Context:
1. On a $7 \times 7$ checkerboard, 14 cells are marked. We will call a pair of cells with a common side interesting if at least one cell in the pair is marked. What is the maximum number of interesting pairs that can be?
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Context:
Evaluate the surface integral $$\int_{S}\frac{dxdy}{z}+\frac{dydz}{x}+\frac{dxdz}{y}$$ over the surface of the sphere $S$ with radius $R$ and outward orientation, without using the Gauss-Ostrogradsky formula.
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Context:
Given positive numbers $a$ and $b$ satisfying $a+b=1$, $c\in R$, find the minimum value of $\frac{3a}{b{c}^{2}+b}+\frac{1}{ab{c}^{2}+ab}+3c^{2}$.
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Context:
Given a square matrix \( A \) of order 2, with \( |A| \neq 0 \) such that \( |A + |A| \text{adj}(A)| = 0 \), where \( |A| \) and \( \text{adj}(A) \) denote the determinant and adjoint/adjugate of matrix \( A \), respectively. Find \( |A - |A| \text{adj}(A)| \).
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Context:
A kite is a quadrilateral whose diagonals are perpendicular. Let kite \(ABCD\) be such that \(\angle B = \angle D = 90^{\circ}\). Let \(M\) and \(N\) be the points of tangency of the incircle of \(ABCD\) to \(AB\) and \(BC\) respectively. Let \(\omega\) be the circle centered at \(C\) and tangent to \(AB\) and \(AD\). Construct another kite \(AB'C'D'\) that is similar to \(ABCD\) and whose incircle is \(\omega\). Let \(N'\) be the point of tangency of \(B'C'\) to \(\omega\). If \(MN' \| AC\), then what is the ratio of \(AB : BC\)?
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Context:
Given $(x-1)^4(x+2)^8 = ax^{12} + a_1x^{11} + \ldots + a_{n}x + a_{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$.
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Context:
Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB=11$, $BC=5$, $CD=19$, and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$?
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Context:
A carpet is on sale with its price reduced by 10% every 15 minutes. Initially, at 9:00 a.m., the carpet costs $10.00. At what time will the price of the carpet fall below $8.00? Express your answer in the format HH:MM a.m./p.m.
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Context:
Emília's tree grows according to the following rule: two weeks after the appearance of a branch, this branch produces a new branch each week and the original branch continues to grow. After five weeks, the tree has five branches, as shown in the figure. How many branches, including the main branch, will the tree have at the end of eight weeks?
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Context:
Let \( B_n \) be the set of all sequences of length \( n \), consisting of zeros and ones. For every two sequences \( a, b \in B_n \) (not necessarily different), define strings \( \varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n \) and \( \delta_0\delta_1\delta_2 \dots \delta_n \) such that \( \varepsilon_0=\delta_0=0 \) and
\[  \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).  \]
Let \( w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n \). Find \( f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} \).
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Context:
Given the function \( f(x) = x^2 + 2x + a \ln x \), it holds for any \( t \geq 1 \) that \( f(2t-1) \geq 2f(t) - 3 \). Determine the range of values for the real number \( a \).
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Context:
Evaluate the integral \( \int_0^{\infty} \frac{rdr}{(r^2 + a^2 - 2ar\cos{(\theta)})^{3/2}} \int_0^{\pi} a\sin{(\theta)}^3 d\theta \) for \(a>0\).
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Context:
Evaluate the double integral \( \int_0^a \int_0^a \frac{dx \, dy}{(x^2+y^2+a^2)^{\frac{3}{2}}} \).
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Context:
Let $w, x, y, z$ be real numbers such that $w+x+y+z =5$, $2 w+4 x+8 y+16 z =7$, $3 w+9 x+27 y+81 z =11$, $4 w+16 x+64 y+256 z =1$. What is the value of $5 w+25 x+125 y+625 z ?$
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Context:
Evaluate the expression \([-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}\), where \(n\) is any integer greater than 1.
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Context:
In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?
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Context:
Let \( {x}_{1} = 1/{20},{x}_{2} = 1/{13} \) ,and
 \( {x}_{n + 2} = \frac{2{x}_{n}{x}_{n + 1}\left( {{x}_{n} + {x}_{n + 1}}\right) }{{x}_{n}^{2} + {x}_{n + 1}^{2}} \)
 for all integers \( n \geq  1 \) . Evaluate \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {1/\left( {{x}_{n} + {x}_{n + 1}}\right) }\right) \) .
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Context:
Evaluate the polynomial congruence \(\sum_{j=(p-1)/2}^{p-1}\left(\frac{j}{p}\right)j(j-1)...(j-(p-1)/2+1) \ (\textrm{mod} \ p)\), where \(p\) is an odd prime.
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Context:
Edmund is saving up money to buy a new computer that costs $750. He has a starting balance of $200 and receives a monthly gift of $50 from his grandparents. Additionally, he gets a part-time job that pays him $10 a day for 3 days a week. He convinces his parents to pay him for extra chores on top of his regular chores. He normally has to do 12 chores a week without any payment. His parents agree to pay him $2 for every extra chore he does during the week, $3 for every car wash, and $5 for every lawn mowing task. If he does 4 chores a day, washes 2 cars a week, and mows 1 lawn a week, how many days will it take him to save up the money needed to buy the computer?
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Context:
In an election between two candidates A and B, the number of valid votes received by A exceeds those received by B by a certain percentage of the total number of votes polled. 20% of the votes polled were invalid and a total of 5720 votes were polled. B got 1859 valid votes. What is the percentage by which A's valid votes exceeded B's valid votes?
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Context:
Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. The equation of line $m$ is in the form $ax + by = 0$. Please find the value of a + b.
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Context:
Use Stoke's Theorem to evaluate $\int_C -y^3dx + x^3dy - z^3dz$ where $C$ is the intersection of the cylinder $x^2 + y^2 = 1$ and the plane $x + y + z = 1$ oriented counterclockwise.
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Context:
In the convex quadrilateral \(ABCD\), the length of side \(AD\) is 4, the length of side \(CD\) is 7, the cosine of angle \(ADC\) is \(\frac{1}{2}\), and the sine of angle \(BCA\) is \(\frac{1}{3}\). Find the length of side \(BC\) given that the circumcircle of triangle \(ABC\) also passes through point \(D\).
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Context:
Given the equation in terms of $x$, $x^2 + 2kx + k^2 + k + 3 = 0$, with roots $x_1$ and $x_2$, find the minimum value of $(x_1 - 1)^2 + (x_2 - 1)^2$.
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Context:
A liter of chlorine costs $10 and is now sold at a certain discount. A box of soap that costs $16 is now sold at 25% off. You will save $26 if you buy 3 liters of chlorine and 5 boxes of soap. What is the discount on a liter of chlorine?
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Context:
Into how many maximum parts can the coordinate plane \( xoy \) be divided by the graphs of 100 quadratic trinomials of the form

