Context:
Find the value of the real number $a$ if the constant term in the expansion of $({(\sqrt[6]{x}-\frac{a}{\sqrt{x}})}^{8})$ is $56$. Express your answer as a real number, including the \pm symbol if necessary.
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Context:
He puts them into three boxes, a red one, a white one, and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?
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Context:
Given two discrete distributions, $X$ with probability mass function $P_X(x) = \frac{1}{3}$ for $x = 1, 2, 3$, and $Y$ with probability mass function $P_Y(y) = \frac{1}{y}$ for $y = 2, 3, 6$, calculate the probability $P(X_1X_2 > Y_1 + Y_2)$, where $X_1$ and $X_2$ are independent samples from $X$, and $Y_1$ and $Y_2$ are independent samples from $Y$. Can this probability be determined algebraically?
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Context:
Evaluate the limit of the double integral as the radius of the circular region approaches 0, given that $f$ is a differentiable function such that $f(0) = 0$:
$$\lim_{r\to 0^{+}} \iint_{x^2 + y^2 \leq r^2} \frac{3}{2\pi r^3} f(\sqrt{x^2+y^2}) \, dx \, dy$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct parse trees for the expression $a+b*c-d$ using the following grammar:

$$\begin{align}
E &\to E + T \mid E - T \mid T \\
T &\to T * F \mid T / F \mid F \\
F &\to (E) \mid id
\end{align}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular pentagon \(ABCDE\), point \(K\) is on side \(AE\) such that the ratio \(AK : KE = 3 : 7\), and point \(L\) is on side \(CD\). Given that \(\angle LAE + \angle KCD = 108^\circ\), determine the ratio \(CL : AB\).
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Context:
Calculate the volume of the solid formed by rotating the region bounded by the graphs of the functions \( y = \arccos \frac{x}{3} \), \( y = \arccos x \), and \( y = 0 \) about the \( y \)-axis.
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Context:
Determine the cardinality of the set of all epsilon numbers less than $\omega_1$. Provide your answer as a cardinal number.
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Context:
Determine the degree of the line bundle over $\mathbb{P}^1$ with transition function $e^{1/z}$, where $z \neq 0, \infty$. What is the degree of this line bundle?
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Context:
Evaluate the limit: \[ \lim_{m\to\infty} \frac{\int_0^{\pi/2}(\sin x)^{2m}dx}{\int_0^{\pi/2}(\sin x)^{2m+1}dx} \]
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Context:
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
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Context:
In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)?
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Context:
Given the input signal $x(t) = \cos(t)$ and impulse response $h(t) = u(t)$, find the output $y(t)$ using Fourier transforms. Express the final time-domain solution $y(t)$.
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Context:
When dividing the polynomial \(x^{1951} - 1\) by \(x^4 + x^3 + 2x^2 + x + 1\), the quotient and remainder are obtained. Find the coefficient of \(x^{14}\) in the quotient.
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Context:
Find a simple expression or a good upper bound for the expected maximum amount of money found, \(E[\max(x_1, \ldots, x_n)]\), when \(n\) individuals find street money. Each person has a probability of \(\dfrac{1}{2^x}\) to find \(x\) dollars (\(1 \leq x \leq n\)), with probabilities being independent. It is known that the expected value of money found by each person is less than 2 dollars. Initial observations suggest it might be around \(\log(n)\).
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Context:
A PhD entrance question of TIFR 2020 asks to find the integer part of the sum $\sum\limits_{n=1}^{9999}\frac 1{n^{\frac 14}}$. Can you help me find the answer?
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Context:
In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$.

