Context:
Determine the maximum number of people that can be arranged in a 2D plane such that no more than 3 people get wet when each person sprays the person closest to them.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{\infty}|\ln n-\ln k|\left(1-\frac{1}{n}\right)^k \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest known natural number $n$ such that every sequence of length $n$ appears in the base 10 expansion of $\pi$. Provide the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the covariant derivative of the scalar density of weight 1, \( \rho=\sqrt{|g|} \), without using the formula for the Christoffel symbols.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the differential form \( a = p_1 dq_1 + p_2 dq_2 - p_1 p_2 dt \) in the space \( \mathbb{R}^5 \) with coordinates \((p_1, p_2, q_1, q_2, t)\). Compute \( da \) and \( da \wedge da \). Evaluate the integral \( \int_S t \, da \wedge da \) where \( S \) is the 4-dimensional surface (with boundary) in \( \mathbb{R}^5 \) defined by the relations: \( p_1^2 + p_2^2 \le 1, \ q_1^2 + q_2^2 \le 2 \) and \( t = 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the line integral \( \int_C ze^{\sqrt{x^2+y^2}} \, ds \) where \( C \) is the curve of intersection of the sphere \( x^2 + y^2 + z^2 = a^2 \) and the plane \( x + y = 0 \), with \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many $4$ element subsets of $\{0,1,2,\dots,20\}$ contain their sum modulo $21$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_A \left(\frac{xe^{x^2+y^2}}{x^2+y^2}-1\right)\mathrm{d}x\mathrm{d}y \) over the region \( A=\{(x,y)\in \mathbb{R}^2: 1 \le x^2+y^2\le 4 \} \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sub-differential of the function \( f(x) = -\sqrt{x} \) at \( x = 0 \), where \( f: [0, 1] \rightarrow \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute $\left\lfloor \sum_{k=0}^{10}\limits  \left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$
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Context:
In an equilateral triangular prism \( S-ABC \) with height \( SO = 3 \) and a base edge length of 6, a perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with the foot of the perpendicular at \( O' \). On \( AO' \), a point \( P \) is chosen such that \( AP:PO' = 8 \). Calculate the area of the cross-section passing through point \( P \) and parallel to the base.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence for the function \( f(z) = \frac{z - z^3}{\sin(\pi z)} \) at the point \( z=0 \).
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Context:
Evaluate the integral: $$\int_{0}^{1} \ln (1-x^2)\ dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(a\) such that the function \(g(u,v) = u_1v_1 + 5u_2v_2 + au_3v_3 + 2(u_1v_2+u_2v_1) + 3(u_1v_3+u_3v_1) + 4(u_2v_3+u_3v_2)\) defines a scalar product in \(\mathbb{R}^3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \( x^{10} = \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \) in the space of 2x2 real matrices, \( M_{2}(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 3x3 matrices over the field of three elements that are conjugate to a diagonal matrix.
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Context:
Determine the values of $b$ for which the integral \( \int_{1}^{\infty} \frac{\ln x}{x^b} \cos x \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limiting value of the expression: $$\sqrt{2 + 3 \sqrt{2 + 4 \sqrt{2 + \cdots \sqrt{2 + (n - 1) \sqrt{2 + n}}}}}$$ as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability that a randomly selected subset of the interval [0, 1] is measurable according to the Lebesgue measure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $L$ be a list of unique elements. Consider a modified algorithm for finding the maximum value in $L$ where we initialize the current maximum to be the first element of $L$. Calculate the expected number of times the current maximum will be reset when we randomly permute the elements of $L$ before running the modified algorithm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the integral \( w = \int_{xy}^{2x-3y} \frac{du}{\ln(u)} \, du \), find \( \frac{\partial y}{\partial x} \) at \( x = 3, y = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of \(|z|\) given that \(|z^2 + 2z \cos \alpha| \leq 1\) for all real values of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $1006 \sin \frac{\pi}{1006}$. Approximating directly by $\pi=3.1415 \ldots$ is worth only 3 points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the solid obtained by revolving the region bounded by \( y = \sin\left(\frac{\pi x}{12}\right) \) and the x-axis about the line \( x = -1 \) using the method of cylindrical shells.
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Context:
Determine the range of \( \alpha \) for which the improper double integral \[ I = \iint_{x^2+y^2>0} \frac{\sin(x^2+y^2)}{(x^2+4y^2)^{\alpha}}\, dx\, dy \] is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(x^2 + C) \) for all \( x \in \mathbb{R} \), where \( C > 0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Magnus and Viswanathan play a series of three games with the following conditions:

- Each player's score in each game is a positive integer, and the scores across all games are six different integers.
- The winner's score in each game is at least 25 points. If the winner scores 25, the opponent's score is at most 23. If the winner scores more than 25, the opponent's score is exactly 2 less than the winner's score.
- Viswanathan wins either the first or the second game, but not both.
- Viswanathan wins the third game with a score of 25.
- Each player's score in the second game is the average of their scores in the first and third games.

Determine Magnus's score in the third game.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \mathbf{F} \) be a vector field defined on \( \mathbb{R}^2 \setminus\{(0,0)\} \) by \( \mathbf{F} (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j \). Consider the curves \( \gamma(t) = (8\cos 2\pi t, 17\sin 2\pi t) \) and \( \alpha(t) = (26\cos 2\pi t, -10\sin 2\pi t) \) for \( t \in [0,1] \). If \( 3\int_{\alpha} \mathbf{F \cdot dr} - 4 \int_{\gamma} \mathbf{F \cdot dr} = 2m\pi \), find the value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate
\[\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Avik holds six identical ropes with their midpoints in his fist, and the ropes' ends lie on opposite sides. Kamrul randomly pairs and ties the ends together from one side and then the other. If the probability of creating a loop by tying all six ropes is \(\frac{a}{b}\), where \(a\) and \(b\) are coprime, find the value of \(a+b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice and Bob are playing a coin-flipping game. Alice wins when the sequence "h, h, h" appears, and Bob wins when the sequence "t, h, t" appears. What is the expected number of coin flips until one of them wins?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \bigtriangleup ^{6} p(x) \big| _{x=0} \), where \( p(x) = \sum_{k=0}^{10} k^{2} x^{ \underline{k} } \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the probability matrix of a Markov chain:
\[ P = \begin{bmatrix}0&1/2&1/2\\1/3&0&2/3\\1/2&1/2&0\end{bmatrix} \]
find the limit:
\[ \lim_{n \to \infty} P(X_n = 2 \mid X_0 = 1) \] analytically.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to cover a 3 × 8 rectangle with 12 identical dominoes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when $99{,}999^{99}$ is divided by $999{,}999$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be an analytic function on the annulus $\{z : 1/2 < |z|<2\}$ except for a simple pole at $z=1$ with residue 1. The Laurent expansions of $f$ on the annuli $\{ z: 1/2<|z|<1\}$ and $\{z: 1<|z|<2\}$ are given by $\sum a_n z^n$ and $\sum b_n z^n$, respectively. Compute $b_n - a_n$ for every integer $n$. Provide your answer as a single expression or value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic bound in terms of $n$ for the summation:
\[ \sum_{i=0}^{n}{\frac{P(n, i)}{n^i}} = \sum_{i=0}^{n}{\frac{n!}{(n-i)! \cdot n^i}}. \] Is it $O(n)$, $O(\log n)$, or another bound?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( k \) such that for any \( k \) vectors \( v_1, v_2, \ldots, v_k \in \mathbb{R}^n \), there exist real numbers \( a_1, a_2, \ldots, a_k \), not all zero, satisfying \( a_1v_1 + a_2v_2 + \cdots + a_kv_k = 0 \) and \( a_1 + a_2 + \cdots + a_k = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any set $S$, let $P(S)$ be its power set, the set of all of its subsets. Over all sets $A$ of $2015$ arbitrary finite sets, let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \\in P(A), T \\in P(P(A))$, $S \\in T$, and $S \\subseteq T$. (Note that by convention, a set may never contain itself.) Find the remainder when $N$ is divided by $1000.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the uniformly most powerful (UMP) test at significance level \( \alpha = 0.05 \) for the hypothesis test concerning a geometric distribution with probability mass function \( P(X=k) = p^k(1-p) \), where \( 0 < p < 1 \). The null hypothesis is \( H_0: p \leq 0.95 \) and the alternative hypothesis is \( H_1: p > 0.95 \). Determine the rejection region for this test.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $n$ for which the binomial expansion of $(A+B)^n$ implies that $A$ and $B$ commute.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the module \(A = (\mathbb{Z}[i]/I^{2014})\otimes_{\mathbb{Z}[i]}(\mathbb{Z}[i]/J^{2014})\), where \(I\) and \(J\) are the principal ideals of \(\mathbb{Z}[i]\) generated by \(7-i\) and \(6i-7\), respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the region $V$ bounded by the paraboloid $z=5-4x^2-4y^2$ and the $xy$-plane. Given that the surface integral of the vector field \( \vec{F}=\nabla\times \vec{G}=2\vec{i}+2y^2\vec{j}+z\vec{k} \) over the circle in the $xy$-plane is 15, find the value of the surface integral over the paraboloid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the center of the polynomial ring $R[x, y]$, where $R$ is a non-commutative ring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a rectangular box with a diagonal of length 10 units, determine the maximum possible perimeter of the box.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k - n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).

Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n + 1$ through $ 2n$ are all off.

Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n + 1$ through $ 2n$ are all off, but where none of the lamps $ n + 1$ through $ 2n$ is ever switched on.

Determine $ \frac {N}{M}$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The answer is \( 2^{k - n} \).

}


Context:
If $G$ is a non-abelian group of order 27, what is the order of its commutator subgroup $G'$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of common normals to the curves given by the equations $x^2 + (y-1)^2 = 1$ and $y^2 = 4x$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given real numbers \(x, y, z\), solve the equation \(4^{\sqrt{5x+9y+4z}} - 68 \cdot 2^{\sqrt{5x+9y+4z}} + 256 = 0\). Find the product of the minimum and maximum values of \(x+y+z\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the topological space $[0,\infty]$ with the order topology. Let $E_n = [n, \infty)$. Determine the limit of $\mu(E_n)$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( A = \lim_{x\to +\infty}\left(\frac{x}{x+1}\right)^x + \lim_{x\to +\infty}\left[\frac{x}{x+1}\right]^x \), where \([x]\) is the floor function of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function satisfying the equation \( f(2xy) + f(f(x + y)) + f(x + y) = xf(y) + yf(x) \) for all real numbers \( x \) and \( y \). Determine the sum of all possible values of \(|f(48)|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$
be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold.
$$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$
Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a non-zero function \( f(x) \) such that \( f(2x) = f'(x) \cdot f''(x) \), where \( f' \) and \( f'' \) are the first and second derivatives of \( f \), respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the ratio of consecutive terms in the sequence $b_n$, where $b_n$ is the number of alternating sequences of length $n$ that do not start or end in 0, as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_W (x^2 + y^2 + z^2) \, dV \), where \( W \) is the region bounded by the planes \( x = 0 \), \( y = 0 \), \( z = 0 \), \( x + y = 1 \), and \( z = x + y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an upper bound for the limit: $$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=2}^{2n} \frac{\sin (\cos (2 \pi \log (k)))}{k}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a $\mathbb{C}$-algebra and $V$ be an irreducible $A$-module with countable dimension. Determine the dimension of $\text{End}(V)$ as an $A$-module.
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Context:
Let $S$ be the set of all permutations of $\\{1, 2, 3, 4, 5\\}$. For $s = (a_1, a_2,a_3,a_4,a_5) \\in S$, define $\\text{nimo}(s)$ to be the sum of all indices $i \\in \\{1, 2, 3, 4\\}$ for which $a_i \u003E a_{i+1}$. For instance, if $s=(2,3,1,5,4)$, then $\\text{nimo}(s)=2+4=6$. Compute \[\sum_{s\in S}2^{\text{nimo}(s)}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of distinct seating plans for 15 people in a row of 15 chairs, where two seating plans are considered the same if they share the same adjacent quadruples. For example, if the people are seated in the order 1, 2, 3, 4, 5, ..., 15, then the quadruples 1234, 2345, 3456, ..., 12131415 cannot be used in other plans. What is the maximum number of distinct seating plans possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are infinitely many boxes - initially one of them contains $n$ balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}\) to an accuracy of \(10^{-3}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest number of sets required to generate the power set of a finite set $A$ where $|A| = n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the kernel of the adjoint representation \( P: SU_2 \to \text{Aut}_\mathbb{R}(E) \), where \( P(A) \) maps \( X \) to \( AXA^{-1} \). Here, \( E \) is the space of traceless Hermitian matrices of the form:
\[
\begin{bmatrix}x_1 & x_2 + ix_3 \\ x_2 - ix_3 & -x_1\end{bmatrix}
\]
with \( x_1, x_2, x_3 \in \mathbb{R} \). Determine the matrices \( A \) such that \( AXA^{-1} = X \) for all \( X \) in \( E \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{2015^{k}}{\sum_{i=0}^{k-1}2015^i\sum_{l=0}^k2015^l} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of all $6$-digit natural numbers such that the sum of their digits is $10$ and each of the digits $0,1,2,3$ occurs at least once in them.
Please reason step by step, and put your final answer within \boxed{}.


Context:
 Let $f:\mathbb{N}\to\mathbb{N}$ be a function such that\[f(n) =\sum_{abc = n | a,b,c \in \mathbb{N}} ab + bc + ca.\]For example, $f(5) = (1 \cdot 1 + 1 \cdot 5 + 5 \cdot 1) + (1 \cdot 5 + 5 \cdot 1 + 1 \cdot 1) + (5 \cdot 1 + 1 \cdot 1 + 1 \cdot 5) = 33$, where we are summing over the triples $(a,b,c) = (1,1,5), (1,5,1)$, and $(5,1,1)$. Find the last three digits of $f(30^{3})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint_S \mathbf{A} \cdot \mathbf{n} \, \mathrm{d}S \) using the divergence theorem, where \( \mathbf{A} = y\mathbf{i} + 2x\mathbf{j} - z\mathbf{k} \) and \( S \) is the surface of the plane \( 2x + y = 6 \) in the first octant, cut off by the plane \( z = 4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the series \( \sum_{n=1}^{\infty} a_n \), where \( a_n = \frac{n^2 + 1}{4n^2 - 3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of the expression \( \text{trace} (\Sigma^{\frac{1}{2}} S^{-1}\Sigma^{\frac{1}{2}}) \) with respect to the matrix \( \Sigma \), where both \( \Sigma \) and \( S \) are positive definite matrices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Lie derivative of the metric tensor \(g_{ab} = \begin{pmatrix} 1 & 0 \\ 0 & x^2 \end{pmatrix}\) along the vector field \(X = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\begin{pmatrix} 0 & 0 \\ 0 & 2x \end{pmatrix}}


