Context:
Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. Using the scoring information, calculate Lily's total score. Express your answer as a single integer.
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Context:
A positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $n = a^b$. How many positive integer divisors of $729000000$ are base-able?
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Context:
Write digits in place of the letters in the equation

\[ \overline{abc}^2 = \overline{adeff} \]

to make it correct. (The overline indicates that the letters represent the digits of a number in the decimal system. Identical letters represent identical digits, and identical digits are labeled with the same letters.)
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Context:
There is a question involving a polynomial added to $2a^{2}-3a+5$. Li Ming mistakenly calculated it as a subtraction and obtained $a^{2}-5a+7$. What should be the correct result?
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Context:
Mr. and Mrs. Smith have three children. They own a family van with a driver's seat, a passenger seat in the front, and two seats in the back. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible if one of the children insists on sitting in the front passenger seat?
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Context:
Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\).
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Context:
Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself?
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Context:
In a class, 20% of the students were absent for an exam. 30% failed by 20 marks and 10% just passed. The remaining students scored an average of 65 marks. The average score of the class is 36. What was the cut off for passing the exam?
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Context:
Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq\{1,2,3, \ldots, 8\}$. (b) $|A \cap B|=|B \cap C|=|C \cap A|=2$. (c) $|A|=|B|=|C|=4$. Here, $|S|$ denotes the number of elements in the set $S$.
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Context:
The numbers $1,2 \cdots 11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
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Context:
The coefficient of $x^2$ in the expansion of $\left(ax-\frac{1}{\sqrt{x}}\right)^8$ is $70$. Find the value of $a$.
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Context:
We measured a distance at $t=26^{\circ} \mathrm{C}$ temperature using a copper rod that is exactly $a=3 \mathrm{~m}$ long at $0^{\circ} \mathrm{C}$. In this way, we found $n=7856.728 \mathrm{~m}$. What is the actual distance if the linear expansion coefficient of copper is $\lambda=0.000017$.
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Context:
If $f(x)=\log_{3a}[(a^{2}-3a)x]$ is a decreasing function on $(-\infty,0)$, then the range of the real number $a$ is ______.
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Context:
Let  $ k$  be a positive integer. Suppose that the integers  $ 1,2,3,\dots,3k \plus{} 1$  are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by  $ 3$  ? Your answer should be in closed form, but may include factorials.
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Context:
Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n?
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Context:
Find the smallest positive integer \( n \) such that in any two-coloring of the complete graph \( K_{n} \), there exist 3 monochromatic triangles that are pairwise edge-disjoint.
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Context:
From 2014 labeled numbers, 100 numbers are to be sampled. If systematic sampling is used, then the sampling interval should be ______.
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Context:
7. Given that the volume of a cylinder is $V$ (a constant), then the minimum value of the total surface area of the cylinder is

保留源文本的换行和格式，直接输出翻译结果。

Note: The last sentence is a note in Chinese, which is not part of the translation. The translation itself is provided above.
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Context:
What is the derivative of $ \log{\left(1 - \frac{r}{r_{s}} \right)} $ with respect to $ \log{(r)} $, where $r_{s}$ is a constant?
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Context:
Find the side of an equilateral triangle that is the orthogonal projection of a triangle with sides $\sqrt{6}, 3,$ and $\sqrt{14}$ onto a certain plane.
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Context:
Let  $n\geq 2$  be a given integer. Initially, we write  $n$  sets on the blackboard and do a sequence of moves as follows: choose two sets  $A$  and  $B$  on the blackboard such that none of them is a subset of the other, and replace  $A$  and  $B$  by  $A\cap B$  and  $A\cup B$ . This is called a  $\textit{move}$ .
Find the maximum number of moves in a sequence for all possible initial sets.
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Context:
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$  that satisfy $$|S| +\ min(S) \cdot \max (S) = 0.$$
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Context:
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
$ax+by=1\,$
$x^2+y^2=50\,$
 has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there?
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Context:
已知抛物线$\Gamma:y^{2}=2px(p>0)$在第一象限与单位圆交于点$A,\Gamma$在$A$处的切线为$l$，过原点的$\odot M$与$l$切于点$A$。当$p$变化时，求$\odot M$的半径$R$的最小值。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given that an object is displaced by the concurrent forces $\mathbf{F}_1 = (\log_2, \log_2)$ and $\mathbf{F}_2 = (\log_5, \log_2)$ to a displacement $\mathbf{S} = (2\log_5,1)$, calculate the work $\mathbf{W}$ done by the concurrent forces on the object. Express your answer as a single numerical value.
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Context:
Consider $f$ is differentiable, what is the value of $$\lim_{n\to \infty} \frac{1}{n^2} \sum_{k=1}^n \frac{f(a+\frac{k}{n^2}) -f(a)}{\frac{k}{n^2}}$$ ?
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Context:
Find the number of even digits in the product of the two 10-digit numbers \(2222222222 \times 9999999999\).
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Context:
Rosa received a perfume bottle in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After two weeks of using the perfume, $0.45$ liters remained in the bottle. What fraction represents the volume of perfume that Rosa has already used?
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Context:
A hundred silver coins are laid down in a line. A wizard can convert a silver coin into a golden one in $3$ seconds. Each adjacent golden coin reduces this time by $1$ second. What is the minimal time required for the wizard to convert all coins to gold?
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Context:
A dishonest dealer professes to sell goods at the cost price but uses different weights for different items. For some items, he uses a weight of 900 grams per kg, while for others, he uses a weight of 850 grams per kg. Calculate his overall gain percentage, considering the total weight and price of the items.
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Context:
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
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Context:
已知无穷数列$\left\{ {{a}_{n}} \right\}$，${{a1}}=1$，${{a}_{2}}=2$，对任意$n\in {{N}^{*}}$，有${{an+2}}={{a}_{n}}$，数列$\left\{ {{bn}} \right\}$满足${{b}_{n+1}}-{{bn}}={{a}_{n}}\left( n\in {{N*}} \right)$，若数列$\left\{ \frac{{{b2n}}}{{{a}_{n}}} \right\}$中的任意一项都在该数列中重复出现无数次，则满足要求的${{b1}}$的值为__．
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Context:
What is the value of the limit $\lim_{n\to\infty} \sqrt[n]{a_n}$ for the sequence $a_n=\sum_{k=0}^{[n/2]} C_{n-k}^k$, where $[n/2]$ denotes the largest integer less than or equal to $n/2$, and $C_{n-k}^k$ represents the binomial coefficient $\frac{(n-k)!}{k!(n-2k)!}$? Note that $a_{2m+1}-a_{2m}=a_{2m-1}$.
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Context:
Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is in the form \(\frac{k-m\sqrt{3}}{n}\), please provide the value of k + m + n.
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Context:
Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called <i>n-ray</i> partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$-ray partitional but not $60$-ray partitional?
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Context:
If  $x=\sqrt2+\sqrt3+\sqrt6$  is a root of  $x^4+ax^3+bx^2+cx+d=0$  where  $a,b,c,d$  are integers, what is the value of  $|a+b+c+d|$ ?
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Context:
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.
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Context:
For a positive integer  $n$ , let  $S_n$  be the total sum of the intervals of  $x$  such that  $\sin 4n x\geq \sin x$  in  $0\leq x\leq \frac{\pi}{2}.$ Find  $\lim_{n\to\infty} S_n.$ 60 point
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Context:
Let \( f(n) = \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \). Calculate \( \sum_{n=2}^{\infty} f(n) \).
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Context:
Calculate the improper integral: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$
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Context:
In 2012, a telecommunications company launched a series of mobile phone numbers. The first seven digits of the card number are fixed, and the last four digits range from "0000" to "9999", totaling 10,000 numbers. The company stipulates that if the last four digits of the card number are arranged in ascending order from smallest to largest, it is called a "Auspicious Dragon Card" and enjoys certain preferential policies. The number of "Auspicious Dragon Cards" in this series is ____.
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Context:
How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit?
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Context:
有 20 张卡片上分别写有数字1,2,…,20, 将它们放入一个盒子内.有4个人从中不放回地各抽取一张卡片,抽到两个较小数字的两人在同一组,抽到两个较大数字的两人在同一组.现其中有两人抽到5、14,则此两人在同一组的概率等于___。原始答案为最简分数形式\frac{m}{n}，请给出m + n的值。
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Context:
Compute the number of ordered pairs \((m,n)\) of positive integers that satisfy the equation \(\text{lcm}(m,n)+\gcd(m,n)=m+n+30\).
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Context:
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen. On one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\) if the radius of the circle is 10 and \(\sin \alpha = \frac{\sqrt{24}}{5}\).
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Context:
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\sin B\sin C}{\sin A}=\frac{3\sqrt{7}}{2}$, $b=4a$, and $a+c=5$, find the area of $\triangle ABC$.
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Context:
Given $b$ and $r \in \{1,2,3,4\}$, the probability that the line $y=x+b$ and the circle $x^{2}+y^{2}=r$ have common points is $\_\_\_\_\_\_$.
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Context:
A city's education committee plans to conduct a survey in seven schools named A, B, C, D, E, F, G. The survey must be completed within a week, from Monday to Friday, and at least one school must be visited each day. School A and B are scheduled for Monday and Tuesday respectively, while School C and D must be scheduled on the same day. School E cannot be scheduled on Friday. What is the total number of different arrangements? (Answer with a numerical value)
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Context:
For how many primes \( p \), there exist unique integers \( r \) and \( s \) such that for every integer \( x \), the congruence \( x^{3} - x + 2 \equiv (x - r)^{2} (x - s) \pmod{p} \) holds? Find the number of such primes.
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Context:
Find the smallest natural number, which divides  $2^{n}+15$  for some natural number  $n$  and can be expressed in the form  $3x^2-4xy+3y^2$  for some integers  $x$  and  $y$ .
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Context:
A $3 \times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.
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Context:
Tim drops a ball off the roof of a 96-foot tall building. The ball has an initial bounce efficiency of 50%, but with each bounce, its efficiency decreases by 5%. Additionally, air resistance causes the ball to lose 2% of its height on each bounce. How high will it bounce on the fifth bounce?
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Context:
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB}) = 11^{\circ}$, and $m(\widehat{EBC}) = 71^{\circ}$. If the answer is $x^{\circ}$, find the value of $x$ for $m(\widehat{DCE})$.
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Context:
Points A and B move in a clockwise direction on the sides of a square with side length 100m. Point A starts from corner A of the square and moves at a speed of 5m/s, and Point B starts from the midpoint of side CD and moves at a speed of 10m/s. After how many seconds will they first be on the same side?
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Context:
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 10$ and $\angle O_{1}PO_{2} = 90^{\circ}$, then $AP = \sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c + d$.
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Context:
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
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Context:
What is the necessary and sufficient condition for the function \( f(x)=\frac{\sqrt{a^{2}}-x^{2}}{|x+a|-a} \) to be an odd function?
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Context:
The monotonic increasing interval of the function $f(x) = -(x-5)|x|$ is __________.
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Context:
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with two foci $F_{1}$ and $F_{2}$. Points $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$. If the area of $\triangle PF_{2}Q$ is $S≥\frac{1}{8}|PQ|^{2}$, then the range of the eccentricity of $C$ is ______.
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Context:
On the side \( PQ \) of triangle \( PQR \), point \( N \) is taken, and on the side \( PR \), point \( L \) is taken, such that \( NQ = LR \). The point \( A \), where segments \( QL \) and \( NR \) intersect, divides the segment \( QL \) in the ratio \( m:n \) from point \( Q \). Find the ratio \( PN : PR \).
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Context:
Let \( n \geq 4 \) be an integer. \( A = \{a_1, a_2, \cdots, a_n\} \) is a set of integers such that \( 2023 \in A \) and the absolute difference between any two elements in \( A \) is a prime number. Determine the number of such sets \( A \).
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Context:
How many roots does the equation

