 x-b>0\}, a, b \in \mathbf{N}$, 且 $A \bigcap B \cap \mathbf{N}=\{2,3,4\}$ ，求整数对 $(a, b)$ 的个数。
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the average of a sample $m$, $4$, $6$, $7$ is $5$, then the variance of this sample is ______.
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Context:
Find a real number \( c \) and a positive number \( L \) for which
 \( \mathop{\lim }\limits_{{r \rightarrow  \infty }}\frac{{r}^{c}{\int }_{0}^{\pi /2}{x}^{r}\sin {xdx}}{{\int }_{0}^{\pi /2}{x}^{r}\cos {xdx}} = L. \)
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Context:
Two trains, A and B, started simultaneously from opposite ends of a 375-mile route and traveled toward each other on parallel tracks. Train A, traveling at a constant rate, completed the 375-mile trip in 36 hours; Train B, traveling at a constant rate, completed the 375-mile trip in 24 hours. Both trains make multiple trips before they finally meet. How many miles had Train A traveled when it met Train B?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine into how many parts a regular tetrahedron is divided by six planes, each passing through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each part if the volume of the tetrahedron is 1.
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Context:
There are 5 people taking a photo, among them, the positions of A, B, and C from left to right must remain unchanged (these three people may not be adjacent). How many total arrangements are there? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two spheres of one radius and two spheres of another radius are arranged so that each sphere touches three other spheres and a given plane. Find the ratio of the radii of the spheres.
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Context:
Let  $P$  be the set of all  $2012$  tuples  $(x_1, x_2, \dots, x_{2012})$ , where  $x_i \in \{1,2,\dots 20\}$  for each  $1\leq i \leq 2012$ . The set  $A \subset P$  is said to be decreasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in A$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \leq x_i (1\leq i \leq 2012)$  also belongs to  $A$ . The set  $B \subset P$  is said to be increasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in B$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \geq x_i (1\leq i \leq 2012)$  also belongs to  $B$ . Find the maximum possible value of  $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where  $A$  and  $B$  are nonempty decreasing and increasing sets ( $\mid \cdot \mid$  denotes the number of elements of the set).
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Context:
How can we find the expected number of tails before achieving a run of three consecutive heads using a biased coin with probability of heads $p$?
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Context:
Jenna sets a goal of reading a certain number of pages a month for the month of September (which has 30 days). She knows that she'll be really busy at work the four weekdays starting on the 13th, so she won't have time to read. She can also read 100 pages on the 23rd, when she'll have a long boring flight. If she reads 20 pages a day every other day, how many pages is her goal for the month?
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Context:
There are \( n + 4 \) labeled points on a plane, where 4 points are the vertices of a square, and the remaining \( n \) points are inside this square. A line segment can be drawn between any two labeled points, but these segments must not contain any other labeled points except at their endpoints, and no two segments can share any points except at their endpoints. Determine the maximum number of such line segments that can be drawn.
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Context:
Triangle $ABC$ has side lengths $13$, $14$, and $15$. Let $E$ be the ellipse that encloses the smallest area which passes through $A, B$, and $C$. The area of $E$ is of the form $\frac{a \sqrt{b}\pi}{c}$ , where $a$ and $c$ are coprime and $b$ has no square factors. Find $a + b + c$.
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Context:
若集合A中的每个元素都可表为1,2，···，9中两个不同的数之积，则集合A中元素个数的最大值为 .
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Context:
A certain operation is performed on a positive integer: if it is even, divide it by 2; if it is odd, add 1. This process continues until the number becomes 1. How many integers become 1 after exactly 10 operations?
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Context:
What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?
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\boxed{It is a negative number ending with a 5.}


