Context:
19. [10] Define $x \star y=\frac{\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute
$$
((\cdots((2007 \star 2006) \star 2005) \star \cdots) \star 1) .
$$
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Context:
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$.
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Context:
In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \).
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Context:
Consider a seller who must sell a single private value good to two potential buyers. The valuations of the buyers are uniformly distributed between 0 and 2. The seller will offer the good using a second-price sealed-bid auction with a reserve price of r. What is the optimal reserve price in this case?
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Context:
Given the function $f\left( x \right)={x}^{2}+{\left( \ln 3x \right)}^{2}-2a(x+3\ln 3x)+10{{a}^{2}}(a\in \mathbf{R})$, determine the value of the real number $a$ for which there exists ${{x}_{0}}$ such that $f\left( {{x}_{0}} \right)\leqslant \dfrac{1}{10}$.
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Context:
In triangle \(ABC\), \(\angle B = 110^\circ\) and \(\angle C = 50^\circ\). On side \(AB\), point \(P\) is chosen such that \(\angle PCB = 30^\circ\), and on side \(AC\), point \(Q\) is chosen such that \(\angle ABQ = 40^\circ\). Find the angle \(\angle QPC\).
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Context:
设对于k=1,2,⋯,n，存在实数x满足如下不等式：$2 ^ { k } < x ^ { k } + x ^ { k + 1 } <$ $2 ^ { k + 1 }$，试求n的最大值.
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Context:
In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. We know that whatever the arrangement of the edges, from every vertex, we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.
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Context:
Annie has 120 barrettes. She has half as many scrunchies as barrettes. She gifted some bobby pins to her friend and now has 50 fewer bobby pins than three times the number of barrettes. She also has twice as many hairbands as the difference between the number of her bobby pins and scrunchies. What percentage of her total hair decorations are bobby pins, rounded to the nearest percent?
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Context:
记 $M_{100}(\mathbb{R})$ 为所有 100 阶实方阵的集合. 设 $S=\left\{A \in M_{100}(\mathbb{R}) \mid A^{4}+A^{2}=\right.$ $O\}$ ，则 S 中最多可以选出 $\qquad$个互不相似的矩阵. (我们称 $A, B \in M_{100}(\mathbb{R})$ 相似,当且仅当 存在可逆的 $P \in M_{100}(\mathbb{R})$ ，使得 $P^{-1} A P=B$ )
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Context:
Find the smallest positive integer \( n \) such that for any given \( n \) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \( R_{1}, R_{2}, R_{3} \) such that \( R_{1} \) can be nested inside \( R_{2} \) and \( R_{2} \) can be nested inside \( R_{3} \).
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Context:
Given $a \gt 1$, if for any $x∈[{\frac{1}{3},+∞})$, the inequality $\frac{1}{{3x}}-x+ln3x≤\frac{1}{{a{e^x}}}+lna$ always holds, then the minimum value of $a$ is ____.
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Context:
Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF\) of this triangle.
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Context:
In triangle \(ABC\), point \(N\) is taken on side \(AC\) such that \(AN = NC\). Side \(AB\) is twice the length of \(BN\) and the angle between \(AB\) and \(BN\) is \(40^\circ\). Find the angle \(ABC\).
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Context:
Solve the system of equations in natural numbers:

\(a^{2} + b - c = 100\),

\(a + b^{2} - c = 124\).
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Context:
Tolya arranged in a row 101 coins of denominations 1, 2, and 3 kopecks. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kopeck coins could Tolya have?
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Context:
The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \( P(-2,0) \) to its asymptote is \( \frac{2 \sqrt{6}}{3} \). If a line with slope \( \frac{\sqrt{2}}{2} \) passes through point \( P \) and intersects the hyperbola at points \( A \) and \( B \), intersects the y-axis at point \( M \), and \( P M \) is the geometric mean of \( P A \) and \( P B \), then the half focal length of the hyperbola is what?
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Context:
Example 4.2.5 How many $1 \times 2$ dominoes must be placed on a $6 \times 6$ chessboard to achieve a saturated cover.
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Context:
In $\triangle ABC$, given $a=x,b=2,B=60^{\circ}$, if $\triangle ABC$ has two solutions, then the range of $x$ is
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Context:
1. A mathematician left point A for point B. After some time, a physicist also left point A for point B. Catching up with the mathematician after 20 km, the physicist, without stopping, continued to point B and turned back. They met again 20 km from B. Then each continued in their respective directions. Upon reaching points A and B respectively, they turned and walked towards each other again. How many kilometers from point A will they meet for the third time, if the distance between points A and B is 100 km?
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Context:
If $a,b,c,d\geq 0$ and $t\geq \frac{2+\sqrt{2}}{8}(a+b+c+d)$, then
$$(t-a)(t-b)(t-c)(t-d)\leq \left(t-\frac{a+b+c+d}{4}\right)^4.$$
Find the minimum value of $k$ such that the inequality holds for all $a,b,c,d\geq 0$ and $t\geq k(a+b+c+d)$.
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Context:
In a ternary tree of height 3, where each node except the root has three children, how many pairs of vertices (u, v) are there such that the distance between u and v is 4? Note that (u, v) and (v, u) are considered the same pair.
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Context:
Determine the asymptotic behavior of the ratio of the $n$th nice prime to the $n$th prime.
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Context:
Determine all values of the parameter \( a \) for which the system of equations \( \left\{\begin{array}{l} |y+x+8|+|y-x+8|=16 \\ (|x|-15)^{2}+(|y|-8)^{2}=a \end{array}\right. \) has exactly two solutions.
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Context:
$\begin{aligned} & \mathrm{For}0\leq t\leq1,\mathrm{define}f(t)=\int_0^{2\pi}|\sin x-t|dx. \\ & \mathrm{Evaluate}\int_0^1f(t)dt. \end{aligned}$
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Context:
A hostess baked a pie for her guests. There might be either $p$ people or $q$ people at the table, where $p$ and $q$ are coprime. What is the minimum number of pieces the pie needs to be cut into in advance so that it can be evenly distributed among the guests in either case?
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Context:
Calculate the value of the expression

$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$
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Context:
Given a regular 2007-gon, find the minimal number $k$ such that among every $k$ vertices of the polygon, there always exist 4 vertices forming a convex quadrilateral where 3 sides of the quadrilateral are also sides of the polygon.
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Context:
Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$.
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Context:
When w is divided by 13, the remainder is a certain number. If w is 3 more than its value and when divided by 11, its remainder is 0. What is the remainder when w is divided by 13?
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Context:
Find the derivative of the trace of the matrix expression \( \operatorname{Trace}\left[\left(AB\right)^{T}Q\left(AB\right)\right] \) with respect to the matrix \( A \), where \( Q = Q^T > 0 \).
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Context:
A car of the new model travels $4 \frac{1}{6}$ kilometers more per liter of gasoline than the car of the old model. At the same time, the fuel consumption per 100 km for the new car is 2 liters less. How many liters of gasoline does the new car consume per 100 km?
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Context:
实数 $a 、 b 、 c$ 满足 $a^{2}+b^{2}+c^{2}=\lambda(\lambda>0)$, 试求