Context:
Given that $T \sim t_7$, find the value of $t$ such that:
\[ P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the roots of the equation \(\cos(12x)=5\sin(3x)+9\tan^2(x)+\frac{1}{\tan^2(x)}\) for \(0\le x\le 2\pi\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), point \(X\) is located on base \(AD\) such that segments \(XB\) and \(XC\) divide the trapezoid into three similar, yet pairwise unequal, non-isosceles triangles. Given that side \(AB\) has a length of 6, determine the value of \(AX \cdot DX\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the area of triangle \( QGH \) is \( 4 \sqrt{2} \) and the area of triangle \( FGH \) is greater than 16, find the angle \( GFH \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the recurrence relation \( t(2^{2^{k}}) = 1 + t(\sqrt{2^{2^k}}) \) in terms of \( k \), i.e., find \( t(k) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the supremum of \(|f(1)|\) for all analytic functions \(f\) mapping from \(\mathbb{C} \setminus \{0\}\) to \(\mathbb{C}\) such that \(|f(z)| \leq 5|z|^{-\sqrt{2}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group of order 120, determine the number of elements of order 12, given that there is at least one Sylow 5-subgroup.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the series: $$\sum_{n=-\infty}^{\infty} \sin \left(8n+\frac{2\pi}{3}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{\gamma}{\Re\left(z\right) \over z - a}\,\mathrm{d}z \) in terms of \(a\), where \(\gamma\) is the unit circle of radius 3 oriented in the positive sense and \(0 < \left\vert a\right\vert < 2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest value of $P(A) = P(B) = P(C)$ such that $P(A \cap B \cap C) > 0.95$, given that the relationship between the events $A$, $B$, and $C$ is unknown.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\), where \(\{x\}\) denotes the fractional part of \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the function \( f(x) \), where \( f(x) \) is a twice differentiable function satisfying the differential equation \( f(x) + f''(x) = -x|\sin x|f'(x) \) for \( x \geq 0 \). Given that \( f(0) = -3 \) and \( f'(0) = 4 \), determine the maximum value of \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit: \[ \lim_{x \to \infty} \frac{\exp(2x)}{\exp(s\log(\cosh(\frac{2x}{s})))} \] for \( s > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square with side length $2n$ is divided into $4n^2$ smaller squares of equal size. A circle with diameter $2n-1$ is inscribed in the larger square. Determine the number of smaller squares that intersect the circumference of the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to 0}\frac{\cos^2(1-\cos^2(1-\cos^2(1-\cos^2(\ldots 1\ldots \cos^2 x))\ldots))}{\sin\left(\pi\left(\frac{\sqrt{x+4}-2}{x}\right)\right)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathbb{P}$ be the uniform distribution on $\Omega = \{1, 2, \ldots, 10\}$. An event $A \subseteq \Omega$ is said to be non-trivial if $0 < P(A) < 1$. Determine the largest integer $k$ such that there exist non-trivial events $A_1, A_2, \ldots, A_k \subseteq \Omega$ that are independent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We say that a four-digit number $\overline{abcd}$ is slippery if the number $a^4+b^3+c^2+d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ slippery, since $2^4 + 0^3 + 2 ^2 + 3 = 23$. How many slippery numbers are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine an asymptotically tight bound for the summation \(\sum_{k=1}^n k^u \cdot v^k\), where \(u\) and \(v\) are constants with \(u \geq 0\) and \(v > 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can 12 students be assigned to 6 pairs of seats (one in front, one behind) such that students in each pair receive the same exam variant, there are 3 variants available, and no two adjacent pairs receive the same variant?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum \( \sum_{k=1}^{2018} a_{k} \), where \( a_{k} \) is the number of integer terms in the sequence \( \log_{2} k, \log_{3} k, \ldots, \log_{2018} k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{C}\frac{e^{\frac{1}{z}}}{z-1}dz \) where \( C: |z-1|=\frac{3}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
1. Find the term that does not contain $x$ in the expansion of $\left(\frac{1}{x}+3 x\right)^{n}$, if the coefficient of the power of $x$ in the tenth term is the largest.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many two digit numbers are the product of two distinct prime numbers ending in the same digit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=x+\sqrt{1-x}$, determine the minimum value of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected time of failure for a machine with two components, X and Y, where Y starts working only after X fails, and the machine fails when Y fails. The joint density function is given by \( f_{X,Y}(x,y)=4e^{-2x}e^{-y} \) for \( 0<x<y< \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $a$ for which the series \( \sum_{n=2}^\infty \left( 1+\frac{1}{3} + \dotsb + \frac{1}{n-1} \right) \frac{\sin (na)}{n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the inequality \( f(c)-b^2 \leq f(a+b)-f(a)+f(c) \leq b^2+f(c) \) for all real numbers \( a, b, \) and \( c. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sequence of natural numbers from 1 to 37, each number, starting from the second, divides the sum of all preceding numbers. The number 37 is placed first. Determine the number that occupies the third position in this sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Joe has 1729 unit cubes, which are initially unpainted. He randomly forms two cubes of side lengths 9 and 10 or 1 and 12, and dips them into white paint. This process is repeated, with the side lengths being chosen randomly each time, until every side of every unit cube is painted. After how many repetitions is the expected number of painted faces closest to half of the total possible painted faces?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_1^3 e^{2x}\left(1+\{2x\}+\{2x\}^2\right)dx \) using the properties of the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of sequences \((a, b, c)\) where \(a, b, c \in \{1, 2, \ldots, 2n\}\) that are called *joke* sequences. A sequence is a joke if its shortest term is odd and if only that smallest term, or no term, is repeated. Express your answer in terms of \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Erin the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a chessboard with 2 rows and 2019 columns, each cell contains a real number. The numbers in the first row are all distinct, and the numbers in the second row are a permutation of the numbers in the first row. In each column, the sum of the two numbers is rational, and the two numbers are different. What is the maximum number of irrational numbers that can be placed on the chessboard?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( \mathbb{E}(N(1,4] \mid N(3,10] = 7) \) where \((N_t, t \geq 0)\) is a Poisson counting process with rate \(\lambda = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set $E$ of all positive integers $n$ such that when divided by $9, 10, 11$ respectively, the remainders (in that order) are all greater than 1 and form a non-constant geometric progression. If $N$ is the largest element of $E$, find the sum of the digits of $N$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{|z|=3} \frac{\mathrm{e}^{1/(1-z)}}{z} \,\mathrm{d}z \) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of $n$ days of walking from the camp, under the following conditions:
$(i)$ Each member of the expedition can pick up at most $3$ liters of water.
$(ii)$ Each member must drink one liter of water every day spent in the desert.
$(iii)$ All the members must return to the camp.
How much water do they need (at least) in order to do that?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two sequences of sets \(\{A_{n}\}\) and \(\{B_{n}\}\) defined as follows:

- \(A_{1} = \varnothing\)
- \(B_{1} = \{0\}\)
- \(A_{n+1} = \{x + 1 \mid x \in B_{n}\}\)
- \(B_{n+1} = (A_{n} \cup B_{n}) \backslash (A_{n} \cap B_{n})\)

Find all positive integers \(n\) such that \(B_{n} = \{0\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions for \( x \in \{1, 2, \ldots, 1000\} \) that satisfy the congruence \( x^2(x+1)^2 \equiv 0 \pmod{1000} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the stochastic integral \( \int_0^4 \Phi(t) \, dW(t) \), where the stochastic process \( \Phi(t) \) is defined as follows:
\[
\Phi(t) = \begin{cases}
2, & \text{if } t \in [0,1] \\
W(1), & \text{if } t \in (1,2] \\
W(1.5), & \text{if } t \in (2,3] \\
0, & \text{if } t \in (3,4]
\end{cases}
\]
Here, \( W(t) \) represents a Brownian motion. Use Ito isometry to find the variance.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( k \) such that \( k! \cdot S_{2016} \) is an integer, where \( S_n \) is the sum of the reciprocals of the non-zero digits of all positive integers up to and including \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the binomial series to find the value of $X+Y-Z$ in the equation: $$\sum_{n=1}^{\infty}{\frac{\prod_{r=0}^{n-1}{(2-r)}}{4^nn!}}=\left(\frac{X}{Y}\right)^2-Z$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence \(\{\alpha_{n}\}\) is an arithmetic sequence with common difference \(\beta\). The sequence \(\{\sin \alpha_{n}\}\) is a geometric sequence with common ratio \(q\). Given that \(\alpha_{1}\) and \(\beta\) are real numbers, determine the value of \(q\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \( I_{\infty}(N) = \lim_{x\to\infty} \int \prod_{k=1}^N \left(k-\frac{k}{x}\right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$k$ is a positive integer,
$R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}$ for $n=2k$
$R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}$ for $n=2k+1$.
A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from $R_{n}$ with the property that any two connected marbles have different numbers on them, we call it nice labeling. If each one of the marbles are written on some numbers from $R_{n}$ with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to $0$, we call it precise labeling.
$n\geq{3}$, if every mechanism that is labeled nicely with $R_{n}$, could be labeled precisely with $R_{m}$, what is the minimal value of $m$?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The minimal value of \( m \) is \( 2n - 1 \).