Context:
Determine the mass percentage of H in a mixture containing Ammonium iodide and Potassium sulfate.
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Context:
Define a sequence $a_{i, j}$ of integers such that $a_{1, n}=n^{n}$ for $n \geq 1$ and $a_{i, j}=a_{i-1, j}+a_{i-1, j+1}$ for all $i, j \geq 1$. Find the last (decimal) digit of $a_{128,1}$.
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Context:
Ten birds land on a  $10$ -meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any  $x$ -meter portion of the wire, there is an  $\tfrac{x}{10}$  probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?
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Context:
A website offers one of three quizzes, A, B, or C, each day. If the majority of visitors pass the quiz, the next day's quiz is chosen randomly among A, B, and C with equal probability. If the majority fail, quiz A is chosen with certainty. The probabilities of passing each quiz are: \( P(A) = \frac{3}{10} \), \( P(B) = \frac{6}{10} \), \( P(C) = \frac{9}{10} \). Given the stationary distribution \( (G, B) = (\frac{3}{7}, \frac{4}{7}) \), find the limiting probability \( \xi_A \) that quiz A is used.
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Context:
The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120$. What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?
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Context:
A cocktail mixture contains three types of alcoholic beverages: 8 liters of 35% alcohol rum, 6 liters of 40% alcohol vodka, and 4 liters of 50% alcohol gin. The rest of the cocktail is water, making a total volume of 30 liters. If 5 liters of water and 3 liters of a new 25% alcohol solution are added to the cocktail, what would be the percentage of alcohol in the final mixture?
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Context:
1. On an island, there live only knights, who always tell the truth, and liars, who always lie. One fine day, 30 islanders sat around a round table. Each of them can see everyone except themselves and their neighbors. Each person in turn said the phrase: "All I see are liars." How many liars were sitting at the table?
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Context:
Farmer James invents a new currency such that for every positive integer $n \le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be \textit{nice} if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
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Context:
Let $\{a_{i}\}_{i \geq 0}$ be a sequence of real numbers defined by $a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}$ for $n \geq 0$. Determine the largest value for $a_{0}$ such that $\{a_{i}\}_{i \geq 0}$ is bounded.
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Context:
134. Solve the equation $y^{\prime \prime}-3 y^{\prime}+2 y=0$.
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Context:
设 $y_n=1\overbrace{22\cdots21}^n$ ，若 $\left(10^{9}-1\right) \mid y_{n}$ ，则正整数 $n$ 的最小值是 $\qquad$.
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Context:
Rose has a spherical plum of radius 2 and a spherical watermelon of radius 8.  She builds a glass sphere around the two fruits to contain them, making the sphere as small as possible.  When she has done this, the volume contained within the sphere, but not in the plum or the watermelon, is $K\pi$.  What is the value of $K$?
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Context:
Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots +  \binom{\binom{40}{2}}{2}\]is divided by $1000$.
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Context:
Given a triangle $ABC$ with angles $A$, $B$, $C$ and their corresponding sides $a$, $b$, $c$, and the area of the triangle is $S_{\triangle ABC}=\frac{\sqrt{3}}{12}a^2$. Determine the range of real numbers $m$ that satisfy the equation $\sin^2 B + \sin^2 C = m \sin B \sin C$.
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Context:
Find the number of permutations of  $\{1,2,...,n\}$  like  $\{a_1,...,a_n\}$  st for each  $1 \leq i \leq n$ :  $$ a_i | 2i $$
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Context:
Find all  $10$ -digit whole numbers  $N$ , such that first  $10$  digits of  $N^2$  coincide with the digits of  $N$  (in the same order).
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Context:
A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second time. How many minutes does it take for the frog to climb from the bottom of the well to the top?
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Context:
Determine the number of field extensions between \( \mathbb{Q}(\zeta^3) \) and \( \mathbb{Q}(\zeta) \), where \( \zeta \) is a primitive 12th root of unity.
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Context:
How many ways are there to arrange the letters $A, A, A, H, H$ in a row so that the sequence $HA$ appears at least once?
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Context:
Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. Initially, $B$ moves more slowly than $A$ until they meet. After meeting, $B$ doubles his speed and next meets $A$ at the starting point. Let $d$ m be the distance traveled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.
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Context:
For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$.
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Context:
In the figure, the triangle $\triangle ABC$ is isosceles, with $\angle BAC = 20^\circ$.