If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain?
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Context:
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\triangle ABC$.
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Context:
In Mr. Jacob's class, $12$ of the $20$ students received a 'B' on the latest exam. If the same proportion of students received a 'B' in Mrs. Cecilia's latest exam, and Mrs. Cecilia originally had $30$ students, but $6$ were absent during the exam, how many students present for Mrs. Cecilia’s exam received a 'B'?
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Context:
Find the coefficient of \(x^{33}\) in the expansion of \((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\).
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Context:
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n}\ d \sigma \) where \( \mathbf{F} = xy\mathbf{i}+ z^2\mathbf{j}+y\mathbf{k} \) and \( S \) is the exterior side of the cylinder \( y^2 + z^2 =9 \), bounded by \( x=0 \), \( x=1 \), \( y=0 \), and \( z=0 \) in the first octant.
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Context:
Given the sets $A=\{-4, 2a-1, a^{2}\}$ and $B=\{a-5, 1-a, 9\}$, and $\{9\}=A\cap B$, find the value of $a$.
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Context:
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$.  There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$.  Find $m+n$.
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Context:
In a polygon with 2013 sides labeled as \(A_1, A_2, \ldots, A_{2013}\), each vertex is marked with a number such that the sum of the numbers on any 9 consecutive vertices is 300. Given that \(A_{13}\) is marked with 13 and \(A_{20}\) is marked with 20, find the number marked on \(A_{2013}\).
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Context:
Given the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $|\varphi| \leq \frac{\pi}{2}$), if $(-\frac{\pi}{4}, 0)$ is the center of symmetry of the graph of $f(x)$, and $x = \frac{\pi}{4}$ is an extremum point of $f(x)$, and $f(x)$ is monotonic in the interval $(\frac{5\pi}{18}, \frac{2\pi}{5})$, then the maximum value of $\omega$ is \_\_\_\_\_.
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Context:
Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. Determine the rank of the free group $\pi_{1}(\hat{X})$. What is the rank of this free group?
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Context:
Among all triangles formed by selecting any 3 vertices from the 8 vertices of a rectangular prism, how many of them are acute-angled triangles?
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Context:
An artist who needs to buy only paint and canvas finds that the cost of canvas has decreased by some percent and the cost of paint has decreased by 60 percent. If paint previously accounted for 4 times as much of the expense of painting as did canvas, the total cost for paint and canvas decreased by 55.99999999999999 percent. By what percent did the cost of canvas decrease?
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Context:
Let $a_1, a_2, \ldots, a_n$ be a permutation of the numbers 1, 2, \ldots, n. Define the sequence number of $a_i$ (for $i=1,2,\ldots,n$) as the count of numbers that are to the left of $a_i$ and smaller than $a_i$. Given that the sequence number of 8 is 2, the sequence number of 7 is 3, and the sequence number of 5 is 3, calculate the number of different permutations. Express your answer as a single integer.
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Context:
Two trains, each composed of 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after their first cars met, a passenger named Sasha, sitting in the compartment of the third car, passed by a passenger named Valera in the opposite train. Moreover, 32 seconds later, the last cars of these trains completely passed each other. In which car was Valera traveling?
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Context:
Calculate the measure of the set \(A = ([-1,2] \times [0,3] \times [-2,4]) \cup ([0,2] \times [1,4] \times [-1,4]) \setminus [-1,1]^3\).
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Context:
The height of a pyramid is 3, and the area of its base is 9. Find the volume of the prism, one base of which is part of the base of the pyramid, and the opposite base is a section of the pyramid created by a plane passing at a distance of 1 from the apex.
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Context:
Find the surface area of the cone defined by \(x^2 + y^2 = z^2\) that lies above the \(xy\)-plane and below the cylinder \(y^2 + z^2 = 16\).
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Context:
Given the equation ax + by + 2c = 0, where c!= 0 and ab - c^2 >= 0, find the maximum value of xy. Express your answer as a single numerical value.
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Context:
Given the parabola $y^2=4x$ with focus $F$, and a moving point $P$ on the parabola, and a point $B(3,4)$, find the minimum value of $|PB|+|PF|$.
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Context:
Find the least ordinal \( \beta > 0 \) such that \( \alpha \beta = \beta \) for a given \( \alpha > 1 \).
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Context:
In trapezoid $ABCD$, $AB < CD$, $AB \perp BC$, $AB \parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD \perp DC$. If
\[\frac{QP}{AP} + \frac{AP}{QP} = \left( \frac{51}{14} \right)^4 - 2,\]
then $\frac{BP}{AP} - \frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$.
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Context:
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条，则由这 3030 条线段能构成不全等的三角形的个数为 $\qquad$ （用数字作答）。
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Context:
Suppose you have a sequence of independent random variables {$X_i, i\geq1$}, such that $$\Bbb P(X_i=i^3 -1)=i^{-3},$$ $$ \Bbb P(X_i=-1)=1-i^{-3}.$$ Find the value of the constant d to which $\frac1n \sum_{i=1}^n X_i$ converges almost surely.
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Context:
On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
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Context:
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, \ldots, 2020$. The students are initially divided into $20$ groups of $101$, with each division into groups equally likely. In each group, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we choose the champion uniformly at random. If the expected value of the champion’s number can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers, determine $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the integral \( \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr \) converges.
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Context:
Given a natural number  $k$ , find the smallest natural number  $C$  such that  $$ \frac C{n+k+1}\binom{2n}{n+k} $$  is an integer for every integer  $n\ge k$ .
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Context:
Given two independent random variables X and Y, where the distribution for X is \( f(x) = 2x \) for \( 0 \leq x \leq 1 \) and 0 otherwise, and the distribution for Y is \( f(y) = \frac{3-y}{3} \) for \( 0 \leq y \leq 3 \), determine the probability that \( X + Y \leq 2 \) given that \( X < Y < 3 - X \).
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Context:
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "400".
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Context:
What is the Euler characteristic of the hyperboloid of one sheet?
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Context:
Find the residue of the function \( \frac{3w-2}{(w-1)^3} \) at its pole of order 3.
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Context:
Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______.
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Context:
A stick of length $10$ is marked with $9$ evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing either right or left. Suddenly, all the ants start walking simultaneously at a rate of $ 1$ unit per second. If two ants collide head-on, they immediately reverse direction (assume that turning takes no time). Ants fall off the stick as soon as they walk past the endpoints (so the two on the end don\u2019t fall off immediately unless they are facing outwards). On average, how long (in seconds) will it take until all of the ants fall off of the stick?
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Context:
The whole numbers from 1 to \( 2k \) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \( k \)?
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Context:
Given point \( A(-2,2) \), point \( B \) is a moving point on the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 \), and \( F \) is the left focus. Find the coordinates of \( B \) when \( |AB| + \frac{5}{3}|BF| \) is minimized.
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Context:
Calculate the area between the function \( f(x) = x \cdot e^{x^2 - 1} \) and its inverse.
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Context:
Given that there are intersections between the circle $x^2+y^2=m$ and the circle $x^2+y^2+6x-8y-11=0$, find the range of the real number $m$.
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Context:
At time  $0$ , an ant is at  $(1,0)$  and a spider is at  $(-1,0)$ .  The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the  $x$ -axis.  It so happens that the ant's horizontal speed is always half the spider's.  What will the shortest distance ever between the ant and the spider be?
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Context:
Given a right triangle \( \triangle ABC \) with hypotenuse \( BC = 2\pi \). A circle with center \( A \) and radius \( r \) is tangent to \( BC \). Let \( S \) be the remaining area of \( \triangle ABC \) after being covered by the circle. Find the maximum value of \( S \). Express your answer as a single value.
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Context:
Let \( \alpha \) be an element in an algebraic closure of \( GF(64) \) such that \( \alpha^4 = \alpha + 1 \). Determine the value of \( r \in \mathbb{N} \) for which the field extension \( GF(64)(\alpha) \) is isomorphic to \( GF(2^r) \).
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Context:
Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$
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Context:
Evaluate the integral for $N > 0$:
$$ \frac{\exp\left(-\frac N2\right)}{2\pi N} \iint \exp\left(-\left( \frac {x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$
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Context:
A secret facility is a rectangle measuring $200 \times 300$ meters. Outside the facility, there is one guard at each of the four corners. An intruder approached the perimeter of the facility from the outside, and all the guards ran towards the intruder using the shortest paths along the external perimeter (the intruder remained stationary). The total distance run by three of the guards to reach the intruder was 850 meters. How many meters did the fourth guard run to reach the intruder?
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Context:
给定正整数 $n(n \geqslant 2)$. 已知 $2 n$个正实数 $a_{1}, a_{2}, \cdots, a_{2 n}$ 满足: $\sum_{k=1}^{n} a_{2 k-1} \sum_{k=1}^{n} a_{2 k}=\prod_{k=1}^{n} a_{2 k-1}+\prod_{k=1}^{n} a_{2 k}$. 求 $S=\sum_{k=1}^{2 n} \frac{a_{k}^{n-1}}{a_{k+1}}$ 的最小值，其中 $a_{2 n+1}=a_{1}$ 。请给出当 n = 2 时的 S 的最小值。
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Context:
Evaluate the sum \( \sum_{k=1}^{\infty} \arctan\left(\frac{1}{k}\right) \).
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Context:
Given that Connie divided a number by $3$ and got $27$, determine the correct number that she should have multiplied by $3$ to get the correct answer.
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Context:
Find the minimum value of \( \frac{n}{T(n)} \) for \( n \) in the range \([10000, 99999]\), where \( T(n) \) denotes the number of positive integer divisors of \( n \).
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Context:
John bought 9.25 m of cloth at a cost price of $45 per metre. How much did John spend on the cloth?
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Context:
Given \( f(x) = x^2 + ax + b \) has two real roots in the interval \([0, 1]\), find the range of values for \( a^2 - 2b \).
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Context:
Determine the interval of convergence for the power series \( \sum_{n=1}^{\infty}\frac{(-1)^{\lfloor \log_2 n\rfloor}}{n}x^n \), including the behavior at the endpoints.
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Context:
Find the radius of curvature of the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$. Provide your answer as a single numerical value.
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Context:
Diana works 10 hours per day on Monday, Wednesday, and Friday, and 15 hours per day on Tuesday and Thursday. She also occasionally takes up additional tasks on weekends, earning a varying amount depending on the task. On one particular week, she earns a total of $1800, with an additional $200 earned from supplementary tasks on Saturday. Furthermore, she also receives a performance-based bonus of $150 for the overall week. There is a major holiday during the week, so Diana earns double her usual hourly rate on Wednesday. Considering all these factors, determine Diana's regular hourly rate.
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Context:
149. Scattered set. In a printing house, a multiplication line of the form $\overline{a b c} \cdot \overline{b c a} \cdot \overline{c a b}^{*}$ was being set; but the set scattered, and the digits of the product got mixed up. As a result, the product was printed as 2342355286. It is known that $a>b>c$ and that the units digit of the printed product is the correct digit 6. Restore the true value of the product.
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Context:
Given the ellipse $C:\dfrac{x^{2}}{4}+\dfrac{y^{2}}{3}=1$ with its left and right vertices as $A$ and $B$, respectively, and $F$ as the right focus of the ellipse $C$. There is a moving point $P$ on the circle $x^{2}+y^{2}=4$, where $P$ is different from points $A$ and $B$. The line $PA$ intersects the ellipse $C$ at point $Q$ (different from point $A$). If the slope of line $QF$ exists, then the range of $\dfrac{k_{{PB}}}{k_{{QF}}}$ is \_\_\_\_.
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Context:
Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They depart together and arrive in Deauville at the same time. However, Monsieur Dubois spent on stops one-third of the time during which his wife continued driving, while Madame Dubois spent on stops one-quarter of the time during which her husband was driving.

What is the ratio of the average speeds of each of their cars?
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Context:
Find a value of $k$ such that the set $S = \{A_1, A_2, A_3\}$, where each $A_i$ is a $2 \times 2$ matrix containing $k$, generates a vector space. Ensure that $k$ appears at least once in each matrix.
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Context:
Evaluate the expression $$(x-y)^5 + 5(x-y)^3 - 2(x-y)^2 + 4(x-y)$$ given that $$\sqrt[3]{\frac{x+y}{x-y}} + \sqrt[3]{\frac{x-y}{x+y}} = x+y$$ and $$\sqrt{xy} = 1.$$
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Context:
Consider the quadratic equation $x^{2}-(r+7) x+r+87=0$ where $r$ is a real number. This equation has two distinct real solutions $x$ which are both negative exactly when $p<r<q$, for some real numbers $p$ and $q$. What is the value of $p^{2}+q^{2}$?
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Context:
Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$  Find $n \times s.$
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Context:
What is the smallest number that can be formed by using natural numbers 2 and 4?
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Context:
There are 100 towns labeled from 1 to 100. The cost to build a bridge between town "i" and town "j" is \( \max\{|i - j|, 4\} \). However, if a bridge is built between town "i" and town "2i", the cost is 2. What is the minimum cost to connect all the towns in a single connected graph?
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Context:
Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?
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Context:
Sasha discovered that there are exactly $n$ functional buttons with digits left on the calculator. It turns out that any natural number from 1 to 99999999 can either be entered using only the functional buttons, or obtained as the sum of two natural numbers, each of which can be entered using only the functional buttons. What is the smallest $n$ for which this is possible?
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Context:
It takes Bryan some minutes to walk from his house to the bus station. Then he rides the bus for 20 minutes. After that, he walks the same amount of time from the bus station to his job. It takes the same amount of time in the morning and the evening. Bryan spends 365 hours per year traveling to and from work, if he works every day. How many minutes does it take Bryan to walk from his house to the bus station and from the bus station to his job?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the group \( G = \langle a, b : a^3 = b^2 = (ab)^3 = e \rangle \).
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Context:
Calculate the closed loop integral \( \int_\gamma Pdx + Qdy \) for the path \( \gamma: (2, 0) \to (-1,1) \to (0, -2) \to (1, 2) \to (-2, -1) \to (2, 0) \) where \( P(x,y)=  \frac{x-y}{x^2+y^2} \) and \( Q(x,y)=  \frac{x+y}{x^2+y^2} \). Use Green's Theorem to determine the value of the integral.
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Context:
Xiao Li was doing a subtraction problem and mistook the tens digit 7 for a 9 and the ones digit 3 for an 8, resulting in a difference of 76. The correct difference is ______.
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Context:
Consider the following flowchart: INPUT $\rightarrow$ Subtract $8 \rightarrow \square \rightarrow$ Divide by $2 \rightarrow \square$ Add $16 \rightarrow$ OUTPUT. If the OUTPUT is 32, what was the INPUT?
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Context:
a jar full of whisky contains 40 % alcohol . a part of this whisky is replaced by another containing 19 % alcohol and now the percentage of alcohol was found to be 26 % . the quantity of whisky replaced is :
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Context:
In a diagram, several circles are connected by line segments. Choose a natural number \( n \) and place different natural numbers in the circles such that:

- If two numbers \( a \) and \( b \) are not connected by a line segment, then \( a - b \) is coprime with \( n \).
- If two numbers \( a \) and \( b \) are connected by a line segment, then \( a - b \) shares a common natural divisor greater than 1 with \( n \).

Determine the smallest natural number \( n \) for which such an arrangement is possible.
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Context:
If
\[
\sum_{n=1}^{\infty}\frac{\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}}{\binom{n+100}{100}} = \frac{p}{q}
\]
for relatively prime positive integers $p,q$, find $p+q$.
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Context:
What is the smallest number of plastic snap-together cubes that can be snapped together such that only receptacle holes are showing, with all protruding snaps hidden? Express your answer as a single integer.
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Context:
Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
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Context:
$2020$ magicians are divided into groups of $2$ for the Lexington Magic Tournament. After every $5$ days, which is the duration of one match, teams are rearranged so no $2$ people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n?$
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Context:
Find the difference between the maximum and minimum values of the function \( f(x) = \sqrt{x^2 + 2x + 2} - \sqrt{x^2 - 3x + 3} \).
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Context:
Consider a market where time is measured in milliseconds, and focus on market buy orders only. The market imbalance can be in one of three states: high (H), medium (M), or low (L). The imbalance changes due to incoming market orders or changes in standing orders. If the imbalance is low, it can remain low or increase to medium, each with probability \( \frac{1}{2} \), and no market buy orders arrive. If the imbalance is medium, it can decrease to low with probability \( \frac{1}{3} \) or increase to high. The increase from medium to high can occur due to a new market buy order with probability \( \frac{1}{2} \) or without a new order with probability \( \frac{1}{6} \). If the imbalance decreases from medium to low, no market buy orders arrive. If the imbalance is high, it can decrease to medium or stay high, each with probability \( \frac{1}{2} \), and a market order arrives with probability \( \frac{2}{3} \).

Betty keeps a unit standing sell order in the exchange, and every market buy order executes her order, earning her 1 cent. What is Betty's long-run profit per unit of time?

Hint: Construct a Markov chain with state space including more than just H, M, and L.
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Context:
设实数 $x_{1}, x_{2}, \cdots, x_{22}$ 满足对任意 $1 \leqslant i \leqslant$ 22, 有 $2^{i-1} \leqslant x_{i} \leqslant 2^{i}$ 。 求
$$
\left(x_{1}+x_{2}+\cdots+x_{22}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{22}}\right)
$$
的最大值。原始答案为$(2^{12}-1-\frac{1}{2^{11}})^{2}$，请给出近似整数值。
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Context:
If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth.
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Context:
There are three boxes of bottles. Box A contains 2 dozen water bottles, 7/4 dozen orange juice bottles, and half a dozen more apple juice bottles than there are water bottles. Box B has 25% more water bottles and 30% fewer orange juice bottles than Box A. Box C has twice as many water bottles and 1.5 times as many apple juice bottles as Box B, with no orange juice bottles. How many bottles are in all three boxes combined?
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Context:
How many bridge hands consist of exactly two 5-card suits, a void, and a remaining suit with a run of 3 cards?
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Context:
Given \(a > b > c > 0\), find the minimum value of \(2a^{2}+ \dfrac {1}{ab}+ \dfrac {1}{a(a-b)}-10ac+25c^{2}\). Express your answer as a single number.
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Context:
In a lake, there is a patch of lily pads. The patch grows at a certain rate. It takes 39 days for the patch to cover the entire lake, and 38 days for the patch to cover half of the lake. What is the growth rate of the patch of lily pads?
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Context:
Determine the number of homomorphisms from the alternating group $A_5$ to the symmetric group $S_5$. Provide your answer as a single integer.
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Context:
A positive integer is called "oddly even" if the sum of its digits is even. Find the sum of the first 2013 oddly even numbers.
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Context:
Jenna is buying concert tickets for her group of 10 friends. She's able to get three of the tickets from the concert website for $50 each before they sell out. A scalper offers to sell her four more tickets for 250% of the normal price, but Jenna convinces him to give her a $15 off discount that total payment. Finally, two of her friends get discounted tickets through their jobs - one for 60% of the normal price and one for 75% of the normal price. Additionally, there is a 10% service fee on each ticket purchased from the concert website and the scalper. How much do Jenna's friends pay for their tickets in total?
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Context:
Evaluate the limit \( \lim_{n \to \infty} \oint_{\gamma} \frac{\cot(\pi z)}{(z+a)^2} \, dz \), where \( \gamma \) is the rectangular path with vertices at \( n+\frac{1}{2}+ni, -n-\frac{1}{2}+ni, -n-\frac{1}{2} -ni, n+\frac{1}{2}-ni, n+\frac{1}{2}+ni \), and \( a \) is a complex constant. Use the Residue Theorem to determine the behavior of this limit.
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Context:
Dragoons take up $1 \times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the $y$-coordinates of their centers are both at most $6$, regardless of any dragoons in between. For example, a dragoon centered at $(4, 5)$ can fire at a dragoon centered at the origin, but a dragoon centered at $(7, 0)$ cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?
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Context:
Given the number of minutes studied by Clara and Mira over a period of one week, calculate the average difference in minutes per day in the number of minutes studied by Mira and Clara.
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Context:
Given a sequence $\{a_n\}$ that satisfies $a_0= \frac{1}{2}$ and $a_{n+1}=a_n+ \frac{a_n^2}{2018}$ for $n=0,1,2,\ldots$, if $a_k < 1 < a_{k+1}$, what is the value of the positive integer $k$?
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Context:
Given natural numbers \( M \) and \( N \), each greater than ten, consisting of the same number of digits, and such that \( M = 3N \). To obtain the number \( M \), it is necessary to add 2 to one of the digits of \( N \), and to each of the remaining digits, add an odd digit. Which digit could the number \( N \) end with? Find all possible answers.
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Context:
The kid cut out a rectangle from grid paper, with each cell having a side length of 1, and calculated its area and perimeter. Carlson then took the scissors and cut out a square from this rectangle, along the grid lines, that was adjacent to the boundary of the rectangle.

- "My rectangle..." the kid sobbed. "There is something strange about this figure!"
- "Nonsense, do not mention it," Carlson said, waving his hand carelessly. "Here you see, in this figure, the perimeter is the same as the area of the rectangle was, and the area is the same as the perimeter was!"

What size square did Carlson cut out?
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Context:
Chuck the llama is tied to the corner of a 2 m by 3 m shed on a 3 m leash. Calculate the area in which Chuck has to play if he can only go around the outside of the shed. Express your answer in square meters (m²), including π.
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Context:
Let \( p(x) = x^2 - x + 1 \). Let \(\alpha\) be a root of \( p(p(p(p(x)))) \). Find the value of
\[
(p(\alpha) - 1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))
\]
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Context:
在正 $\triangle A B C$ 的边 $B C, C A, A B$ 上有内分点 $D, E, F$ 将边分成 $3:(n-3)(n>6)$, 线段 $A D, B E, C F$ 相交所成的 $\triangle P Q R(B E$ 交 $A D$ 于 $P$, 交 $F C$ 于 $Q)$ 是 $\triangle A B C$ 的面积的 $\frac{4}{49}$时.求 $n$ 的值.
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Context:
已知复数 $z_{1}, z_{2}, z_{3}$ 使得 $\frac{z_{1}}{z_{2}}$ 为纯虚数, $\left|z_{1}\right|=\left|z_{2}\right|=1,\left|z_{1}+z_{2}+z_{3}\right|=1$, 则 $\left|z_{3}\right|$ 的最小值是以 $\sqrt{m}-n$ 的形式给出，请提供m+n的值。
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Context:
What is the maximum possible value of $\int_{0}^{1}(f(x))^3dx$ if $f(x)$ is a function such that $-1\leq f(x)\leq 1$ for all $x \in [0,1]$, and $\int_{0}^{1}f(x)dx=0$?
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Context:
Calculate the mass of the region \( \Omega \) in the first quadrant, bounded by the curves \( x^2 + y^2 = 4 \), \( x^2 + y^2 = 9 \), \( x^2 - y^2 = 1 \), and \( x^2 - y^2 = 4 \). The density at each point \((x,y)\in \Omega\) is equal to the product of the distances of the point to the coordinate axes. Use a change of variables to find the mass of \( \Omega \).
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Context:
For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote
    \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying
    \[a_1+a_2+\cdots+a_{10}=20.\]
    Compute the number of positive integer divisors of $N$.