$$
\overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0
$$

have, where \( f(x) = |x| - 1 \)?
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Context:
Problem 2. Given 10 equal segments in a plane. All intersection points are colored. Each colored point divides each of the given segments it lies on in the ratio $3: 4$. What is the maximum number of colored points?
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Context:
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$.
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Context:
A triangle with angles \( A, B, C \) satisfies the following conditions:
\[
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \frac{12}{7},
\]
and
\[
\sin A \sin B \sin C = \frac{12}{25}.
\]

Given that \( \sin C \) takes on three possible values \( s_1, s_2 \), and \( s_3 \), find the value of \( 100 s_1 s_2 s_3 \).
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Context:
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, \frac{a_{n+1}-a_{n}}{a_{n}}=\frac{a_{n+2}-a_{n+1}}{a_{n+2}}\left(n \in N^{*}\right)$, 若 $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{6} a_{7}=3$, 则 $a_{2024}=$. 请给出原始答案为frac{m}{n}格式时m + n的值。
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Context:
At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other participant exactly once. A match winner got one point, the loser got none, and in case of a draw, both participants received half a point each. By the end of the tournament, it turned out that in matches against economists, each participant gained half of all their points. How many lawyers participated in the tournament? Provide the smallest possible number as the answer.
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Context:
Identify the algebra $H\otimes_{\mathbb{R}} \mathbb{C}$ in terms of a familiar algebraic structure, where $H$ denotes the Hamiltonian quaternions, $\mathbb{C}$ represents the complex numbers, and $\mathbb{R}$ stands for the real numbers.
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Context:
Find the surface area of an ideal tetrahedron in hyperbolic space $\mathbb{H}^3$ as a function of its vertices, which are parametrized by four complex numbers $z_1, z_2, z_3, z_4$ on the boundary $\mathbb{CP}^1$. Express the surface area in terms of $z_1, z_2, z_3, z_4$.
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Context:
Determine the ratio of the circumference to the diameter of a circle on a sphere, given that the circle is defined by an angle \( \theta \) from the pole. The radius of the sphere is \( r \). Express the ratio \( R(\theta) \) in terms of \( \theta \).
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Context:
A colony of bacteria grows exponentially, with an initial population of 10 bacteria. The time it takes for the population to double is an exponential random variable with parameter $\lambda = 2$. What is the expected time it takes for the population to reach 1000 bacteria?
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Context:
Determine the number of non-equivalent colorings of the vertices of a regular hexagon using the colors red, white, and blue.
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Context:
In a right triangle \( \triangle ABC \), the lengths of the two legs are 5 and 12. Two arcs are drawn with centers at points \( A \) and \( B \) and radii 12 and 5, respectively. These arcs intersect the hypotenuse at points \( M \) and \( N \). Calculate the length of the segment \( MN \). Express your answer as a single number.
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Context:
The area bounded by the curve \( y = e^{x^2} \), the x-axis, and the lines \( x = 1 \) and \( x = 2 \) is given to be \( a \) square units. Find the area bounded by the curve \( y = \sqrt{\ln x} \), the y-axis, and the lines \( y = e \) and \( y = e^4 \) in terms of \( a \).
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Context:
Dima and Serezha decided to hold a race on a circular lake. They start simultaneously from the same point. Serezha rides a motorboat at a constant speed of 20 km/h and somehow crosses the lake (not necessarily via the diameter) in 30 minutes. Meanwhile, Dima runs along the shore of the lake for the first 15 minutes at a speed of 6 km/h, and then for the remaining time, he moves at a constant speed by boat to the other shore, parallel to Serezha's trajectory. After simultaneously arriving at the opposite shore, the boys run towards each other. During the entire race, the boys run at constant and equal speeds. How much time after they part will they meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$\int_0^\infty \frac{\sin^4(x)}{x^2} \, dx$$
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Context:
Calculate (1.0025)^10 accurate to five decimal places. Express your answer as a decimal with five places.
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Context:
Given the set $A={}$ of triangles with the longest side length of $7$ and all three side lengths being positive integers, determine the number of true subsets of set $A$.
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Context:
Evaluate the following limit:
$$\lim_{r\rightarrow \infty}r\cdot \bigg(\frac{2}{\pi}\bigg)^{r+1}\cdot \int^{\frac{\pi}{2}}_{0}e^{r\log(x)}\sin(x)\,dx$$
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Context:
What is the sum of the real roots of the equation $(x\sqrt{x})^x = x^{x\sqrt{x}}$? Provide your answer in the form of a simplified fraction \frac{k}{m}, and then give the value of k + m.
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Context:
In a $46 \times 46$ grid, a man can move only straight up or to the right with the following constraints: when moving right, he must move at least 4 consecutive coordinates; when moving up, he must move at least 12 consecutive coordinates. Determine the number of coordinates that are unreachable by the man under these movement rules.
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Context:
Determine the supremum of the set of values $A$ such that \( I_n = \int_{0}^{1}\cos\left(\frac{1}{x}+nx\right)\,dx = O(n^{-A}) \) as \( n \to +\infty \).
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Context:
Given two circles $(x^{2}+y^{2}=4)$ and $(x^{2}+y^{2}-2mx+m^{2}-1=0)$, find the real number $m$ such that they are externally tangent.
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Context:
Determine the number of terms required in the series representation of $e = \sum^∞_{k=0}{1\over k!}$ to ensure that the error is at most $0.06$ units in the $20$th decimal place.
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Context:
Peri travels 1 m each day at a constant rate and without pause except for every tenth day when Peri takes a 24-hour rest. Starting on Monday, Peri has to travel 90 m to visit Granny. On which day of the week does Peri arrive at Granny's? Express your answer as the day of the week (e.g. Monday, Tuesday, etc.).
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Context:
Aida made three cubes with positive integer side lengths $a,b,c$. They were too small for her, so she divided them into unit cubes and attempted to construct a cube of side $a+b+c$. Unfortunately, she was $648$ blocks off. How many possibilities of the ordered triple $\\left(a,b,c\\right)$ are there?2017 CCA Math Bonanza Team Round
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Context:
Find the distance from the point of tangency of two circles with radii \( R \) and \( r \) to their common external tangent.
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Context:
Determine the number of distinct Jordan normal forms (up to the order of the Jordan blocks on the diagonals) for matrices $B \in M_5 (\mathbb{C})$ with the characteristic polynomial $x_B(\lambda)=-(\lambda-2)^2(\lambda-3)^3$. Consider the algebraic and geometric multiplicities of the eigenvalues.
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Context:
There are two-digit numbers written on a board. Each number is composite, but any two numbers are relatively prime. What is the maximum number of such numbers that can be written?
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Context:
A cylinder has a height of 6 and a radius of 3, with the centers of its bases at (0,0,0) and (0,0,6). Determine the volume of the smallest convex polyhedron that encloses all integer lattice points inside the cylinder.
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Context:
In triangle $\triangle ABC$, $AB=3$, $\sin B=m\cdot \sin A$ $(m\geqslant 2)$. The maximum value of the area of $\triangle ABC$ is ______.
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Context:
Solve the inequality f(x+1) > f(2x), where f(x) = ln(sqrt(x^2+1)) + x^2 + 2. Express the solution set as an interval using standard interval notation.
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Context:
Positive real numbers \( a, b, c \) are in a geometric progression \((q \neq 1)\), and \( \log _{a} b, \log _{b} c, \log _{c} a \) are in an arithmetic progression. Find the common difference \( d \).
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Context:
The speed of a boat in still water is 18 km/hr and the rate of current is 4 km/hr. The boat travelled downstream for some time and covered a distance of 5.133333333333334 km. How long was the boat travelling downstream?
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Context:
In △AOB, there are 4 points on edge OA and 5 points on edge OB, including point O, making a total of 10 points. The number of triangles formed by taking these 10 points as vertices is $\_\_\_\_\_\_$.
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Context:
Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. The area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
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Context:
Anna and Zsófi take turns rolling a die, and they always add the rolled number to the sum of the numbers rolled so far. The one who, after their roll, makes the sum first divisible by 4 wins. If Anna starts, what is the probability that she wins?
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Context:
Let  $p=2^{16}+1$  be a prime. A sequence of  $2^{16}$  positive integers  $\{a_n\}$  is *monotonically bounded* if  $1\leq a_i\leq i$  for all  $1\leq i\leq 2^{16}$ . We say that a term  $a_k$  in the sequence with  $2\leq k\leq 2^{16}-1$  is a *mountain* if  $a_k$  is greater than both  $a_{k-1}$  and  $a_{k+1}$ . Evan writes out all possible monotonically bounded sequences. Let  $N$  be the total number of mountain terms over all such sequences he writes. Find the remainder when  $N$  is divided by  $p$ .