Given that $BC = BD = BE$, determine the measure of the angle $\angle BDE$.
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Context:
In the given diagram, each Chinese character inside a circle represents a number between 1 and 9. Different characters correspond to different numbers. The sum of the numbers at the three vertices of each small triangle is equal. If the sum of the 7 numbers is 12, what number does "杯" (cup) represent?
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Context:
Olya drew $N$ different lines on a plane, any two of which intersect. It turned out that among any 15 lines, there are always two that form an angle of $60^{\circ}$ between them. What is the largest possible value of $N$ for which this is possible?
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Context:
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
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Context:
Determine the number of distinct ways to arrange 8 indistinguishable red beads and 32 indistinguishable blue beads into a necklace such that there are at least 2 blue beads between any two red beads.
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Context:
Let  $A_1B_1C_1$  be a triangle with  $A_1B_1 = 16, B_1C_1 = 14,$  and  $C_1A_1 = 10$ . Given a positive integer  $i$  and a triangle  $A_iB_iC_i$  with circumcenter  $O_i$ , define triangle  $A_{i+1}B_{i+1}C_{i+1}$  in the following way:

(a)  $A_{i+1}$  is on side  $B_iC_i$  such that  $C_iA_{i+1}=2B_iA_{i+1}$ .
(b)  $B_{i+1}\neq C_i$  is the intersection of line  $A_iC_i$  with the circumcircle of  $O_iA_{i+1}C_i$ .
(c)  $C_{i+1}\neq B_i$  is the intersection of line  $A_iB_i$  with the circumcircle of  $O_iA_{i+1}B_i$ .

Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]

Note:  $[K]$  denotes the area of  $K$ .

*Proposed by Yang Liu*
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Context:
An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that from each island no more than 5 bridges lead out, and among any 7 islands there must be two connected by a bridge. What is the maximum value that $N$ can take?
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Context:
if the product of the integers w, x, y, and z is 720, and if 0 < w < x < y < z < 20, what is the value of w + z?
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Context:
求具有下述性质的最小正数 c：对任意整数 n\geq4，以及集合 A\subseteq\{1,2,\cdots,n\}，若 |A|>cn，则存在函数 f:A\rightarrow\{-1,1\}，满足\left|\ds{a\in A}{}f(a)\cdot a\right|\leq1.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
2. We consider sports tournaments with $n \geqslant 4$ teams where each pair of teams plays against each other at most once. We call such a tournament balanced if every quartet of teams plays exactly three matches among themselves. Not all teams play against each other.

Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.
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Context:
Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected from the other $47$ points.
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Context:
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$.
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Context:
Let  $ \mathcal{R}$  be the region consisting of the set of points in the coordinate plane that satisfy both  $ |8 \minus{} x| \plus{} y \le 10$  and  $ 3y \minus{} x \ge 15$ . When  $ \mathcal{R}$  is revolved around the line whose equation is  $ 3y \minus{} x \equal{} 15$ , the volume of the resulting solid is  $ \frac {m\pi}{n\sqrt {p}}$ , where  $ m$ ,  $ n$ , and  $ p$  are positive integers,  $ m$  and  $ n$  are relatively prime, and  $ p$  is not divisible by the square of any prime. Find  $ m \plus{} n \plus{} p$ .
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Context:
在平面直角坐标系中，设$\vec{e}$是一个单位向量，向量$\vec{a}$满足$\vec{a} \cdot \vec{e}=2$，并且$|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数$t$成立。请提供$|\vec{a}|$的取值范围左端点的两倍与右端点之差。
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Context:
In a college's independent admission test for minor languages, the school received 5 recommendation spots, of which there are 2 for Korean, 2 for Japanese, and 1 for Russian. Both the Japanese and Korean spots require female participants. After selection, the school nominated 3 females and 2 males for a total of 5 recommended candidates. Find the number of different recommendation methods.
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Context:
The rank of a rational number \( q \) is the unique \( k \) for which \( q = \frac{1}{a_1} + \cdots + \frac{1}{a_k} \), where each \( a_i \) is the smallest positive integer such that \( q \geq \frac{1}{a_1} + \cdots + \frac{1}{a_i} \). Let \( q \) be the largest rational number less than \( \frac{1}{4} \) with rank 3, and suppose the expression for \( q \) is \( \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} \). Find the ordered triple \( \left(a_1, a_2, a_3\right) \).
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Context:
In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ such that $\angle BAD = \angle DAC$. Let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $\triangle ABC$. Find the length of segment $AE$. \( \text{(Express your answer as a number.)} \)
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Context:
Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more consecutive 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition.
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Context:
\(a, b, c,\) and \(d\) are four non-zero numbers such that the quotients