}


Context:
Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016}-N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)\in\Bbb R^2$. Determine the probability that $W_t$ hits the positive half of the $x$-axis before it hits the negative part.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int\int_{R} y^2 \, dx \, dy \) using polar coordinates, where \( R \) is the region in the first quadrant bounded by the curves \( x^2 + y^2 = 16 \) and \( x^2 + y^2 = 4x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 2-3 trees of depth 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $T: V \to V$ be a linear transformation with minimal polynomial $x^{13}(x-1)$. Determine the minimal polynomial of the transformation $T^4$. \( \text{Provide your answer as a polynomial.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the imaginary part of the holomorphic function \( f(z) = z\overline{z} + iv(x,y) = x^2 + y^2 + iv(x,y) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of prime implicants for a conditional statement in a Karnaugh map with \( n \) variables?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of Sylow 2-subgroups of the symmetric group $S_5$. Given that $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, use the Sylow theorems to find the number of Sylow 2-subgroups.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box contains 674 apples, 674 oranges, and 674 pears. Determine the number of ways to pack these fruits into two boxes such that each box contains all three types of fruits and the product of the number of apples, oranges, and pears in each box is the same.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square $ABCD$ with a side length of $2$ is rotated around $BC$ to form a cylinder. Find the surface area of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( F(x, y) = 2x^4 + 3y^3 + 5xy \) and the condition \( F(x, y) = 10 \), find the derivative \( y'(x) \) at the point \( x = 1 \) and \( y = 1 \). Assume a small change in \( x \) from 1 to 0.999. Use the Implicit Function Theorem to determine how \( y \) changes with respect to \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
$$\int_0^13x^2dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A board game is played on a hexagonal grid of 19 tiles. A 'traveler' token starts on the center tile. Each turn, a die is rolled to determine which neighboring tile the traveler moves to, with all six directions being equally likely. The game ends on the turn that the traveler leaves the board. What is the expected number of turns until the game ends?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $k$ for which the set of vectors $S = \{1+kx, k + x + x^2 + x^3, kx^2 + x^3 \}$ is linearly independent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set $L$ of binary strings of length less than or equal to $9$, and for a string $w$ define $w^{+}$ to be the set $\{w,w^2,w^3,\ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x,y\in L$ such that $x^{+}\cap y^{+}\neq \varnothing$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the measure of the set \(S = \{w \in \{0,1\}^\omega \mid \text{for every } N \in \mathbb{N}, w \text{ has a prefix of the form } pp \text{ with } |p| \ge N \}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of right-angled triangles with integer sides that have an inradius of 2009.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of \( n \) such that the Chebyshev interpolation of \( f(x) = e^x \) on the interval \([-1, 1]\) has an error bound less than \(10^{-10}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence of the power series \( \sum_{n=-\infty}^{\infty} 3^{-|n|} z^{2n} \), where \( z \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $f(k)$ is a strictly increasing function from $\mathbb{Z}^+$ to $\mathbb{Z}^+$ satisfying $f(f(k)) = 3k$, find the sum $f(1) + f(9) + f(99)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A necklace consists of 100 beads, which are colored red, blue, and green. In any sequence of five consecutive beads, there is at least one blue bead, and in any sequence of seven consecutive beads, there is at least one red bead. Given that the beads are arranged cyclically, what is the maximum number of green beads that can be in this necklace?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum: $$\frac{1}{2^{99}}\sum_{k=1}^{50} \sum_{r=0}^{k-1}\binom{50}{k}\binom{49}{r}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected time $T_n$ for a bacterium, which divides exponentially with rate $\lambda = 1$, to reach a population of $n$ bacteria. Assume the bacterium initially divides into two at time zero, and each subsequent division time is an exponential random variable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least square approximation of the function \( p(x) = x^{1/3} \) in the subspace \( W = \text{span}\{1, x, x^2 - \frac{1}{3}\} \) of the inner product space \( V = \mathcal{C}[-1, 1] \) with the inner product \( \langle f, g \rangle = \int_{-1}^{1} f(x)g(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n}\left(\sqrt[n]{\int_0^{\pi/4}e^{nx}(\tan^{n-1}x+\tan^nx+\tan^{n+1}x)dx}-1\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are relatively prime positive integers $m$ and $n$ so that\[\dfrac{\dfrac{1}{2}}{\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}+\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}}}=\dfrac{m}{n}.\]Find $m+2n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an $n \times n$ matrix with elements randomly chosen from a uniform distribution over $[-1,1]$. Determine the probability that this matrix is diagonalizable over $\mathbb{R}$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U$ be the open unit disk, and let $f: U \rightarrow U$ be a holomorphic function such that $f(1/2) = 1/3$. Determine the maximum value of $|f'(1/2)|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} \frac{\sqrt[n]{a_2a_3\cdots a_{n+1}}}{a_{n+1}} \), where \( \alpha > 1 \) and \( a_n \) is the largest natural number \( k \) such that \( k! \leq \alpha^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim\limits_{x\rightarrow\infty} \frac{e^{x/e^x}-1}{\frac{x}{e^x}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The temperature outside a building is stable at $0^{\circ}$ Celsius. At time $t = 0$, the temperature inside the building is $18^{\circ}$ Celsius. The building has a time constant of 4 hours. At $t = \frac{1}{3}$ hour, a heat source is turned on, generating heat at an intensity of 60000 BTU/hr for 1/2 hour. Given that the heat capacity $\gamma$ of the building is $\frac{1}{6}$ degree per thousand BTU, find the temperature $T(t)$ inside the building at $t = 1$ hour. Use the equation $\frac{dT(t)}{dt} = K(M(t) - T(t)) + \gamma H(t)$, where $H(t)$ describes the production of heat, and express $H(t)$ using unit step functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of growth of the sum \(\sum_{k=1}^n\frac{1}{\sqrt{k+2}}\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest common divisor (GCD) of the complex numbers \(4 + 22i\) and \(17 + i\) in the Gaussian integers \(\mathbb{Z}[i]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the summation convention to simplify the expression \( g^{1}_{11} + g^{1}_{12} + g^{1}_{21} + g^{1}_{22} + g^{2}_{11} + g^{2}_{12} + g^{2}_{21} + g^{2}_{22} \) and determine the value of \( n \) necessary for this simplification.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ be an element in $GF(2^9)$ such that $a^9 + a^8 = -1$. Determine the order of the subgroup $H = \langle a \rangle$ in $GF(2^9)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a normal random variable with mean $0$ and variance $1$. Let $Y$ be a Bernoulli random variable with $p = \frac{1}{2}$, and suppose that $X$ and $Y$ are independent. Define $Z = (Y + 1)X$. Calculate the conditional probability $P(Y = 1 \mid Z = 3)$. Provide your answer as a decimal rounded to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint \frac{z+2}{z^2+9} \, dz \) using the residue at infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_5$ be the remainder when the positive integer $x$ is divided by 5. Compute the sum of all elements in the solution set of the equation: $$x^5(x_5)^5 - x^6 - (x_5)^6 + x(x_5) = 0.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the group $G$ with the presentation $\langle a,b \mid a^{16}=b^6=1,bab^{-1}=a^3\rangle$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer \( a \) such that \( x^4 + a^2 \) is not a prime number for any integer \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), the bases \(AB\) and \(CD\) measure 55 and 31 units, respectively. The diagonals of the trapezoid are perpendicular to each other. Calculate the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{\textbf{X} \rightarrow \infty} \left(\sum_i (\ln(X_i))^a\right)^{1/a} \) where \( a \) is a negative constant. Determine whether the limit is 0 or infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. (10 points) On a plane, there are 5 different lines, and these 5 lines form $m$ intersection points. How many different values can $m$ have?