[i]2021 CCA Math Bonanza Individual Round #14[/i]
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Context:
A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set?
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Context:
Define a finite sequence $\left( s_i \right)_{1\le i\le 2004}$ with initial conditions $s_0 + 2 = s_1 + 1 = s_2 = 2$ and the recurrence relation:
$$ s_n = 1 + s_{n-1} + s_{n-2} - s_{n-3}. $$
Calculate its last element.
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Context:
Calculate the dimension of the space of divisors on the Jacobian $J$ of a smooth compact complex curve $C_g$ (of genus $g$) that are linearly equivalent to the theta divisor $\Theta$. Specifically, determine $\dim( H^0(J,\mathcal{O}(\Theta)))$.
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Context:
Given two random matrices $A$ and $B$ in $\mathbb{R}^{n\times n}$, determine the probability that the matrix algebra generated by $A$ and $B$ is the full matrix algebra $\mathbb{R}^{n\times n}$.
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Context:
Let $X$ and $Y$ be independent random variables, each following a normal distribution $N(1,1)$. Calculate the probability $P(2-X < Y < X)$. Express your answer as a single probability value.
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Context:
A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide the answer to the nearest percent.
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Context:
On a rectangular table of size $x$ cm $\times 80$ cm, identical sheets of paper of size 5 cm $\times 8$ cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter above and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length of $x$ in centimeters?
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Context:
What is the lowest prime number that is thirteen more than a cube?
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Context:
Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use?
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Context:
At the beginning of the school year, Jinshi Middle School has 17 classrooms. Teacher Dong Helong takes 17 keys to open the doors, knowing that each key can only open one door, but not knowing which key matches which door. What is the maximum number of attempts he needs to make to open all 17 locked doors?
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Context:
Find the smallest upper bound for the expression:
\[ A(N) = \max_{a_1, a_2, \ldots, a_N \in [0,1]} \sum_{n=1}^{N} a_n^{1/2} \prod_{s=1}^{n-1} (1-a_s)^{1/2} \]
where an empty product is defined to be 1.
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Context:
Find the mass percentage of O in Dinitrogen pentoxide
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Context:
In an isosceles triangle, the base \(a\) and the angle \(\alpha\) at the base are given. Find the length of the median drawn to the lateral side.
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\boxed{\frac{a}{4}\sqrt{9 + \tan^2 \alpha}}


Context:
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{2\sin A - \sin C}{\sin C} = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{{a}^{2} + {c}^{2} - {b}^{2}}$, then the maximum value of $\sin ^{2}A + \sin ^{2}C$ is ______.
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Context:
It is known that \( A \) is the largest number that is the product of several natural numbers whose sum is 2011.

What is the largest power of three that divides \( A \)?
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Context:
A “palindromic integer” is an integer that remains the same when its digits are reversed. There are 200 palindromic integers of a certain digit length that are both even and greater than 500000. What is the digit length of these palindromic integers?
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Context:
A pharmaceutical company needs to optimize the cultivation temperature for a certain medicinal liquid through bioassay. The experimental range is set from 29℃ to 63℃, with an accuracy requirement of ±1℃. When using the fractional method for optimization, the minimum number of experiments required to ensure the best cultivation temperature is found is ______.
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Context:
Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is:
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Context:
Find an upper bound for the series \( \sum_{i=1}^{n}\left(\frac{n}{i}\right)^{i} \).
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Context:
Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle{CDP}$.
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Context:
What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen can attack at least one other queen?
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Context:
A set of composite numbers from the set $\{1,2,3,4, \ldots, 2016\}$ is called good if any two numbers in this set do not have common divisors (other than 1). What is the maximum number of numbers that a good set can have?
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Context:
Given a parabola C: y²=x and a point M (1, -1) on it. Points A and B are two moving points on the parabola C, and $\overrightarrow {MA}\cdot \overrightarrow {MB}$=0. The maximum value of the distance from point M to line AB is _______.
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Context:
A tank has two outlet pipes - outlet pipe A can empty the tank when it is full in 5 hours, and outlet pipe B can empty the tank when it is full in 8 hours. If the inlet pipe is kept open, letting water in at a rate of 4 litres/min, and both outlet pipes A and B are open simultaneously, the combined system would take 3 hours longer to empty the tank. Find the capacity of the tank.
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Context:
A square $ABCD$ has side length $5$. Let $E$ be a point $20$ units from point $D$. For every point $F$ inside the square, rotate $E$ about $F$ $120$ degrees counterclockwise. Determine the area of the region formed by all of the image points of the rotation.
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Context:
Let $a_1, a_2, \cdots, a_{n}$ be real numbers such that $a_1^2 + a_2^2 + \cdots + a_{n}^2 = 1$. Determine the maximum value of the expression $$\large |a_1 - a_3| + |a_3 - a_5| + \cdots + |a_{2n-1} - a_{2n+1}| + |a_{2n+1} - a_1|$$ where $n \in \mathbb{N}$ and $n \geq 2$. Provide your answer as a single number.
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Context:
In the Cartesian coordinate system, consider the ellipse \(C: \frac{x^{2}}{27}+\frac{y^{2}}{18}=1\) with its right focus \(F\). A line \(l\) passing through \(F\) intersects the ellipse \(C\) at points \(A\) and \(B\). Determine if there exists a fixed point \(P\) on the \(x\)-axis such that the dot product \(\overrightarrow{PA} \cdot \overrightarrow{PB}\) remains constant as the line \(l\) rotates about \(F\).
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Context:
Find the limit of the series as \( n \) approaches infinity:
\[ x(n) = \sum_{k=1}^n \frac{k^2}{n^3+k^2} \]
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Context:
Let $G$ be a group and $H$ be a subgroup of finite index $n$ in $G$. Determine if there exists a normal subgroup $N$ of $G$ contained in $H$ such that the index $[G:N]$ is finite. If such a subgroup exists, provide an upper bound for $[G:N]$. Provide your answer as a specific number or expression.
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Context:
Evaluate the integral \( I = \int_{-\infty}^{\infty} \frac{\sin(ax)}{x(\pi^2-a^2x^2)}\,dx \) for \( a > 0 \).
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Context:
In a round-robin tournament with four football teams (each pair of teams plays one match), the winner of each match earns 3 points, the loser earns 0 points, and in the event of a draw, both teams earn 1 point. After the tournament, it was found that no team won all their matches, and all four teams had different scores. Determine the minimum possible number of draws in all matches. Express your answer as a single integer.
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Context:
Evaluate the limit: \( \lim_{n\to\infty} \frac{1}{n}\int_{0}^n \frac{dx}{5+4\cos(x)} \).
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Context:
Calculate the variance of the random variable $Y$, which represents the number of draws needed to replace all black balls with white balls in a bag initially containing 2 white balls and 2 black balls. In each draw, a white ball is returned to the bag, while a black ball is replaced by a white ball. Find $\operatorname{var}(Y)$. \( \boxed{?} \)
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Context:
$A$ standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Calculate the expected sojourn time for cars that decide to take gas at a station where cars arrive according to a Poisson process with rate \( \frac{1}{3} \). An arriving car finding \( n \) cars at the station leaves with probability \( q_n = \frac{n}{4} \) and joins the queue with probability \( 1 - q_n \), for \( n = 0, 1, 2, 3, 4 \). Cars are served in order of arrival, and the service time is \( \text{Exponential}(\frac{1}{3}) \).
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Context:
Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$.
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Context:
In the  $ x \minus{} y$  plane, find the area of the region bounded by the parameterized curve as follows.