\[ y = a_{n} x^{2} + b_{n} x + c_{n} \quad (n = 1, 2, \ldots, 100) ? \]
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Context:
Given that $\sin \alpha = 3 \sin \left(\alpha + \frac{\pi}{6}\right)$, find the value of $\tan \left(\alpha + \frac{\pi}{12}\right)$.
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Context:
Evaluate the limit of the expression \( a_{n+1}\left[1+\left(\frac{a_{n+1}}{a_n}-1\right)\right]^n \) as \( n \to \infty \), given that \( a_n \to b < \infty \) and \( L = \lim_{n \to \infty} (a_{n+1}-a_n)n \) exists.
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Context:
Find the smallest natural number \( n \) such that in any set of \( n \) different natural numbers that do not exceed 1000, it is always possible to choose two numbers where the larger number is not divisible by the smaller one.
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Context:
 When I was six, my sister was half my age and my brother was twice my age. In five years, what will the sum of their ages be?
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Context:
Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\sqrt{4a^2 + b^2}}
$$
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Context:
Find the real number \( t \) such that the sequence of integers not in the sequence of floors of multiples of \( \pi \) is given by the floor of \( t \cdot n \). Provide \( t \) to the nearest thousandth.
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Context:
How many natural numbers are there whose square and cube together require 10 digits to describe?
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Context:
Determine the number of subgroups in the group $\Bbb Z_{q} \times \Bbb Z_{q}$, where $q$ is a prime number.
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Context:
a rectangular paper when folded into two congruent parts had a perimeter of 34 cm foer each part folded along one set of sides and the same is 38 cm . when folded along the other set of sides . what is the area of the paper ?
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Context:
Determine whether the limit \( \lim_{n\to\infty} \left(\frac{\log(p_{n+1})}{\log(p_n)}\right)^n \) exists, and if it does, find its value.
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Context:
What is the maximum number of non-overlapping groups into which all the 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:
若实数 $x, y$ 满足 $x^{2}+2 \cos y=1$, 请给出 $x-\cos y$ 的取值范围端点和的平方的值。
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Context:
设椭圆 $\Gamma$ 的两个焦点是 $F_{1}, F_{2}$, 过点 $F_{1}$ 的直线与 $\Gamma$ 交于点 $P, Q$. 若 $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$,且 $3\left|P F_{1}\right|=4\left|Q F_{1}\right|$, 则椭圆 $\Gamma$ 的短轴与长轴的比值为。原始答案为\(\frac{m \sqrt{n}}{k}\)格式，请给出m + n + k的值。
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\operatorname{arctan} 3} \frac{4+\operatorname{tan} x}{2 \sin ^{2} x+18 \cos ^{2} x} d x
$$
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Context:
In the TV series "The Secret of Santa Barbara," there are 20 characters. In each episode, one of the following events takes place: a certain character learns the Secret, a certain character learns that someone knows the Secret, or a certain character learns that someone does not know the Secret. What is the maximum number of episodes the series can continue?
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Context:
In the diagram, \(PRTY\) and \(WRSU\) are squares. Point \(Q\) is on \(PR\) and point \(X\) is on \(TY\) so that \(PQXY\) is a rectangle. Also, point \(T\) is on \(SU\), point \(W\) is on \(QX\), and point \(V\) is the point of intersection of \(UW\) and \(TY\), as shown. If the area of rectangle \(PQXY\) is 30, calculate the length of \(ST\). Express your answer as a single number, rounded to two decimal places if necessary.
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Context:
1. Given a set of data consisting of seven positive integers, the only mode is 6, and the median is 4. Then the minimum value of the sum of these seven positive integers is $\qquad$ .
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Context:
Juan wants to calculate the area of a large circular garden, whose actual diameter is 50 meters. Unfortunately, his measurement device has an accuracy error of up to 30%. Compute the largest possible percent error, in percent, in Juan’s computed area of the circle in square meters.
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Context:
Calculate the volume, \( V(M_1) \), of the solid \( M_1 \) in \( \mathbb{R}^3 \) defined by:
\[ M_1 = \{(x,y,z) : (x,y) \in D, \ 0 \leq z \leq yx^2 + y^3 \} \]
where the domain \( D \) is:
\[ D = \{(x, y) : |x| \leq y, \ 1 \leq x^2 + y^2 \leq 4 \} \]
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Context:
Determine the value of the derivative \( \frac{d z^*}{dz} \), where \( z^* \) denotes the conjugate of the complex variable \( z \).
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Context:
Let \( n \) (\( n \geq 2 \)) be a given integer. Find the smallest real number \(\lambda\) such that for any real numbers \( x_{1}, x_{2}, \cdots, x_{n} \in [0, 1] \), there exist \(\varepsilon_{1}, \varepsilon_{2}, \cdots, \varepsilon_{n} \in \{0, 1\}\) satisfying that for any \(1 \leq i \leq j \leq n\), the following holds:
$$
\left|\sum_{k=i}^{j}\left(\varepsilon_{k}-x_{k}\right)\right| \leq \lambda.
$$
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Context:
在半径为 10 的圆周 C 上任给 63 个点，设以这些点为顶点且三边长都大于 9 的三角形的个数为 S. 求 S 的最大值.
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Context:
Determine the number of ways to arrange the components of a vector $v = (a, b, c)$ in $\mathbb{R}^{3}$ such that the linear span of the resulting vectors creates a vector subspace of dimension two in $\mathbb{R}^{3}$.
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Context:
Determine the smallest cardinality of a base for the topological space \((X, \tau)\), where \(X = \mathbb{R}^2\) and \(\tau = \{\emptyset, \mathbb{R}^2, \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < r \}, r \in \mathbb{R}, r > 0\}\).
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Context:
How many Hamiltonian cycles are there in the cube graph $Q_3$?
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Context:
Let $m$ be the integer part of $\sqrt{13}$, and let $n$ be the decimal part of $\sqrt{13}$. Find the value of $(m-n)^{2}$.
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Context:
In $\triangle ABC$, when $a^{2}+c^{2}-b^{2}=\sqrt{3} ac$, the angle $B=$____.
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Context:
Evaluate the triple integral $$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$$ where V is the unit sphere.
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Context:
Given the quadratic equations \(x^2 - 10ax - 11b = 0\) with roots \(c\) and \(d\), and \(x^2 - 10cx - 11d = 0\) with roots \(a\) and \(b\), find the value of \(a + b + c + d\).
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Context:
对 $n=5$, 令 $L_{n}=\frac{1}{n!} \frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{2}-1\right)^{n}, S=\int_{-1}^{1}\left|L_{n}\right|^{2} \mathrm{~d} x$ ，则 $[|S|]=$ $\qquad$.
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Context:
Non-negative real numbers $a_1, a_2, \ldots, a_n$ satisfy $a_1 + a_2 + \ldots + a_n = k$, where $k$ is a constant. Determine the maximum value of the expression $$a_1a_2 + a_2a_3 + \ldots + a_{n-1}a_n + a_na_1.$$
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Context:
Evaluate the limit using real analysis methods: $$\lim_{ n \rightarrow +\infty} \frac{1}{\Gamma(n)} \int_0^n x^{n-1}e^{-x}\, {\rm d}x$$ where \(\Gamma(n)\) is the gamma function.
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Context:
Sisyphus has three boxes of stones. He moves stones one by one between the boxes. Each time he moves a stone, Zeus gives him coins equal to the difference between the number of stones in the destination box and the source box (excluding the moved stone). If the difference is negative, Sisyphus returns the corresponding amount to Zeus. After all stones are returned to their initial boxes, what is the greatest possible earning Sisyphus can achieve?
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Context:
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12 square units, what is the area of the equilateral triangle? Express your answer in square units.
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Context:
已知$\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {a}、\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {b}$是平面内两个互相垂直的单位向量，且此平面内另一向量$\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c}$在满足$\left( 3\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {a}+\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c} \right)\left( 4\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {b}-\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c} \right)=0$，均能使$\left| \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c}-\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {b} \right|\le k$成立 ，则$k$的最小值是_________.
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Context:
If $f(x) = g(x^2)$ where $g(x)$ is a twice-differentiable function, and $f''(0) = 4$, what is $g''(0)$?
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Context:
There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible sum?
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Context:
A bowl of fruit holds 18 peaches. Some of the peaches are ripe and two more ripen every day, but on the third day three are eaten. After five days, there are 7 more ripe peaches than unripe peaches in the bowl. How many peaches were ripe initially?
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Context:
Find the cardinality of the set \( \{ (x,y) \in \mathbb{R}^2 \; | \; y > x > 0, \; x^x = y^y \} \).
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Context:
If points $A(x_{1}$,$-1)$, $B(x_{2}$,$3)$, and $C(x_{3}$,$5)$ are all on the graph of the inverse proportion function $y=-\frac{4}{x}$, then the relationship between $x_{1}$, $x_{2}$, and $x_{3}$ is ______.
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Context:
Calculate the surface area of the sphere defined by \(x^2 + y^2 + z^2 = 1\) that lies above the disc \((x-1/2)^2 + y^2 \le 1/4\) in the xy-plane.
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Context:
Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake?
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Context:
The lateral sides \(AB\) and \(CD\) of trapezoid \(ABCD\) are equal to 2 and 3, and the angles at vertices \(A\) and \(D\) are acute. The angle bisectors of angles \(A\) and \(B\) of the trapezoid intersect at point \(M\), and the angle bisectors of angles \(C\) and \(D\) intersect at point \(N\).