*Proposed by Michael Ren*
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Context:
Given that point $A(-2,3)$ lies on the axis of parabola $C$: $y^{2}=2px$, and the line passing through point $A$ is tangent to $C$ at point $B$ in the first quadrant. Let $F$ be the focus of $C$. Then, $|BF|=$ _____ .
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Context:
In $\triangle ABC$, $AB=10$, $AC=8$, and $BC=6$. Circle $P$ passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection of circle $P$ with sides $AC$ and $BC$ (excluding $C$). The length of segment $QR$ is
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Context:
Primes like $2, 3, 5, 7$ are natural numbers greater than 1 that can only be divided by 1 and themselves. We split 2015 into the sum of 100 prime numbers, requiring that the largest of these prime numbers be as small as possible. What is this largest prime number?
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Context:
Usain is walking for exercise by zigzagging across a \(100\)-meter by \(30\)-meter rectangular field, beginning at point \(A\) and ending on the segment \(\overline{BC}\). He wants to increase the distance walked by zigzagging as shown in the figure below \((APQRS)\). What angle \(\theta = \angle PAB=\angle QPC=\angle RQB=\cdots\) will produce a path length that is \(120\) meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.) The original answer is in the form \(\arccos\frac{k}{m}\), please give the value of \(k+m\).
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Context:
A scientific expedition team went upstream to investigate an ecological zone along a river. They set off at a speed of 17km per day, traveled upstream for several days to reach their destination, then spent several days investigating the ecological zone. After completing their task, they returned at a speed of 25km per day. On the 60th day after departure, the expedition team traveled 24km and then returned to the starting point. The question is: How many days did the scientific expedition team spend investigating the ecological zone?
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Context:
Suppose \(\triangle A B C\) has lengths \(A B=5, B C=8\), and \(C A=7\), and let \(\omega\) be the circumcircle of \(\triangle A B C\). Let \(X\) be the second intersection of the external angle bisector of \(\angle B\) with \(\omega\), and let \(Y\) be the foot of the perpendicular from \(X\) to \(B C\). Find the length of \(Y C\).
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Context:
Two circumferences of radius  $1$  that do not intersect,  $c_1$  and  $c_2$ , are placed inside an angle whose vertex is  $O$ .  $c_1$  is tangent to one of the rays of the angle, while  $c_2$  is tangent to the other ray. One of the common internal tangents of  $c_1$  and  $c_2$  passes through  $O$ , and the other one intersects the rays of the angle at points  $A$  and  $B$ , with  $AO=BO$ . Find the distance of point  $A$  to the line  $OB$ .
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Context:
9.5 What is the largest number of non-overlapping groups into which all integers from 1 to 20 can be divided so that the sum of the numbers in each group is a perfect square?
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Context:
Evaluate the limit: \[ \lim_{x\to 1} \frac{\sin(x^2-1)}{x-1} \] using the known limit \( \alpha = \lim_{x\to 0} \frac{\sin(x)}{x} \). Do not use L'Hôpital's rule.
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Context:
Identify an exceptional prime below 100 that does not satisfy the three conditions outlined by Kummer for the impossibility of solving Fermat's equation.
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Context:
Calculate
$$
\lim_{n\rightarrow\infty}n\int_0^{\infty} e^{-x}\sqrt[n]{e^x - 1 -\frac{x}{1!} - \frac{x^2}{2!} - \dots -\frac{x^n}{n!}}\,dx.
$$
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Context:
Evaluate the triple integral \(\iiint (x+y+z) \, dx\,dy\,dz\) over the region \(D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 9 \wedge z > y\right \}\).
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Context:
Evaluate the integral: $$\int_{0}^{\infty}\text{sech}^2(x+\tan(x))dx$$
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Context:
In triangle \(\triangle ABC\), \(AB = 16\), \(BC = 30\), and \(\angle ABC = 90^\circ\). On the circumcircle of \(\triangle ABC\), let \(P\) be the midpoint of arc \(AC\) that does not contain \(B\). Points \(X\) and \(Y\) lie on lines \(AB\) and \(BC\), respectively, such that \(PX \perp AB\) and \(PY \perp BC\). Calculate \(XY^2\).
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Context:
Find an upper bound for the recurrence relation \( T(x) \leq 4 T(\left \lfloor{\frac{x}{2}} \right \rfloor) + x \) using the master theorem, where \( x \in \mathbb{N} \).
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Context:
Evaluate the integral: \[ \int_{0}^{\frac{\pi}{4}} (\tan^{9}x + 5\tan^{7}x + 5\tan^{5}x + \tan^{3}x) \, dx \]
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Context:
Find the smallest positive integer $n$ such that the error of the Taylor polynomial of degree $n$ for the sine function, centered at $x=0$ and evaluated at $x=0.1$, is less than $10^{-10}$. What is this integer $n$?
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Context:
Arrange the numbers 21, 31, 41, 51, 61, 71, and 81 such that the sum of every four consecutive numbers is divisible by 3. How many such arrangements are possible?
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Context:
Po picks 100 points \( {P}_{1},{P}_{2},\cdots ,{P}_{100} \) on a circle independently and uniformly at random. He then draws the line segment
 connecting \( {P}_{1}{P}_{2},{P}_{2}{P}_{3},\ldots ,{P}_{100}{P}_{1} \) . Find the expected number of regions that have all sides bounded by straight lines.
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Context:
7. Arrange the ten digits from 0 to 9 into a ten-digit number without repetition and with the first digit not being zero. The number of such ten-digit numbers that can be divided by 99 is $\qquad$.
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Context:
Let \( E \) be the elliptic curve defined by \( y^2 = x^3 + 1 \) over the finite field \( F_{599} \). Using Hasse's theorem, determine the upper bound for the order of \( E(F_{599}) \).
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Context:
Determine whether the problem of scheduling classes to minimize penalty and soft constraint violations is NP-hard or NP-complete.
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Context:
Evaluate the limit: $$\lim_{x \to \infty} 2\sum _{n=1}^{\infty}\left(\frac{\sqrt{x^2+(2n-1)^2}}{x^2}\right).$$
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Context:
Calculate the mass of the region \( R \) enclosed by the hyperbolas \( x^2 - y^2 = 1 \), \( x^2 - y^2 = 6 \), \( xy = 3 \), and \( xy = 8 \) in the first quadrant. The density \( \delta(x,y) \) is proportional to the square of the distance from the origin, with a proportionality constant of 2. Use the change of variables \( u = x^2 - y^2 \) and \( v = xy \) to evaluate the integral \( m = \int \int_R \delta(x,y) \, dA \).
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Context:
Consider performing a Lagrange interpolation for the points $x_0$, $x_0 + \varepsilon$, and $x_1$. As $\varepsilon \to 0$, what happens to the interpolation polynomial $P(x) = y_0L_0(x) + y_1L_1(x) + y_2L_2(x)$? Determine the degree of the resulting polynomial.
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Context:
Determine the cardinality of the ring \( R = \frac{\mathbb{Z}[x]}{((x^{2}+x+1)(x^{3}+x+1))} \), where \( I \) is the ideal generated by 2 in \( R \).
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Context:
Given the sets \( A=\{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B=\{(x, y) \mid |xy| + 1 = |x| + |y|\} \), if \( A \cap B \) forms the vertices of a regular octagon in the plane, find the value of \( a \).
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Context:
A point $P$ is picked uniformly over a disc with radius $1$ and center at the origin. Subsequently, a new point $Q$ is picked uniformly over a disc with radius $|P|$ and center at the origin. Compute the expected value of $|Q|$. Note: A random point $(X,Y)$ uniformly distributed over a disc with radius $r > 0$ has density $f_{X,Y}(x,y) = \frac{1}{\pi r^{2}}$ for $x^{2} + y^{2} \leq r^{2}$.
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Context:
Determine the value of $p$ for which the inequality \( \lVert f\rVert_{L^\frac{np}{n-p}(\mathbb{R}^n)}\le c \lVert\nabla f\rVert_{L^p(\mathbb{R}^n)} \) holds for all functions \( f \in C_0^\infty(\mathbb{R}^n) \), where \( n \in \mathbb{N} \), \( c > 0 \), and \( C_0^\infty(\mathbb{R}^n) \) denotes the space of infinitely differentiable functions with compact support in \( \mathbb{R}^n \).
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Context:
Identify the Lie group $G$ corresponding to the Lie algebra:

$$\text{Lie}(G)=\left\{\begin{bmatrix}0 & a & u\\ -a & 0 & v\\0&0&0\end{bmatrix}; a,u,v\in\mathbb{R}\right\}$$
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Context:
Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty}\limits li nS_nS_{n+1}.$
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Context:
Let \( D \) be a point on side \( \left\lbrack  {AB}\right\rbrack \) of triangle \( {ABC} \) with \( \left| {AB}\right|  = \left| {AC}\right| \) such that \( \left\lbrack  {CD}\right\rbrack \) is
 an angle bisector and \( m\left( \overset{⏜}{ABC}\right)  = {40}^{ \circ  } \) . Let \( F \) be a point on the extension of \( \left\lbrack  {AB}\right\rbrack \) afte
 \( B \) such that \( \left| {BC}\right|  = \left| {AF}\right| \) . Let \( E \) be the midpoint of \( \left\lbrack  {CF}\right\rbrack \) . If \( G \) is the intersection of
 lines \( {ED} \) and \( {AC} \) ,what is \( m\left( \widehat{FBG}\right) \) ?

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Context:
Evaluate the integral \( \int_{-1}^1 \frac{e^x}{\sqrt{1-x^2}} \, dx \).
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Context:
Find the value of $w$ in Cartesian form if $z = e^w$ is a complex number such that $|z| = 1$ and $\arg(z) = (2k + 1)\pi$, where $k$ is an integer.
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Context:
Find the inverse of the polynomial \(x^2 + 1\) in the ring \(\frac{\mathbb{Z}_3[x]}{x^3 + 2x + 1}\).
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Context:
Jonathan finds out that his ideal match is Sara Lark, but to improve his odds of finding a girlfriend, he is willing to date any girl whose name is an anagram of "Sara Lark," and whose name consists of both a first and last name of at least one letter. How many such anagrams are there?
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Context:
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?
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Context:
Let $A$ and $B$ be square matrices of size $n$ with rank $n-1$. Determine the rank of the matrix $AB - BA$. Express your answer as a single integer.
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Context:
Evaluate the limit \( \lim_{t \to 0} \int_B \frac{l(t,x,y)}{t^2} \, dA \), where \( l(t,x,y) \) is the length of the arc of the circle with radius \( t \) and center \((x,y)\) that lies outside the ball \( B \) in \( \mathbb{R}^2 \).
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Context:
Determine all non-negative integers \( n \) for which there exists a non-periodic function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x+1) + f(x-1) = \sqrt{n} f(x) \) for all \( x \).
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Context:
Let $a,\ b$ be given positive constants.

Evaluate

\[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\]
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Context:
Determine the time complexity of the algorithm for the Traveling Salesman Problem (TSP) variant, given that the precomputation step to find the nearest reachable node on the spanning tree is optimized to have a complexity of \(O(\log(N))\).
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Context:
The 79 trainees of the Animath workshop each choose an activity for the free afternoon among 5 offered activities. It is known that:

- The swimming pool was at least as popular as soccer.
- The students went shopping in groups of 5.
- No more than 4 students played cards.
- At most one student stayed in their room.

Determine the number of different lists of student counts for each activity that could have been written.
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Context:
Calculate the value of \( \triangle^5[(1-x)(1-x^2)(1+x^2)] \).
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Context:
The circles \( S_1 \) and \( S_2 \) with centers \( O_1 \) and \( O_2 \) respectively touch each other externally; a line touches the circles at distinct points \( A \) and \( B \) respectively. It is known that the intersection point of the diagonals of quadrilateral \( O_1 A B O_2 \) lies on one of the circles. Find the ratio of the radii of the circles.
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Context:
Find the limit of the ratio of consecutive coefficients in the power series representation of a function that has a simple pole at a non-zero complex number.
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Context:
Determine the smallest possible degree of the polynomial $f(x)$ in the expression \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^3-x} =\frac{f(x)}{g(x)} \), where $p(x)$ is a polynomial of degree less than 100 and not divisible by $x^3-x$, and $f(x)$ and $g(x)$ are polynomials.
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Context:
Determine the values of \( n \) for which insertion sort, which runs in \( 8n^2 \) steps, is faster than merge sort, which runs in \( 64n\log_2n \) steps.
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Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{\int_{0}^1\left(x^2-x-2\right)^n dx}{\int_{0}^1\left(4x^2-2x-2\right)^n dx}$$
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Context:
Compute the Euler characteristic of the $n$-sphere $S^n$ for $n \ge 2$ using the standard triangulation of the $(n+1)$-simplex.
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Context:
Determine the dimension of the vector space $V$ defined by $$ V = \{P \in \mathbb{R}_n [X] : P (X^3)-( X^{n+1} + 1) P (X) =0\} $$ for $n \in \mathbb {N}$. \( \mathbb{R}_n [X] \) denotes the space of real polynomials of degree at most \( n \).
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Context:
A large university has a 13-day registration period for students. The number of elapsed days before a randomly selected student registers follows a continuous distribution with a density function that is symmetric about day 6.5 and proportional to \( \frac{1}{t+1} \) between days 0 and 6.5. A student registers at the 60th percentile of this distribution. Calculate the number of elapsed days in the registration period for this student.
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Context:
Sonya the frog chooses a point uniformly at random lying within the square \( \left\lbrack  {0,6}\right\rbrack   \times  \left\lbrack  {0,6}\right\rbrack \) in the coordinate plane and hop
 that point. She then randomly chooses a distance uniformly at random from \( \left\lbrack  {0,1}\right\rbrack \) and a direction uniformly at random fron
 \{north, south east, west\}. All he choices are independent. She now hops the distance in the chosen direction. What is the
 probability that she lands outside the square?
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Context:
Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?
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Context:
Calculate the contour integral \( \int_{2-i\infty}^{2+i\infty} \frac{1}{t} \, \mathrm{d}t \).
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Context:
The circle with radius $4$ passing through $A$ and $B$ is tangent to the circle with radius $3$ passing through $A$ and $C$, where $|AB|=2$. If the line $BC$ is tangent to the second circle, what is $|BC|$?
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Context:
Given a constant $C$ where $0 < C < \frac{\pi}{2}$, determine the values of $A_1, A_2, \ldots, A_n > 0$ that minimize the sum $\sum_{i=1}^n \tan(A_i)$ subject to the constraint $\sum_{i=1}^n A_i = C$. Provide the values of $A_1, A_2, \ldots, A_n$ that achieve this minimum.
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Context:
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
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Context:
Determine the values of \(\alpha\) for which the double integral \(\int_{0}^{1}\left(\int_{1}^{1+x} \frac{1}{(x-y)^{\alpha}} \ dy\right)dx\) converges.
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Context:
Evaluate the line integral \( \oint_C P \, dx + Q \, dy \) using Green's theorem, where \( P(x,y) = \frac{y}{x^2+y^2} \) and \( Q(x,y) = \frac{-x}{x^2+y^2} \), and \( C \) is a piecewise smooth Jordan curve contained in \( S = \{(x,y) | x^2 + y^2 > 0\} \) such that the origin \((0,0)\) lies interior to \( C \).
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Context:
In a family of 4, each member exercises at least one of the 7 deadly sins, with no sin being shared between two distinct members. Determine the number of ways to assign the sins to the family members such that no single member exercises both greed and gluttony.
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Context:
Calculate the arc length of the curve defined by the parametric equations $x(t) = \frac{\cos(3t)}{t^3}$ and $y(t) = \frac{\sin(3t)}{t^3}$ for $t \geq \frac{1}{\sqrt{3}}$. Use the arc length formula $l = \int_{t_1}^{t_2} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$ to find the total arc length. Determine the appropriate limits of integration.
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Context:
Let $a$ be a prime number, and let $b$ be a positive integer such that $a \equiv b \equiv 1 \pmod{4}$. Given the biconditionals:

$$(A \equiv 1 \pmod{4}) \iff (a \equiv b \pmod{24})$$

$$(A \equiv 3 \pmod{4}) \iff (a \equiv b + 4 \pmod{8})$$

If $a \equiv b \equiv 5 \pmod{12}$, determine the value of $A$ modulo 4.
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Context:
Given that there is at least one solution to the equation \(a^{2} + 2b^{2} = p^{11}q^{13}\), where \(a, b \in \mathbb{Z}\) and \(p, q\) are odd prime numbers, determine the total number of integer solutions.
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Context:
Consider a hat containing slips numbered from \(kr+1\) to \((k+1)r\), where \(k\) is a positive integer. What is the expected number of slips that need to be drawn from the hat until all numbers drawn are unique?
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Context:
Evaluate the integral \( \int_{-\infty}^\infty (f(x+1)-f(x)) \, dx \) given that \( \lim_{x \rightarrow \infty} f(x) = 2 \) and \( \lim_{x \rightarrow -\infty} f(x) = 1 \). Assume that the function \( f(x) \) is Riemann-integrable over every bounded interval.
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Context:
Determine the values of \( s \) for which the integral \( \int_{\mathbb{R}^n}\left(\frac{1}{1+|x|^2}\right)^{s}dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum expected number of bits required to encode sequences of length 200 from the typical set $S$, where $S$ consists of all sequences $x$ with at most three 1s. Assume $x \in \{0,1\}^{200}$ and $P(0) = 0.99$. Use $\log_{2}|S|$ to encode such sequences.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the standard notation for the space \(\mathcal{H}_2\), which contains stochastic processes that are adapted and have finite \(L^2\) norm?
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Context:
A 10-sided die with numbers 1 to 10 is rolled $n$ times. Determine the smallest value of $n$ such that the probability of rolling each of the numbers 1 to 5 at least once is 50%.
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Context:
Find all possible values of \(M\) such that the expression \(M^n + 12^{2n-1}\) is divisible by 133 for all positive integers \(n\).
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Context:
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distances from \( O \) to the vertices \( A \), \( B \), and \( C \) are 6, 4, and 8, respectively. Find the area of triangle \( ABC \).
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Context:
Determine the value of \( \lim_{N \to \infty} \frac{f(N)}{N} \), where \( f(N) \) is the number of integers \( 1 \leq n \leq N \) such that the number of divisors of \( n \), denoted \( \tau(n) \), is greater than \( \ln{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the result of the Fourier Transform of the conjugate of a function in the Fourier plane, i.e., FFT($T^*(fx,fy)$)?
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Context:
Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the kernel of the ring homomorphism \( \varphi : R[u,v] \to R[x,1/x] \) defined by \( \varphi(p(u,v)) = p(x,1/x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the function \( v(x,t) \) as \( t \to \infty \), where \( v(x,t) \) is a bounded solution of the heat equation \( v_t = v_{xx} \) with the initial condition \( v(x,0) = \frac{e^{-x^2}-1}{e^{-x^2}+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal area of a rhombus circumscribed around the region bounded by the curves $y = \cos(x)$ and $y = -\cos(x)$ for $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. The rhombus is formed by drawing a tangent line to the upper curve at a point $A = (x, \cos x)$, where $x \in \left(0, \frac{\pi}{2}\right)$, and reflecting the intersection points of this tangent line with the axes to form the vertices of the rhombus. What is the minimal area of such a rhombus?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the graph diameter of the 1-skeleton of the omnitruncated $E_8$ polytope?
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Context:
Consider a block matrix \( C \) defined as follows, where \( m, n \in \mathbb{N} \), \( J \) is an all-ones matrix, and \( A \) is an \( n \times n \) matrix with rank \( n \), such that \( A \neq J, I, J-I \):

\[ C = \begin{pmatrix} (J-I)_{m\times m} & J_{m\times n}\\ J^T_{n\times m} & A_{n\times n}\end{pmatrix} \]

Determine the rank of the matrix \( C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of group homomorphisms from \(\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z\) to \(S_3\).
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Context:
Solve Burgers' equation \(u_t + uu_x = 0\) with the initial condition \(u(x,-1) = x^2\). Find the value of \(u(1,2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the integral \( I_{n} = \int_{\mathbb{R}^{n}} \exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x \) in terms of \( I_1 \), where \( \left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots x_{n}^{2}\right)^{\frac{1}{2}} \) is the Euclidean norm.
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Context:
Evaluate the limit: $$\lim_{x\to0}\frac{\Phi^{-1}(1-x)}{\Phi^{-1}(1-x/n)}$$ where $\Phi^{-1}$ denotes the inverse cumulative distribution function (CDF) of the standard normal distribution, and $n>0$.
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Context:
Find a generator for the cyclic group of units in \( \mathbb{Z}/(\mathbb{2017Z}) \).
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Context:
Find the number of non-negative integer solutions to the equation \(a+b+c+d+e=79\) given the constraints \(a \geq 7\), \(b \leq 34\), and \(3 \leq c \leq 41\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{-1}^1 x \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{q\to 1^-}\frac{\pi K'} {K} \cdot\frac{K} {2\pi^2}\cdot(E-k'^2K)$$ where $K, K', E$ are elliptic integrals and $k, k'$ are moduli corresponding to the nome $q$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_R (x+y) \, dA \) where \( R \) is the trapezoidal region with vertices at \((0,0), (5,0), (5/2, 5/2), (5/2, -5/2)\). Use the transformation \( x = 2u + 3v \) and \( y = 2u - 3v \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $2n$ people, each having exactly $k$ friends, find the smallest positive integer $k$ such that it is always possible to divide the $2n$ people into two groups of $n$ people each, where every person in each group has at least one friend in their own group.
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Context:
In triangle $ABC, AB=\sqrt{30}, AC=\sqrt{6},$ and $BC=\sqrt{15}.$  There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ and $\angle ADB$ is a right angle.  The ratio \[ \frac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)} \] can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers.  Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the function \( f(x) \) such that the differential equation \( xy'' + 2y' - xy = f(x) \) corresponds to the solution \( xy = Ae^x + Be^{-x} + x^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the expression as \( n \) approaches infinity: $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least possible sum of the distances from a point E inside a rectangle ABCD to its vertices, given that these distances are all distinct integers.
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Context:
What is the minimum number of states required for a non-deterministic finite automaton (NFA) to recognize the complement of the set of binary strings of length $i$?
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Context:
Consider a population that, in each generation, has the following probabilities: it dies out with probability 0.2, stays constant with probability 0.5, and doubles with probability 0.3. What is the expected long-term behavior of this population?
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Context:
Determine the Lipschitz constant of the radial retraction function for a norm that does not originate from an inner product.
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Context:
Calculate the probability that when 36 people are randomly split into 12 groups of 3, and then split again into 12 groups of 3, at least one group will have the same members as in the first grouping.
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Context:
A triangle is composed of circular cells arranged in 5784 rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells, so that each cell belongs to exactly one pair. A pair of adjacent cells is called diagonal if the two cells in it aren't in the same row. What is the minimum possible number of diagonal pairs in this division?
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Context:
Given four non-coplanar points \(A, B, C, D\) in space where the distances between any two points are distinct, consider a plane \(\alpha\) such that the distances from three of the points to \(\alpha\) are equal, and the distance from the fourth point to \(\alpha\) is twice the distance of one of the three points. Determine the number of such planes \(\alpha\).
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Context:
Determine the number of continuous functions \( f(x) \) defined on \([0,1]\) that satisfy the following conditions: \( I_1 = \int_0^1 f(x) \, dx = 1 \), \( I_2 = \int_0^1 x f(x) \, dx = a \), and \( I_3 = \int_0^1 x^2 f(x) \, dx = a^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A closed orientable surface $R$ has a genus of 5. If there is a degree 2 covering map from $R$ to another surface $S$, what is the genus of the surface $S$?
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Context:
Determine the number of different subgroups of order 9 that are isomorphic to $C_3 \times C_3$ in the symmetric group $S_6$. Provide your answer as a single integer.
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Context:
Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the time complexity of the recurrence relation: \[ T(n) = n + \sum_{k=1}^{n-1} [T(n-k) + T(k)] \] with the base case \( T(1) = 1 \). Find the order of \( T(n) \) without considering a term for \( T(0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface area of the portion of the cylinder \(x^2 + y^2 = 2ay\) that is contained within the sphere \(x^2 + y^2 + z^2 = 4a^2\), given that \(a > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangular prism formed by the planes $P_1: 2x + y + z = 2$, $P_2: x + y - z = 3$, and $P_3: x + 2z = 2$, the normal vectors of the planes are co-planar. Determine the area of the base of the prism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the eigenvalues of the operator \( T: \ell^2 \to \ell^2 \) defined by \( T(x_1, x_2, x_3, \ldots) = \left(0, x_1, \frac{x_2}{2}, \frac{x_3}{3}, \ldots\right) \), where \( \ell^2 \) is the space of square-summable sequences.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( m = 2007^{2008} \). Determine the number of natural numbers \( n \) such that \( n < m \) and \( n(2n + 1)(5n + 2) \) is divisible by \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \oint_{\varGamma} \dfrac{\partial \ln r}{\partial \vec{n}} \mathrm{d}s \) where \( \varGamma: x^2+y^2+z^2=R^2 \) and \( r=\sqrt{x^2+y^2+z^2} \). Here, \( \vec{n} \) is the outward unit normal vector on the surface \( \varGamma \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the integral as \( k \) approaches infinity:

\[ \lim_{k\rightarrow\infty}\int_{-\infty}^\infty\frac{1}{1+x^2+\cdots+x^{2k}}\ \text{dx} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A planar, 3-regular graph has a proper edge coloring with 3 colors. Every alternating path of red and yellow edges with a length of 6 forms a cycle, and every alternating path of yellow and blue edges with a length of 6 forms a cycle. However, every alternating path of red and blue edges with a length of 4 forms a cycle. How many vertices does this graph have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Square \(CASH\) and regular pentagon \(MONEY\) are both inscribed in a circle. Given that they do not share a vertex, determine the number of intersection points between these two polygons.
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Context:
Consider the space \(\mathbb{R}^3\). Let \(L\) be the set of all lines and \(P\) be the set of all planes in \(\mathbb{R}^3\). Define topologies \(\tau_1\) and \(\tau_2\) on \(\mathbb{R}^3\) generated by the subbases \(L\) and \(P\), respectively. Determine the relationship between \(\tau_1\) and \(\tau_2\). Are these topologies equivalent, or is one finer than the other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the specific solution to the differential equation \( y' = \frac{y(y + 2x - 1)}{x + y} \) given the general solution \( y^2 + 2xy - c e^{2x} = 0 \) and the initial condition \( y(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$ \mathrm{p}19.$How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{1} \frac{\ln x}{x^2 + 1} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all subgroups of the multiplicative group of non-zero real numbers, \(\mathbb{R}^*\), that have an index of 2.
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Context:
A ball with mass 0.15 kg is thrown upward with an initial velocity of 20 m/s from the roof of a building 30 m high. The force due to air resistance is given by \(|v|/30\), directed opposite to the velocity, where \(v\) is the velocity in m/s. Determine the time at which the ball hits the ground.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( x, y, \) and \( z \) are real numbers satisfying \( x^2 + z^2 = 1 \) and \( y^2 + 2y(x + z) = 6 \), find the maximum value of \( y(z - x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the group $\left(\mathbb{Z}_{/(2^k\mathbb{Z})}\right)^*$ that have an order of $1$, $2$, or $4$. Provide your answer as a single number.
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Context:
Consider all the natural numbers from $000000$ to $999999$. A number is called "lucky" if the sum of its first three digits equals the sum of its last three digits. A number is called "medium" if the sum of all its digits equals $27$. Find the difference between the number of lucky numbers and medium numbers.
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Context:
Determine the values of $a$ for which the improper integral $$\int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx$$ converges.
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Context:
Suppose $X_1, \ldots, X_{100}$ are independent and identically distributed as $N(\mu,1)$, but only the values of $X_i$ that are less than 0 are recorded. If 40 observations are less than 0, find the maximum likelihood estimator (MLE) of $\mu$.
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Context:
What is the minimum number of direct connections required to ensure that any group of 8 or fewer workstations out of 12 can access different servers simultaneously, given that there are 7 servers available?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the automorphism group of the group \(\mathbb{Z}_2 \oplus \mathbb{Z}_2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a single element in the ring \( \mathbb{Q}[x] \) that can generate both \( 2x^2 \) and \( 2x^3 \).
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Context:
Let $S = \{m + \frac{1}{n} \mid m, n \in \mathbb{N}\}$. Determine the derived set (set of cluster points) of $S$.
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Context:
Given positive real numbers $a, b, c$ such that $a^2 + b^2 + c^2 = 3$, determine the minimum value of the expression \[ P = \frac{1}{(a-b)^2} + \frac{1}{(b-c)^2} + \frac{1}{(c-a)^2}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of possible derived trees for the expression $1-2-(3-4)*5*6$ using the following grammar:

$$\begin{align}V_n&=\{expr,term,factor,number\}\\ V_t&= \{(,),-,*,0...9\}\\ P&=\left \{ \begin{aligned} expr&\to expr-expr\;\mid\;term\\ term&\to term*factor\;\mid\;factor\\ factor&\to number \;\mid\; (expr) \\ number&\to 0|1|2|3|4|5|6|7|8|9 \end{aligned} \right \}\\ S&=expr \end{align} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: \( \lim_{x \to +\infty} f(x) \), where \( f(x) = \frac{1}{\Gamma(x)} \int_x^{+\infty} t^{x-1} e^{-t} \, dt \) for all \( x \in \Bbb R_+^* \).
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Context:
Consider the matrix \( A = I_9 - 2u^Tu \) with \( u = \frac{1}{3}[1, 1, 1, 1, 1, 1, 1, 1, 1] \). If \( p \) and \( q \) are two distinct eigenvalues of \( A \), find \( |p-q| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the expression: $L_1^2L_1^*L_1 \cap L_1^*L_1^2\Sigma^*$.
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Context:
Determine the base $d$ of a numbering system that allows writing the largest number of numbers using a given number of digits, where digits in different positions are considered different. Consider only sets of $m$ digits, where $m$ is divisible by both bases being compared.
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Context:
Find a maximal ideal in \( \mathbb{Z}[x] \) that properly contains the ideal \((x-1)\).
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Context:
Evaluate the limit: $$L=\lim_{x\to 0}\frac{\sin(\ln(x+1))-\ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)}$$
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Context:
Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \int_r (xy^2z^2-1, x^2yz^2, x^2y^2z)\cdot ds \) where \( r = r_1 \cup r_2 \). The parametric representation of \( r_1 \) is \( \sigma_1 (t) = (\sin t, \cos t, t-\frac{\pi}{2}), t \in [0,\frac{\pi}{2}] \) and the parametric representation of \( r_2 \) is \( \sigma_2 (t) = (\cos^3 \phi, \sin^3 \phi, 0), \phi \in [0,\pi] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C \vec{F}(x,y) \cdot \,d\vec{r} \) where \( \vec{F}(x,y) = \begin{bmatrix} 5y \\ 10x \end{bmatrix} \) and \( C \) is the quarter-circle arc centered at \((0,2)\) going from the origin to \((2,2)\). Use polar coordinates to solve the integral.
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Context:
Evaluate the integral \( \int_0^{2\pi} \cos(\cos(z))\cosh(\sin(z)) \, dz \) using a corollary of Cauchy's Integral Formula or Gauss's Mean Value Theorem.
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Context:
Determine the Galois group of the polynomial \((x^2-1)(x^2-2)\ldots(x^2-p+1)\) over the finite field \(\mathbb{Z}_p\), where \(p\) is an odd prime.
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Context:
Evaluate $a+2b+3c$ given that \( \lim_{h \to 0}\frac { \int _{ 0 }^{ h }{ f(x)dx-h(af(0)+bf(h/3)+cf(h)) } }{ { h }^{ 4 } } \) is a finite non-zero number.
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Context:
$A, B, C, D$ all lie on a circle with $\overline{AB} = \overline{BC} = \overline{CD}$. If the distance between any two of these points is a positive integer, what is the smallest possible perimeter of quadrilateral $ABCD$?
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Context:
In a queuing system, there are two facilities, A and B, each with a single server. The service times are exponentially distributed with mean service times of 1 minute at facility A and 4 minutes at facility B. Sixty customers per hour arrive at the facilities, and each customer independently chooses a facility to minimize their expected time in the system. What is the long-run probability that a random customer chooses facility A?
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Context:
A square with sides initially measuring $4m$ increases at a constant rate of $3m$ per second. One corner of the square moves along a line $L$ at a speed of $2m/s$ for $5$ seconds. During this time, the square makes one complete revolution about $L$, revolving at a constant rate and remaining always perpendicular to $L$. Calculate the volume of the solid generated by this motion.
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Context:
Find the exact area under the curve defined by the equation \([x+y] + [x-y] = 5\) with the constraints \(x \geq y\), \(x \geq 0\), and \(y \geq 0\), where \([\cdot]\) denotes the greatest integer function.
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Context:
The set of real numbers $ x$ for which
\[ \frac{1}{x-2009}+\frac{1}{x-2010}+\frac{1}{x-2011}\ge 1\]
is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Factor the polynomial $2t^4 + t^2 + 2$ into prime polynomials over the finite field $\mathbb{F}_3$. Provide the factorization.
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Context:
Assume the Generalized Continuum Hypothesis (GCH) holds and \(\kappa\) is a regular cardinal. Determine the number of non-isomorphic partial orders of size less than \(\kappa\).
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Context:
Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \), which satisfy \( x \leq y \leq z \), \( x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine whether \( x + y \) is greater than, less than, or equal to \( z \). Fill in the blank with ">", "<" or "=".
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the length of the graph of the function \( y = f(x) \) over the interval \([0, 3]\), where \( f(0) = 0 \) and \( f'(x) = \frac{\ln(x+2)}{x^2+1} \) for \( x > -2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \frac{\sqrt{n}}{1 + n \ln(1 + x^2)} \, dx \).
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Context:
Find the form of matrices that commute with all matrices of the form $E_{ij}$, where $E_{ij}$ is the matrix with a $1$ in the $(i,j)$ entry and $0$s elsewhere.
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Context:
Given an AR(1) model with 144 observations, where \( y_t = \phi y_{t-1} + \epsilon_t \) and \( \epsilon_t \) is white noise with mean zero and variance \( \sigma^2 \), find an estimate for \( \phi \). The initial and final observations are \( y_1 = -1.7 \) and \( y_{144} = -2.1 \). Additionally, \( \sum_{t=2}^{144} y_t y_{t-1} = -128.6 \) and \( \sum_{t=1}^{144} y_t^2 = 246.4 \). Use the formula for the first sample autocorrelation: \[ \hat{\phi} = \frac{\sum_{t=2}^{144} (y_t - \overline{y})(y_{t-1} - \overline{y})}{\sum_{t=1}^{144} (y_{t-1} - \overline{y})^2} \] to find \( \hat{\phi} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of equivalence classes of propositional terms, using a set of 5 propositional variables, that evaluate to TRUE under exactly 13 distinct valuations.
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Context:
For how many angles $x$, in radians, satisfying $0\le x<2\pi$ do we have $\sin(14x)=\cos(68x)$?
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Context:
Provide an example of a sequence of numbers \( v_n \) such that its mean tends to a limit, but \( v_n \) itself does not converge to a limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.
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Context:
You toss a weighted coin 100 times. The probability of getting heads, $p$, is either 0.1 or 0.01, but you don't know which. Determine the decision boundary for the number of heads obtained that would lead to the maximum likelihood estimate (MLE) of $p$ being 0.1.
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Context:
Calculate the cardinality of the product \( \prod_{0<\alpha<\omega_1}\alpha \).
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Context:
Suppose \( P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0 \) where \( a_0, a_1, \ldots, a_{n-1} \) are real numbers for \( n \geq 1 \) (monic \( n \)th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all real numbers \( x,y \), determine the minimum possible value of \( P(2024) \).
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Context:
Four glasses are arranged in a row: the first and third contain orange juice, while the second and fourth are empty. Each time, you can take a full glass and pour all its contents into one of the two empty glasses. What is the expected number of pourings required for the first and third glasses to be empty, while the second and fourth glasses are full?
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Context:
Raymond wants to travel in a car with $3$ other (distinguishable) people. The car has $5$ seats: a driver’s seat, a passenger seat, and a row of $3$ seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car?
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Context:
Determine the order of the group $G = \{A \in M_2(\mathbb{Z}_p): \mathrm{det}A = \pm 1 \}$, where $p > 2$ is a prime number.
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Context:
Evaluate the sum \( a_k = \sum_{j=\lceil k/2\rceil}^k {j \choose k-j} \) in closed form using a generating function.
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Context:
Determine the range of values for $b$ such that for all solutions $y(x)$ of the differential equation $y'' + y' + by = 0$, the limit \( \lim_{x \to \infty} \frac{y(x)}{e^x} = 0 \) is satisfied.
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Context:
Find the least positive integer \( k \) such that \( 10^k \equiv 1 \pmod{2003} \).
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Context:
Evaluate the integral \( \int_\Omega xye^{x^2-y^2}dxdy \) over the domain \( \Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\} \).
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Context:
Mary chose an even 4-digit number \(n\). She wrote down all the divisors of \(n\) in increasing order from left to right: \(1, 2, \ldots, \frac{n}{2}, n\). At some moment Mary wrote 323 as a divisor of \(n\). What is the smallest possible value of the next divisor written to the right of 323?
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Context:
Determine the number of non-singular matrices of the form \(A=\begin{bmatrix} 1 & a & b\\ w&1 &c \\ w^2 &w & 1 \end{bmatrix}\) where \(a, b, c\) are elements of the set \(S=\{w, w^2\}\), and \(w\) is a complex cube root of unity with \(w \neq 1\).
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Context:
Let \( G = (V, E) \) be a simple connected graph with exactly one cycle. The graph has four vertices of degree 2, five vertices of degree 3, three vertices of degree 4, and one vertex of degree 5. If the maximum degree \( \Delta = 5 \), determine the number of pendant vertices in \( G \).
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Context:
Determine the conditions on the complex numbers $a$, $b$, and $c$ such that the matrix \(\begin{bmatrix}2&0&0\\a&2&0\\b&c&-1\end{bmatrix}\) has a minimal polynomial of \((x-2)(x+1)\).
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Context:
Evaluate the double integral \( \iint_L (xy - x^2) \, dx - (x+y)^2 \, dy \) where \( L \) is the triangle with vertices \( O(0,0) \), \( A(3,1) \), and \( B(2,2) \). Solve this without using Green's Theorem.
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Context:
Eric rolls a ten-sided die (with sides labeled $1$ through $10$) repeatedly until it lands on $3, 5$, or $7$. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
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Context:
Determine the values of $x \in \mathbb{R}$ for which the series \( \sum_{n=1}^\infty \frac{n^{nx}}{n!} \) converges.
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Context:
Determine the domain of the function \( f(x) = \int_{1}^{x} \frac{e^t}{t} \, dt \).
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Context:
Find the cumulative distribution function (CDF) of the absolute value of the difference between two independent random variables, each uniformly distributed on the interval [0,1].
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Context:
Given that triangle $PQR$ is a right triangle, each side being the diameter of a semicircle, the area of the semicircle on $\overline{PQ}$ is $18\pi$, and the arc of the semicircle on $\overline{PR}$ has length $10\pi$, calculate the radius of the semicircle on $\overline{QR}$.
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Context:
A Martian writes down the equation \(x^2 - 19x + 76 = 0\). When asked to write down the difference between the larger and smaller root, the Martian writes 9. How many fingers do Martians have?
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Context:
Evaluate the triple integral \( \iiint_D |z| \, dx \, dy \, dz \) over the domain \( D = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2-16 \leq z \leq 4 - \sqrt{x^2+y^2} \} \).
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Context:
Evaluate the integral \( \int_1^4 \frac{x^2}{\sqrt{1-x^3}}\, dx. \)
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Context:
Given a random sample \(X_1\) and \(X_2\) of size \(n=2\) from an exponential distribution with parameter \(\theta\), find the efficiency of the estimator \(2Y_1\) relative to \(\bar{X}\), where \(Y_1\) is the first order statistic. Both \(2Y_1\) and \(\bar{X}\) are unbiased estimators of \(\theta\). Calculate the variance of \(2Y_1\) and use it to determine the relative efficiency \(\frac{V(\bar{X})}{V(2Y_1)}\), given that \(V(\bar{X})=\frac{\theta^2}{2}\).
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Context:
In square ABCD, M and N are the midpoints of BC and CD respectively. Find sin(theta), where theta is the angle between AM and AN. Express your answer as a numerical value.
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Context:
For how many pairs of integers $(m, n)$ with $0 < m \le n \le 50$ do there exist precisely four triples of integers $(x, y, z)$ satisfying the following system?
\[\begin{cases} x^2 + y+ z = m \\ x + y^2 + z = n\end{cases}\]
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Context:
Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}+\sqrt[3]{x})}\ dx$.
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Context:
Determine the number of steps required to reduce \( n = 2016^{155} \) to 1 using the algorithm where, at each step, the largest divisor of the current number (other than the number itself) is subtracted from it.
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Context:
Determine the order of the entire function \(f(z) = \cos\sqrt{z}\).
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Context:
Given an exponential distribution with the PDF of the minimum of a sample as \( f_{\min}(x) = n\cdot\lambda\cdot(e^{-\lambda x})^n \), calculate the probability \( P\left(\frac{\lambda}{n} \le \frac{\ln(3)}{\min(x)}\right) \).
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Context:
Given the function \( z(x,y) \) defined implicitly by the equation \( F(x,y,z) = 5x + 2y + 5z + 5\cos(5z) + 2 = 0 \), find the mixed partial derivative \( \frac{\partial^2 z}{\partial x \partial y} \) at the point \( \left(\frac{\pi}{5}, \frac{3}{2}, \frac{\pi}{5}\right) \).
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Context:
Given three independent events $E_1, E_2, E_3$, the probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$, and only $E_3$ occurs is $\gamma$. The probability $p$ that none of the events occurs satisfies the equations $(\alpha-2\beta)p=\alpha\beta$ and $(\beta-3\gamma)p=2\beta\gamma$. If the probability of occurrence of $E_1$ is $X$ and the probability of occurrence of $E_3$ is $Y$, find the ratio $\frac{X}{Y}$.
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Context:
Minimize the expression \( \{tr^Trt^T\}_{ii} \) subject to the constraint \( r^Tt = 1 \), where \( r, t \in \mathbb{R}^{n \times 1} \). Assume that the diagonal elements of \( t \) are ordered in nondecreasing order.
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Context:
You flip a fair coin more than 11 times. Calculate the probability that there is at least one occurrence of a sequence of 6 consecutive heads immediately followed by 6 consecutive tails.
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Context:
Find the limit of the expression as $x$ approaches $0$ from the right: $$\lim_{x \to 0^+} \frac{\sum_{n=1}^\infty \frac{\sin(nx)}{n^2}}{\sin(x)}$$
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Context:
Find the minimum value of the integral \( \int_{0}^{\infty} |x^3 - a - bx - cx^2|^2 e^{-x} \, dx \) with respect to the real numbers \(a, b, c\).
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Context:
Determine the cardinality of the set $X$, where $X$ is the set of all relations $R$ over $\mathbb{N}$ such that the transitive closure $R^{*}$ equals $\mathbb{N} \times \mathbb{N}$. What is the cardinality of $X$?
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Context:
Let $a$ be a positive real number. In the coordinate plane, denote by $S$ the area of the region bounded by the curve $y = \sin x$ for $0 \leq x \leq \pi$ and the $x$-axis. Denote by $T$ the area of the region bounded by the curves $y = \sin x$ and $y = a \cos x$ for $0 \leq x \leq \frac{\pi}{2}$ and the $x$-axis. Find the value of $a$ such that the ratio $S:T = 3:1$. \( \boxed{a} \)
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Context:
Given distinct prime numbers $p$ and $q$, find the number of positive integer solutions to the equation:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{pq} \]
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Context:
Find the greatest common divisor (GCD) of expressions of the form \(p^6 - 7p^2 + 6\), where \(p\) is a prime number greater than or equal to 11.
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Context:
Given the equations $x + y + z = 0$, $x^2 + y^2 + z^2 = 1$, and $x^4 + y^4 + z^4 = 2$, find the value of $x^5 + y^5 + z^5$.
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Context:
Determine the values of \( p \) for which the double series \( \sum_{i, j=1}^\infty a_{i+j}^p \) converges, given that \( (a_i)_{i=1}^\infty \) is a sequence of positive numbers such that \( \sum_{i=1}^\infty a_i < \infty \). Specifically, investigate the case when \( p = 2 \).
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Context:
In a grid representing numbers in base 8, select numbers such that there is exactly one number with each possible last digit (0-7) and one with each possible second digit (0-7). Note that numbers ending with 0 represent a value 8 greater than their residue class. Calculate the sum of these selected numbers.
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Context:
If the curve $y=\frac{a}{x}$ ($x>0$) has a common tangent line with the curve $y=2\ln x$, then the range of values for $a$ is ______.
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Context:
Let  $T$  be the answer to question  $18$ . Rectangle  $ZOMR$  has  $ZO = 2T$  and  $ZR = T$ . Point  $B$  lies on segment  $ZO$ ,  $O'$  lies on segment  $OM$ , and  $E$  lies on segment  $RM$  such that  $BR = BE = EO'$ , and  $\angle BEO' = 90^o$ . Compute  $2(ZO + O'M + ER)$ .