$$
\frac{a}{5}, \quad \frac{-b}{7a}, \quad \frac{11}{abc}, \quad \text{and} \quad \frac{-18}{abcd}
$$

are all positive. Determine the signs of \(a, b, c,\) and \(d\).
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Context:
A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?
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Context:
A scientist walks through a forest and records the heights of 5 trees in integers. She notices that the height of each tree is either double or half the height of the tree to its right. Unfortunately, some data was lost due to rain. Based on her observations, she is able to determine the missing values. The height of the second tree is 11 meters. Calculate the average height of these trees in meters, rounded to one decimal place.
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Context:
15. Given the set $A=\{1,2,3,4\}$, every second, Andel randomly picks a number from $A$ (repetition is allowed), and stops picking when the sum of the last two numbers picked is a prime number. What is the probability that the last number picked is “1”?
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Context:
Find the smallest positive integer $n$ such that there exists a sequence of all rational numbers $a_1, a_2, \dots$ where no subsequence of length $n$ forms an arithmetic progression.
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Context:
Let $A_{10}$ denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by $x$ and $y$, respectively. Given that the product of the radii of these two circles is $15 / 2$, and that the distance between their centers is $A_{10}$, determine $y^{2}-x^{2}$.
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Context:
Determine the number of ring homomorphisms from the ring of real numbers to itself that satisfy the condition $f(0) = 0$.
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Context:
Let a sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$ be defined by $a_{0}=\sqrt{2}, a_{1}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 1$. The sequence of remainders when $a_{0}, a_{1}, a_{2}, \cdots$ are divided by 2014 is eventually periodic with some minimal period $p$ (meaning that $a_{m}=a_{m+p}$ for all sufficiently large integers $m$, and $p$ is the smallest such positive integer). Find $p$.
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Context:
Find the inverse Laplace transform of the function: $$\mathcal{L}^{-1}\left\{\dfrac{1}{(s^2+a^2)^2}\right\}$$

Given the identity: $tf(t)\longleftrightarrow -\frac{\mathrm d}{\mathrm ds}F(s)$ and $\int_{0}^{t}g(\tau)d\tau\longleftrightarrow \frac{1}{s}G(s)$, where $f(t)$ and $g(t)$ are the inverse Laplace transforms of $F(s)$ and $G(s)$, respectively.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{\sin (a t)-a t \cos (a t)}{2 a^3}}