 $ \left\{\begin{array}{ll} x \equal{} \cos 2t & \quad
y \equal{} t\sin t & \quad \end{array} \right.\ (0\leq t\leq 2\pi)$
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Context:
A metal sphere with a radius of $\sqrt[z]{16}$ is melted and recast into a cone whose lateral surface area is three times the area of its base. Find the height of the cone.
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Context:
Given that $a^{-1}ba = b^2$ and $|a| = 3$ in a group, find the order of the element $b$, where $b \neq e$ and $b$ is not of order 2.
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Context:
Compute the following expression:
\[ 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4)))))))) \]
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Context:
Determine the rate of divergence of the series \( \sum_{j=0}^{\infty} \frac{1}{\sqrt{j}} \) as \( n \to \infty \). Express the rate of divergence of the partial sums \( S_n \) in the form \( O(n^{\alpha}) \) for some \( \alpha > 0 \).
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Context:
In the diagram, $ABCDEFG$ is a regular heptagon (a $7$-sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Determine the greatest positive integer \( x \) such that the expression \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square.
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Context:
Determine the minimum number of symbols required to open a lock with 4 symbols (A, B, C, D) and a 3-symbol password, given that the lock does not "rest" after each attempt. Repeated symbols are allowed.
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Context:
Evaluate the limit: $$ \lim_{n \to \infty} \left( \int_1^{1+\frac{1}{n}} \left\{ [(x+1)\ln x +1] e^{x (e^x \ln x +1)} + n \right\} \, dx \right)^n $$
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Context:
Sandy buys an old scooter for $800 and spends $200 on its repairs. In addition, she pays $50 for a new set of tires and $100 for a paint job. The scooter's value also depreciates by 10% per year. If Sandy sells the scooter for $1,200 after 2 years, what is the gain percent?
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Context:
Find the function $f(x)$ if the arc length of the curve $y=f(t)$ on the interval $0 \le t \le x$ is given by \( L(x) = \ln(\sec x + \tan x) \) for every $x$ on $0 \le x \le 1$, and the curve passes through the origin.
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Context:
Four distinct natural numbers, one of which is an even prime number, have the following properties:
- The sum of any two numbers is a multiple of 2.
- The sum of any three numbers is a multiple of 3.
- The sum of all four numbers is a multiple of 4.

Find the smallest possible sum of these four numbers.
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Context:
In triangle \( \triangle ABC \), the medians \( AE \), \( BF \), and \( CD \) intersect at point \( M \). Given that points \( E \), \( C \), \( F \), and \( M \) are concyclic, and that \( CD = n \), find the length of segment \( AB \).
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Context:
Evaluate the integral \( \int_{-\infty}^{+\infty} \text{sinc}(8t) \text{sinc}^2(t) \cos(8\pi t) \, dt \) where the sinc function is defined as \( \text{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{\pi t} \quad &\text{for } t \neq 0, \\ 1 \quad &\text{for } t = 0. \end{cases} \)
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Context:
A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.
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Context:
In a simple undirected graph $G$, the vertex set $V$ is partitioned into $A \cup B$ where $|B| = 30$. Each vertex in $A$ is connected to exactly 3 vertices in $B$. Each vertex in $B$ is connected to exactly 4 other vertices in $B$. Additionally, for any two vertices in $B$ that are connected by an edge, there exists a vertex in $A$ that is connected to both of them. Determine the minimum number of vertices in $A$. \( \text{Provide your answer as a single integer.} \)
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Context:
Calculate the area of the region that lies inside the polar curve \( r = 1 + \cos\theta \) and outside the polar curve \( r = 2\cos\theta \).
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Context:
The distance between any two lateral edges of an inclined triangular prism is \( a \). The lateral edge is equal to \( l \) and is inclined at an angle of \( 60^\circ \) to the base plane. Find the total surface area of the prism.
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Context:
Determine the number of automorphisms of $Z_4 ⊕ Z_2$ that map the element (1, 0) to an element of order 4 and the element (0, 1) to an involution.
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Context:
Find a particular solution to the differential equation: \[ y'' + 3y' + 2y = \exp(\mathrm{e}^x) \]
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Context:
Evaluate the limit:

\[ \lim_{{x \to 0}} \frac{x-\sin\sin\cdots\sin x}{x^3}, \]

where \( \sin\sin\cdots\sin x \) represents the composition of the sine function \( n \) times.
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Context:
What is the maximum number of cells in an $8 \times 8$ square that can be colored such that the centers of any four colored cells do not form the vertices of a rectangle with sides parallel to the edges of the square?
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Context:
Determine the asymptotic relationship between the functions \( f(n) = 2^n n \) and \( g(n) = \frac{3^n}{n^2} \) by evaluating the limit \( \lim_{n \to \infty} \frac{f(n)}{g(n)} \).
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Context:
Given the vectors \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2\hat{i}+2\hat{j}+\hat{k}, \vec{c}=5\hat{i}+\hat{j}-\hat{k}\), find the area of the region formed by the position vector \(\vec{r}\) that satisfies the equation \(\hat{r}\cdot \vec{a}=5\) and \(|\vec{r}-\vec{b}|+|\vec{r}-\vec{c}|=4\). Determine the area closest to an integer.
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Context:
Determine the cardinality of the collection of open subsets in \(\mathbb{R}^2\).
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Context:
A polyhedron is enclosed by eight faces, each of which is a regular polygon. Four of these faces lie in the same plane, forming a square. If the edge length of the polyhedron is $2$, and all its vertices lie on the sphere $O$, then the surface area of $O$ is ______.
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Context:
Suppose you have a basket containing a single coin marked with the number 1. At each time interval of noon minus \( \frac{1}{k} \) minutes, where \( k \) is a positive integer, the coin numbered \( k \) is removed from the basket and replaced with two new coins numbered \( 2k \) and \( 2k+1 \). How many coins are in the basket at exactly noon?
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Context:
A merchant has two types of items in his inventory:

Item A is marked up by 60% and then offered a discount of 20% on the marked price. There are 30 units of Item A in stock, each with a unit cost price of $10.

Item B is marked up by 80% and then offered a discount of 10% on the marked price. There are 20 units of Item B in stock, each with a unit cost price of $18.

After selling all the units of both items at their respective discounted prices, what is the overall profit percentage made by the merchant?
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Context:
Let \( A_n = \frac{3}{4} - \left(\frac{3}{4}\right)^2 + \left(\frac{3}{4}\right)^3 - \cdots + (-1)^{n-1}\left(\frac{3}{4}\right)^n \) and \( B_n = 1 - A_n \). Find the smallest integer \( n_0 \) such that \( B_n > A_n \) for all \( n \geq n_0 \).
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Context:
If the price of a book is first decreased by 50% and then increased by 60%, what is the net change in the price?
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Context:
Let \( A \in \mathbb{R}^{5 \times 5} \) and \( \alpha \in \mathbb{R} \). Consider the linear system \( Ax = \begin{pmatrix} 1 \\ -1 \\ a^2 + a \\ 1 \\ -1 \end{pmatrix} \), with the following general solution: \( x = \begin{pmatrix} t + u + 1 \\ s + t - u \\ s - t + 1 \\ -s - t + u \\ -s - u - 1 \end{pmatrix} \). Also, \( A \) has the column vectors \( \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ a^2 - 3 \end{pmatrix} \) and \( \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\ a^2 - 2 \end{pmatrix} \). Determine the value of \( \alpha \) for which \( A^2 = 0 \).
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Context:
In the triangle $ABC$, the angle $\angle ACB = 120^\circ$ and $AC + BC = 2$. What are the possible lengths for the side $AB$?
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Context:
The sequence {an} is an arithmetic sequence with a non-zero common difference, and ai ∈ [0, 4], 1 ≤ i ≤ 2019. The function f(x) = 3sin( $$\frac {π}{4}$$x - $$\frac {π}{2}$$). If f(a1) + f(a2) + f(a3) + ... + f(a2019) = 0, then a1 + a2 + a3 + ... + a2019 = _____.
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Context:
Let \(ABC\) be a triangle with \(AB=13, BC=14\), and \(CA=15\). Pick points \(Q\) and \(R\) on \(AC\) and \(AB\) such that \(\angle CBQ=\angle BCR=90^{\circ}\). There exist two points \(P_{1} \neq P_{2}\) in the plane of \(ABC\) such that \(\triangle P_{1}QR, \triangle P_{2}QR\), and \(\triangle ABC\) are similar (with vertices in order). Compute the sum of the distances from \(P_{1}\) to \(BC\) and \(P_{2}\) to \(BC\).
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Context:
Given vectors $\overrightarrow{a}=(1, \sqrt {1+\sin 40^{\circ}})$, $\overrightarrow{b}=( \frac {1}{\sin 65^{\circ}},x)$ are collinear, find the value of the real number $x$. Express your answer as a single numerical value.
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Context:
设数列 $\left\{a_{n}\right\}$ 满足
$a_{0}=\frac{1}{2}, a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2012}(n=0,1, \cdots)$.
求整数 $k$, 使得 $a_{k}<1<a_{k+1}$.
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Context:
On a plane, there are points A(-2,1), B(1,4), C(4,-3). Point D lies on line AB such that AD/BD = 1/2. Extending DC to E makes CE/ED = 1/4. What are the coordinates of point E?
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Context:
Compute the residue field of the affine scheme \( Y = \operatorname{Spec}\left(\frac{\mathbb{Z}[x, y]}{(x^3 - y^2)}\right) \) at the point \((1,1)\).
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Context:
Determine the Krull dimension of the ring \( B = A[x,y,z]/\langle xy, xz \rangle \), where \( A \) is a Noetherian commutative ring.
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Context:
A person left for a walk from Appleminster to Boniham at noon, while his friend left from Boniham to Appleminster at 2 PM the same day. They met each other on the way at five minutes past four and then simultaneously reached their respective destinations. When did they finish their journey?
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Context:
Example 6 Three equal cylindrical surfaces are pairwise tangent, and their axes are perpendicular to each other. If the radius of each cylindrical surface is $r$, find the radius of the smallest sphere that is tangent to these three cylindrical surfaces.