$$
f=\min \left\{(a-b)^{2}, \quad(b-c)^{2}, \quad(c-a)^{2}\right\}
$$

的最大值.
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Context:
Find the value of \(a_2 + 11a_3 + 70a_4\) for which the function \(f(x) = x^3 + 3(a-7)x^2 + 3(a^2-9)x - 1\) has a positive point of maximum in the interval \((a_1, a_2) \cup (a_3, a_4)\).
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Context:
The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$.
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Context:
Calculate the time-averaged power of the continuous signal given by: \[ z(t) = \sqrt{E} \sum_{k=-\infty}^{+\infty} A_k g(t-kT) \] where \( A_k = 1 \) if \(|k|\) is a prime number and \( A_k = -1 \) otherwise, and \( g(t) = 1 \) for \( 0 \leq t \leq T \) and \( g(t) = 0 \) otherwise. \( T \) is the fundamental period. The power of a continuous signal is defined by: \[ \lim_{T\to\infty} \frac{1}{2T} \cdot \int_{-T}^{T} |x(t)|^2 dt \]
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Context:
4. (7 points) Linlin copied a repeating decimal incorrectly as 0.123456. If the digits are correct but the repeating part indicator was mistakenly left out, how many possible original repeating decimals could there be?
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Context:
一个不均匀的骰子，掷出 1,2,3,4,5,6 点的概率依次成等差数列. 独立地先后掷该骰子两次，所得的点数分别为 a,b. 若事件"a+b=7"发生的概率为 \df{1}{7}，则事件"a=b''发生的概率为__________.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
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Context:
The numbers \(1,2,3, \ldots, n^{2}\) are arranged in a square table. A random number is selected from this table, and the row and column containing this number are crossed out. Then, from the remaining numbers, another number is selected, and again, the row and column containing this number are crossed out. This process continues until only one number remains in the table, which is automatically included among the selected numbers. What is the sum of all the numbers selected in this way?
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Context:
Identify the Banach space that is isometrically isomorphic to the quotient space $c/Y$, where $c$ is the space of convergent sequences and $Y$ is the subspace of sequences converging to zero.
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Context:
Two $4 \times 4$ squares are randomly placed on an $8 \times 8$ chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap?
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Context:
a and b are two multiples of a certain number, and q is the set of consecutive integers between a and b, inclusive. q contains 11 multiples of that number. There are 21 multiples of 7 in q. What is the number that a and b are multiples of?
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Context:
In one of the hotels on the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. The following information is known about the guests:

1. If two women know each other, then their husbands also know each other, and vice versa.
2. Each sauna can be used by either men only or women only (there is no limit on the number of people using a sauna at once, as long as they are of a single gender).
3. Each woman is only willing to share a sauna with women whom she knows.
4. Each man is only willing to share a sauna with men whom he does not know.