Context:
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn using two rectangles $ABCD$ and $EFGH$, whose sides are parallel to the edges of the sheet. It is given that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$.
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Context:
One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. If the maximum possible median number of cans of soda bought per customer on that day is $x$, what is the value of $10x$?
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Context:
If $x_1, x_2, x_3, x_4$ are the roots of the polynomial $x^4 - x^3 + 2x^2 + 3x + 1$, find the value of the expression:
\[ \frac{\sum x_i^3 + \sum (x_i x_j x_k)^3}{\sum (x_i x_j)^3 - (x_1 x_2 x_3 x_4)^3} \]
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Context:
Let \(A \cup B \cup C = \{1, 2, 3, 4, 5, 6\}\), and \(A \cap B = \{1, 2\}\). Additionally, \(\{1, 2, 3, 4\} \subseteq B \cup C\). Determine the number of distinct groups \((A, B, C)\) that meet these conditions. (Note: The order of \(A\), \(B\), and \(C\) matters, so different orderings are considered different groups.)
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Context:
Find the greatest common divisor of all numbers of the form  $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$  where  $a,b,c$  are integers.
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Context:
The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided by $1001$.
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Context:
Given that \( f(x) \) is a function defined on \(\mathbf{R}\), and for any \(x, y \in \mathbf{R}\), \( f(x) + f(y) = 2 f\left(\frac{x+y}{2}\right) f\left(\frac{x-y}{2}\right) \) holds true. It is also known that \( f\left(\frac{\pi}{4}\right) = 0 \), but \( f(x) \) is not identically zero, and \( f(0) > 0 \). Find the period of the function \( f(x) \). Express your answer as a single numerical value.
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Context:
Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals
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Context:
In Ms. Johnson's class, each student averages two days absent out of thirty school days. What is the probability that out of any three students chosen at random, exactly two students will be absent and one will be present on a Monday, given that on Mondays the absence rate increases by 10%? Express your answer as a percent rounded to the nearest tenth.
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Context:
What is the molecular weight of 7 moles of Aluminum iodide?
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Context:
Let  $\phi(n)$  denote the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ . Over all integers  $1\le n \le 100$ , find the maximum value of  $\phi(n^2+2n)-\phi(n^2)$ .

*Proposed by Vincent Huang*
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Context:
A random triangle is produced as follows. A pair of standard dice is rolled independently three times to get three random numbers between 2 and 12, inclusive, by adding the numbers that come up on each pair rolled. Call these three random numbers $a$, $b$, and $t$. The random triangle has two sides of lengths $a$ and $b$ with the angle between them measuring $15(t - 1)$ degrees. What is the probability that the triangle is a right triangle?
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Context:
Alice and Bob are each given $2000 to invest. Alice puts all of her money in the stock market and doubles her money. Bob invests in real estate and makes five times more money than he invested. How much more money does Bob have now than Alice?
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Context:
On the Arctic Circle ($60^{\circ}$), there are two locations $A$ and $B$. The arc length between them on this latitude circle is equal to $\frac{\pi R}{2}$ ($R$ is the radius of the Earth). What is the spherical distance between locations $A$ and $B$?
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Context:
Exercise 12. Let $M N P Q$ be a rhombus such that $M P=180$. Suppose there exists a square $A B C D$ such that $A B=18, A \in[M N], B \in[N P], C \in[P Q]$ and $D \in[Q M]$. Determine the area of $M N P Q$.
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Context:
A rhombus, which is divided by its diagonal into two equilateral triangles, has an inscribed circle with a radius of 2. Find the side of the rhombus.
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Context:
Calculate the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $3$ and $\mathcal{L}$ gives $T$ a $(p,q,r)$-polarization.
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Context:
Determine the exact value of the series
\[\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.\]
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Context:
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers.  Determine the $100$th smallest element of $S$.
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Context:
In the diagram, right triangles PSQ and RSQ are shown. Given QR = 25, SR = 20, and PS = 8, find the perimeter of triangle PQR. Express your answer as a whole number.
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Context:
Determine the values of $k$ for which the Fourier cosine transform of $e^{-x^k}$, defined as \[ g_k(z) = \int_0^\infty \cos(zx) e^{-x^k} \, dx, \] is non-negative and decreasing for $z \in \mathbb{R}^{+}$. Provide your answer as a set of values for $k$.
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Context:
Tarek wants to shade two further squares on the given diagram so that the resulting pattern has a single axis of symmetry. Calculate the total number of different ways he can complete his pattern to achieve this symmetry. Express your answer as a whole number.
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Context:
Evaluate the expression $\frac{C}{D}$, where $C=\sum_{k=1}^{2000}\frac{1}{(2k-1)(2k)}$ and $D=\sum_{k=0}^{999}\frac{1}{(2000+k)(2999-k)}$.
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Context:
Given 2022 points uniformly distributed on a circle, determine the maximum number of arcs that can be drawn between these points such that no arc is completely included in another, the lengths of arcs are pairwise different, and the smallest and biggest arcs do not overlap or share the same endpoint.
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Context:
设黑板上写有128个1，每一步可以擦去黑板上的任意两个数a和b并写上ab+这样做了127次之后，只剩下1个数，将这样剩下的数的最大可能值记作A，试求A 的末位数字. （1992年圣彼得堡市代表队选拔试题）
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Context:
Consider $2003^{2009}$+$2009^{2017}$+$2015^{2025}$ in base 3, we get the very long string $a_{n} a_{n-1}... a_{1} a_{0}$. What are the values of the last 3 digits $a_2, a_1, a_0$?
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Context:
圆内接四边形 $A B C D$ 中, $A B=136, B C=80, C D=150, D A=102$, 求它的外接圆直径。
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Context:
Determine the cardinality of the union of a collection $T = \{E_{\alpha}:\alpha \in I\}$, where $I$ is an uncountable directed set, each $E_{\alpha}$ is a countably infinite set, and $E_{\alpha} \subset E_{\beta}$ for $\alpha \le \beta$ with $T$ totally ordered by set inclusion.
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Context:
Determine the exponent that should accompany $i$ in the expression $i^n$ to achieve the most optimized change of direction without altering the modulus.
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Context:
There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number?
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Context:
Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. Find Lily's score. Express your answer as a numerical value.
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Context:
In a three-day MLB game between Team A and Team B, the ratio of misses to hits and types of hits varies each day for both teams. On the first day, Team A has twice as many misses as hits (with 60 misses), while Team B has three times as many misses as hits (with 90 misses). For each team, half of their hits on the first day are home runs and the other half are singles.