保留源文本的换行和格式，翻译结果如下：

9. (10 points) On a plane, there are 5 different lines, and these 5 lines form $m$ intersection points. How many different values can $m$ have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a sphere of radius 1, there are four non-coplanar points A, B, C, D, and AB = CD = x, BC = DA = y, CA = BD = z. Calculate the value of x^2 + y^2 + z^2. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of points that must be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知一个正三棱柱的各条棱长均为 $3$, 则其外接球的体积的$\frac{2 \sqrt{21}}{\pi}$为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
a ring x of diameter 16 inches is placed inside another ring y of diameter of 18 inches . how much fraction of the ring x surface is not covered by the ring y ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $E(n)$  denote the largest integer  $k$  such that  $5^k$  divides  $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$  Calculate  $$ \lim_{n\to \infty} \frac{E(n)}{n^2 }. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the derivative of a function f(x) is f'(x), if for any x in R, f'(x) - f(x) < 1, and f(0) = 2022, find the solution set of the inequality f(x) + 1 > 2023e^x.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $A=\{x|-1<x<3\}$ and $B=\{x|-1<x<m+1\}$. If $x \in A$ is a necessary but not sufficient condition for $x \in B$, then the range of the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An exam paper contains 4 multiple-choice questions, each with three options (A), (B), and (C). Several students take the exam, and it is found that for any group of 3 students, there is always at least one question where their answers are all different. Furthermore, each student answers every question. What is the maximum number of students who can take the exam?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\left\{a_{n}\right\}$ where all terms are non-negative real numbers, and it satisfies: for any integer $n \geq 2$, $a_{n+1} = a_{n} - a_{n-1} + n$. If $a_{2} a_{2022} = 1$, find the maximum possible value of $a_{1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circular road runs around a circular garden. The sum of the circumferences of the inner and outer circles is 88 meters, and the radius of the inner circle is one-third of the outer circle. What is the width of the road?
Please reason step by step, and put your final answer within \boxed{}.


Context:
9.6. Waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing the watermelon on one scale pan with one or two weights on the other pan (possibly identical). In the process, the seller wrote down on a piece of paper the weights he used. What is the smallest number of different numbers that could have appeared in his notes, if the weight of each weight is an integer number of kilograms?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of a right triangle where one leg is $1/3$ longer than the other leg and $1/3$ shorter than the hypotenuse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let $n$ be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and $m$ be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair $(n, m)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Whole numbers that read the same from left to right and right to left are called symmetrical. For example, the number 513315 is symmetrical, whereas 513325 is not. How many six-digit symmetrical numbers exist such that adding 110 to them leaves them symmetrical?
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Context:
A rotating beacon is located 2 miles out in the water. Let $A$ be the point on the shore that is closest to the beacon. As the beacon rotates at $10 \mathrm{rev} / \mathrm{min}$, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. Find the speed at which the point where the beam hits the shore is moving when the beam is lighting up a point 2 miles along the shore from the point $A$ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{$80 \pi \mathrm{mi} / \mathrm{min}$}


Context:
On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? If Monday is day 1, what is the number of the day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
All the diagonals of a convex heptagon divide it into several non-overlapping smaller convex polygons. What is the maximum possible number of sides of these smaller polygons? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many decimal places of accuracy does the following approximate equality hold?

$$
495 \sqrt{2}-388 \sqrt{3} \approx 28
$$

that is, how many zeros can we write after 28 following the decimal point, so that the resulting number is still the rounded value of the correct left-hand side?
Please reason step by step, and put your final answer within \boxed{}.


Context:
若函数 $f(x)=-\frac{1}{2} x^{2}+\frac{13}{2}$ 在区间 $[a, b]$ 上的最小值为 $2 a$, 最大值为 $2 b$, 求 $[a, b]$. 标准答案格式为$[a,b]=[m,n]$ 或 $\Big[-a-\sqrt{b},\frac{c}{d}\Big]$，请提供区间 $[a, b]$ 的端点m+n+a+b+c+d和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \geq 3 \) be an integer. What is the largest possible number of interior angles greater than \( 180^\circ \) in an \( n \)-gon in the plane, given that the \( n \)-gon does not intersect itself and all its sides have the same length?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
\[\cos^6 0^\circ + \cos^6 1^\circ + \cos^6 2^\circ + \dots + \cos^6 90^\circ.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
一个 $10 \times 10$ 的数表在初始状态下每个格内均填有数字1，现在对数表进行如下操作：选择某一行或某一列，并将该行（列）中所有格子内的数变为其相反数。对于非负整数 N ，如果可经过有限次操作，使得最终数表中 -1 的个数恰为 N ，则称 N 是 "好的"，则好的 N 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest real number  $k$  such that, for any positive  $a,b,c$  with  $a^{2}>bc$ ,  $(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Josh gets together with 12 friends. Everyone including him puts 5 dollars into a pot. First place gets 65% of the money, while second and third place split the remaining 35%. How much money does third place get?
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Context:
For what smallest natural number \( n \) can the numbers from 1 to \( n \) be arranged in a circle such that each number is either greater than all of the 40 numbers following it in the clockwise direction or smaller than all of the 30 numbers following it in the clockwise direction?
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Context:
A square $ABCD$ with side length 4 is folded along $BD$ to form a dihedral angle of $60^{\circ}$, then the distance between the midpoint of $BC$ and point $A$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $\{a_n\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\{a_n\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with its right focus at $F(c,0)$, points $M$ and $N$ are on hyperbola $C$, $O$ is the coordinate origin. If quadrilateral $OFMN$ is a parallelogram with an area of $bc$, determine the eccentricity of hyperbola $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_1: y = x + \frac{1}{2}a$ and $l_2: y = (a^2-3)x + 1$, if $l_1 \parallel l_2$, then the value of $a$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
两个圆内切于 $K$, 大圆的弦 $A B$ 与小圆切于 $L$, 已知 $A K: B K=2: 5, A L=10$ ，求 $B L$ 的长。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the square root of $\dfrac{10!}{210}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the number of different triangles that can be obtained when each side of a square is divided into 8 equal parts, and using the divisions as vertices (not including the vertices of the square)?
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Context:
There are 37 students who need to cross a river. There is a small boat at the ferry that can carry up to 5 people. To get all of them across the river, what is the minimum number of times the boat must be used? Note that a round trip counts as one time. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vector equation $\overrightarrow{OA} + \overrightarrow{OB} + \sqrt{2}\overrightarrow{OC} = \overrightarrow{0}$, where O is the circumcenter of triangle ABC, find the exact measure of angle C in radians.
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Context:
Given that line $l$: $x+ay-1=0 (a\in R)$ is the symmetric axis of circle $C$: $x^2+y^2-4x-2y+1=0$, and it passes through point $A(-4,a)$. Draw a tangent line from point $A$ to circle $C$, with the tangent point being $B$. Calculate the length of $|AB|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a$  be a positive real number,  $n$  a positive integer, and define the *power tower*  $a\uparrow n$  recursively with  $a\uparrow 1=a$ , and  $a\uparrow(i+1)=a^{a\uparrow i}$  for  $i=1,2,3,\ldots$ .  For example, we have  $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has  $155$  digits.  For each positive integer  $k$ , let  $x_k$  denote the unique positive real number solution of the equation  $x\uparrow k=10\uparrow (k+1)$ .  Which is larger:  $x_{42}$  or  $x_{43}$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sign pyramid, a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The pyramid has four levels, and the bottom row has four cells. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the wins of five baseball teams as displayed in the graph, identify the number of games won by the Patriots. The clues are as follows:
1. The Tigers won more than the Eagles.
2. The Patriots won more than the Cubs, but fewer than the Mounties.
3. The Cubs won more than 15 games.
4. The Falcons won more games than the Eagles but fewer than the Mounties.

How many games did the Patriots win?

Graph data:
Teams (in unknown order): 10, 18, 22, 27, 33 games.
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Context:
At the school, there are three times as many girls as boys in the Biology class. The Physics class has a 2:3 ratio of boys to girls. The Chemistry class has 270 students, with an equal number of boys and girls. If the Biology class has half as many students as the Physics class, and the total number of students in all three classes is 1000, how many boys are in the Biology class?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle BCA=\gamma$. Find $\angle IPJ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task 12/78