The length of segment \(MN\) is 4. Find the radius of the circle inscribed in triangle \(ABD\) if the area of trapezoid \(ABCD\) is \(\frac{26\sqrt{2}}{3}\).
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Context:
A car radiator has a 6-liter capacity. If the liquid in the radiator is a certain percentage of antifreeze, 1 liter of liquid must be replaced with 100% antifreeze to bring the mixture up to a 50% solution. What is the initial percentage of antifreeze in the radiator?
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Context:
On a rectangular sheet of paper, a picture is drawn in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$, where the sides are parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
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Context:
Determine the maximum area of a $1000 \times 1000$ grid of unit squares that can be covered by eels, where an eel is a polyomino formed by a path of unit squares making two turns in opposite directions.
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Context:
James injured his ankle and decides to slowly start working back up to his previous running goals and then surpass them.  Before the injury, he was able to run 100 miles per week.  He wants to get up to 20% more than that total in 280 days and each week he will increase miles walked in the week by the same amount.  How many miles does he need to add per week?
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Context:
The sides of a triangle have the combined equation \(x^2 - 3y^2 - 2xy + 8y - 4 = 0.\) The third side, which is variable, always passes through the point \((-5, -1)\). If the range of values of the slope of the third line, such that the origin is an interior point of the triangle, lies in the interval \((a, b)\), find the value of \(a + \frac{1}{b^2}\).
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Context:
A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively.
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Context:
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
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Context:
A man saves a certain portion of his income during a year and spends the remaining portion on his personal expenses. Next year, his income increases by 25% and inflation causes his expenses to increase by 10%. In addition to this, he receives a tax rebate equal to 5% of his previous year's income, which he adds to his savings. As a result, his savings increase by 100%. If his total expenditure over the two years is double his expenditure in the first year, taking into account inflation, and assuming his tax rate remains constant, what percentage of his income in the first year did he save?
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Context:
What is the probability that the sum of the squares of four random numbers chosen uniformly from the interval [0, 1] is less than one?
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Context:
If the line $y=kx+1$ intersects the circle $x^2+y^2+kx+my-4=0$ at points M and N, and M and N are symmetric with respect to the line $x+y-1=0$, then the value of $k-m$ is \_\_\_\_\_\_.
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Context:
Let $x,y,z>0$ such that: $x+y+z=1$. Using Lagrange multipliers, find the minimum value of the expression:

$$P=(\frac{x+1}{x})^3.\frac{z^2}{z^2+1}+(\frac{y+1}{y})^3.\frac{x^2}{x^2+1}+(\frac{z+1}{z})^3.\frac{y^2}{y^2+1}$$
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Context:
Convex quadrilateral ABCD is given, and lines are extended from points A, B, C, and D to points B', C', D', and A' respectively, such that AB = BB' = 6, BC = CC' = 7, CD = DD' = 8, and DA = AA' = 9. If the area of quadrilateral ABCD is 10, calculate the area of quadrilateral A'B'C'D'. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the Burgers equation \( u_t + uu_x = 0 \) with the initial condition
\[ u(x,0) = \phi(x) = \begin{cases}
      2 & x \leq \frac{\pi}{2} \\
      \sin(x) + 1 & \frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \\
      0 & x \geq \frac{3\pi}{2}
   \end{cases} \].
Determine the breaking time \( t_B \), which is the time at which the characteristic curves intersect.
Please reason step by step, and put your final answer within \boxed{}.


Context:
从正九边形中任取三个顶点构成三角形, 则正九边形的中心在三角形内的概率为 $\qquad$.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given point M(-2, 1), let N(x0, 1). If there exists a point P on the circle ⊙O: x² + y² = 1 such that ∠MNP = 60°, determine the range of x0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center of mass) of $\triangle ABC$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit: \lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}. The original answer is in the format $\frac{m}{n}$, please give the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola $y^{2}=4x$, a line with a slope of $\frac{\pi}{4}$ intersects the parabola at points $P$ and $Q$, and $O$ is the origin of the coordinate system. Find the area of triangle $POQ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\Gamma} \pi e^{\pi\bar{z}} \, dz \) where \( \Gamma \) is the square with vertices at \( 0, 1, 1+i, i \) oriented anticlockwise.
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Context:
Isabel has $204 in her piggy bank. She spent 40% of the amount on a toy for herself. Then, with the remaining money, she spent 50% on a book for her brother and saved 20% for a charity donation. After that, she spent 30% of the remaining amount on a gift for her friend and decided to save 10% of what was left for her future expenses. How much money, in dollars, was left over in her piggy bank after all transactions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit (if it exists) of the sequence $\{a_n\}$ defined by $a_1 = 1$ and the recursion formula $a_{n+1} = \frac{3n-1}{3n} a_n + \frac{1}{n^2}$ for all $n \geq 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The inclination angle of the line $\sqrt{3}x+y-1=0$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer  $n$ , consider a triangular array with entries  $a_{ij}$  where  $i$  ranges from  $1$  to  $n$  and  $j$  ranges from  $1$  to  $n-i+1$ . The entries of the array are all either  $0$  or  $1$ , and, for all  $i > 1$  and any associated  $j$  ,  $a_{ij}$  is  $0$  if  $a_{i-1,j} = a_{i-1,j+1}$ , and  $a_{ij}$  is  $1$  otherwise. Let  $S$  denote the set of binary sequences of length  $n$ , and define a map  $f \colon S \to S$  via  $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$ . Determine the number of fixed points of  $f$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample?
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Context:
Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be the midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_2$ be the radius of $(AEF)$. If $\frac{R_1^2}{R_2^2} = a - \sqrt{b + \sqrt{c}}$ where $a, b, c$ are integers, find $a^{b^{c}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Big Joe is the tallest player on the basketball team. He is one foot taller than Ben, who is one foot taller than Larry, who is one foot taller than Frank, who is taller than Pepe. Pepe is 4 feet six inches tall, and Big Joe is 8 feet tall. How much taller is Frank than Pepe, in feet?
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Context:
On a  $300 \times 300$  board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook.  At what least  $k$ , it is possible to state that there is at least one rook in each  $k\times k$  square ?
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Context:
Find $G'(0)$ if \[ G(x) = \int_{\cos x}^{x^2}\left( \int_0^{t^2}\sqrt{1+s^4} \ ds \right) dt. \]
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Context:
Find the largest positive integer $k$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:

- The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
- It is possible to color the edges of $\mathcal{P}$ with $k$ colors such that for every color $c$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the color $c$.
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Context:
Ann and Max play a game on a $100 \times 100$ board. Ann writes an integer from 1 to 10,000 in each square of the board, using each number exactly once. Max starts by placing a token on a square in the leftmost column and moves it to the rightmost column, moving to adjacent squares by side or vertex. Max pays Ann the number of coins equal to the number in each visited square. If both players use their best strategies, how much money will Max pay Ann?
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Context:
已知点 A,F_1,F_2 分别为椭圆 \df{x^2}{4}+\df{y^2}{3}=1 的左顶点和左、右焦点. 过点 A 作斜率为 k 的直线交椭圆于另一点 B，连接 BF_2 并延长交椭圆于点 C. 若 F_1C\bot AB，则 k^2 的值为__________.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
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
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Context:
Let $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the number $\{x + x^2 + \cdots + x^{20}\}$, where $\{a\}$ denotes the fractional part of $a$.
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Context:
有 $n$ 个人，已知他们中的任意两人至多通电话一次，他们中的任意 $n-2$ 个人之间通电话的次数相等, 都是 $3 k$ 次, 其中 $k$ 是自然数, 求 $n$ 的可能值的和.
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Context:
Given that $\sin \alpha = \frac{4\sqrt{3}}{7}$ and $\cos(\alpha + \beta) = -\frac{11}{14}$, where both $\alpha$ and $\beta$ are acute angles, find the value of $\cos \beta$.
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Context:
A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle.
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Context:
A square $OPQR$ is inscribed in $\triangle ABC$. The areas of $\triangle AOR$, $\triangle BOP$, and $\triangle CRQ$ are given as $S_{1}=1$, $S_{2}=3$, and $S_{3}=1$, respectively. Find the side length of the square $OPQR$. Express your answer as a single number.
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Context:
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
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Context:
Calculate the line integral $$\int_C(x-y)dz$$ where curve $C$ is the intersection of the paraboloid $z=x^2+y^2$ and the plane $z-2x-2y=1$, oriented counterclockwise when viewed from the point $(0,0,10)$.
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Context:
8 identical books are to be divided into three piles. There are a total of $\boxed{\text{answer}}$ different ways to do this.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The cubic polynomial
\[8x^3 - 3x^2 - 3x - 1 = 0\]has a real root of the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},$ where $a,$ $b,$ and $c$ are positive integers.  Find $a + b + c.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the divergence of the vector

$$
\mathbf{a}=\varphi(r) \mathbf{r}^{0}=\frac{\varphi(r)}{r} \mathbf{r}
$$

where \( r = |r| \) is the distance from the origin to the variable point \( M(x, y, z) \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \frac{\varphi(r)}{r} + \varphi'(r)}