On the second day, Team A has four times as many misses as hits (with 68 misses), while Team B has two times as many misses as hits (with 56 misses). On this day, one-third of each team's hits are doubles and the remainder are singles.

On the third day, the ratio of misses to hits for both teams is 5:1. Team A has 100 misses, and Team B has 120 misses. For third day hits of both teams, one-fourth are triples, one-fourth are home runs, and the remaining hits are singles.

Calculate the combined total of misses, singles, doubles, triples, and home runs for Team A and Team B made during the entire three-day game.
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Context:
Determine all integers  $n$  such that both of the numbers:  $$ |n^3 - 4n^2 + 3n - 35| \text{ and }  |n^2 + 4n + 8| $$  are both prime numbers.
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Context:
on a sum of money , the simple interest for 2 years is rs . 326 , while the compound interest is rs . 340 , the rate of interest being the same in both the cases . the rate of interest is
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Context:
Find $x$ such that the sum $\sum_{k=1}^{2014} k^k$ is congruent to $x$ modulo $10$.
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Context:
Determine the asymptotic growth rate of the recurrence relation \(T(n) = 2T\left(\frac{n}{3}\right) + T\left(\frac{n}{4}\right) + n^2\) using the Akra-Bazzi theorem.
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Context:
Let $P(x)$ be a polynomial of degree at most 3 such that $P(x)=\frac{1}{1+x+x^{2}}$ for $x=1,2,3,4$. What is $P(5) ?$
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Context:
Evaluate the infinite series ratio:

$$\frac{1+\frac{2^2}{2!}+\frac{2^4}{3!}+\frac{2^6}{4!}+\ldots}{1+\frac{1}{2!}+\frac{2}{3!}+\frac{2^2}{4!}+\ldots}$$