What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?
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Context:
How many sequences of real numbers $a_1, a_2, \ldots, a_9$ satisfy
\[ |a_1 - 1| = |a_2 - a_1| = \cdots = |a_9 - a_8| = |1 - a_9| = 1? \]
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Context:
Example 1 Given 20 numbers $1,2, \cdots, 20$, two people, A and B, take turns placing a “+” or “-” sign in front of these numbers (the order of placement is not limited). After 20 steps, the absolute value $S$ of the resulting algebraic sum is calculated. A wants to make $S$ as small as possible, while B wants to make $S$ as large as possible. What is the largest $S$ that B can guarantee?
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Context:
给定整数 $n \geqslant 3$ 。求最小的正整数 $k$, 使得存在一个 $k$ 元集合 $A$ 和 $n$ 个两两不同的实数 $x_{1}, x_{2}, \cdots, x_{n}$, 满足 $x_{1}+x_{2}, x_{2}+x_{3}, \cdots$, $x_{n-1}+x_{n}, x_{n}+x_{1}$ 均属于 $A$
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Context:
Calculate the integral \( \int_T f \) over the region \( T \) in \( \mathbb{R}^3 \) with vertices \((1,0,1), (0,2,1), (2,1,0), (1,-1,2)\), where \( f(x,y,z) = y - 2x + z \). Use a linear transformation to simplify the bounds of integration.
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Context:
Let ${f'}\left(x\right)$ be the derivative of the function $f\left(x\right)$, and ${f'}\left(x\right) \lt f\left(x\right)\left(x\in R\right)$, $f\left(1\right)=e$, then the solution set of the inequality $f\left(\ln x\right) \gt x$ is ______.
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Context:
A cup is filled with a saline solution of $15\%$ concentration. There are three iron balls of different sizes: large, medium, and small, with their volumes in the ratio 10:5:3. Initially, the small ball is submerged in the cup of saline causing $10\%$ of the saline solution to overflow. Then the small ball is removed. Next, the medium ball is submerged in the cup and then removed. This is followed by submerging and removing the large ball. Finally, pure water is added to the cup until it is full again. What is the concentration of the saline solution in the cup at this point?
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Context:
Compute the largest squarefree positive integer \( n \) such that \( \mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}} \) is a unique factorization domain.
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Context:
In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$.
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Context:
Evaluate the integral \( \int_{\gamma} f(z) \, dz \), where \( f(z) = \frac{1}{[(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)]} \) and \( \gamma \) is the polygon with vertices \( 0, 2, 2+2i, 2i, 0 \).
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Context:
Let $\mathbf{F}$ be a vector field defined on $\mathbb{R}^2 \setminus\{(0,0)\}$ by $$\mathbf{F}(x,y) = \frac{y}{x^2+y^2}i - \frac{x}{x^2+y^2}j.$$ Let $\gamma, \alpha: [0,1] \to \mathbb{R}^2$ be defined by $$\gamma(t) = (8\cos 2\pi t, 17\sin 2\pi t)$$ and $$\alpha(t) = (26\cos 2\pi t, -10\sin 2\pi t).$$ If $$3\int_{\alpha} \mathbf{F \cdot dr} - 4 \int_{\gamma} \mathbf{F \cdot dr} = 2m\pi,$$ find the value of $m$.
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Context:
8. Let \( A B C \) be a triangle with sides \( A B = 6 \), \( B C = 10 \), and \( C A = 8 \). Let \( M \) and \( N \) be the midpoints of \( B A \) and \( B C \), respectively. Choose the point \( Y \) on ray \( C M \) so that the circumcircle of triangle \( A M Y \) is tangent to \( A N \). Find the area of triangle \( N A Y \).
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Context:
Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$.
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Context:
Determine the minimum number of real roots of the equation \((x^2+px+3q)(x^2+rx+q)(-x^2+sx+2q)=0\) for real numbers \(p, q, r, s\), without expanding the expression.
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Context:
What percentage error do we make if we approximate the side of a regular heptagon by taking half of the chord corresponding to the $120^\circ$ central angle?
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Context:
In a round-robin chess tournament with 12 players, each player creates 12 lists. The first list contains only themselves, and the $(k+1)$-th list contains the players from the $k$-th list and those who were defeated by any of the players in the $k$-th list. It is known that the 12th list of each player is different from their 11th list. Determine the number of draws that occurred in the tournament.
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Context:
Let  $x, y,z$  satisfy the following inequalities  $\begin{cases} | x + 2y - 3z| \le 6
| x - 2y + 3z|  \le  6
| x - 2y - 3z|  \le 6
| x + 2y + 3z|  \le  6 \end{cases}$ Determine the greatest value of  $M = |x| + |y| + |z|$ .
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Context:
On a two-dimensional coordinate plane, the line a = x^2 - x^3 touches the x-axis in 2 places. What is the second term of the equation representing the line?
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Context:
已知平面直角坐标系中的点集L=\{(x,y)\ |\ x,y\in\{1,2,\cdots,100\}\}.设 A 是由平面上若干个凸多边形组成的集合，满足：A 中每个凸多边形的所有顶点均属于 L，并且 L 中的每个点都恰好是 A 中一个凸多边形的顶点.求 A 中所有凸多边形的面积之和的最小可能值.
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Context:
Find the smallest radius \( r \) such that there is at least one local maximum point of the function \( f \) inside or on the boundary of a circle of radius \( r \) centered at \( (x_0, f(x_0)) \), given the conditions \( f(x_0) \ge f(x_0+h) \) and \( f(x_0) \ge f(x_0-h) \).
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Context:
Determine the minimum number of colors required to paint all the cells of a 6 by 6 square grid such that each row, column, and diagonal of the square has all cells with different colors. A diagonal is defined as all rows of at least two cells going diagonally from one edge of the square to the other at an angle of $45^\circ$ or $135^\circ$ to the horizontal.
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Context:
Calculate the probability \( \mathbb{P}(Z_1 > 0, Z_1 + Z_2 < 0) \) where \( Z_1 \) and \( Z_2 \) are independent standard normal random variables.
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Context:
A bicycle is traveling along a path with a 5-degree incline. The front wheel has a perimeter of 30 cm, and the back wheel has a perimeter of 20 cm. The surface is such that the front wheel experiences a slip ratio of 5%, while the back wheel experiences a slip ratio of 7%. If the front wheel revolves 240 times, how many revolutions will the back wheel take to cover the same distance along the incline?
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Context:
Determine the number of points within the interval \([0, 6\pi]\) where the function \( f(x) \) is continuous, given that \( f(x) = \tan x \) for rational \( x \) and \( f(x) = x^2 + 1 \) for irrational \( x \).
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Context:
Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers?
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Context:
In a plane, point C is located 300 meters due east of point B. A tower, not on the line connecting B and C, is observed from points B and C with vertical angles of 45 degrees and 60 degrees, respectively. From point D, which is 500 meters west of B, the tower is observed with a vertical angle of 30 degrees. Calculate the height of the tower.
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Context:
Select 4 students to participate in a relay race from a group of 8 students, which includes 6 boys and 2 girls, using stratified sampling by gender. If girls are not to be the last runner, find the number of different arrangements. Express your answer as a numerical value.
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Context:
A mixture of $30$ liters of paint is $25\%$ red tint, $30\%$ yellow tint and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?
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Context:
Let $\gamma : [0,1] \to \mathbb{C}$ be the contour given by $\gamma(t) = t + it$. Suppose $f : \mathbb{C} \to \mathbb{C}$ is a complex differentiable function which satisfies $|f(z)| \le 3$ for all $z$ such that $|z| \le 100$. Determine the maximum possible value of \(\int_\gamma f(z)\,dz\).
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Context:
At a ball, there are \(n\) married couples. In each couple, the husband and wife are of the exact same height, but no two couples share the same height. When the waltz begins, all attendees randomly pair up: each man dances with a randomly chosen woman. Find the expected value of the random variable \(X\), which represents the number of men who are shorter than their partners.
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Context:
Let  $ d_1,d_2 \ldots, d_r$  be the positive divisors of  $ n$
 $ 1\equal{}d_1<d_2< \ldots <d_r\equal{}n$
If  $ (d_7)^2 \plus{} (d_{15})^2\equal{} (d_{16})^2$  find all posible values of  $ d_{17}$
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Context:
How many positive integers less than or equal to $150$ have exactly three distinct prime factors?
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Context:
Evaluate the limit \( \lim_{n \to \infty} n^2 \sum_{k=1}^n \frac{1}{(n^2+k^2)^2} \) using the concept of Riemann sums.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive real numbers \(a\) for which the series \( \sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A} a^{k}} \) is convergent, where \(\mathcal{F}\) is the family of all nonempty finite subsets of \(\mathbb{N} \cup \{0\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can you select an even number of green shirts from 50 drawers, each containing a blue shirt, a red shirt, a green shirt, and a yellow shirt, considering that the order of non-green shirts matters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the diagram, the houses of Xiaohong, Xiaolan, and Xiaolu are located along a circular road where one can only travel in a counterclockwise direction. It takes Xiaohong 30 meters to walk to Xiaolan's house, Xiaolu 26 meters to walk to Xiaohong's house, and Xiaolan 28 meters to walk to Xiaolu's house. Calculate the distance it takes Xiaolan to walk to Xiaohong's house in the reverse direction. Express your answer in meters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If the area $S= \frac {1}{4}(b^{2}+c^{2}-a^{2})$, then $\angle A=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The  $\emph{Stooge sort}$  is a particularly inefficient recursive sorting algorithm defined as follows: given an array  $A$  of size  $n$ , we swap the first and last elements if they are out of order; we then (if  $n\ge3$ ) Stooge sort the first  $\lceil\tfrac{2n}3\rceil$  elements, then the last  $\lceil\tfrac{2n}3\rceil$ , then the first  $\lceil\tfrac{2n}3\rceil$  elements again. Given that this runs in  $O(n^\alpha)$ , where  $\alpha$  is minimal, find the value of  $(243/32)^\alpha$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can you divide the set of eight numbers \( \{ 2,3,\cdots ,9\} \) into 4 pairs such that no pair of numbers has \( \gcd \)
 equal to 2 ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matrix $A \in \mathbb{R}^{m \times n}$ with full rank and $m \le n$, solve the optimization problem:
\[ \mathop{\arg\max}\limits_{\substack{v \in \mathbb{R}^m \\ \| v \| = 1}} v^T A A^T v \]
using the singular value decomposition (SVD) of $A$. Determine the vector $v$ that maximizes the expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression $a/b$, where $a = \sum_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right)$ and $b = \prod_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right)$. Determine the value of $a/b$ without using graphing calculators.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Agent 1 and Agent 2 each have two tasks to complete. The time it takes Agent 1 to complete Task 1 and Task 2 follows an exponential distribution with means of 6 and 4, respectively. Similarly, the time it takes Agent 2 to complete Task 1 and Task 2 follows an exponential distribution with means of 3 and 1, respectively. Task 2 for each agent can only be started after Task 1 is completed. Assuming all task completion times are independent, calculate the probability that Agent 1 completes Task 2 before Agent 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a conference of 50 participants, each person shakes hands with everyone else, except for the committee members and the participants who have a red badge. The committee consists of 10 participants, and there are 5 participants with red badges. Determine the maximum number of handshakes that can occur without any circular handshakes involving the committee members or sequences that involve participants with red badges shaking hands with each other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the minimal polynomial of a matrix \( A \) be \( f(x)=(x-c_1)^{n_1}\cdots(x-c_k)^{n_k} \). Determine the minimal polynomial of the matrix \( \begin{pmatrix}A & 2I \\0 & A \\\end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $g: H \to D(0,1)$ be a holomorphic function, where $H$ is the right half-plane minus the y-axis. Given that $g(1) = 0$ and $|g(z)| < 1$ for all $z \in H$, determine an upper bound for $|g'(1)|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a random variable X follows a t-distribution with n degrees of freedom, what is the probability distribution of the random variable X^2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条，则由这 3030 条线段能构成不全等的三角形的个数为 （用数字作答）。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G = \mathbb{Z} \oplus \mathbb{Z}$ and $H = \langle (2,2) \rangle$. Determine the order of the factor group $G/H$. \( \text{Note: } H = \{ (x,y) \mid x,y \in 2\mathbb{Z} \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the expression \( \frac{( b \times (a \times b)) \cdot (b \times (a\times b))}{(a\times b) \cdot (a\times b)} \), where \(a\) and \(b\) are 3-dimensional vectors, and express it without a denominator.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The championship finals of the Chinese Volleyball Super League adopts a seven-game four-win system, which means that if one team wins four games first, that team will be the overall champion, and the competition will end. Now team A and team B are in the finals. Since the two teams have similar strengths, the probability of each team winning each game is $\frac{1}{2}$. According to past data, the ticket revenue for the first game is 5 million yuan, and the ticket revenue for each subsequent game increases by 100,000 yuan compared to the previous game. What is the probability that the total ticket revenue for the finals is exactly 45 million yuan?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The angle between the line $x+ \sqrt {3}y+2=0$ and the line $x+1=0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, where $\angle A = 100^\circ$ and $\angle B = \angle C = 40^\circ$, the side $AB$ is extended to a point $D$ such that $AD = BC$. Determine the measure of $\angle BCD$ without using trigonometry.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train passes a platform in 25 seconds. The same train passes a man standing on the platform in 20 seconds. The speed of the train is some km/hr. The length of the platform is 75.006 meters. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_\gamma \frac{\mathrm{Im} (z)}{z - \alpha} dz \), where \( \alpha \) is a complex number such that \( 0 < |\alpha| < 2 \) and \( \gamma \) is the circle oriented in the positive sense, centered at the origin with radius 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that:
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$.
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$.

Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $MATH$  be a square with  $MA = 1$ . Point  $B$  lies on  $AT$  such that  $\angle MBT = 3.5 \angle BMT$ . What is the area of  $\vartriangle BMT$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(x\) satisfies \(\log _{5x} (2x) = \log _{625x} (8x)\), find the value of \(\log _{2} x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the Hilbert space \( l^2 = \{ (x_1, x_2, x_3, \dots) : x_n \in \mathbb{R}, \, \sum_{n=1}^{\infty} x_n^2 < \infty \} \) with the norm \( \lvert \lvert (x_1, x_2, x_3, \dots) \rvert \rvert_2 = \left( \sum_{n=1}^{\infty} x_n^2 \right)^{\frac{1}{2}} \), consider the subspace \( M \subset l^2 \) defined by \( \sum_{n=1}^{\infty} \frac{x_n}{4^n} = 0 \). The orthogonal complement of \( M \) is denoted as \( M^{\perp} \). Given the vector \( v = \left(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \right) \in l^2 \), its orthogonal projection onto \( M^{\perp} \) is \( \alpha \left(\sum_{n=1}^{\infty} \frac{1}{n4^n}\right) \left(\frac{1}{4}, \frac{1}{4^2}, \frac{1}{4^3}, \dots \right) \) for some \( \alpha \in \mathbb{R} \). Find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the constant \( k \) such that the cubic equation \( x^3 - 12x + k = 0 \) has distinct roots in the interval \([0, 2]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n$, let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of $n$ for which $f(n) \le 300$. <b>Note:</b> $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell borders (the hole does not contain boundary cells). If you cut this figure along all the horizontal grid lines, you get 20 strips that are 1 cell wide. How many strips will you get if you cut it along all the vertical grid lines instead? (A $1 \times 1$ square is also considered a strip!)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the third-order differential equation \(y''' + 7y'' + 16y' + 10y = 0\) and knowing that \(y = e^{-x}\) is a solution, find another solution to the equation without using a calculator.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A person walks at 14 km/hr instead of a slower speed, and as a result, he would have walked 20 km more. The actual distance travelled by him is 50 km. What is the slower speed he usually walks at?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit as \( k \to \infty \):
\[ \lim_{k\rightarrow \infty} 2k(m+2k)\frac{e^{\frac{-m}{2k}}(m+2k)^{\frac{m}{2k}}}{(k!)^{\frac{1}{k}}\Gamma\left(k+\frac{p}{2}\right)^{\frac{1}{k}}c^{\frac{m}{2k}}} \]
where \(m, p \in \mathbb{N}\) and \(c > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a committee to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$, $b$, $c$ be three distinct real numbers such that the equation $x^2+ax+1=0$ and $x^2+bx+c=0$ have one common real root, and the equation $x^2+x+a=0$ and $x^2+cx+b=0$ also have one common real root. Find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S \) be the set of all points \((x_1, x_2, x_3, \dots, x_{2017})\) in \(\mathbb{R}^{2017}\) satisfying \(|x_i|+|x_j|\leq 1\) for any \(1\leq i< j\leq 2017\). The volume of \( S \) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(100m+n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pyramid has a square base with sides of length $1$ and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse Fourier Transform of the function $2\pi \delta(x-\omega)$, where $\omega$ is a real parameter and $x \in \mathbb{R}$. Express your answer in terms of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the infinite series: \[ \sin(x) - \frac{1}{2}\sin(2x) + \frac{1}{3}\sin(3x) - \frac{1}{4}\sin(4x) + \cdots \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, \cdots, X_n$ be $n$ random variables (not necessarily independent) such that $E[X_i] > E[X_j]$ whenever $i < j$. What is a lower bound on the following probability: $P[ (X_1 > X_2) \wedge (X_1 > X_3) \wedge \cdots \wedge (X_1 > X_n) ]$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a circle with 800 points labeled in sequence clockwise as \(1, 2, \ldots, 800\), dividing the circle into 800 arcs, initially, one point is painted red. Subsequently, additional points are painted red according to the following rule: if the \(k\)-th point is already red, the next point to be painted red is found by moving clockwise \(k\) arcs from \(k\). What is the maximum number of red points that can be obtained on the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the universal covering space of a closed surface of genus 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tosha travels from point A to point B via point C. From A to C, Tosha travels at an average speed of 75 km/h, and from C to B, Tosha travels at an average speed of 145 km/h. The total time taken for the journey from A to B is 4 hours 48 minutes. The next day, Tosha travels back at an average speed of 100 km/h. The journey from B to C takes Tosha 2 hours, and from C to A, Tosha travels at an average speed of 70 km/h. Find the distance between points B and C.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $2010$ people sitting around a round table. First, we give one person $x$ a candy. Next, we give candies to the $1^{st}$ person, $1+2^{nd}$ person, $1+2+3^{rd}$ person, $\cdots$, and $1+2+\cdots+2009^{th}$ person clockwise from $x$. Find the number of people who get at least one candy.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the set of subsequential limits of the sequence \( \{x_n\} = \left\{1, \frac{1}{10}, \frac{2}{10},\cdots,\frac{9}{10}, \frac{1}{10^2}, \frac{2}{10^2},\cdots,\frac{99}{10^2}, \cdots, \frac{1}{10^n}, \frac{2}{10^n}\cdots \frac{10^n-1}{10^n}, \cdots\right\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a uniformly distributed random variable $X$ with support $[0,1]$, consider the function:

$$f(X) = \frac{1}{n^2}\sum_{l=1}^n \left[\frac{1}{2}\left(|x_l - \frac{1}{2}| - |1 - x_l|\right)\right]$$

where $n$ is a positive integer, and $x_l$ denotes the $l$th draw from $X$. Determine an upper bound for $f(X)$ using mathematical inequalities, such as Markov's or Chebyshev's inequality. Note that $x_l$ is nonnegative, but $f(X)$ may not be.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 4 sections A, B, C, and D in a class, consisting of 36, 44, 50, and 30 students respectively. The average weight of students in section A is 40 kg, in section B is 35 kg, in section C is 42 kg, and in section D is 38 kg. Furthermore, 5 students in section A have an additional 10 kg of average weight, and 7 students in section B have 8 kg less average weight compared to their sections' averages. Find the average weight of the entire class considering these additional differences in student weights.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the series \( T = 1+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\frac{1}{23}+\cdots \), where each term is the reciprocal of a prime number that is one more than a multiple of 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pupil's two marks were wrongly entered: the first as 83 instead of 63 and the second as 75 instead of 85. Due to these errors, the average marks for the class got increased by half (1/2) when considering a weighted average of 3:2 for the first and the second marks, respectively. Find the number of pupils in the class.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to 0} \frac{\tan( [-\pi^2]x^2)- \tan^2( [-\pi^2]x^2)}{\sin^2x} \] where \([\cdot]\) denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mara has 90 crayons and 15 percent of her crayons are pink. Luna has 120 crayons and 25 percent of them are pink. Sara has 45 crayons, with 5 percent of them being pink. In total, how many pink crayons do Mara, Luna, and Sara have together?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 6 different books: one math book, two English books, and three music books. They are to be arranged in a row such that the English books are not adjacent, and the music books are also not adjacent. Find the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the expansion of $(3+x)^5+(2-x)^6$, the binomial coefficient of the term containing $x^4$ is the term of the sequence $\{a_n\}$ with the general term $a_n=2n+2$. Which term of the sequence is it? Express your answer as the term number (e.g., 6th, 7th, etc.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $Z_n\sim \chi^2(n)$. What is the limiting distribution of $V_n= \dfrac{\sqrt{Z_n+2}-\sqrt{n+2}}{\sqrt{2}}?$
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Context:
Find the limit of the sum \( \displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that \( n \) is a positive integer and that the set \( S \) contains exactly \( n \) distinct positive integers. If the mean of the elements of \( S \) is equal to \( \frac{2}{5} \) of the largest element of \( S \) and is also equal to \( \frac{7}{4} \) of the smallest element of \( S \), determine the minimum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the numbers \(1, 2, 3, \cdots, 7\), select several numbers such that the sum of the even numbers is equal to the sum of the odd numbers. How many ways are there to satisfy this condition?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Count the number of non-negative integer solutions to the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 23$$ such that $0 \leq x_i \leq 9$ for all $i$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
通项为$a _ { n } = b [ \sqrt { n + c } ] + d$的数列，逐次算得各项是