 untranslated text remains the same as the source text in terms of line breaks and format.
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Context:
Determine the number of positive integral solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = x_1 \cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5\).
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Context:
Let \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\).
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Context:
Determine the interval of convergence for the power series \( \sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ \ln n} \).
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Context:
The product \( 29 \cdot 11 \), and the numbers 1059, 1417, and 2312, are each divided by \( d \). If the remainder is always \( r \), where \( d \) is an integer greater than 1, calculate \( d - r \). Express your answer as a single integer.
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Context:
Given the expression $2^{50} + 2^{51} + 2^{52} + \ldots + 2^{99} + 2^{100}$, find an equivalent algebraic representation in terms of 'a', where $2^{50} = a$. Express your answer as an algebraic expression in terms of 'a'.
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Context:
10. Right triangle $X Y Z$ has right angle at $Y$ and $X Y=228, Y Z=$ 2004. Angle $Y$ is trisected, and the angle trisectors intersect $X Z$ at $P$ and $Q$ so that $X, P, Q, Z$ lie on $X Z$ in that order. Find the value of $(P Y+Y Z)(Q Y+X Y)$.
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Context:
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\) is:
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Context:
We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?
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Context:
Given the probability measure \( \mathbf{P}_X [A] := \frac{1}{2} \delta_0 (A) + \frac{1}{2} \int_{A \cap (0, \infty)} e^{-t} dt \) for \( A \in \mathcal{B} (\mathbb{R}) \), find the expectation \( \mathbf{E} [X] \).
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Context:
Fill the numbers $1, 2, \cdots, 36$ into a $6 \times 6$ grid with each cell containing one number, such that each row is in ascending order from left to right. What is the minimum possible sum of the six numbers in the third column?
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Context:
Calculate the total number of different scheduling methods between "archery" and "mathematics" with at most one art in between, given that there are six sessions in a day, one for each art. Express your answer as a single integer.
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Context:
A triangle $H$ is inscribed in a regular hexagon $S$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$?
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Context:
Determine the largest topology on a set $X$ that contains a given family of subsets $\mathcal{S}$.
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Context:
24. Find the number of 7-digit integers formed by some or all of the five digits, namely, $0,1,2,3$, and 4 , such that these integers contain none of the three blocks 22,33 and 44 .
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Context:
A small weight is suspended in the air on a spring. When this weight is fully submerged in kerosene while still on the same spring, the deformation of the spring remains the same. Determine the density of the material of the weight. The density of kerosene is $\rho=800 \ \mathrm{kg/m}^3$.
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Context:
In a single round-robin chess competition, where the winner of each match receives 2 points, the loser gets 0 points, and a draw results in 1 point for each player, the champion has a unique condition: they have more points than any other participant but have won fewer matches than any other participant. What is the minimum number of participants required for this condition to be possible?
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Context:
Evaluate the following improper integral: $\displaystyle\lim_{n\to\infty} \displaystyle\int_0^\infty \dfrac{n\sin x}{1+n^2x^2}\ dx$
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Context:
Exercise: Compute $$\lim_{n\to\infty}\int_\limits{0}^{1}\frac{nx}{1+n^2x^4}dx.$$
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Context:
Calculate the expectation of the expression \( \max\{U_{1}, U_{2}\} \times \max\{U_{1}, U_{3}\} \), where \( U_{1}, U_{2}, U_{3} \) are independent random variables each uniformly distributed over the interval \( (0, 1) \).
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Context:
On a planet, there are $2^{N}$ countries $(N \geqslant 5)$. Each country has a flag composed of a line of $N$ squares of side 1, each being yellow or blue. The flags are all distinct from each other. A set of $N$ flags is diverse if they can be placed in a certain order to form an $N \times N$ square such that the $N$ squares on the main diagonal are of the same color. Determine the smallest integer $M$ such that any set of $M$ flags contains a diverse set of $N$ flags.

## Directed Graphs
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Context:
In triangle \(ABC\), the bisectors \(BL\) and \(AE\) of angles \(ABC\) and \(BAC\) respectively intersect at point \(O\). It is known that \(AB = BL\), the perimeter of triangle \(ABC\) is 28, and \(BO = 2OL\). Find \(AB\).
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Context:
A Pretti number is a seven-digit positive integer with the following properties:

- The integer formed by its leftmost three digits is a perfect square.
- The integer formed by its rightmost four digits is a perfect cube.
- Its ten thousands digit and ones (units) digit are equal.
- Its thousands digit is not zero.

How many Pretti numbers are there?
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Context:
The roots of a quadratic equation with a mistaken constant term are \(-17\) and \(15\), and with a mistaken leading coefficient, the roots are \(8\) and \(-3\). Find the actual roots of the quadratic equation.
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Context:
Let $@$ be a binary operation on the natural numbers satisfying the properties that, for all  $a, b,$ and $c$, $(a + b)@c  = (a@c) + (b@c)$ and $a@(b + c) = (a@b)@c. $  Given that  $5@5=160$,   find the value of  $7@7$
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Context:
Determine the minimum number of times a biased die must be rolled so that the probability of rolling a 6 can be estimated with an uncertainty of 0.01 and a confidence level of at least 0.9. Use the Central Limit Theorem to approximate this number.
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Context:
Find the volume of the common part of two cubes if one of them is obtained by rotating the other cube by $90^\circ$ around an axis passing through the midline of one of its faces. The edge of the cube is equal to $a$.
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Context:
Find the simplest example of numbers $n$ and $k$ such that $k$ divides $n$ ($k|n$), but there exists a group $G$ of order $n$ that does not have any element of order $k$.
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Context:
Consider the following time series processes:
\begin{align*}
\epsilon_t &\sim i.i.d.(0,1) \\
x_t &= \alpha x_{t-1}+\epsilon_t \\
y_t &= y_{t-1}+x_t \\
z_t &= z_{t-1}+y_t
\end{align*}
Assume $\alpha = 0.8$. Determine the order of integration $d$ for the process $z_t$, such that $z_t$ is an $I(d)$-process.
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Context:
For $n \in \mathbb{N}$, consider the functions $g_n(x) = x^{1/n}$ and $f_n(a) = \int_0^1 (1 - x^a)^n \, dx$. If the series $\sum_{a=2}^\infty \left( \lim_{n \to \infty} \frac{g_n(f_n(a))}{a!} \right)$ converges to a constant $\alpha$, find the value of $\alpha$.
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Context:
Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$.
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Context:
As shown in the figure, in $\triangle ABC$, $BC > AC$ and $\angle A = 60^\circ$. Points $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively. If $PC$ bisects $\angle ACB$ and $PD$ bisects $\angle ADE$, then $\angle DPC$ equals $\qquad$.
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Context:
Consider a sequence of non-negative integers starting from $0$, where each subsequent term is either $0$ or the previous term plus $1$, taken modulo $500$. The sequence terminates when the number $499$ appears exactly twice. Determine the number of such sequences that contain exactly $1000$ instances of the number $0$. What is the remainder when this count is divided by $500$?
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Context:
In a family of 40 people, 16 people eat only vegetarian, 12 people eat only non-vegetarian, 8 people eat both vegetarian and non-vegetarian, 3 people are pescatarians who eat fish but not meat, and 1 person is a vegan who eats neither animal products nor by-products. How many people in the family eat vegetarian food?
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Context:
Compute the number of digits in $2015!$.
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Context:
Evaluate the limit: \[ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}} \] where \( P_{n} \) is the product of the numbers in the \( n \)-th row of Pascal's Triangle.
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Context:
Find the set of complex numbers $z$ for which the series \( \sum\limits_{n=1}^{\infty}(1-\cos\frac{1}{n})z^n \) converges.
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\boxed{\{ z \in \mathbb{C} \mid |z| \leq 1 \}}