Context:
Among five points $A, B, C, D, E$ in space, any four points are non-coplanar and the distances between any two points are distinct. For each tetrahedron formed by selecting four out of the five points, the shortest edge is colored black. After all the edges have been colored, the number of black segments emanating from a point is called the "wish number" of that point. What is the maximum value of the wish number among these points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{0}^{2\pi} \frac{\cos(3\theta)}{5-4\cos(\theta)}d\theta \] without using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the series:
$$\sum_{n=2}^\infty\frac{({2}^{n})(n-1)(n!)}{(2n)!}$$
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Context:
Let $Z_1, Z_2, \ldots, Z_7$ be independent standard normal random variables, each distributed as $N(0,1)$. Calculate the probability that the following inequality holds:
\[ P(Z_1^2 + Z_2^2 + Z_3^2 \leq 4.943(Z_4^2 + Z_5^2 + Z_6^2 + Z_7^2)) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_1,\ldots,n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^{4}=B^{7}=ABA^{-1}B=1,A^{2}\neq1$, and $B\neq1$.
 How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?
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Context:
Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns to take 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, while still following the rules of the game. The computer plays according to the following algorithm: on each turn, it takes as many stones as necessary to put itself in the most advantageous position. Petya always starts the game. What is the probability that Petya will win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers \(1, 2, \cdots, 9\) are randomly placed into a \(3 \times 3\) grid such that each cell contains exactly one number and each number is used exactly once. Calculate the probability that the sum of the numbers in each row and each column is odd. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $(x+1)^5 - x^5 = a + a_1 (x+4)^4 x + a_2 (x+1)^3 x^2 + a_3 (x+1)^2 x^3 + a_4 (x+1) x^4$, and $a_1$ (where $i=0,1,\ldots,4$) are constants, then find the value of $a_1 + a_3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知实数 $x_{1}, x_{2}, \cdots, x_{2018}$ 满足：对任意的 $1 \leqslant i<j \leqslant 2018$, 均有 $x_{i}+x_{j} \geqslant(-1)^{i+j}$. 求 $\sum\limits_{i=1}^{2018} i x_{i}$ 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
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac{1}{4}$ of the way to his destination, and in $45$ minutes he will be exactly $\frac{1}{3}$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an equilateral triangle with vertices R, S, and T, and midpoints of its sides X, Y, and Z, how many noncongruent triangles can be formed using any three of these six points as vertices? Provide your answer as a whole number.
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Context:
18 people play in a tournament of 17 rounds. There are 9 games in each round and each person plays in one game in each round. Each person plays every other person just once in the tournament. What is the largest \( n \) such that however the tournament is arranged we can find 4 people amongst whom only one game has been played by the end of round \( n \)?
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Context:
Provide an example of a non-commutative ring that has only the trivial ideals \( \{0\} \) and the ring itself, but is not a field.
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Context:
Find the maximum value of the integral $\int_0^1 (f(x))^3 dx$ if $|f(x)| \leq 1$ and $\int_0^1 f(x)dx = 0$. Express this maximum value as the fraction $\frac{p}{q}$, where $p$ and $q$ are coprime integers.
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Context:
Find the maximum value of the parameter \( b \) for which the inequality \( b \sqrt{b}\left(x^{2}-10 x+25\right)+\frac{\sqrt{b}}{\left(x^{2}-10 x+25\right)} \leq \frac{1}{5} \cdot \sqrt[4]{b^{3}} \cdot \left| \sin \frac{\pi x}{10} \right| \) has at least one solution.
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Context:
Given a parabola $C$: $y^{2}=4x$ with a point $M(4,-4)$ on it. Points $A$ and $B$ are two moving points on the parabola such that $\overrightarrow{MA} \cdot \overrightarrow{MB}=0$. Determine the maximum distance from point $M$ to line $AB$.
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Context:
What is the term for a partially ordered set that is isomorphic to its own dual?
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Context:
Let $F$ and $E$ be fields with $F < E$. Suppose $V$ is a vector space over $F$ with $\mathrm{dim}_F(V) = n$. Consider $V^E = V \otimes_F E$, viewing $V$ as a vector space over $E$. What is $\mathrm{dim}_E(V^E)$?
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Context:
A and B play a game on a $2^{100} \times 100$ grid. They take turns filling in symbols in the first row, with A going first. In each round, A chooses an empty cell in the first row and fills it with an "$\times$"; then, B chooses an empty cell in the first row and fills it with an "○". Once all the cells in the first row are filled, they start filling in the symbols in the second row, and so on, until all the cells in the grid are filled.

A's goal is to have as many distinct rows as possible in the grid, while B's goal is to have as few distinct rows as possible. If both use their optimal strategies, how many distinct rows will there be in the grid in the end?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the hyperbola such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = 0$ and $|\overrightarrow{PF\_1}| \cdot |\overrightarrow{PF\_2}| = 2ac (c$ is the semi-focal distance$)$, find the eccentricity of the hyperbola.
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Context:
A circle passes through vertices \(A\), \(B\), and \(C\) of a parallelogram \(ABCD\) with sides \(AB = 5\) and \(BC = 2\) and intersects the line containing the diagonal \(BD\) at point \(E\). It is known that \(BE = 8\). Find \(BD\).
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Context:
Elijah is painting his L-shaped living room, which consists of a main rectangle and an adjacent square. The main rectangular part of the room has two walls that are 5 meters wide and two walls that are 3 meters wide, while the square part of the room has walls that are 4 meters wide. The height of the room is 2.5 meters. Elijah needs enough masking tape to cover the entire width of the walls and windows, as well as tape off the door frames located on one of the 3-meter walls.

He decides to apply two layers of masking tape in order to ensure proper coverage. There are 3 windows in the living room with dimensions of 1.5 meters by 1 meter and a door frame that is 2 meters high and 1 meter wide. Elijah can order the exact amount of masking tape he needs. Considering the dimensions of the room, doors, and windows, how many meters of masking tape does Elijah need to order?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sphere of radius \( r \) touches all the lateral faces of a triangular pyramid at the midpoints of the sides of its base. The segment connecting the apex of the pyramid with the center of the sphere is divided into two equal parts by the point of intersection with the base of the pyramid. Find the volume of the pyramid.
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Context:
Given that the real numbers $a$, $b$, and $c$ form an arithmetic sequence, and the projection point $M$ of point $P$($-3$, $0$) on the moving line $ax + by + c = 0$ ($a$ and $b$ are not simultaneously zero). If the coordinates of point $N$ are ($2$, $3$), find the range of $|MN|$.
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Context:
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers  $x$ .

*2019 CCA Math Bonanza Lightning Round #3.4*
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Context:
Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$.
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Context:
Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, if $a_2a_4a_6a_8=120$ and
$$\frac {1}{a_{4}a_{6}a_{8}}+ \frac {1}{a_{2}a_{6}a_{8}}+ \frac {1}{a_{2}a_{4}a_{8}}+ \frac {1}{a_{2}a_{4}a_{6}}= \frac {7}{60},$$
find the value of $S_9$.
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Context:
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, where the left and right vertices are $A$ and $B$ respectively, the right focus is $F$, $P$ is a point on the ellipse, the line $AP$ intersects the line at point $M$, the angle bisector of $\angle PFB$ intersects the line $x=a$ at point $N$. If $PF\perp AB$, the area of triangle $MAB$ is $6$ times the area of triangle $NFB$, then the eccentricity of the ellipse $C$ is ____.
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Context:
Determine the values of \( \alpha \in \mathbb{R} \) such that the function \((x,y) \mapsto (1-xy)^{\alpha}\) is integrable on the region \((0,1)\times (0,1)\).
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Context:
Let \(T_1\) be a triangle with side lengths \(2011\), \(2012\), and \(2013\). For \(n \geq 1\), if \(T_n = \triangle ABC\) and \(D, E\), and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB\), \(BC\), and \(AC\), respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE\), and \(CF\), if it exists. Find the perimeter of the last triangle in the sequence \(\left(T_n\right)\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company employs 3 workers, each with a tenure following an exponential distribution with a mean of 1 year before quitting. The time to hire a replacement worker is also exponentially distributed with a mean of 0.1 year, independently for each position. The daily revenue based on the number of workers is as follows: 3 workers - $1000, 2 workers - $800, and 0 or 1 worker - $500. Calculate the company's expected long-run average daily revenue.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求 $\lim\limits _{n \rightarrow+\infty} \sum_{k=0}^{n-1} \frac{120}{\sqrt{n^{2}+k n}}$. 原始答案为 $m(\sqrt{n}-q)$ 的形式，请给出m+n+q的值。
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Context:
Determine the sum of the infinite series $$\sum_{n=1}^{\infty} \frac{f(n)}{n(n+1)}$$, where $f(n)$ represents the number of ones in the binary representation of the integer $n$.
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Context:
Let \(a > 1\) be a positive real number and \(n \geq 2\) be a natural number. The equation \(\lfloor a x \rfloor = x\) has exactly \(n\) different solutions. Find the range of values for \(a\).
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Context:
Cathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Cathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$, 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:
Let $A B C$ be an acute scalene triangle with circumcenter $O$ and centroid $G$. Given that $A G O$ is a right triangle, $A O=9$, and $B C=15$, let $S$ be the sum of all possible values for the area of triangle $A G O$. Compute $S^{2}$.
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Context:
The power function $y=(m^2-m-1)x^{1-m}$ is decreasing for $x \in (0, +\infty)$. Find the value of $m$.
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Context:
Determine the value of $$\sum_{k=1}^{2011} \frac{k-1}{k!(2011-k)!}$$
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Context:
Let $w = e^{i\cdot \pi/2022}$. Determine the value of the sum $T = w^1 + w^3 + w^5 + ... + w^{2021}$.
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Context:
Given a fluid with a velocity field and density, calculate the rate of change of density in the Lagrangian frame, where the initial density \( \rho_0 = 2 \) and \( A = 1 \).
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Context:
7.