When asked how old he is, someone answers that in the year $x^{2}$ he was exactly $x$ years old. In which year was he born?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the binomial series expansion of $\displaystyle (1-x)^{\frac{1}{2}}$ as $a_{0}+a_{1}x+a_{2}x^2+\cdots$ and the condition that $a_{0}+a_{1}+a_{2}+\cdots+a_{10}=\frac{\binom{20}{10}}{k^{10}}$, find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知椭圆$C:\frac{{{x}^{2}}}{8}+\frac{{{y2}}}{2}=1$与圆$M:{{x}^{2}}+{{y2}}+2\sqrt{2}+2-{{r}^{2}}=0(0< r<\sqrt{2})$，过椭圆$C$的上顶点$P$作圆$M$的两条切线分别与椭圆$C$相交于$A,B$两点（不同于$P$点），则直线$PA$与直线$PB$的斜率之积等于__________．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$.
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Context:
The sum of all solutions to the equation \( 2(x-1) \sin \pi x + 1 = 0 \) in the interval \([-2, 4]\) is equal to _____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the optimal amount of a material to add lies between 100g and 1100g, if we use the 0.618 method to arrange the experiment, and the first and second test points are $x_1$ and $x_2$ ($x_1 > x_2$) respectively, then if $x_2$ is the better point, the third test point $x_3$ should be ____ g (Please answer with a number).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other intersection point of line \( AF \) with the circumcircle of \( \triangle ACE \); \( P \) is the other intersection point of line \( AD \) with the circumcircle of \( \triangle AMN \). Find the length of \( AP \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
During vacation, Jimmy visits different beaches on an island, so he changes his place of lodging from time to time. The first 3 days he stays in a hostel, where he is charged $15 per night. However, there is a 10% service tax that is added to the nightly charge. The fourth and fifth days he stays in a cabin where he shares expenses with 2 of his friends, and they are charged $45 total per night, with a 7% government tax charged on top of the base cost. On the third day, Jimmy gets a discount code from a fellow traveler, which gives him a 15% discount on the cabin for the 5th day. How much did Jimmy spend on lodging during his vacation, including taxes and considering the discount he received?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four cousins Alan, Bob, Carl, and Dan are 3, 8, 12, and 14 years old, although not necessarily in that order. Alan is younger than Carl. The sum of the ages of Alan and Dan is divisible by 5. The sum of the ages of Carl and Dan is divisible by 5. What is the sum of the ages of Alan and Bob?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If for a natural number \( n (\geqslant 2) \), there exist integers \( a_{1}, a_{2}, \cdots, a_{n} \) such that
\[
a_{1}+a_{2}+\cdots+a_{n}=a_{1} \cdot a_{2} \cdot \cdots \cdot a_{n}=1990,
\]
find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=2$, and $\frac{1}{2}na_{n+2}$ is the arithmetic mean of $(n+2)a_n$ and $\lambda (n^2+2n)$, if the sequence $\{a_n\}$ is increasing, then the range of the real number $\lambda$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数 $y=f(x)$ 的图像既关于点 $(1,1)$ 中心对称, 又关于直线 $x+y=0$轴对称。若 $x \in(0,1)$ 时, $f(x)=\log _{2}(x+1)$, 则 $f\left(\log _{2} 10\right)$ 的值为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the function $f(x)=A\sin(\omega x+\varphi)$ ($A$, $\omega$, $\varphi$ are constants, $A > 0$, $\omega > 0$). If $f(x)$ is monotonic in the interval $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$ and $f\left(\frac{\pi}{2}\right)=f\left(\frac{2\pi}{3}\right)=-f\left(\frac{\pi}{6}\right)$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P \in [AC]$, $Q \in [BC]$, $R \in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K \in [BC]$, $M \in [AB]$, and $L \in [AC]$. Find $|AB|+|AC|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are given that \[17! = 355687ab8096000\] for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above.
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Context:
Two trains of different lengths run at the speed of 36 kmph and 18 kmph in opposite directions in parallel tracks. The time which they take to cross each other is 46.66293363197611 seconds. If the other train is 400 m long, how long is the first train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum number of solutions of $(k'(x))^2+k(x)k''(x)=0$ in $[a,e]$ for a twice differentiable function $k:\Bbb R\to\Bbb R$ such that $ k(a)=k(b)=0 ,k(c)=k(d)=1 , k(e)=2$, where $a<b<c<d<e$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $O, A, B, C$ 为空间内四点, $\overrightarrow{O A}=\boldsymbol{a}, \overrightarrow{O B}=\boldsymbol{b}, \overrightarrow{O C}=\boldsymbol{c}$, 且 $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{a}, \boldsymbol{a} \cdot \boldsymbol{b}=\boldsymbol{b} \cdot \boldsymbol{c}=\boldsymbol{c} \cdot \boldsymbol{a}$ $=-1$, 则 $|\boldsymbol{a}|+|\boldsymbol{b}|+|\boldsymbol{c}|=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
What are the spherical coordinates of point M with the Cartesian coordinates $(-1, -1, \sqrt{2})$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The decreasing interval of the function $f(x) = -x^2 - x + 4$ ($x \in \mathbb{R}$) is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知矩形 $A B C D$ 中, $A B=2, B C=1$, 折叠使点 $A, C$ 重合, 折痕为 $M N$, 打开平面 $A D M N$, 使二面角 $A-M N-C$ 为 $\frac{\pi}{3}$, 求直线 $M N$ 与直线 $A C$ 的距离，原始答案为 $\frac{\sqrt{m}}{n}$ 的格式，请给出 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius $3$ centered at $A$ and $B,$ and circles of radius $2$ centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}{p},$ where $k, m, n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
\( \triangle ABC \) has side lengths \( 13 \), \( 14 \), and \( 15 \). Let the feet of the altitudes from \( A \), \( B \), and \( C \) be \( D \), \( E \), and \( F \), respectively. The circumcircle of \( \triangle DEF \) intersects \( AD \), \( BE \), and \( CF \) at \( I \), \( J \), and \( K \) respectively. What is the area of \( \triangle IJK \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$, and $(5,0)$. Find the slope of the line through the origin that divides the area of this region exactly in half. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a betting game, you can win or lose a quantity $x$. The probability of winning a single bet is constant, $p$. You start with a wealth of $x$, which you bet in the first bet. What is the probability of losing all the money, i.e. of ruin, in an infinite number of bets, as a function of $p$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of $\frac{\tan 22.5^{\circ}}{1 - \tan^2 22.5^{\circ}}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $k=1, 2, \ldots, 2011$, we denote $S_k = \frac{1}{k} + \frac{1}{k+1} + \cdots + \frac{1}{2011}$. Compute the sum $S_1 + S_1^2 + S_2^2 + \cdots + S_{2011}^2$.
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Context:
Given that $\operatorname{tg} \theta$ and $\operatorname{ctg} \theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\alpha < \theta < \frac{5 \pi}{4}$, find the value of $\cos \theta - \sin \theta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
P.I. Find the largest natural number from which it is impossible to obtain a number divisible by 11 by erasing digits.