Simplify the expression to find the numerical value of the ratio.
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Context:
In the triangular pyramid \(P-ABC\), \(\triangle PAC\) and \(\triangle ABC\) are both equilateral triangles with side length 6. If the dihedral angle \(P-AC-B\) is \(120^\circ\), find the surface area of the circumscribed sphere around \(P-ABC\).
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Context:
Cary is an engineer in charge of designing an irrigation system for three farmers. Farmer Bob grows 3 acres of corn, 9 acres of cotton, 12 acres of beans, and 4 acres of wheat. Farmer Brenda grows 6 acres of corn, 7 acres of cotton, 14 acres of beans, and 5 acres of wheat. Farmer Bernie grows 2 acres of corn, 12 acres of cotton, and 3 acres of wheat. If corn takes 20 gallons of water an acre, cotton takes 65 gallons of water an acre, beans take three times as much water as corn, and wheat takes 45 gallons of water an acre, what percentage of the total water used will go to Farmer Bob's farm?
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Context:
There are  $n\geq 3$  cities in a country and between any two cities  $A$  and  $B$ , there is either a one way road from  $A$  to  $B$ , or a one way road from  $B$  to  $A$  (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define  $d(A,B)$  to be the minimum number of roads you must go through to go from city  $A$  to  $B$ . Consider all possible ways to build the roads. Find the minimum possible average value of  $d(A,B)$  over all possible ordered pairs of distinct cities in the country.
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Context:
At a family gathering, Jane’s mom prepared a dessert platter with a variety of bite-size treats. There were 15 cinnamon swirls, 24 chocolate brownie bites, and 18 fruit tartlets. In total, 8 people including Jane ate an equal number of each dessert item. How many pieces of each dessert item did Jane eat?
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Context:
Write down the natural numbers starting from 1, consecutively until reaching the 198th digit. Find the remainder when this number is divided by 9. Express your answer as a single integer.
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Context:
Fabián has four cards, each with a different positive integer less than 10 written on it. The numbers are written in different colors, and the following conditions hold:

- The product of the green and yellow numbers is the green number.
- The blue number is the same as the red number.
- The product of the red and blue numbers is a two-digit number written with the green and yellow digits (in this order).