In triangle $A B C$, the bisector $A L$ ( $L \in B C$ ) is drawn, and $M$ and $N$ are points on the other two bisectors (or their extensions) such that $M A=M L$ and $N A=N L, \angle B A C=50^{\circ}$.

Find the measure of $\angle M A N$ in degrees.
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Context:
Given that point $P\left(x,y\right)$ is located in the second quadrant, and $y\leqslant x+4$, where $x$ and $y$ are integers, write down the coordinates of a point $P$ that satisfies the above conditions.
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Context:
How many solutions does the equation \( \frac{\lfloor x \rfloor}{\{ x \}} = \frac{2007x}{2008} \) have?
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Context:
Given the line equation ax + by + c = 0, where ab < 0 and bc < 0, determine the quadrant that the line does not pass through. Express your answer as the name of the quadrant (e.g., First, Second, Third, Fourth).
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Context:
Given the sequence $\left\{a_{n}\right\}$ satisfying $a_{1}=1$ and $a_{n+1}=a_{n}+\frac{1}{2 a_{n}}$, find $\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)$.
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Context:
Higher Secondary P4

If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n^3} \sum_{i=1}^n \left(e^{\frac{2i}{n}} + \frac{4i^2}{n^2}\right)$$
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Context:
Calculate the line integral \( \int_C \vec{B} \cdot d\vec{r} \), where \( \vec{B} = \frac{\cos(\alpha)}{p^2}e_p + \left(\frac{\sin(\alpha)}{p^2}+\frac{1}{ap}\right)e_\alpha + 0e_z \) and the curve \( C \) is parameterized by \( \vec{r} = (b\cos(s),b\sin(s),b) \) with \( b>0 \) and \( s: 0\rightarrow2\pi \).
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Context:
Calculate the mass of a plate with surface density \( \mu = \frac{x}{y^5} \), bounded by the curves \( \frac{x^2}{16} + y^2 = 1 \), \( \frac{x^2}{16} + y^2 = 3 \), \( y = \frac{x}{4} \), and \( x = 0 \) where \( y \geq \frac{x}{4} \) and \( x \geq 0 \).
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Context:
A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful?
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Context:
Simplify the expression into a closed form: $$\frac{1}{6}\sum_{n=0}^{\infty} \sum_{k=0}^{n} \left[{n \choose k}\left(\frac{1}{3}\right)^{k}\left(\frac{1}{2}\right)^{n-k} (2+5k+3(n-k))\right]$$
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Context:
Each cell of a \(3 \times 3\) square contains a natural number. All the numbers are distinct and not equal to one. It is known that the number in each cell is a divisor of the product of all the numbers in the cells adjacent to it (by side). Find the maximum possible number of prime numbers among those written in the cells.
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Context:
Find the probability that if the digits $0,1,2, \ldots, 9$ are placed in random order into the empty places in the row of digits

$$
5-383-8-2-936-5-8-203-9-3-76
$$

the resulting number will be divisible by 396.
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Context:
There are two people each wearing red and yellow clothes, and one person wearing blue clothes. Now, these five people are to be arranged in a row, with the requirement that people wearing the same color clothes cannot be adjacent. The total number of different arrangements is ___.
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Context:
Circle $\omega_1$ has radius $7$ and center $C_1$. Circle $\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\omega_1$ and $\omega_2$ pass through $A_1$ on $\omega_1$ and $A_2$ on $\omega_2$, and let a common external tangent of $\omega_1$ and $\omega_2$ pass through $B_1$ on $\omega_1$ and $B_2$ on $\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$.
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Context:
One day, School A bought 56 kilograms of fruit candy at 8.06 yuan per kilogram. A few days later, School B also needed to buy the same 56 kilograms of fruit candy, but it happened that there was a promotional event, and the price of fruit candy was reduced by 0.56 yuan per kilogram. Additionally, they received 5% extra fruit candy for free. How much less did School B spend compared to School A?
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Context:
Evaluate the integral: $$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx.$$
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Context:
Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$, there exist at least $k$ consecutive composite integers in the set \{$n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$\}.
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Context:
将正九边形的每个顶点等概率地涂上红、蓝两种颜色之一, 则存在三个同色的顶点构成锐角三角形的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Find an isomorphism between the localization of the ring $\mathbb{Z}_{12}$ at the set $S=\{4,8\}$ and a familiar ring.
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Context:
Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f, a, b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$, and $a, b \in \mathbb{N}_0$ that satisfies the following conditions:

- $f(1) = 2$
- $f(a) + f(b) \leq 2\sqrt{f(a)}$
- For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b + 100)$.
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Context:
Suppose that $Y$ is a positive continuous random variable with probability density function (pdf) $f_Y(t) = t f_X(t)$, where $f_X(t)$ is the pdf of a positive continuous random variable $X$. Given that $\operatorname{var}(Y) = 2$, determine $\operatorname{var}(X)$. \( \text{Provide your answer as a numerical value.} \)
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Context:
Determine whether the series \( \sum\limits_{n=1}^\infty q^{\sqrt{n}} \) converges for \( q > 0 \).
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Context:
Let $K$ be an algebraically closed field of characteristic $0$. Determine the polynomial condition for $a \in K$ such that the polynomial $f(x) = x^3 - 3ax + 1$ has distinct roots.
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Context:
Calculate the line integral of the vector field along the curve \( y = x^2 \) from \( x = 0 \) to \( x = 1 \).
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Context:
Compute the limit: $$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx$$
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Context:
Find the chromatic number of a graph $H$ where the vertices represent non-empty subsets of $[n]$, and two vertices are connected if their corresponding subsets have a non-empty intersection.
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Context:
Suppose X, Y, and Z are independent exponential random variables with rate \( \lambda \). Calculate the probability that \( X \le 2Y \le 3Z \).
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Context:
Determine the number of real solutions to the differential equation \( y'' = -4yy' \).
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Context:
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is:
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Context:
Find the minimal polynomial of the linear transformation $T_p: M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ defined by $T_p(X) = P^{-1}XP$, where $P = \begin{pmatrix} i & 2 \\ -1 & -i \end{pmatrix}$. Determine the minimal polynomial of $T_p$.
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Context:
Let $G = \langle g_1, g_2 \rangle$ be a group, and let $H = \langle g_1 g_2 g_1^{-1} g_2^{-1} \rangle$ be a subgroup of $G$. Determine the normal subgroup of $G$ generated by $H$.
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Context:
What is the term used to describe a topological space where every proper filter can be refined by a Cauchy filter, which is a stronger condition than being totally bounded?
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Context:
Compute the limit: $$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$$
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Context:
What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43?
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Context:
Compute $i^{-100}+i^{-99}+i^{-98}+\cdots+i^{-1}+i^0+i^1+\cdots+i^{99}+i^{100}$.
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Context:
Rabbits Peter and Pauline have three offspring--Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
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Context:
Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.
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Context:
Let \( \mathbb{Z}_{\ge 0} \) denote the set of nonnegative integers. Define a function \( f:\mathbb{Z}_{\ge 0} \to\mathbb{Z} \) with \( f(0)=1 \) and \[ f(n)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \] for all \( n \ge 1 \). Determine the number of nonnegative integers \( n \) such that the hexadecimal (base \( 16 \)) representation of \( f(n) \) contains no more than 2500 digits.
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Context:
Given $0 \leqslant \alpha \leqslant \frac{\pi}{2}, 0<\beta \leqslant \frac{\pi}{2}$, find the minimum value of $\cos ^{2} \alpha \sin \beta+\frac{1}{\sin \beta}$. Express your answer as a single number.
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Context:
The altitudes of an acute-angled triangle \(ABC\) drawn from points \(B\) and \(C\) are extended to intersect with the circumcircle at points \(B_1\) and \(C_1\). It turned out that the segment \(B_1C_1\) passes through the center of the circumcircle. Find the angle \(BAC\).
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Context:
If a square is divided into parts by three lines, there is always a polygon among the resulting parts whose diameter is not less than $\sqrt{13}$. What is the minimum side length of the square? (The diameter of a polygon is the distance between its two farthest vertices.)
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Context:
Let $V = \mathbb{R}^n$ with a non-standard inner product defined by a metric matrix $G$. Let $L$ be the linear operator on $V$ represented by the matrix $A$. Find the matrix representation of the adjoint operator $L^A$ in terms of the matrix $A$ and the metric matrix $G$.
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Context:
In triangle \( ABC \), the measure of angle \( \angle B \) is \( 16^\circ \). Let \( I \) be the incenter and \( E \) be the excenter relative to side \( BC \). If \( IE = 2 \times AC \), calculate the measure of angle \( \angle ACB \).
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Context:
Tian Ji and the King of Qi had a horse race. Tian Ji's top horse is better than the King of Qi's middle horse but worse than the King of Qi's top horse. Tian Ji's middle horse is better than the King of Qi's bottom horse but worse than the King of Qi's middle horse. Tian Ji's bottom horse is worse than the King of Qi's bottom horse. Now, if a horse is randomly selected from each side for a race, what is the probability that the King of Qi's horse wins? Express your answer as a simplified fraction.
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Context:
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:

$$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$

[i]Proposed by Mykyta Kharin[/i]
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Context:
The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$.
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Context:
A train of a certain length is running at a speed of 55 km/hr. It crossed a platform of length 520 m in 64.79481641468682 seconds. What is the length of the train?
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Context:
Given the numbers \( x_{1}, \ldots, x_{n} \in\left(0, \frac{\pi}{2}\right) \), find the maximum value of the expression

\[
A=\frac{\cos ^{2} x_{1}+\ldots+\cos ^{2} x_{n}}{\sqrt{n}+\sqrt{\operatorname{ctg}^{4} x_{1}+\ldots+\operatorname{ctg}^{4} x_{n}}}
\]
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Context:
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \);..., circle \( C_{1} \) passes through exactly 1 point in \( M \). What is the minimum number of points in \( M \)? Express your answer as a single integer.
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Context:
Given \(5^p + 5^3 = 140\), \(3^r + 21 = 48\), and \(4^s + 4^3 = 280\), find the product of \(p\), \(r\), and \(s\).
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Context:
Given an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\) with left focus \(F\). Points \(P_{1}, P_{2}, \cdots, P_{n}\) are \(n\) points arranged in a clockwise order on the ellipse such that \(\angle P_{1}FP_{2} = \angle P_{2}FP_{3} = \angle P_{3}FP_{4} = \cdots = \angle P_{n}FP_{1}\). Find the sum of the reciprocals of the distances of these \(n\) points to the left directrix \(x = -\frac{a^{2}}{c} \ (c=\sqrt{a^{2}-b^{2}})\) expressed in terms of \(a, b, c, n\).
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Context:
A new model car travels 4.2 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption for the new model is 2 liters less per 100 km. How many liters of gasoline per 100 km does the new car consume? Round your answer to the nearest hundredth if necessary.
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Context:
Find the smallest integer M such that if a number N is divisible by all integers from 1 to M, then N must also be divisible by M + 1, M + 2, M + 3, and M + 4.
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Context:
Three balls are lying on a table, each pair touching each other. The radii of the balls form a geometric progression with a common ratio \( q \neq 1 \). The radius of the middle ball is 2012. Calculate the ratio of the sum of the squares of the sides of the triangle formed by the points of contact of the balls with the table to the sum of the sides of the triangle formed by the centers of the balls.
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Context:
Find the minimum value of the function \( f(x,y,z) = xy + 2xz + 3yz \) subject to the constraint \( xyz = 6 \) and \( x \ge 0 \), \( y \ge 0 \), \( z \ge 0 \). Use the method of Lagrange multipliers.
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Context:
Given \( z \in \mathbf{C} \) (a complex number), \( \arg \left(z^{2}-4\right) = \frac{5 \pi}{6} \) and \( \arg \left(z^{2}+4\right) = \frac{\pi}{3} \), find the value of \( z \).
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Context:
Determine the dimension of the quotient space \(c_0/W\), where \(c_0\) is the space of sequences converging to zero, and \(W\) is the subspace of \(c_0\) defined by \(W = \{ (y_n) \in c_0 : y_1 + y_2 + y_3 = 0 \}.\)
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Context:
In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line is drawn perpendicular to the hypotenuse, dividing the triangle into two equal areas. In what ratio does this line divide the hypotenuse?
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Context:
Let \( \phi : G \to H \) be a group homomorphism with kernel \( K \). If \( \phi(g) = h \), find the fiber over \( h \) in \( G \), i.e., the set \( \phi^{-1}(h) \).
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Context:
Find the largest positive integer \( n \) such that the system of equations
$$
(x+1)^{2}+y_{1}^{2}=(x+2)^{2}+y_{2}^{2}=\cdots=(x+k)^{2}+y_{k}^{2}=\cdots=(x+n)^{2}+y_{n}^{2}
$$
has integer solutions \(\left(x, y_{1}, y_{2}, \cdots, y_{n}\right)\).
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Context:
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
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Context:
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of $\angle E$?
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Context:
Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $S$.
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Context:
In the x-y plane, there is a right triangle ABC (∠B = 90°). If the length of AC is 225 and the slope of the line segment AC is 4/3, what is the length of AB?
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Context:
Given a triangle $ABC$ with internal angles $A$, $B$, $C$ and the lengths of the sides opposite these angles are $a$, $b$, and $c$ respectively. If $a^{2}=b^{2}+c^{2}-2bc\sin A$, then the magnitude of internal angle $A$ is \_\_\_\_\_\_.
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Context:
Elfs and trolls are seated at a round table, with a total of 60 creatures. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everyone claims to sit between an elf and a troll, but exactly two elfs made a mistake. How many trolls are there at this table?
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Context:
Simplify and find the value of: $5x^2y-[3xy^2-2(3xy^2-\frac{7}{2}x^2y)]$, where $|2+y|+(x-1)^{2}=0$.
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Context:
Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[ \text{Area}(WXYZ) = \gamma \omega - \delta \omega^2. \]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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Context:
In the Cartesian coordinate system $xOy$, the area of the closed figure formed by the curve $$y= \frac {1}{x} (x > 0)$$, the line $y=x$, and the line $y=3$ is \_\_\_\_\_\_.
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Context:
Given triangle ABC, vectors $\overrightarrow {m}$ = ($\sqrt {3}$sinA, sinB) and $\overrightarrow {n}$ = (cosB, $\sqrt {3}$cosA). If the dot product of $\overrightarrow {m}$ and $\overrightarrow {n}$ is equal to 1 + cos(A + B), find the value of angle C.
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Context:
Determine the number of permutations of the digits 1 through 9 such that the blocks 12, 34, and 567 do not appear.
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Context:
Determine whether the limit \( \lim_{n \rightarrow \infty} \sum_{k = 1}^\infty \frac{(1-\epsilon(n))^k}{k} \) exists and is finite, given that \( \epsilon(n) \rightarrow 0 \) as \( n \rightarrow \infty \).
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Context:
In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4 \times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2 \times 2$ squares.)
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Context:
Matthew, Shawn, and their friend Amber played swimming-pool-basketball. Each basket was worth either 2 points or 3 points. Matthew scored 10 points, Shawn scored 8 points, and Amber scored 6 points. If Matthew scored 3 baskets that were worth 3 points each, and Amber scored 2 baskets that were worth 3 points each, how many baskets in total were made during this game, considering the combination of 2-point and 3-point baskets?
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Context:
Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, what is the value of $a$?
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Context:
If the function $y=\sin \omega x$ is monotonically increasing in the interval $[0,2\pi]$, then the range of the real number $\omega$ is \_\_\_\_\_\_.
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Context:
If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:
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Context:
Given \(x_{i} \in \mathbf{R}_{+}\) for \(i=1,2,3,4,5\), find the minimum value of:

\[f = \frac{x_{1}+x_{3}}{x_{5}+2 x_{2}+3 x_{4}} + \frac{x_{2}+x_{4}}{x_{1}+2 x_{3}+3 x_{5}} + \frac{x_{3}+x_{5}}{x_{2}+2 x_{4}+3 x_{1}} + \frac{x_{4}+x_{1}}{x_{3}+2 x_{5}+3 x_{2}} + \frac{x_{5}+x_{2}}{x_{4}+2 x_{1}+3 x_{3}}.\]
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Context:
In a swimming pool, there are 3 valves A, B, and C. With all three valves open, the pool can be filled with water in 40 minutes. Valve A alone would fill the pool in 3 hours, and valve B can fill the pool in 4 hours. If valve C emits 75 cubic meters of water more than valve A every minute, and valve B emits 60 cubic meters of water more than valve A every minute, what is the capacity (T) of the pool?
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Context:
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is selected. On one side of \(AO\) on the circumference, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\), given that the radius of the circle is \(28\), and \(\sin \alpha = \frac{\sqrt{45}}{7}\).
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Context:
Given an isosceles triangle where one of the legs and the base are the roots of the equation $x^2 - 6x + a = 0$, find the range of values for $a$ when there is only one such triangle.
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Context:
Consider a volleyball net with sides 10 and 20, divided into $10 \times 20$ square cells, where each cell is further divided into four triangular cells by its diagonals: the vertices of the cell are connected to the center of the cell with strings. The nodes of the net are the vertices and centers of the cells. What is the maximum number of strings connecting neighboring nodes that can be cut so that the net does not fall apart into separate pieces?
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Context:
Determine the number of analytic functions that satisfy the condition \( f\left(\frac{1}{n\pi}\right)=1 \) for \( n\in\mathbb{N} \) within a domain \( D \), where \( D \) is a disk centered at \( z = a \) with radius \( r \), and \( 0 \notin D \).
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Context:
Let $V$ be the vector space $M_{n \times n}(\mathbb{R})$, and let $B : V \times V \to \mathbb{R}$ be the bilinear form defined by $B(X,Y) = \text{tr}(XY^T)$. Calculate the signature of $B$. Here, $\text{tr}$ denotes the trace of a matrix.
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Context:
Compute the differential form $d\beta$ for the two-form $\beta = z(y^2-x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz$ on three-dimensional real space.
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Context:
Evaluate the limit: \[ \lim_{R \to \infty} \int_0^R \frac{\cos x}{x^2} \, dx \]
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Context:
Use the Ritz method to find an approximate value of the smallest eigenvalue of the kernel \( K(x, t) = x t \) over the interval \([0, 1]\).
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Context:
In a chess tournament, there are $100$ players. Each player plays one game with every other player. For a win, $1$ point is awarded; for a loss, $0$ points are given; and for a draw, both players receive $0.5$ points.

Ion scored more points than any other player. Mihai lost only one game but accumulated fewer points than any other player. Determine all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.
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Context:
Point \( A \) lies on the side \( LM \) of triangle \( KLM \) with a \( 120^\circ \) angle at vertex \( K \). The circles inscribed in triangles \( AKL \) and \( AKM \) have centers \( F \) and \( O \), respectively. Find the radius of the circumcircle of triangle \( FKO \) if \( AO = 2 \) and \( AF = 7 \).
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Context:
Given that $X_t$ is a Brownian bridge, compute the expected value $E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}]$, where $u$ is a real number.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $2\sin A\sin B < -\cos(2B+C)$, what must be the relationship among $a$, $b$, and $c$?
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Context:
Suppose $X_1, X_2, \ldots, X_n$ are non-constant real random variables such that $X_1 + X_2 + \cdots + X_n = 0$. Determine the largest possible independence depth of the set $\{X_1, X_2, \ldots, X_n\}$. Note that the independence depth is defined as the maximal number $k$ such that any $k$ distinct elements of the set are mutually independent.
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Context:
Determine the cardinality of the set of all continuous surjections from a non-empty open path-connected set $D \subseteq \mathbb{R}^n$ to a non-empty open path-connected set $E \subseteq \mathbb{R}^n$, given that $D$ and $E$ are homeomorphic.
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Context:
Determine the condition under which the matrix expression \((P+Q-PQ-QP)\) is a projection matrix, given that \(P\) and \(Q\) are projection matrices.
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Context:
Form a six-digit number without repeating digits from 1, 2, 3, 4, 5, 6, with the requirement that odd numbers are not adjacent, and 4 is not in the fourth position. Calculate the total number of such six-digit numbers.
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Context:
Find the maximum value of the expression \(A = \left(3(x_{1} + \ldots + x_{n}) - 5(y_{1} + \ldots + y_{n})\right) \cdot \left(5(x_{1} + \ldots + x_{n}) + 3(y_{1} + \ldots + y_{n})\right)\) given that \(x_{1}^{2} + \ldots + x_{n}^{2} + y_{1}^{2} + \ldots + y_{n}^{2} \leq 1\).
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Context:
From the digits $1, 2, \ldots, 9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows: $123456789$, $123456798$, $\ldots$, $987654321$. What is the $100000^{th}$ number?
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Context:
Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$? If the answer is $x^\circ$, what is the value of $x$?
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Context:
Let $\pi_1, \pi_2$ be distinct irreducible representations of a finite group $G$, both of degree $d$, with corresponding characters $\chi_1, \chi_2$. Given the orthogonality relation $$ \sum_{g \in G} \chi_1(g) \chi_2(g)^* = 0, $$ determine if the following matrix sum is the zero matrix for $\pi_1 \neq \pi_2$: $$ \sum_{g \in G} \pi_1(g) \pi_2(g^{-1})^\top. $$
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Context:
Find the distance between the foci of the hyperbola
\[\frac{x^2}{50} - \frac{y^2}{22} = 2.\]
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Context:
A submarine has three navigational devices, each with an exponential failure time. The means of these failure times are 1 year, 1.5 years, and 3 years, respectively. The submarine can remain at sea as long as at least two devices are working. Calculate the average length of time the submarine can remain at sea.
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Context:
Evaluate the integral of the function \( f(z) = \tan z \) around a rectangular contour in the complex plane. The rectangle extends from \( 0 \) to \( 2\pi \) in the real direction and from \( -0.5i \) to \( 0.5i \) in the imaginary direction. Determine the value of this integral.
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Context:
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of 120° between them, and $|\overrightarrow {a}| = |\overrightarrow {b}| = 1$, vector $\overrightarrow {c}$ is collinear with $\overrightarrow {a} + \overrightarrow {b}$. Calculate the minimum value of $|\overrightarrow {a} + \overrightarrow {c}|$. Express your answer as an exact value.
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Context:
In a hexagon described around a circle, five consecutive sides are given as $a, b, c, d, e$. Find the sixth side.
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Context:
If $f(x)=\frac{1}{(1-x^{2011})^{\frac{1}{2011}}}$, find the number of real solutions to the equation $f(f(f(f(x)))) = \frac{1}{\sqrt{2}}$.
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Context:
Some boys went to water trees. There were 29 trees. Each of them watered the equal amount of trees, and 2 trees were left. How many boys went to water the trees?
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Context:
In a regular $1000$-gon, all the diagonals are drawn. What is the maximum number of diagonals that can be chosen such that among any three selected diagonals, at least two have the same length?
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Context:
求最小的正整数 m，使得对任意整数 A,B，存在多项式 f(x)，满足 f(-2)=A,f(-3)=B，且 f(x) 的各项系数都是不超过 m 的非负整数.
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Context:
Given an integer \( n \geq 3 \), determine the largest real number \( M \) such that for any sequence of positive real numbers \( x_{1}, x_{2}, \ldots, x_{n} \), there exists a permutation \( y_{1}, y_{2}, \ldots, y_{n} \) satisfying the inequality:
\[
\sum_{i=1}^{n} \frac{y_{i}^{2}}{y_{i+1}^{2} - y_{i+1} y_{i+2} + y_{i+2}^{2}} \geq M,
\]
where \( y_{n+1} = y_{1} \) and \( y_{n+2} = y_{2} \).
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Context:
Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1, \dots, 30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1 + A_2 + \dots + A_{30})$.
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Context:
How many ways are there to divide the set of numbers \(\left\{2^{0}, 2^{1}, 2^{2}, \cdots, 2^{2005}\right\}\) into two non-empty, disjoint subsets \(A\) and \(B\) such that the equation \(x^{2} - S(A)x + S(B) = 0\) has integer roots, where \(S(M)\) denotes the sum of all elements in the set \(M\)?
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Context:
If \( 4^3 + 4^r + 4^4 \) is a perfect square and \( r \) is a positive integer, find the minimum value of \( r \).
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Context:
Determine the values of $b \in \mathbb{R}$ for which the function \[ h(x,y) = \begin{cases} \dfrac{x^b y^2}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \] is continuous.
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Context:
(My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the *flux* of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?
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Context:
Find the integer that does not divide $n^{2225} - n^{2005}$ for every integer value of $n$.
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Context:
Maddie went to a beauty store and bought some beauty products. She bought 3 different makeup palettes which cost $15 each, with a 20% discount on the total cost of the palettes. She also bought 4 lipsticks which cost $2.50 each, with a "buy 3 get 1 free" promotion. Additionally, she purchased 3 boxes of hair color which cost $4 each. Maddie had a 10% off coupon for one box of hair color, and she also redeemed 100 reward points to get $5 off her total purchase.