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


Context:
The math teacher of the ninth grade teaches an average of 6 tutoring classes per month. If the classes are taught by female teachers, each female teacher should teach 15 classes per month. If the classes are taught only by male teachers, how many classes should each male teacher teach per month? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. Point $M$ divides the diagonal $A C$ of square $A B C D$ in the ratio $M C: A M=1: 4$. A line passing through point $M$ divides the square into two parts, the areas of which are in the ratio $1: 11$. In what ratio does this line divide the perimeter of the square?
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Context:
It is known that unequal numbers $a$, $b$, and $c$ are successive members of an arithmetic progression. All of them are greater than $1000$ and all are squares of natural numbers. Find the smallest possible value of $b$.
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Context:
Walter fell from the eighth platform of some scaffolding and fell past David after falling 4 meters. If he fell for an additional three more times that depth before hitting the ground, and the platforms are evenly spaced out in height, what platform was David on?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $m\in R$, the moving straight line passing through the fixed point $A$ with equation $x+my-2=0$ intersects the moving straight line passing through the fixed point $B$ with equation $mx-y+4=0$ at point $P\left(x,y\right)$. Find the maximum value of $|PA|\cdot |PB|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $m$, $n$, $a$, and $k$ be positive integers with $k > 1$ such that the equation \(5^m + 63n + 49 = a^k\) holds. Find the minimum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $x > 2$, $y > 0$ and they satisfy $2^{x \cdot 2^{y}} = 16$, find the minimum value of $\frac {2}{x-2}$ + $\frac {2}{y}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two sets of real numbers \( A = \{a_{1}, a_{2}, \cdots, a_{100} \} \) and \( B = \{b_{1}, b_{2}, \cdots, b_{50} \} \), find the number of mappings \( f \) from \( A \) to \( B \) that make every element in \( B \) have a preimage, and satisfy the condition:
$$
f(a_{1}) \leq f(a_{2}) \leq \cdots \leq f(a_{100})
$$
Express your answer as a combination (e.g., C(n, k)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of $n$ such that this sequence of circles can be constructed through circle $C_n$ where the $n$ circles are all tangent to line $A$ at points to the right of $P$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The line $l: x = my + n\ (n > 0)$ passes through point $A(4, 4)$. If the diameter of the circumcircle of the feasible region is 8, then the value of the real number $n$ is _________.
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Context:
Evaluate the infinite series $$\sum^{\infty}_{k=1}\frac{(-1)^{k-1}}{k}\sum^{\infty}_{r=0}\frac{1}{k2^{r}+1}$$
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Context:
For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is:
\[ (-1)^{\alpha(i-1, j-1)}. \]
Compute the remainder when $\det M$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tom has a list of nine integers: 1, 2, 3, 4, 5, 6, 7, 8, and 9. He creates a second list by adding 2 to some of the integers in the first list and by adding 5 to all of the other integers in the first list. What is the smallest number of different integers he can obtain in the second list?
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Context:
In triangle $ABC$, let sides $a$, $b$, and $c$ be opposite to angles $A$, $B$, and $C$ respectively. Given that $\sin A = 2\sin B$ and $a + b = \sqrt{3}c$, find the magnitude of angle $C$.
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Context:
In △ABC, the sides opposite to angles A, B, C are a, b, c, respectively, and the altitude on side BC is $\frac{a}{2}$. When $\frac{c}{b} + \frac{b}{c}$ reaches its maximum value, the value of angle A is _____.
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Context:
Let $p$, $q$, $r$, $s$, $t$, and $u$ be positive integers with $p+q+r+s+t+u = 2023$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, $s+t$ and $t+u$. What is the smallest possible value of $N$?
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Context:
Find the number of real solutions of the equation
\[
\frac{x}{50} = \sin x.
\]
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Context:
$14 \cdot 36$ In the sequence $\left[\frac{1^{2}}{1980}\right],\left[\frac{2^{2}}{1980}\right],\left[\frac{3^{2}}{1980}\right], \cdots,\left[\frac{1980^{2}}{1980}\right]$, how many different numbers are there?
(6th All-Russian Mathematical Olympiad, 1980)
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Context:
Let $\omega$ be an incircle of triangle $ABC$. Let $D$ be a point on segment $BC$, which is tangent to $\omega$. Let $X$ be an intersection of $AD$ and $\omega$ against $D$. If $AX : XD : BC = 1 : 3 : 10$, a radius of $\omega$ is $1$, find the length of segment $XD$. Note that $YZ$ expresses the length of segment $YZ$.
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Context:
On a computer screen is the single character a. The computer has two keys: c (copy) and \( p \) (paste), which may be pressed in any sequence. Pressing \( p \) increases the number of a's on screen by the number that were there the last time \( c \) was pressed. \( c \) doesn't change the number of a's on screen. Determine the fewest number of keystrokes required to attain at least 2018 a's on screen. (Note: pressing \( p \) before the first press of \( c \) does nothing).
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Context:
A dealer offers a cash discount of 15% and still makes a certain profit when he further allows 25 articles to be sold at the cost price of 20 articles to a particular sticky bargainer. His articles were listed 100% above the cost price. What is the profit percentage made by the dealer?
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Context:
## Task 2.

In a certain archipelago, there are 2017 islands named $1, 2, \ldots, 2017$. Two agencies, Red Dragon and Blue Eye, are negotiating the schedule of ferry routes between the islands. For each pair of islands, exactly one agency will organize a ferry route, and this route will only operate from the island with the smaller number to the island with the larger number.

A ferry route schedule is good if there do not exist two islands with labels $A < B$ such that it is possible to travel from island $A$ to island $B$ using only ferries from Red Dragon, and also using only ferries from Blue Eye.

Determine the total number of good ferry route schedules.
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Context:
Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$.
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Context:
Given the condition p: $|x|>1$ and condition q: $x<m$, if $\lnot p$ is a necessary but not sufficient condition for $\lnot q$, then the range of the real number $m$ is $\_\_\_\_\_\_.$
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Context:
a 270 m long train running at the speed of 120 km / hr crosses another train running in opposite direction at the speed of 80 km / hr in 9 sec . what is the length of the other train ?
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Context:
A bus leaves from city $A$ to city $B$, which is 240 km away from $A$, at a speed of 40 km/h. At the same time, a car leaves from $B$ heading towards $A$ at a speed of $v$ km/h. Half an hour after meeting the bus, the car turns back towards $B$ without reaching city $A$ and continues moving towards $B$ at the same speed. Determine all values of $v$ for which the car arrives in city $B$ before the bus.
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Context:
In a triangle \( \triangle ABC \), \(a\), \(b\), and \(c\) are the sides opposite to angles \(A\), \(B\), and \(C\) respectively, with \(B= \dfrac {2\pi}{3}\). If \(a^{2}+c^{2}=4ac\), then find the value of \( \dfrac {\sin (A+C)}{\sin A\sin C} \).
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Context:
For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.
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Context:
A rectangular tiled patio is composed of 30 square tiles. The rectangular patio will be rearranged so that there will be 2 fewer columns of tiles and 4 more rows of tiles. After the change in layout, the patio will still have 30 tiles, and it will still be rectangular. How many rows are in the tile patio before the change in layout?
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Context:
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1}  \] is itself an integer.

[i]Laurentiu Panaitopol, Romania[/i]
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Context:
A car traveling with 9/13 of its actual speed covers 120 km in 2 hours, 19 minutes, and 36 seconds. Find the actual speed of the car.
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Context:
.4 + .02 + .006 =
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Context:
In Italy, they produce clocks in which the hour hand completes one full rotation per day, and the minute hand completes 24 rotations per day. As usual, the minute hand is longer than the hour hand. Consider all positions of the two hands and the zero marking on the Italian clock that coincide with those on a regular clock. How many such positions exist on the Italian clock within a day? (The zero marking on the Italian clock represents 24 hours, while on a regular clock it represents 12 hours.)
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Context:
Find the  $2019$ th strictly positive integer  $n$  such that  $\binom{2n}{n}$  is not divisible by  $5$ .
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Context:
The difference between the cube and the square of a number has the form $a b c a b c$ (in the decimal system). What is this number?
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Context:
We consider 2021 lines in the plane, no two of which are parallel and no three of which are concurrent. Let E be the set of their intersection points. We want to assign a color to each point in E such that any two points on the same line, whose connecting segment contains no other point of E, have different colors. What is the minimum number of colors needed to achieve such a coloring?
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Context:
$记\triangle ABC的内角A，B，C的对边分别为a,b,c,已知\sin C \cdot \sin(A-B) = \sin B \cdot \sin(C-A).$
$若a=5，cos A=\frac{25}{31}，求\triangle ABC的周长.$
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Context:
Let  $S$  be the set of 81 points  $(x, y)$  such that  $x$  and  $y$  are integers from  $-4$  through  $4$ .  Let  $A$ ,  $B$ , and  $C$  be random points chosen independently from  $S$ , with each of the 81 points being equally likely.  (The points  $A$ ,  $B$ , and  $C$  do not have to be different.)  Let  $K$  be the area of the (possibly degenerate) triangle  $ABC$ .  What is the expected value (average value) of  $K^2$  ?
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Context:
A fenced, rectangular field measures $24~meters$ by $52~ meters$. An agricultural researcher has $1994$ meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the $1994$ meters of fence?
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Context:
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$
Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$
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Context:
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that $f(f(1995)) = 95$, $f(xy) = f(x)f(y)$, and $f(x) \le x$ for all $x, y$. Find all possible values of $f(1995)$.
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Context:
Given the function $f(x)=2\sin ( \frac {π}{6}x+ \frac {π}{3}) (2 < x < 10)$, suppose its graph intersects with the $x$-axis at point $A$, and the line $l$ passing through point $A$ intersects with the graph of $f(x)$ at points $B$ and $C$. If $O$ is the coordinate origin, find the value of $( \overrightarrow{OB}+ \overrightarrow{OC})⋅ \overrightarrow{OA}$.
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Context:
In the number \(2 * 0 * 1 * 6 * 0 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0,1,2,3,4,5,6,7,8\) (digits can repeat) such that the resulting 10-digit number is divisible by 18. How many ways can this be done?
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Context:
Points  $ K$ ,  $ L$ ,  $ M$ , and  $ N$  lie in the plane of the square  $ ABCD$  so that  $ AKB$ ,  $ BLC$ ,  $ CMD$ , and  $ DNA$  are equilateral triangles. If  $ ABCD$  has an area of  $ 16$ , find the area of  $ KLMN$.
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Context:
A circle of radius $R$, passing through the vertices $A, B$ and $C$ of a rectangular trapezoid $A B C D$ ( $\angle A=\angle B=90^{\circ}$ ) intersects the segments $A D$ and $C D$ at points $M$ and $N$, respectively, such that $A M: A D=C N: C D=1: 3$. Find the area of the trapezoid.
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Context:
Four college students are applying for jobs at three companies, with each student being employed by at most one company. How many ways are there for each company to employ at least one student? Express your answer as a whole number.
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Context:
What three-digit positive integer is one more than a multiple of 3, 4, 5, 6, and 7?
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Context:
A certain school held a radio gymnastics competition, with scoring criteria including spirit, neatness, and standard of movements. The total score is calculated in a ratio of $2:3:5$ for the three criteria. It is known that Class 8-1 scored $8$ points, $9$ points, and $10$ points for the three criteria respectively. What is the total score of Class 8-1 in this competition?
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Context:
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{36 - 4\sqrt{5}} \sin x + 2) \cdot (3 + 2\sqrt{10 - \sqrt{5}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.
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Context:
Pedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:

1. The sum of the fractions is equal to $2$.
2. The sum of the numerators of the fractions is equal to $1000$.

In how many ways can Pedro do this?
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Context:
Mitya is going to fold a square sheet of paper $A B C D$. Mitya calls a fold beautiful if the side $A B$ intersects the side $C D$ and the four resulting right triangles are equal.

Before that, Vanya selects a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold that passes through point $F$.
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Context:
Alex is working on a science project and has taken pictures of all of the plants in a park near her home. She wants to find out how many of each plant she has pictures of, but she has 1200 pictures, and counting them all will take her a long time. Her teacher suggested she use a program on the computer that can automatically tell her what plant is in each picture and count them for her. However, the processing speed of the program varies depending on the type of plant and has a specific rate per hour.

- For tree pictures, it takes 1.5 minutes to process each picture, with a rate of 35 pictures per hour.
- For flower pictures, it takes 2.5 minutes to process each picture, with a rate of 20 pictures per hour.
- For grass pictures, it takes 1 minute to process each picture, with a rate of 50 pictures per hour.
- For shrub pictures, it takes 3 minutes to process each picture, with a rate of 15 pictures per hour.
- For fern pictures, it takes 2 minutes to process each picture, with a rate of 25 pictures per hour.

There are 300 pictures of trees, 400 pictures of flowers, 250 pictures of grass, 150 pictures of shrubs, and 100 pictures of ferns. Calculate the total processing time for all pictures in hours considering the processing rate per hour for each plant category.
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Context:
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$,

$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$

and

$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$

How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?
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Context:
Given a quadratic function $f(x)=ax^{2}+bx+c$ ($a,b,c \in \mathbb{R}$), when $x \in [0,1]$, $|f(x)| \leqslant 1$, then the maximum value of $(a+b)c$ is \_\_\_\_\_\_.
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Context:
Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\).
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Context:
Calculate the radius of convergence for the power series $$\sum_{n=1}^\infty (\frac{z-i} {z+2i})^n$$ using the Cauchy criterion.
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Context:
Given an arc length of π cm with a corresponding central angle of $\frac{π}{4}$, the area of the sector containing this arc is __________ cm².
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Context:
Compute the integral $\int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)}$ using residue theory from complex analysis.
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Context:
Consider the sequences \( \left(a_{n}\right) \) and \( \left(b_{n}\right) \) defined by
\[
a_{1}=3, \quad b_{1}=100, \quad a_{n+1}=3^{a_{n}}, \quad b_{n+1}=100^{b_{n}}.
\]
Find the smallest integer \( m \) for which \( b_{m} > a_{100} \).
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Context:
The maximum value of the function
$$
f(x)=\sqrt{3} \sin 2 x+2 \sin x+4 \sqrt{3} \cos x
$$
is .
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Context:
The base $MQ$ of trapezoid $MNPQ$ ($MQ \parallel NP, MQ > NP$) is the diameter of a circle which is tangent to line $MN$ at point $M$ and intersects side $PQ$ at point $K$. Additionally, $PQ = 4 \sqrt{3} KQ$. The radius of the circle is $R$, and $\angle NQM = 60^{\circ}$. Find the area of the trapezoid.
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Context:
## Condition of the problem

Calculate the definite integral:

$$
\int_{0}^{\operatorname{arctg} \frac{1}{3}} \frac{8+\operatorname{tg} x}{18 \sin ^{2} x+2 \cos ^{2} x} d x
$$
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Context:
Given the circle equation x^2+y^2+2x-m=0, find the range of m such that point P(1,1) is outside the circle. Express your answer as an interval in standard interval notation.
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Context:
$\mathbf{F 1 7}$ (39-6, Bulgaria) Let $\mathbf{N}^{*}$ be the set of all positive integers, and the function $f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ satisfies: for any $s$ and $t$ in $\mathbf{N}^{*}$, we have
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2},
$$

Determine the smallest possible value of $f(1998)$ among all such functions $f$.
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Context:
The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face?
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Context:
Given the function f(x) = ax^2 - c, satisfying the conditions: -4 ≤ f(1) ≤ -1 and -1 ≤ f(2) ≤ 5, determine the range for f(3). Express your answer as an interval (lower bound ≤ f(3) ≤ upper bound).
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Context:
What is the smallest three-digit number in Pascal's triangle?
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Context:
How many different solutions does the congruence $x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}$ have? Provide your answer as an integer.
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Context:
If the graph of the function $y = \frac{kx^2}{k^2 + k - 2}$ is a hyperbola, and it is located in the second and fourth quadrants, then find the value of $k$. Express your answer as a single value.
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Context:
Let \( \triangle ABC \) be a triangle with \( AB = 13 \), \( BC = 14 \), and \( CA = 15 \). Let \( O \) be the circumcenter of \( \triangle ABC \). Find the distance between the circumcenters of \( \triangle AOB \) and \( \triangle AOC \).
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Context:
Find the limit of the sequence $y_n$ defined by:

$$y_n=\frac{1}{\sqrt{n^2-1}}+\frac{1}{\sqrt{n^2-2}}+\cdots+\frac{1}{\sqrt{(n-1)^2}}$$
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Context:
Given sets $A=\{x|x\geq|x^2-2x|\}$, $B=\{x|x^2-2ax+a\leq0\}$, if $A\cap B=B$, find the range of the real number $a$.
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Context:
Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs.
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Context:
8. The planar region $S=\left\{(x, y) \mid x, y \in\left[0, \frac{\pi}{2}\right], \sin ^{2} x-\sin x \sin y+\sin ^{2} y \leqslant \frac{3}{4}\right\}$ has an area equal to
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Context:
In front of you is a segment-digit display. To show time on a digital clock, each digit uses seven segments, each of which can be illuminated or not; the illuminated segments form a digit, as shown in the figure. For example, zero uses six segments, one uses two segments, and so on. The digital clock displays only hours and minutes. How many minutes throughout the day use more illuminated segments than the subsequent minute? (The day starts at 00:00 and ends at 23:59) (20 points)
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Context:
Calculate the integral $\displaystyle \int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx$.
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Context:
How many acute angles can be drawn with a set of triangle rulers?
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Context:
Let's find the smallest positive integers $x$ and $y$ for which the product of $180$ and $x$ is a fourth power and the product of $180$ and $y$ is a sixth power. Find the sum of $x$ and $y$.
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Context:
Freddie and his team are collecting blankets for three days to be donated to the Children Shelter Organization. There are 15 people on the team on the first day. On the first day, each of them gave 2 blankets. For the second day, 5 new members joined the team, and they tripled the number of blankets they collected on the first day by asking door-to-door. On the second day, the new members contributed 4 blankets each, while the original members continued to contribute 2 blankets each. On the last day, they set up boxes at schools and got a total of 22 blankets, but they also received 30 more blankets through their online outreach efforts. How many blankets did they collect for the three days for donation?
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