Context:
In a complete graph with 5 vertices, labeled \( \{1, 2, 3, 4, 5\} \), how many Hamiltonian cycles include both the edges \( \{1, 2\} \) and \( \{3, 4\} \)? Assume that a Hamiltonian cycle is defined solely by the collection of edges it contains, without regard to orientation or starting point.
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Context:
# Problem 6. (3 points)

In how many ways can natural numbers from 1 to 9 be arranged in a $3 \times 3$ square table so that the sum of the numbers in each row and each column is even? (Numbers can repeat)
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Context:
A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. Find the probability that the frog's final position is no more than $1$ meter from its starting position. The original answer is in the form \(\frac{k}{m}\). Please give the value of \(k + m\).
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Context:
Determine the number of solutions to the equation \(x^{3} - \lfloor x^{3} \rfloor = (x - \lfloor x \rfloor)^{3}\) within the interval \([1, 20]\).
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Context:
Given the inequality $\ln (x+1)-(a+2)x\leqslant b-2$ that always holds, find the minimum value of $\frac {b-3}{a+2}$.
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Context:
Given the sequence \( a_{n} = 1 + n^{3} \) where the sequence is \(\{2, 9, 28, 65, \ldots\} \) and \( \delta_{n} = \operatorname{gcd}(a_{n+1}, a_{n}) \), find the maximum value that \(\delta_{n}\) can take.
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Context:
The graphs of the quadratic functions $y=x^2-2x+2$ and $y=-x^2+ax+b$ ($a>0$, $b>0$) have tangent lines at one of their intersection points that are perpendicular to each other. Find the minimum value of $\frac{1}{a} + \frac{4}{b}$.
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Context:
一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 $p(0 \leq p \leq 1)$ 的概率消失，有 $\frac{1-p}{3}$的概率保持不变，有 $\frac{1-p}{3}$ 的概率分裂成两个，有 $\frac{1-p}{3}$ 的概率分裂成三个.对所有新产生的生物每天也会发生上述事件.假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 $\frac{1}{2}$ ，则 $p$ 至多为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Rectangular tiles each of size 25 cm by 65 cm must be laid horizontally on a rectangular floor of size 150 cm by a certain width, such that the tiles do not overlap and they are placed with edges jutting against each other on all edges. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is 36. What is the width of the floor?
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Context:
Let $ABC$ be a triangle with circumcenter $O$ such that $AC=7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.
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Context:
Determine the number of points among M(1, 1), N(1, 2), P(2, 1), Q(2, 2), and G(2, $\frac{1}{2}$) that satisfy the condition of being a "lucky point," where a "lucky point" is defined as a point that lies on both the graph of a logarithmic function and an exponential function. Provide your answer as a whole number.
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Context:
Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 2:7.$  Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some constants $t$ and $u.$  Enter the ordered pair $(t,u).$

[asy]
unitsize(1 cm);

pair A, B, P;

A = (0,0);
B = (5,1);
P = interp(A,B,2/9);

draw(A--B);

dot("$A$", A, S);
dot("$B$", B, S);
dot("$P$", P, S);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{7}{9}, \frac{2}{9} \right)}