1,3,3,3,5,5,5,5,5,···

其中每一个正奇数m恰好连续出现m次.上述b，c，d是待定的整数.求b+c+d 的值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在透明的球袋中有 1 个黑球和 2 个白球.接下来从袋中摸球，每次摸出一个球，然后放回袋中，并继续往袋中再放入 2 个与摸出球同色的球.记 $S_{n}$ 为前 $n$ 次中摸出黑球的次数. 已知第 2 次摸出的球是黑球，记 $S_{100}$ 的期望为 $E$ ，则 $[E]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( \frac{dx}{dy} \cdot \frac{d^2y}{dx^2} = k \cdot \frac{dy}{dx} \) for a constant \( k \), and the gradient \( \frac{dy}{dx} = 1 \) at the point \( (0,1) \), find the value of the gradient \( \frac{dy}{dx} \) at the point where \( y = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum

$$
\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots+\frac{1}{(3n-1)(3n+2)}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train of a certain length is running at a speed of 55 km/hr. It crossed a platform of length 300 m in 35.99712023038157 seconds. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a continuous non-constant function $f: \mathbb R \to \mathbb R$ that satisfies $(e^x-1)f(2x) = (e^{2x}-1)f(x)$ and $f'(0) = 1$, determine $\lim_{x\to 0}\left(\frac{f(x)}{x}\right)^{\frac 1x}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all pairs of positive integers $ (x,y)$ such that
\[ x^y = y^{x - y}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c, x$ be reals with $(a+b)(b+c)(c+a) \neq 0$ that satisfy $$\frac{a^{2}}{a+b}=\frac{a^{2}}{a+c}+20, \quad \frac{b^{2}}{b+c}=\frac{b^{2}}{b+a}+14, \quad \text { and } \quad \frac{c^{2}}{c+a}=\frac{c^{2}}{c+b}+x$$ Compute $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\sin 3x$, determine the horizontal shift required to obtain the graph of the function $y=\sin \left(3x+\frac{\pi }{4}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On each square of a $9 \times 9$ board, there is a chip. At a certain signal, each chip jumps to one of the squares diagonally adjacent to the one it is currently on. Find the minimum number of empty squares after this movement.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The price of a certain model of mobile phone in January of this year was $a$ yuan per unit. After that, the price decreased by 3% every month. Using the geometric sequence formula, calculate the price of this mobile phone in October of this year. Express your answer in yuan, including the variable $a$, and use the boxed notation for your final answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $B_{k}(n)$ be the largest possible number of elements in a 2-separable $k$-configuration of a set with $2n$ elements $(2 \leq k \leq n)$. Find a closed-form expression (i.e. an expression not involving any sums or products with a variable number of terms) for $B_{k}(n)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twenty people, including \( A, B, \) and \( C \), sit randomly at a round table. What is the probability that at least two of \( A, B, \) and \( C \) sit next to each other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if the curve given by the equation \(x^3 + y^3 = 3xy\) has an asymptote of the form \(x + y = k\). If it does, find the value of \(k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $$\begin{array}{@{}c@{\;}c@{}c@{}c} && &P_b \\ & \times & & P_b \\ \cline{2-4} & & Q & P_b, \\ \end{array} $$where $P$, $Q$, and $b$ represent three distinct digits 1-9. If $Q=\frac{P}{2}$, and $P$ is two less than $b$, what is the value of the digit $P$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all real numbers $a$ such that the equation
\[x^3 - ax^2 - 2ax + a^2 - 1 = 0\]has exactly one real solution in $x.$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( -\infty, \frac{3}{4} \right)}


Context:
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest real number $k$ such that \[x_1^2 + x_2^2 + \dots + x_{11}^2 \geq kx_6^2\] whenever $x_1, x_2, \ldots, x_{11}$ are real numbers such that $x_1 + x_2 + \cdots + x_{11} = 0$ and $x_6$ is the median of $x_1, x_2, \ldots, x_{11}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
C6

What is the least positive integer $k$ such that, in every convex 101 -gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

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Context:
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, where $S_n = (-1)^n a_n - \frac{1}{2^n}$, $n \in \mathbb{N}^{*}$. Calculate the value of $S_1 + S_2 + \ldots + S_{100}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold:
    1. each color is used at least once;
    2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational.
Find the least possible value of $N$.

[i]~Sutanay Bhattacharya[/i]

Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular piece of land is to be fenced in using two kinds of fencing. Two opposite sides will be fenced using standard fencing that costs $\$6/\text{m}$, while the other two sides will require heavy-duty fencing that costs $\$9/\text{m}$. What are the dimensions of the rectangular lot of greatest area that can be fenced in for a cost of $\$9000$?

Please reason step by step, and put your final answer within \boxed{}.

\boxed{250 \text{ meters by } 375 \text{ meters}}


Context:
Determine whether the limit \( \lim_{n\rightarrow\infty} \frac{P(n)}{A(n)} \) exists, where \( P(n) \) is the count of perfect numbers less than or equal to \( n \), and \( A(n) \) is the count of amicable pairs with both numbers less than \( n \). If the limit exists, find its value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : |B_t| = x \}.$$ Compute, for $\lambda \in \mathbb{R}$, $$ \mathbb{E} \bigg(\exp \bigg\{ \lambda B_\tau -\frac{\lambda ^2}{2} \tau \bigg\} \bigg).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether or not there are two digits other than \( a,b \) such that the number \( \overline{ab} \) is a multiple of the number \( \overline{ba} \) (both
 written in decimal notation).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain bacteria colony grows at a specific rate every day for 25 days, at which point it reaches the limit of its habitat and can no longer grow. If two bacteria colonies start growing simultaneously, they will take 24 days to reach the habitat's limit. What is the growth rate of the bacteria colony?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.

Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round to whole degrees. What is the maximum possible deviation of the obtained value from the original temperature in Celsius degrees?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在 \triangle ABC 中，\angle A>90^\circ，\angle B=20^\circ，作 AD\bot AB 交 BC 于点 D. 已知 AB=1, CD=4，设 S 为 \triangle ABC 的面积，则 S^2 写成最简分数之后，分子分母之和为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the odd function $f(x)$ is defined on the domain $\mathbb{R}$ and satisfies $f(x) = f(2 - x)$ for any real number $x$. When $x \in (0, 1)$, $f(x) = 2^x + 1$. Find the value of $f(\log_{\frac{1}{2}} \frac{1}{15})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Let $a_{1}, a_{2}, a_{3}, a_{4}$ be 4 distinct numbers from $1,2, \cdots, 100$, satisfying
$$
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\right)=\left(a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}\right)^{2},
$$