Determine these four numbers.
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Context:
Evaluate the integral $$\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$$
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Context:
A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is:
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Context:
A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$?
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Context:
We dilute one liter of $a\%$ alcohol with water to $b\%$. A portion of this mixture is set aside, and the remaining mixture is further diluted until it becomes $c\%$. How much water was used in total, given that the amount of the resulting $c\%$ mixture is equal to the amount of the set-aside $b\%$ mixture?
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Context:
Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A.
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Context:
What is the molecular weight of 8 moles of Al2O3
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Context:
Determine the values of \( \alpha \in \mathbb{R} \) for which the improper integral \( \int_{0}^{+\infty}\frac{1}{(\arctan x)^\alpha} \, dx \) converges.
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Context:
What is the mass percentage of Ca in Calcium hydroxide?
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Context:
The sum of the squares of the projections of the sides of an equilateral triangle onto any axis is constant.
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Context:
Compute the line integral \( \int_C \vec{F} \cdot d\vec{s} \) where \( \vec{F} = A_0(x\hat{y} - y\hat{x}) \), along a circular path, counterclockwise, about the origin with radius 4.
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Context:
In the Cartesian coordinate plane $xOy$, two circles $C_1$ and $C_2$ are defined as follows: $C_1: x^2+(y-t)^2=4$ and $C_2: (x-2)^2+y^2=14$. If there exists a point $P$ on circle $C_1$ such that a tangent line to circle $C_2$ passes through $P$ at point $Q$ and $PO = \sqrt{2}PQ$, determine the range of values for the real number $t$.
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Context:
An aluminum and a copper part have the same volume. The density of aluminum is \(\rho_{A} = 2700 \, \mathrm{kg/m}^{3}\), and the density of copper is \(\rho_{M} = 8900 \, \mathrm{kg/m}^{3}\). Find the mass of the copper part if it is known that the masses of the parts differ by \(\Delta m = 60 \, \mathrm{g}\).
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Context:
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 + \ldots + 10^{2013}$. If one writes down the number $S$, how often does the digit $5$ occur in the result?
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Context:
Let $a$ and $b$ be positive real numbers with $a \ge b$. Let $\rho$ be the maximum possible value of $\frac{a}{b}$ for which the system of equations
$$a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2$$
has a solution in $(x,y)$ satisfying $0 \le x < a$ and $0 \le y < b$. Then $\rho^2$ can be expressed as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Given triangle  $ ABC$ . Point  $ O$  is the center of the excircle touching the side  $ BC$ . Point  $ O_1$  is the reflection of  $ O$  in  $ BC$ . Determine angle  $ A$  if  $ O_1$  lies on the circumcircle of  $ ABC$ .
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Context:
Given the digits 1, 2, 3, 4, 5, 6 to form a six-digit number (without repeating any digit), requiring that any two adjacent digits have different parity, and 1 and 2 are adjacent, determine the number of such six-digit numbers.
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Context:
Find all prime numbers  $p$  for which the number  $p^2+11$  has less than  $11$  divisors.
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Context:
A man takes $37 to eat pizza in a restaurant. The restaurant offers a discount that increases by multiples of 4 for every additional pizza eaten. For 1 pizza there is 0% off, for 2 pizzas there is 4% off, for 3 pizzas there is 8% off. The pizza costs a certain amount without discount. The man eats 3 pizzas. What is the cost of the pizza without discount?
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Context:
A runner, two cyclists, and a motorcyclist are moving with constant speeds on a circular highway. The runner and one of the cyclists are moving in one direction, while the motorcyclist and the other cyclist are moving in the opposite direction. The runner meets the second cyclist every 12 minutes, the first cyclist overtakes the runner every 20 minutes, and the motorcyclist overtakes the second cyclist every 5 minutes. Determine how often the motorcyclist meets the first cyclist.
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Context:
Sasha discovered that there are exactly \( n \) working buttons with digits on the calculator. It turned out that any natural number from 1 to 99,999,999 can either be entered using only the working buttons or obtained as the sum of two natural numbers, each of which can be entered using only the working buttons. What is the smallest \( n \) for which this is possible?
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Context:
Determine the positive value for the slope of an asymptote of the hyperbola described by the equation:
\[
\sqrt{(x-2)^2 + (y-3)^2} - \sqrt{(x-2)^2 + (y+1)^2} = 4
\]
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Context:
The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \( PQ \) of the larger circle is a tangent to both unshaded circles. The length of \( PQ \) is 6 units. What is the area, in square units, of the shaded region?
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Context:
设双曲线 $C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$. 若 $C_{2}$ 的短轴长与 $C_{1}$ 的实轴长的比值等于 $C_{2}$ 的离心率，求 $C_{1}$ 在 $C_{2}$ 的一条准线上截得线段的长。
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Context:
A seller has weights of 1, 2, 4, 8, 16, and 32 grams (one of each) and a balance scale. A 25 gram candy and three of the weights were placed on the first pan, while the remaining three weights were placed on the second pan, such that the scale balanced. Indicate the weights of all three weights on the second pan.
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Context:
记数集 A=\{a_1,a_2,\cdots,a_n\} 中所有元素的算术平均值为 P(A). 若 B 是 A 的非空子集，且 P(B)=P(A)，则称 B 是 A 的一个"均衡子集". 求数集 M=\{1,2,3,4,5,6,7,8,9\} 的均衡子集的个数.
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Context:
If \( a = \log 25 \) and \( b = \log 49 \), compute
\[
5^{a/b} + 7^{b/a}.
\]
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Context:
Determine the signature of the quadratic form \( f(x, y) = xy \).
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Context:
Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\overline{B C}$ and $\overline{D A}$. Compute $P Q^{2}$.
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Context:
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.
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