Context:
Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
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Context:
Let \( A = \{1, 2, \ldots, 2011\} \). Find the number of functions \( f \) from \( A \) to \( A \) that satisfy \( f(n) \leq n \) for all \( n \) in \( A \) and attain exactly 2010 distinct values.
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Context:
Two opposite edges of a tetrahedron are equal to \( a \), two other opposite edges are equal to \( b \), and the remaining two opposite edges are equal to \( c \). Find the radius of the circumscribed sphere.
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Context:
Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulting fraction can have?
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Context:
Determine the number of distinct ellipses that can be formed by selecting three different numbers from -3, -2, -1, 1, 2, 3 to serve as coefficients of an elliptic equation $ax^2+by^2+c=0$.
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Context:
Let \( q \) be a 7th root of unity. Calculate \(\frac{q}{1+q^{2}} + \frac{q^{2}}{1+q^{4}} + \frac{q^{3}}{1+q^{6}} \).
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Context:
Out of 1,500 senior high school students, 70% passed their English exams, 80% passed their Mathematics exams, and 65% passed their Science exams. To get their degree, a student must pass in all three subjects. Assuming that retaking the exams isn't an option, how many students failed to get their degree?
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Context:
After removing the two jokers from a deck of playing cards, a card is drawn at random. What is the probability of drawing a diamond or an Ace?
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Context:
Determine the limit of the sequence \( a_n = \frac{3 \cdot 5 \cdot 7 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2n} \) as \( n \to \infty \).
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Context:
Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers.
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Context:
将 4 个数 $1,9,8,8$ 写成一行并进行以下操作：对每一对相邻的数都作一次减法, 即用右边的数减左边的数, 然后将所得的数写在两数之间, 算是完成了一次操作. 然后再对这 7 个数所排成的一行进行同样的操作, 如此继续下去, 共操作 100 次, 求最后所得到的一行数的和.
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Context:
Find the smallest possible non-zero difference $|a-b|$ between two distinct positive integers $a$ and $b$ such that the expressions $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are all integers.
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Context:
Determine the number of digits that can replace □ in the multiplication □4×32 to make the product a three-digit number. Express your answer as a whole number.
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Context:
Determine the total number of "magic" circles for a given integer \( n \). A "magic" circle is defined as a circle with \( 2^n \) items on its periphery, where each item holds either 0 or 1. For each item, consider the sequence of \( n \) items to the right. The set of these sequences must consist of all possible \( n \)-length combinations of 0 and 1. What is the number of such "magic" circles for a given \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the following sum as \(x\) approaches 0 from the right: \[\lim_{x\to 0^{+}}\,\,\sum_{n = 2}^{\infty}{(-1)^{n} \over \ln^{x}(n+1)}\]
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Context:
Evaluate the double integral: $$\int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x$$
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Context:
Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$.
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Context:
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30?
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Context:
Find the values of \( k \) for which the matrix \( A = \begin{bmatrix} 1 & 1 & k \\ 1 & 1 & k \\ 1 & 1 & k \end{bmatrix} \) has three linearly independent eigenvectors.
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Context:
Let $C$ be a code with a distance $d = 2t + u + 1$. Determine the maximum number of errors that $C$ can detect if it is used simultaneously to correct $t$ errors.
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Context:
One third of a solution that was 8% sugar by weight was replaced by a second solution, resulting in a solution that was 20% sugar by weight. The second solution was what percent sugar by weight?
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Context:
Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is $12$. Given that $EG^2 = p - q\sqrt {r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p + q + r.$
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Context:
Let a be four years older than b who is a half-sibling to c. If b is twice as old as c, and the total of their ages in months is 720, then how old is b in years, considering that a is three quarters of a year older than his half-sibling d?
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Context:
Find $x$ if \[ \frac{1}{\frac{1}{\frac{1}{\frac{1}{x} + \frac{1}{2}} + \frac{1}{\frac{1}{x} + \frac{1}{2}}} + \frac{1}{\frac{1}{\frac{1}{x} + \frac{1}{2}} + \frac{1}{\frac{1}{x} + \frac{1}{2}}}} = \frac{x}{36}. \]
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Context:
Find the units digit of the decimal expansion of\[(15 + \sqrt{220})^{19} + (15 + \sqrt{220})^{82}.\]
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Context:
Use the Central Limit Theorem to approximate the 0.95-quantile \( q_{0.05}(\mathcal{A}_n) \) for the sample size \( n=100 \), given that \( q_{0.05}(\mathcal{A}_1)=3.84 \). Assume \( \mathcal{A}_1 \sim \chi^2_1 \) and \( n \cdot \mathcal{A}_n \sim \chi^2_n \).
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Context:
There are $5$ pots of different chrysanthemums, among which $2$ pots are white, $2$ pots are yellow, and $1$ pot is red. They need to be arranged in a row such that the red chrysanthemum is in the middle, the white chrysanthemums are not adjacent, and the yellow chrysanthemums are not adjacent. How many different arrangements are there? (Answer with a number.)
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Context:
Calculate the volume of the region bounded by the surfaces \( z = (x^2 + y^2)^2 \) and \( z = x \).
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Context:
Determine the domain of the function \( \ln(x)^{\ln(x)} \).
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Context:
Determine the number of elements of order 10 in the group \( C_{15} \times C_{12} \times C_2 \).
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Context:
Find the expected area of the projection of a unit cube onto a plane under isotropic random direction of projection using a different approach than Arnold's solution.
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Context:
The area of triangle $ABC$ is 1. Let $A_1, B_1, C_1$ be the midpoints of the sides $BC, CA, AB$ respectively. Points $K, L, M$ are taken on the segments $AB_1, CA_1, BC_1$ respectively. What is the minimum area of the common part of triangles $KLM$ and $A_1B_1C_1$?
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Context:
A one-foot stick is marked in 1/4 and 1/5 portions. How many total markings will there be, including the end points?
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Context:
Let \( G \) be a finite group and \( g, h \in G - \{1\} \) such that \( g^{-1}hg = h^2 \). Given that \( o(g) = 5 \) and \( o(h) \) is an odd integer, find the order of \( h \).
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Context:
设 a_n=\left(\df{1+q}{1-q}\right)\cdot\left(\df{1+q^2}{1-q^2}\right)^{\frac{1}{2}}\cdot\left(\df{1+q^4}{1-q^4}\right)^{\frac{1}{4}}\cdot\cdots\cdot\left(\df{1+q^{2^n}}{1-q^{2^n}}\right)^{\frac{1}{2^n}}，其中 q=\df{11}{13}. 记 a=\displaystyle\lim_{n\rightarrow+\infty}a_n，则 [a]=__________.
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Context:
某个国家有 2023 个岛和 2022 座桥, 任意一座桥连接两个不同的岛, 任意两个岛之间至多有一座桥相连, 且可以从任意一个岛通过若干座桥到达其他任何岛. 若三个岛中的某个岛与另两个岛都有桥连接, 则称这三个岛组成"岛群", 已知任两个"岛群"中都有相同的岛, 那么恰有一座桥的岛最少有多少个?
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Context:
Given point $A$ is a fixed point on the circle $O: x^{2}+y^{2}=4$, and point $B$ is a moving point on the circle $O$. If it satisfies $| \overrightarrow{AO}+ \overrightarrow{BO}|=| \overrightarrow{AO}- \overrightarrow{BO}|$, then $\overrightarrow{AO} \cdot \overrightarrow{AB}=$ \_\_\_\_\_\_ .
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Context:
On the segment $OA$ of length $L$ on the number line $Ox$, two points, $B(x)$ and $C(y)$, are randomly placed. Find the probability that from the three resulting segments a triangle can be formed.
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Context:
Given that triangle \( \triangle ABC \) satisfies \( \frac{\angle A}{3} = \frac{\angle B}{4} = \frac{\angle C}{5} \), find the value of \( \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{|BC|^2} \).
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Context:
In each cell of a $5 \times 5$ grid, a natural number is written with invisible ink. It is known that the sum of all the numbers equals 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the grid?
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Context:
Compute the surface integral \( \iint_S F \cdot dS \) where \( F = \langle y, z, -x \rangle \) and \( S \) is the portion of the paraboloid \( z = x^2 + y^2 \) that lies below the plane \( z = 4 \).
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Context:
An infinite sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the recurrence $$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$ for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\cdots+a_{100}$.
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Context:
Determine the number of non-zero ring homomorphisms from the ring \( \mathbb{Z}_{30} \) to the ring \( \mathbb{Z}_{42} \).
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Context:
Determine the minimum number of factors that must be removed from the product of all integers from 1 to 99, denoted as \( 99! = 1 \cdot 2 \cdot \ldots \cdot 99 \), so that the product of the remaining factors ends in the digit 2.
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Context:
How many elements of the set of all $2 \times 2$ matrices with entries in the field with 7 elements are similar to the matrix $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$?
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Context:
The teacher secretly thought of a three-digit number $S$. Students $A$, $B$, $C$, and $D$ tried to guess the number by suggesting $541$, $837$, $291$, and $846$ respectively. The teacher informed them that each of them got exactly one digit correct, and it was in the correct position. What is the number $S$?
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Context:
A rhombus \(ABCD\) and a triangle \(ABC\), containing its longer diagonal, are inscribed with circles. Find the ratio of the radii of these circles, if the acute angle of the rhombus is \(\alpha\).
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Context:
The function $f(x)$ satisfies
\[f(x) - 3 f\left(\frac{1}{x}\right) = 3^x\]
for all $x \neq 0$. Find $f(2)$.
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Context:
In the expansion of $( \sqrt {x}+ \frac {1}{x})^{n}$, the sum of coefficients of all quadratic terms is $64$. What is the constant term in the expansion?
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Context:
Let $N = \prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.
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Context:
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?
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Context:
Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.
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Context:
On the hypotenuse \( AB \) of a right triangle \( ABC \), square \( ABDE \) is constructed externally with \( AC=2 \) and \( BC=5 \). In what ratio does the angle bisector of angle \( C \) divide side \( DE \)?
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Context:
Determine the minimum number of isoclinic rotations required to transform an arbitrary 2-plane in \( \mathbb{R}^4 \), passing through the origin, into any of the Euclidean orthonormal 2-planes in a standard Cartesian 4D coordinate system.
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Context:
Let $y=f(x)$ be a function defined on $\mathbb{R}$ with a period of $1$. If $g(x) = f(x) - 2x$ has a range of $[-2, 6]$ on the interval $[2, 3]$, find the range of $g(x)$ on $[-2017, 2017]$.
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Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{de} f$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
设 C 是单位圆周，z_1,z_2,\cdots,z_{240} 是 C 上的复数（可以相同），满足：(1) 对 C 上任意长为 \pi 的开弧 \Gamma，至多有 200 个 j 使得 z_j\in\Gamma；(2) 对 C 上任意长为 \df{\pi}{3} 的开弧 \gamma，至多有 120 个 j 使得 z_j\in\gamma.求 |z_1+z_2+\cdots+z_{240}| 的最大值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
There are 2015 coins on the table. Two players play the following game: they take turns; on their turn, the first player can take any odd number of coins from 1 to 99, while the second player can take any even number of coins from 2 to 100. The player who cannot make a move loses. How many coins should player number 1 take on their first move to guarantee a win with an unchanged strategy?
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Context:
A particle starts from rest at $t=0$, $x=0$, and comes to rest again at $x=1$, $t=1$. Let the instantaneous acceleration be $a(t)$. Determine the largest constant $c$ such that $\sup_{t \in [0,1]} |a(t)| \geq c$ is guaranteed.
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Context:
Let $T$ be a tree with exactly two vertices of degree $7$ and exactly $20$ vertices of degree $100$. What is the minimum possible number of vertices in a tree $T$ that satisfies those restrictions?
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Context:
Given the quadratic function f(x) = x^2 - ax + 1, find the necessary and sufficient condition on 'a' for the inequality f(x) < 0 to have a solution with respect to x. Express your answer as an inequality involving 'a'.
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Context:
A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus $3$ stops behind. After how many minutes will Zia board the bus?
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Context:
the average of 6 integers is 63 , and none of these integers is greater than 100 . if the average of 3 of the integers is 65 , what is the least possible value of one of the other two integers ?
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Context:
Let $g_n:[0,1] \to \mathbb{R}$ be defined by \( g_n(x) = \frac{nx}{1+n^2 x^q} \) for \( q > 0 \). Determine the values of \( q \) for which the sequence \( g_n \) converges uniformly to its point-wise limit.
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Context:
Find a function \( g: \mathbb{N} \to \mathbb{P}(\mathbb{N}) \) such that the set \( A = \{ n \in \mathbb{N} \mid n \notin g(n) \} \) is finite.
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Context:
Let $f(x)=\mathrm e^{-x}\sin(x)g(x)$, where $g:\mathbb{R^+}\to (-M, M)$ for some positive real $M$. Determine the number of solutions to the equation $|f(n_1)|+|f(n_2)|+\cdots+|f(n_k)|=1$, where all $n_i$ are natural numbers.
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Context:
Consider i.i.d. random variables $X_{1}, X_{2}, \ldots$, such that $X_{i}$ has a uniform distribution over the interval $(0, 1)$. Find the limit:
\[\lim_{n \to \infty} P\left(\sum_{i=1}^{n} X_i \leq 2 \sum_{i=1}^n{X_{i}^2}\right).\]
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Context:
Calculate the line integral \( \iint_{S_1\cup S_2} \text{curl}F\cdot dS \) for the surfaces \( S_1=\{(x,y,z):x^2+y^2=1, 0\le z \le 1\} \) and \( S_2=\{(x,y,z): x^2+y^2 +(z-1)^2 =1, z\ge 1\} \), where \( F(x,y,z)=(zx+z^2y+x, z^2yx+y, z^4x^2) \). Apply Stokes' theorem to evaluate the integral.
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Context:
Evaluate the definite integral: $$\int_{0}^{\infty}\frac{4x\ln (x)}{x^4+4x^2+1}dx$$
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Context:
Evaluate the limit \( \lim_{x\to 0^+}\sum_{n=2020}^{\infty}\frac{(-1)^n}{\log^x(n)} \).
Please reason step by step, and put your final answer within \boxed{}.