then the number of such ordered quadruples $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a tetrahedron \(ABCD\) with the following edge lengths:
\[
AB = 2, \quad AD = \frac{11}{2}, \quad BC = 8, \quad CD = \frac{19}{2}
\]
find the sine of the angle between the skew lines \(AC\) and \(BD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Juan needs 10 half-gallons of milk to bake 40 cookies with. How many half-gallons of milk does Juan need to bake 200 dozen cookies?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that a random variable $X$ follows a standard normal distribution ($X\sim N(0,1)$) and $Y = X^2$, and knowing that the density function of $Y$ matches that of a Chi-squared distribution with 1 degree of freedom, can you conclude that $Y$ is a Chi-squared distributed random variable with 1 degree of freedom ($Y\sim \chi^2_1$)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the problem of finding the partition function of a 3D Ising model is in the complexity class #P or NP.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\omega = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2 + d$, where $0 \leq a, b, c \leq 1$ and $d \in \{0, 1\}$. Find the area of $T$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the integral \( \int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx \) converges as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Laplace transform of the function \( g(t) = \int_{0}^t \frac{1}{e^s\sqrt{s}}ds \) and evaluate it at \( t = 0 \).
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Context:
The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
由 1,2,\cdots,9 这九个正整数构成的所有圆排列中，任意相邻两数之积均不超过 60 的圆排列的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or  $\textit{sell}$  one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If, in the expression $ x^2 - 3$, $ x$ increases or decreases by a positive amount of $ a$, the expression changes by an amount:
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面上，整点（纵、横坐标都是整数的点）到直线 $y=\frac{5}{3} x+\frac{4}{5}$ 的距离中的最小值是多少？原始答案的形式为\frac{\sqrt{n}}{m}，请给出n+m的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $x_{1}$  be a positive real number and for every integer  $n \geq 1$  let  $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$ .
If  $x_{5} = 43$ , what is the sum of digits of the largest prime factors of  $x_{6}$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
C and D are points on a circle with diameter AB such that \( \angle AQB = 2\angle COD \). The tangents at C and D intersect at point P. Given that the circle has a radius of 1, determine the distance from point P to the center of the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence of random variables $X_1, X_2, \ldots, X_n, \ldots$ where $X_n \sim \text{Uniform}(0,n)$. Define $Y_n = e^{-X_n}$. Determine whether the sequence $Y_n$ converges in probability, and if it does, find its limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man nearing retirement wants to invest some money in a fund that compounds annually at a rate of 6 percent, so that in five years, he will have at least $\$100,000$.  How much money must he invest now to make that happen?  (Give your answer to the nearest dollar.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( g(x) = \frac{2}{x^2 - 8x + 17} \). Determine the difference between the largest and smallest possible values of the parameter \( a \) such that the function is defined for all real numbers \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\gamma} f(z) \, dz \), where \( f(z) = \pi e^{\pi \bar{z}} \) and \( \gamma \) is the square with vertices \( 0, 1, 1+i, \) and \( i \) in the counterclockwise direction. Determine if the integral is zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an undirected, weighted, connected graph with distinct edge weights, if there are multiple minimum spanning trees, is it possible for there to be multiple shortest paths between two vertices? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. Find the number of such different patterns.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the limit \( \lim_{n\rightarrow\infty}\int_0^\infty \frac{\sin(x^n)}{x^n}\,dx \) exists, and if it does, find its value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer is called \textit{superprime} if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find number of solutions in non-negative reals to the following equations:
\begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many bricks each measuring 20 cm x 10 cm x 8 cm will be needed to build a wall 10 m x 8 m x 24.5 m?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$P$ is the intersection point of diagonals of cyclic quadrilateral $ABCD$. The circumcenters of $\triangle APB$ and $\triangle CPD$ lie on the circumcircle of $ABCD$. If $AC + BD = 18$, find the area of $ABCD$. The original answer is in the form \(\frac{k\sqrt{m}}{n}\), please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bogan laid out 10 maggots for her pet beetle. The beetle only ate 1 and Bogan had to throw out the rest. Later that day, she tried feeding again and the beetle ate 3. If Bogan served 20 maggots in total, how many did she attempt to feed the beetle the second time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The non-zero digits \( p, q \) and \( r \) are used to make up the three-digit number \( pqr \), the two-digit number \( qr \), and the one-digit number \( r \). The sum of these numbers is 912. Find the value of \( q \). Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that they form a rectangle $ABEF$. Rectangle $KLMN$ is positioned such that its vertices $K, L, M$, and $N$ lie on segments $CD, DF, FE$, and $EC$, respectively. It is given that rectangles $ABEF$ and $KLMN$ are congruent with $AB = MN$. Find the length of segment $AL$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A ball, sliding on a smooth horizontal surface, catches up with a block that was moving on the same surface. The speed of the ball is perpendicular to the face of the block it hits. The mass of the ball is much smaller than the mass of the block. After an elastic collision, the ball slides on the surface in the opposite direction with a speed that is 2 times less than its initial speed.

Find the ratio of the velocities of the ball and the block before the collision.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If Billy rode his bike 17 times, John rode his bike twice as many times, their mother rode her bike 10 times more than John, their cousin Amy rode her bike 3 times more than the square root of the sum of John's and Billy's rides, and their friend Sam rode his bike a total of half their mother's biking frequency minus 5, how many times did they ride their bikes in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ Find the remainder when $N$ is divided by $5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)$ be a quadratic polynomial with a positive leading coefficient. Define $g(x) = 1 - \frac{2}{x-5}$ with the domain $x < 5$. For any real number $t < 3$, let $h(t)$ be the minimum of $f(g(x))$ for $t \leq x \leq t+2$. It is given that $h(t) = f(g(t+2))$ when $t < 1$, $h(t) = 6$ for $1 \leq t < 3$, and $h(-1) = 7$. Determine the value of $f(5)$. Assume that continuity cannot be used in the solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $m$ that minimizes the expression $\binom{m}{k} + \binom{n-m}{k}$, given that $n$ and $k$ are fixed integers and $m = n/2$ if $n$ is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence defined by $a_1 = 0$, $a_2 = 1$, and for $n \geq 3$, $a_n = (n-1)(a_{n-1} + a_{n-2})$. Determine the remainder when $a_{2019}$ is divided by 2019.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are three bags:
1. Bag A contains 3 green candies and 1 red candy.
2. Bag B contains 2 green candies and 2 red candies.
3. Bag C contains 1 green candy and 3 red candies.

A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy.

- If the first candy is green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag.
- If the first candy is red, the child randomly selects a second candy from the same bag as the first candy.

If the probability that the second candy is green is given by the fraction $\frac{m}{n}$ in lowest terms, find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of selfish subsets of the set \( \{1, 2, \ldots, n\} \), where a subset is defined as selfish if it contains its own cardinality and all its other elements are greater than its cardinality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive integer solutions \((x, m, n)\) for the equation \( x^{m} = 2^{2n+1} + 2^{n} + 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the set of numbers $x > 1$ for which the set $\{\sin(x^n) \mid n \in \mathbb{N}\}$ is dense in $[-1, 1]$ is meager or has cardinality $2^{\aleph_0}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On their way driving to Los Angeles, Benjamin and James see lots of billboard ads. They spot two types of billboards: electronic and traditional. Curious, Benjamin starts to count only the electronic billboards they see within specific periods each hour. In the first hour, he notes the billboards during a 25-minute period and counts 15 electronic billboards. They stop for a 15-minute break within this first hour. In the second hour, he only counts during a 45-minute period and sees 31 electronic billboards. In the third hour, during their 35-minute lunch break, Benjamin counts 10 electronic billboards from the restaurant. After resuming their journey, he counts for a 20-minute period and sees 12 electronic billboards. James asks Benjamin the average number of electronic billboards they've seen per hour, considering the varying counting periods and breaks they took during the trip. What should Benjamin tell him?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of \( n \) is it possible to divide any given angle into \( n \) equal parts using only a compass and a straight edge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( T:V\rightarrow W \) be a linear map of inner product spaces with \( T^\ast \) the dual map. Calculate the function \( f(\lambda) = \operatorname{tr}e^{-\lambda T^\ast T} - \operatorname{tr}e^{-\lambda TT^\ast} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the given figure, the line segment \( KM_1 \) intersects the extension of \( AB \) at point \( N \). Determine the measure of angle \( \angle DNA \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Is there a numeral system where numbers can have negative values without using a negative sign? If so, determine the base of such a system.
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Context:
Two numbers between $0$ and $1$ on a number line are to be chosen at random. What is the probability that the second number chosen will exceed the first number chosen by a distance greater than $\frac 14$ unit on the number line? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can 4 distinct beads be placed on a bracelet that has a knot in it?  (Beads cannot slide past the knot.  Two ways are considered the same if one is a reflection of the other, since we can flip the bracelet over.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_t=B_t-tB_1$, where $B_t$ represents standard Brownian motion for $0\leq t\leq 1$. For $0\leq s<t\leq 1$, find $$\mathbb{E}(X_s|X_t=1).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a list of numbers from 1 to 1000, there are 499,500 unique pairs. If you draw 8 numbers at once, what is the minimum number of tries needed to pick all unique pairs, allowing for repetitions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a geometric surface and three points A, B, and C on it, with known geodesic distances AB and AC, determine if the geodesic distance BC satisfies the triangle inequality: AC ≤ AB + BC.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An urn contains 4 blue and 4 red marbles. Initially, a marble is drawn (without looking) and removed from the urn. Then, a marble is drawn from the urn, its color recorded, and put back in the urn. This process is repeated 1000 times. Let event D be that between 450 and 550 blue marbles are drawn, and let event E be that between 379 and 479 blue marbles are drawn. Which event is more likely, D or E?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \in \mathcal{S}(\mathbb{R}^3) \) be a real function. Consider the integral \[ I_f = \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{f(x)f(y)}{|x-y|^2} \, dy \, dx. \] Determine whether it is true that \( I_f \geq 0 \) for a generic function \( f \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $A B C$, points $M$ and $N$ are the midpoints of $A B$ and $A C$, respectively, and points $P$ and $Q$ trisect $B C$. Given that $A, M, N, P$, and $Q$ lie on a circle and $B C=1$, compute the area of triangle $A B C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T  =  TNFTPP$. As $n$ ranges over the integers, the expression $n^4  -  898n^2  +  T  -  2160$ evaluates to just one prime number. Find this prime.

Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of a rectangle remains unchanged when it is made $2\frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2\frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is:

Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a square matrix $A$ with real entries and a generalized eigenvector $v$ of rank $r$ for the eigenvalue $\lambda$, where $v$ is non-null. It is known that $v$ is also a generalized eigenvector of $A^n$ with rank $s \leq r$. If $\lambda \neq 0$, determine whether $s = r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if the function \( f(x) = x^2 \) is Riemann-Stieltjes integrable with respect to \( g(x) = 3x \) on the interval \([0,1]\), given that \( g(1) = 4 \). If it is integrable, find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. Find the number of heavy-tailed permutations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the limit \( \lim_{x \to \infty} \ln\left(\frac{x}{m}\right) \sum_{p \le x} \frac{1}{\ln p} \) converges, where \( p \) denotes all primes and \( m \) is the largest prime less than \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $f(x) = x^2+6x^2+12x+6$, solve the equation $f(f(f(x))) = 0.$

Please reason step by step, and put your final answer within \boxed{}.


Context:
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and$$S(P(n)) = P(S(n)).$$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The only solution is \( P(x) = x \).}


Context:
Solve the partial differential equation \( \frac{\partial z}{\partial x} \cdot \frac{\partial z}{\partial y} = 1 \) with the initial condition \( z(0,0) = 0 \). Determine the value of \( z(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+$ $f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the structure of the group \( G = \mathbb{Z} \times \mathbb{Z} / H \) where \( H = \langle (4, -3) \rangle \), and decide if it is torsion-free.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a geometric surface representing a real-world topography, such as a mountain range, and three points A, B, and C on this surface, where the geodesic distances $AB$ and $AC$ are known, determine if the geodesic distance $BC$ necessarily satisfies the triangle inequality, i.e., is $AC \leq AB + BC$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the limit \( \lim _{t \to \infty} t \cdot \int_t^\infty \frac {\sin^2 (u)}{u^2} \, \mathrm{d}u \) exists.
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Context:
4. In the Olympionov family, it is a tradition to especially celebrate the day when a person turns as many years old as the sum of the digits of their birth year. Kolya Olympionov had such a celebration in 2013, and Tolya Olympionov had one in 2014. Who is older and by how many years?
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Context:
Determine the cardinality of the $\sigma$-algebra generated by the sets $X_i := \{(x_1, x_2, ..., x_n): x_i < \frac{1}{2}\}$ for $i = 1, 2, ..., n$ in the space $X = [0, 1]^n$. Is the cardinality of this $\sigma$-algebra $2^n$?
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Context:
Determine whether the result $x(t) = Ce^t$ holds for solutions $x \in \mathcal{D}'(\mathbb{R})$ of the differential equation $x' = x$ in the theory of distributions.
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Context:
Let \(\mathcal{F}\) be the family of holomorphic functions \(f\) on the open unit disk such that \(\Re f > 0\) for all \(z\) and \(f(0) = 1\). Compute \(\alpha = \sup\{|f'(0)| : f \in \mathcal{F}\}\). Determine whether or not the supremum \(\alpha\) is attained.
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Context:
Determine if the limit \( \lim_{x \to x_0} f(x) \) exists, where \( f(x) = \begin{cases} h(x) & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} \) and \( h(x) \) is a continuous function.
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Context:
Is the genus of $CP^N$ equal to 0 for all $N$?
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Context:
The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$,

$$a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.$$

What is $|a_{2009}|$?

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Context:
## Zadatak B-4.6.

Zadane su elipsa s jednadžbom $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ i hiperbola kojoj su žarišta u tjemenima elipse na velikoj osi, a tjemena u žarištima elipse. Kolika je površina šesterokuta kojemu su vrhovi tjemena elipse na maloj osi i sjecišta zadanih krivulja?

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Context:
NT3 Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.

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Context:
How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?
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Context:
Assume that the time between emissions from a radioactive source is independent and exponentially distributed with rate one. Each emission is detected by a Geiger counter with probability \( p \). Compute the density of the distribution of the time between detections of particles and identify the distribution by name.
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Context:
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?

(Walther Janous)
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Context:
Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$).
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Context:
In a Euclidean manifold $\mathbb{R}^n$, given two tangent vectors at a point $p$, $v_p = v^i \frac{\partial}{\partial x^i}_p$ and $w_p = w^j \frac{\partial}{\partial x^j}_p$, define the dot product between $v_p$ and $w_p$ and provide its mathematical expression.
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Context:
A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. What fraction of the equilateral triangle's area is covered by the infinite series of squares?
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Context:
用1～4这四个数字分别组成两个两位数，使这两个两位数的乘积最大。
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Context:
Find the inverse Laplace transform of \( F(s) = \frac{1}{s(\cosh{\sqrt{2s}}-1)} \). If an analytic expression is not possible, provide an asymptotic expansion for \( f(t) = \mathcal{L}^{-1}[F](t) \) as \( t \to \infty \).
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Context:
Given points $P = (1: -2: 3)$, $Q = (2: 2: -1)$, and $R = (3: 0: 2)$ on a line $g$ in $\\mathbb{P}^{2}\mathbb{R}$, find a coordinate for $g$ such that $(P, Q, R)$ has coordinates $(0, 1, ∞)$. Determine the number of possible coordinates for $g$.
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Context:
Let \(g\) be a continuously differentiable function on \(\mathbb{R}^3\). Consider the closed ball \(B = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1\}\). Since \(g\) is continuous on \(B\), it attains its maximum value at some point \(u \in B\), with \(g(u) < \infty\). If \(u\) lies on the boundary of the ball (\(x^2 + y^2 + z^2 = 1\)), there exists an \(a \in \mathbb{R}\) such that \(\nabla g(u) = au\). Determine the sign of \(a\).
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Context:
Let \(k\) be an algebraically closed field, and let \(q(x) = \frac{r(x)}{s(x)}\) be a nonzero rational function in \(k(x)\), where \(r\) and \(s\) are coprime polynomials and \(d = \deg r \ge \deg s\). Determine whether the degree of the field extension \([k(x):k(q(x))]\) is equal to \(d\).
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Context:
Find the splitting field and its degree over \( \mathbb{Q} \) for the polynomial \( f(x) = x^4 + 2 \).
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Context:
Let $f_n$ be Riemann integrable on $[a,b]$ for all $n$, and let $f, g \in C[a, b]$ be continuous functions. Suppose $f_n \rightarrow f$ in $L^2$ and $f_n \rightarrow g$ pointwise. Is it necessarily true that $f = g$?
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Context:
Let $D \to A$ be a finite type algebra with $D$ a domain. Suppose $V \subset \operatorname{Spec}A$ is open and dense. Determine if $\dim V = \dim A$.
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Context:
Let $g$ be a non-negative function defined on the interval $[0, \pi]$. Given that \( \int_0^x \cos g'(t) \, dt = \int_0^x g(t) \, dt \) for \(0 \leq x \leq \pi\) and \(g(0) = 0\), determine the possible values of \(g\left(\frac{\pi}{2}\right)\).
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Context:
Let \( D \) be a point inside the acute triangle \( \triangle ABC \). Given that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \), find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \).
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Context:
Let \( \{a_n\}_{n\geq1} \) be an absolutely convergent sequence. Determine whether the sequence \( K_n = \frac{1}{\ln(n^2+1)}\sum_{k=1}^{n}a_k\frac{3k^3-2k}{7-k^3}\sin k \) is convergent.
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Context:
Exercise 4. Consider an $8 \times 8$ chessboard whose squares are alternately colored white and black. An infernal rook is a piece that can attack squares of its own color on its row, as well as squares of the opposite color on its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?
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Context:
if an object travels 200 feet in 2 seconds , what is the object ' s approximate speed in miles per hour ? ( note : 1 mile = 5280 feet )
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Context:
A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \mathrm{~cm}^{2}$.
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Context:
When Lauren was born on January 1, 1990, her grandparents put $\$1000$ in a savings account in her name. The account earned $7.5\%$ annual interest compounded quarterly every three months. To the nearest dollar, how much money was in her account when she turned two?
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Context:
Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a linear and positively homogeneous function of degree zero. If $f$ vanishes on a plane, does this imply that $f$ is zero everywhere or it has a discontinuity at a direction parallel to that plane? Answer with 'yes' or 'no'.
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Context:
Let $G$ be a group, and let $H$ and $K$ be distinct subgroups of $G$. If $H \cup K$ is also a subgroup of $G$, does it necessarily follow that either $H \subset K$ or $K \subset H$? Answer 'yes' or 'no'.
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Context:
Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Suppose $A: X \to X$ is a bounded operator that maps $E$ into itself, and the spectral radius of $A$ restricted to $E$ is a finite $r > 0$. Is the spectral radius of $A$ on $X$ also equal to $r$?
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Context:
Let $X$ be a Banach space and $X^*$ its dual space. Suppose $u \in X$ satisfies $\langle f, u \rangle_{X^* \times X} = 0$ for all $f \in X^*$. Is it necessarily true that $\|u\|_X = 0$?
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Context:
Let $K=\mathbb{R}(X)$ be a rational function field, and let $F= \mathbb{R}(X^4 -\frac{1}{X^4 })$. Find the Galois closure $L$ of $K$ over $F$ and determine the degree $[L:F]$. Provide your answer as a single number.
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Context:
Determine the number of possible Jordan canonical forms and similarity classes for a $5 \times 5$ matrix with the minimal polynomial $p(x) = (x-2)(x^2+1)^2$ over $\mathbb{R}$. Provide your answer as a single number.
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Context:
Every cell of $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than $-3$?