The beauty store had a limited-time offer where customers get a 5% discount if their subtotal is between $30 and $50, or a 10% discount if their subtotal is over $50. The sales tax applied to the purchase was 5% on the first $25 of the subtotal and 8% on the remaining amount. How much did Maddie pay in total?
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Context:
Suppose that $3\triangle_4=\triangle2_{11}$, where $\triangle$ represents a digit 0-9. What is the value of the missing digit?
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Context:
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \overrightarrow{MD}$ (where $\lambda > 0$, and $\lambda \neq 1$). If the slope of $AB$ is always $- \frac{1}{2}$ when $\lambda$ changes, find the eccentricity of the ellipse $E$.
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Context:
Suppose that \(g(x)\) is a function such that
\[g(xy) + 2x = xg(y) + g(x)\] for all real numbers \(x\) and \(y.\) If \(g(-1) = 3\) and \(g(1) = 1\), then compute \(g(-101).\)
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Context:
One side of a rectangle measures $5 \text{ cm}$, and the other side measures $4 \text{ dm}$ and $8 \text{ cm}$. Each of the longer sides is divided into equal parts by three points. The midpoint of the bottom side is connected to the first point on the top side. Determine the area of the smaller resulting quadrilateral.
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Context:
Simplify $\tan \alpha \tan 2 \alpha + \tan 2 \alpha \tan 3 \alpha + \cdots + \tan (n-1) \alpha \tan n \alpha$.
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\boxed{\frac{\tan n\alpha}{\tan \alpha} - n}


Context:
Solve for $x,$ where
\[\frac{x}{x - a} + \frac{x - b}{x - a - b} = \frac{x - a}{x - 2a} + \frac{x + a - b}{x - b}.\]Assume that $2a > x > b > a > 0.$
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Context:
How many zeroes of $f(z) = z^5 + 3z^4 + 6$ are within $|z| = 3$?
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Context:
Calculate: $({\frac{{531}}{{135}}+\frac{{579}}{{357}}+\frac{{753}}{{975}}})×({\frac{{579}}{{357}}+\frac{{753}}{{975}}+\frac{{135}}{{531}}})-({\frac{{531}}{{135}}+\frac{{579}}{{357}}+\frac{{753}}{{975}}+\frac{{135}}{{531}}})×({\frac{{579}}{{357}}+\frac{{753}}{{975}})$.
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Context:
Alice has three daughters, each of whom has two daughters, each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)
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Context:
A circle $\omega$ with center $O$ is tangent to the sides $BA$ and $BC$ of the acute angle $ABC$ at points $M$ and $N$, respectively. A line passing through point $M$ parallel to $BC$ intersects the ray $BO$ at point $K$. A point $T$ is chosen on the ray $MN$ such that $\angle MTK = \frac{1}{2} \angle ABC$. It turns out that the line $KT$ is tangent to $\omega$. Find the area of triangle $OKT$ if $BM = a$.
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Context:
In a two-player game played on a 5×5 grid, two tokens, one red and one blue, start in random positions on the board. On each turn, a player can move either token one or two units to the right, or to the leftmost square of the above row. The last player who is able to move wins. Of the 625 possible starting positions for the two tokens, how many are winning positions for the first player if they play optimally?
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Context:
A sequence of pairwise distinct positive integers is called averaging if each term after the first two is the average of the previous two terms. Let $M$ be the maximum possible number of terms in an averaging sequence in which every term is less than or equal to $2022$ and let $N$ be the number of such distinct sequences (every term less than or equal to $2022$) with exactly $M$ terms. What is $M+N?$ (Two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are said to be distinct if $a_i \neq b_i$ for some integer $1 \leq i \leq n$).
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Context:
Given the equation \( k x^2 + \frac{|x|}{x+6} = 0 \) has four distinct real solutions for \( x \). Determine the range of values for the real number \( k \).
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Context:
find all positive integers $n$ for which $1^3+2^3+\\cdots+{16}^3+{17}^n$ is a perfect square.
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Context:
Origami. Mitya is going to fold a square sheet of paper $ABCD$. Mitya calls a fold beautiful if the side $AB$ intersects the side $CD$ and the four resulting right triangles are equal.

Before this, Vanya selects a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold passing through the point $F$.
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Context:
设数列 $\left\{a_{n}\right\}$ 是等差数列，数列 $\left\{b_{n}\right\}$ 满足 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \cdots$ 。
（2）设数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 的公差均是 $d \neq 0$ ，并且存在正整数 $s, t$ ，使得 $a_{s}+b_{t}$ 是整数，求 $\left|a_{1}\right|$ 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:
Among the four-digit numbers formed by the digits 0, 1, 2,..., 9 without repetition, calculate the number of cases where the absolute difference between the units digit and the hundreds digit equals 8. Express your answer as a single numerical value.
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Context:
Evaluate the integral \[ \frac{1}{\pi^2} \int_{0}^{\infty} \frac{(\ln{x})^2}{\sqrt{x}(1-x)^2} \, \mathrm{d}x. \]
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Context:
Find the largest possible value of $a + b$ that is less than or equal to $2007$, where $a$ and $b$ are relatively prime, and there exists a positive integer $n$ such that:
\[
\frac{2^3 - 1}{2^3 + 1} \cdot \frac{3^3 - 1}{3^3 + 1} \cdot \frac{4^3 - 1}{4^3 + 1} \cdots \frac{n^3 - 1}{n^3 + 1} = \frac{a}{b}.
\]
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Context:
Determine the interval of convergence for the series \(\sum_{n=1}^\infty \frac{(2n-1)(x+2)^n}{5^n n^2}.\)
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Context:
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the distance between two adjacent parallel lines?
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Context:
What is a necessary condition for a matrix to be invertible?
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Context:
Find the length of the latera recta of the ellipse given by the equation \(3x^2+3y^2+2xy-12x+12y+4=0\) with one focus at \((1,-1)\) and directrix \(x-y+2=0\).
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Context:
A $39$-tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers $a$, $b$ (these maxima aren't necessarily achieved using the same tuple of real numbers). Find $a + b$.

[i]Proposed by Evan Chang[/i]
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Context:
In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $c - a$ is equal to the height $h$ from $A$ to side $BC$, then $\left(\cos \frac{A}{2} - \sin \frac{A}{2}\right) \cdot \left(\sin \frac{C}{2} + \cos \frac{C}{2}\right) =$ _______.
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Context:
Find the value of N such that the number of odd integers between 4 and N is equal to the number of even integers between 1 and 103. Express your answer as a single integer.
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Context:
Select three different digits from $0, 1, \cdots, 9$ to form a four-digit number (one of the digits may appear twice), such as 5224. How many such four-digit numbers are there?
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Context:
Find the minimum possible value of $x+y-2xy$ under the conditions $x+y \geq 3xy$, $2(x+y) \geq 1+3xy$, and $0 \leq x \leq y \leq 1$.
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Context:
Find the limit of the following sequence using Stolz's theorem: $$\lim _{n \rightarrow \infty}\left(\frac{2}{2^{2}-1}\right)^{\frac{1}{2^{n-1}}} \left (\frac{2^{2}}{2^{3}-1}\right)^{\frac{1}{2^{n-2}}} \cdots\left(\frac{2^{n-1}}{2^{n}-1}\right)^{\frac{1}{2}}$$
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Context:
The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
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Context:
Find the sum of the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\left( \frac { n+2 }{ { n }^{ 5 } } \right) } $ correct to three decimal places.
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Context:
求函数$\frac { 1 - R C s } { s ( 1 + R C s ) }$的拉普拉斯逆变换。
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Context:
There are several solid and hollow circles arranged in a certain pattern as follows: ●○●●○●●●○●○●●○●●●… Among the first 2001 circles, find the number of hollow circles.
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Context:
Given condition $P$: The real number $x$ satisfies $(x-a)(x-3a) < 0$, where $a > 0$; condition $q$: The real number $x$ satisfies $8 < 2^{x+1} \leqslant 16$. If $q$ is a sufficient but not necessary condition for $p$, find the range of the real number $a$.
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Context:
Negate the proposition '$\exists x\geqslant 3$, $x^{2}-2x+3 \lt 0$'. Provide your answer in the form of the negated proposition, using appropriate logical quantifiers and mathematical notation.
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\boxed{$\forall x\geqslant 3$, $x^{2}-2x+3\geqslant 0$}


Context:
1. The greatest common divisor (GCD) of natural numbers $m$ and $n$ is 1. What is the greatest possible value of the GCD of the numbers $m+2000n$ and $n+2000m?$
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Context:
In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. If the answer is $x^\circ$, what is the value of $x$ for $\angle MHC$?
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Context:
Given: $\sqrt{1+\frac{1}{{1}^{2}}+\frac{1}{{2}^{2}}}=1\frac{1}{2}$, $\sqrt{1+\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}}=1\frac{1}{6}$, $\sqrt{1+\frac{1}{{3}^{2}}+\frac{1}{{4}^{2}}}=1\frac{1}{12}$. According to this pattern, $\sqrt{1+\frac{1}{{n}^{2}}+\frac{1}{(n+1)^{2}}}=\_\_\_\_\_\_$.
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