PS. You had better calculate it in terms of  $T$ .
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Context:
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(9,3)$, respectively. What is its area?
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Context:
Find the radius of convergence for the power series: $$\sum_{i=0}^{\infty}\left({(-1)^{i}\over 2^i(i!)}\left(\prod_{j=1}^{i}\left(2j-1\right)\right)\left(-18x^4\right)^i\right)$$
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Context:
Given a sample set: $1, 3, 4, a, 7$. The mean of this sample set is $4$. Find the standard deviation of the sample set.
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Context:
Given the sequence {a_n} satisfying a_1 = 1/2, a_{n+1} = 3a_n + 1, find the sum S_{2016} of the first 2016 terms of the sequence. Express your answer as a fraction or integer.
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Context:
Calculate the volume of the solid formed by revolving the region bounded by the curve $x = y^2$ and the line $x = 9$ about the x-axis. The cross-sections perpendicular to the x-axis are isosceles right triangles with the hypotenuse as the base.
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Context:
Given that the Mathematics College Entrance Examination scores distribution ξ closely follows the normal distribution N(100, 5^2), and P(ξ < 110) = 0.96, find the value of P(90 < ξ < 100). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places.
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Context:
Calculate the residue of the function \( f(z) = \sin\left(\frac{z^2+1}{z}\right) \) at \( z = 0 \).
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Context:
Determine the value of \(a > 2\) such that \(\int_{0}^{1}\frac{x^8-x^2}{x^{12}+ax^6+1}dx = -\frac{\pi}{24}\).
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Context:
How many positive integer solutions less than 71 are there for the equation $x^7 \equiv a \pmod{71}$, given that $a$ is an integer not divisible by 71?
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Context:
长为正整数 $l$ 的木棒每次都可锯成长均为正整数的 2 段，可连续进行这种操作，但要求任何时刻所有木棒中的最长者长小于最短者长的 2 倍。例如，一根长为 6 的木棒只能截成长均为 3 的两段；一根长为 7 的木棒只能截成长分别为 $2,2,3$ 的三段。试问:长为 30 的木棒最多可锯成多少段?
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Context:
Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form.
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Context:
Let \( n = 2^{40}5^{15} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)?
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Context:
Consider the region defined by \((x,y,z)\) such that \(x^2+y^2<1, x>0, 0 \le z \le 5\). This describes one half of a cylinder. A wedge is cut from this half-cylinder along the plane \(z=2x\). Find the volume of the wedge.
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Context:
Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{1}{n^2} \ln \left(1 + \frac{x}{n}\right) \) converges.
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Context:
Two curves are given by the equations \( y=2 \sin \frac{5 \pi}{6}x \) and \( y=\alpha x^2-3 \alpha x+2\alpha +1 \). If these curves touch each other at some point within the interval \( 0 \leq x \leq \frac{18}{5} \), find the value of \( \frac{2 \sqrt{3} \alpha}{5 \pi} \).
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Context:
Given an open set $M \subset \mathbb{R}^3$ and a continuous function $\rho: M \to \mathbb{R}$ representing a charge density, consider the vector field $v: M \to \mathbb{R}^3$ defined by

$$v(x) = \int_{r\in M} \frac{\rho(r)(x-r)}{\|x-r\|^3} \, dr$$

where the integral is taken component-wise. Compute the divergence of $v$, $\nabla \cdot v$. Express your answer in terms of $\rho$. \(n = 3\)
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Context:
Determine the greatest real number \( C \) ,such that for every positive integer \( n \geq  2 \) ,there exists \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \in  \left\lbrack  {-1,1}\right\rbrack \) ,so that
 \( \mathop{\prod }\limits_{{1 \leq  i < j \leq  n}}\left( {{x}_{i} - {x}_{j}}\right)  \geq  {C}^{\frac{n\left( {n - 1}\right) }{2}} \)
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Context:
A toy consists of 9 differently sized rings and a tapering pole. The rings can be stacked on the pole in any order until the top of the pole is reached, at which point they must be in decreasing order of size. How many different configurations of the rings can be created without reaching the top of the pole?
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Context:
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?
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Context:
Evaluate the following definite integral.
\[ 2^{2009}\frac {\int_0^1 x^{1004}(1 - x)^{1004}\ dx}{\int_0^1 x^{1004}(1 - x^{2010})^{1004}\ dx}\]
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Context:
The digits $1, 2, \cdots, 9$ are arranged in random order to form a nine-digit number. What is the probability that this number is divisible by $11$?
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Context:
Compute the line integral \( \int (x^2-y^2) \, dx +(x^2+y^2) \, dy \) over the closed curve formed by the curves \( y=x^2+1 \) from \((1,2)\) to \((2,5)\), and \( y=2x+2 \) from \((2,5)\) to \((1,2)\), using Green's theorem.
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Context:
A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers $\{z \in \mathbb{Z} \mid -2011 < z < 2011\}$ are arithmetic and harmonic?

**Remark:** The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as:

\[A(a,b) = \frac{a+b}{2} \quad \text{and} \quad H(a,b) = \frac{2ab}{a+b} = \frac{2}{\frac{1}{a} + \frac{1}{b}},\]

respectively, where $H(a,b)$ is not defined for some $a$, $b$.
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Context:
Evaluate the limit: \[ \lim_{(x, y, z) \to (0, 0, 0)} \frac{x^2 y^2 z^2}{(x^4 + y^4 + z^4)^{a/4}}. \]
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