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Context:
Evaluate the surface integral \( \iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy \) over the part of the plane \( x+y+z=2 \) in the first octant, where the normal vector \( n \) satisfies \( n \cdot (0,1,0) \geq 0 \).
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Context:
Let  $P(x)$  be a polynomial with integer coefficients and roots  $1997$  and  $2010$ . Suppose further that  $|P(2005)|<10$ . Determine what integer values  $P(2005)$  can get.
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Context:
Given a financial market with a bond $A$, a stock $S$, and call options $f$ on the stock, consider two times $0$ and $T$ with two possible outcomes for $S$ at time $T$. The bond and stock values are $A(0)=100$, $A(T)=120$, $S(0)=50$, $S^u(T)=70$, $S^d(T)=30$. Construct a portfolio consisting of shares of $S$ and call options $f$ with strike $K=50$ and expiration $T$ such that the portfolio value at time $T$ is $120$ regardless of the stock's movement. What is the price of the call option?
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Context:
Find all non-decreasing functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ satisfying the equation:

$$f\left(\sum_{i=1}^{2014} a_{i}^{2014}\right) = \frac{1}{2014^{2013}}\sum_{i=1}^{2014} \left(f(a_{i})\right)^{2014}$$

for all $a_i \in \mathbb{Z}^+$.
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Context:
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if  $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$ ?
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Context:
An aluminum part and a copper part have the same volume. The density of aluminum is $\rho_{A} = 2700 \, \text{kg/m}^3$, and the density of copper is $\rho_{M} = 8900 \, \text{kg/m}^3$. Find the mass of the aluminum part, given that the masses of the parts differ by $\Delta m = 60 \, \text{g}$.
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Context:
Two robots are programmed to communicate numbers using different bases. The first robot states: “I communicate in base 10,
 which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I
 find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of
 which integer?
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Context:
Let $ \left(a_{n} \right)_{n = 1}^{\infty }$ be a sequence on real numbers such that $ a{}_{n + 1} = a_{n} a_{n + 2}$ for every $ n\ge 1$. The number of elements in the set $ \left\{a_{n} : n\ge 1\right\}$ cannot be
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Context:
In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o$, $\angle  CBE = 51^o$. Find the measure of angle $\angle  BED$.
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Context:
Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \frac{\mathbb{F}_2[x]}{\langle x^3+x^2+x+1 \rangle} \).
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Context:
Determine if the derivative of the function \( F(x) = \int_0^x |\cos(1/u)| \, du \) exists at \( x = 0 \) and find \( F'(0) \).
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Context:
Evaluate the integral \( \int_0^\infty\int_0^\infty e^{-sxy}\cos(x)\,dx\,dy \) for \( s>0 \) using the double integration method.
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Context:
Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\frac{1}{2}$, what is its speed?
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Context:
What is the minimum number of moves required to transform a $100 \times 100$ table, initially all white, into a checkerboard pattern by choosing any 99 cells from one row or one column and flipping their colors in each move?
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Context:
Find the optimal value of \( k_n \) in terms of \( n \) such that for any positive integer \( n \), the inequality \( \left(\sum_{i=1}^{3n} a_{i}\right)^{3} \geqslant k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2n+i}\right) \) holds for all sequences \( 0 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{3n} \).
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Context:
Determine whether the image of the function $f(z) = 2z^{14}\cos^2z$ is $\mathbb{C}$ or $\mathbb{C} \setminus \{a\}$ for some $a \in \mathbb{C}$.
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Context:
Determine the number of differentiable functions \( y:(-\infty, \infty) \rightarrow [0, \infty) \) that satisfy the differential equation \( y' = 2 \sqrt{|y|} \) with the initial condition \( y(0)=0 \).
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Context:
If the graph of the function $f(x)=2\sin (2x+\varphi )$ $(0 < \varphi < \frac{\pi}{2})$ passes through the point $(0,\sqrt{3})$, then the decreasing interval of the function $f(x)$ on $[0,\pi ]$ is.
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Context:
$f(x)$ is a nonconstant polynomial. Given that $f(f(x)) + f(x) = f(x)^2$, compute $f(3)$.
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Context:
Let $I$ be an ideal in the polynomial ring $R = k[x,y]$. Consider the algebraic closure $\overline{k}$ of $k$ and the ring $S = \overline{k}[x,y]$. Determine whether $IS \cap R = I$.
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Context:
For which values of \( a \) are the flows of \( \overrightarrow{x}' = A \overrightarrow{x} \) and \( \overrightarrow{y}' = B \overrightarrow{y} \) topologically conjugate, given that \( A \) is a hyperbolic matrix such that all solutions of \( \overrightarrow{x}' = A \overrightarrow{x} \) tend to the origin as \( t \to \infty \), and \( B = \begin{bmatrix}a-3 & 5 \\ -2 & a+3\end{bmatrix} \)?
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Context:
If \(L\) is a language and the language \(\tilde{L} = \{x^k \mid x \in L, k \in \mathbb{N}\}\) is regular, does it necessarily follow that \(L\) is regular? Answer 'yes' or 'no'.
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Context:
If \(2^7\cos^5x \cdot \sin^3x = a\sin8x - b\sin 6x + c\sin 4x + d\sin 2x\) where \(x\) is real, find the value of \(a^4 + b^4 + c^4 + d^4\).
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Context:
Hoping to join Chad and Jordan in the Exeter Space Station, there are $2014$ prospective astronauts of various nationalities. It is given that $1006$ of the astronaut applicants are American and that there are a total of $64$ countries represented among the applicants. The applicants are to group into $1007$ pairs with no pair consisting of two applicants of the same nationality. Over all possible distributions of nationalities, what is the maximum number of possible ways to make the $1007$ pairs of applicants? Express your answer in the form $a \cdot b!$, where $a$ and $b$ are positive integers and $a$ is not divisible by $b + 1$.

Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ... \cdot  2 \cdot  1$.

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Context:
In a math competition, all teams must consist of between $12$ and $15$ members, inclusive. Mr. Beluhov has $n > 0$ students and he realizes that he cannot form teams so that each of his students is on exactly one team. Find the sum of all possible values of $n$.

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Context:
For $i=1,2,\ldots,7$, Zadam Heng chooses a positive integer $m_i$ at random such that each positive integer $k$ is chosen with probability $\frac{2^i-1}{2^{ik}}$. If $m_1+2m_2+\ldots+7m_7\neq35$, Zadam keeps rechoosing the $m_i$ until this equality holds. Given that he eventually stops, what is the probability that $m_4=1$ when Zadam stops?

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Context:
在●○●○○●○○○●○○○○●○○○○○中，空心圈出现的频率是多少？
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Context:
Determine the number of different bases that can be formed from the set of all vectors in \(\mathbb{C}^3\) where each coordinate is either 0 or 1.
Please reason step by step, and put your final answer within \boxed{}.