inclusionAI/Ring-lite-rl-data huggingface dataset:
A parabola is given by the equation $y^{2}=2px (p > 0)$, with a focus at point $F$. A line with a slope angle of $\frac{\pi}{4}$ passes through the focus $F$, intersecting the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through the point $(0, 2)$, calculate the value of $p$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{5}}
Determine the values of \( \beta \) for which the differential equation \( \tau\frac{dx}{dt}=-x+\frac{e^{\beta(x-\frac{1}{2})}}{1+e^{\beta(x-\frac{1}{2})}} \) has two additional fixed points beyond \( x=0.5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > 4}
The lines \(x=\frac{1}{4}y+a\) and \(y=\frac{1}{4}x+b\) intersect at the point \((1,2)\). The original answer is in the form \(\frac{k}{m}\). Find the value of \(k+m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In rectangle $JKLM$, points $N$ and $P$ bisect $\overline{JK}$ and points $Q$ and $R$ bisect $\overline{LM}$. Rectangle $JKLM$ has dimensions $JK = 6$ and $JL = 4$. Find the area of quadrilateral $STUV$ formed by connecting midpoints of sides in the following manner: $S$ is the midpoint of $\overline{JN}$, $T$ is the midpoint of $\overline{KP}$, $U$ is the midpoint of $\overline{LQ}$, and $V$ is the midpoint of $\overline{MR}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A company purchases 400 tons of a certain type of goods annually. Each purchase is of $x$ tons, and the freight cost is 40,000 yuan per shipment. The annual total storage cost is 4$x$ million yuan. To minimize the sum of the annual freight cost and the total storage cost, find the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A robot is standing on the bottom left vertex $(0,0)$ of a $5 \times 5$ grid and wants to go to $(5,5)$, only moving to the right $(a,b) \mapsto (a+1,b)$ or upward $(a,b) \mapsto (a,b+1)$. However, this robot is not programmed perfectly and sometimes takes the upper-left diagonal path $(a,b) \mapsto (a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0 \leq a,b \leq 5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1650}
Consider the set \( S = \{A \in \mathbb{Q}^{k \times k} : A^6 = I, A^n \neq I \text{ for all } 0 < n < 6\} \). How many distinct orbits does \( S \) have under conjugation by \( GL_k(\mathbb{Q}) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of $x$ such that $\gcd(a + x, 3a + x - 1) = 2x + 1$, where $a$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
26. A sphere passes through the vertices of one face of a cube and touches the sides of the opposite face of the cube. Find the ratio of the volumes of the sphere and the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41\pi\sqrt{41}}{384}}
How many independent sets of size 3 does the Petersen graph have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the integral: $$\int_0^{\frac{\pi}{2}}\frac{\tan{x}}{\tan(\frac{x}{2})}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Consider the polynomial 
\[P(x) = x^3 + 3x^2 + 6x + 10.\]
Let its three roots be \(a\), \(b\), and \(c\). Define \(Q(x)\) to be the monic cubic polynomial with roots \(ab\), \(bc\), and \(ca\). Compute \(|Q(1)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
The area of a triangle \(ABC\) is \(40 \text{ cm}^2\). Points \(D, E\) and \(F\) are on sides \(AB, BC\) and \(CA\) respectively. If \(AD = 3 \text{ cm}, DB = 5 \text{ cm}\), and the area of triangle \(ABE\) is equal to the area of quadrilateral \(DBEF\), find the area of triangle \(AEC\) in \(\text{cm}^2\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the integral \( \int_{0}^{2\pi} e^{\cos(\varphi)} \cos(\varphi - \sin(\varphi)) \, d\varphi \) using the method of residues.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Define $ n!!$ to be $ n(n-2)(n-4) \cdots3 \cdot1$ for $ n$ odd and $   n\left( {n - 2}\right) \left( {n - 4}\right) \ldots 4 \cdot  2  $ for $ n$ even. When $ \sum\limits_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $\frac{ab}{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
401
}
Find the positive integer $n$ that satisfi\fes the equation \( {n}^{2} - \left\lfloor  \sqrt{n}\right\rfloor   = {2018} \)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n=45
}
Consider the function \( S(n) \) defined for a positive integer \( n \) as follows:
\[
  S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, .
\]
(All logarithms are base 10.) Determine how many integers \( n \) from 1 to 2011 (inclusive) satisfy the condition \( S(S(n)) = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Determine the smallest number \( n \) such that the set of integers can be partitioned into \( n \) non-overlapping subsets, none of which contain an infinite arithmetic progression. Is it possible to achieve this with \( n = 2 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given an odd function $f(x)$ that is twice differentiable, with $f(-1)=f(2)=f(3)=-1$, express $f(x)$ in the form $f(x)=a_{2n+1}x^{2n+1}+a_{2n-1}x^{2n-1}+\\cdots+a_{1}x$, where $a_{2n+1}, a_{2n-1}, \\ldots, a_{1}$ are constants. How many real solutions does the differential equation $(f'(x))^2+f''(x)=0$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $f:(0,5)\to\mathbb{R}-\mathbb{Q}$ be a continuous function such that $f(2)=\pi$. Determine whether $f(\pi)=\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
On an $8 \times 8$ board, two identical pieces are placed on cells a1 and c3. Peter and Vasya take turns playing, with Peter starting first. On their turn, a player chooses one of the pieces and moves it either vertically up or horizontally to the right by any number of cells. The player who makes a move to cell h8 wins. Determine which player can always ensure a win regardless of the opponent's moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
A horse runs along a circular path with a speed of 20 km/hr. A fence is positioned along the tangent to the circle at the point where the horse starts. A lantern is placed at the center of the circle, casting the horse's shadow on the fence. Calculate the speed of the shadow along the fence in km/hr when the horse has covered \(\frac{1}{8}\) of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
For $\theta<\pi$, find the maximum real number $\theta$ such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0, \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2046\pi}{2047}}
Among the $3 \times 3$ invertible matrices with entries from the field $\mathbb{Z}/3\mathbb{Z}$, how many matrices are similar to the matrix \( \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117}
The real numbers \(x\) and \(y\) satisfy the following system of equations: \[ \begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2} \end{cases} \] Find the numerical value of the ratio \(\frac{y}{x}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine whether the limit \( \lim_{x \to \infty} \frac{\ln(x)^a}{\ln(\ln(x))^{a+b}} \) diverges to infinity for any positive real number \(a\) and integer \(b \geq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the values of \( b > 0 \) for which the infinite series \( \sum_{n=0}^\infty \left(\frac{n}{b(n-3)}\right)^n \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Find a formula for \( S(n) \), which represents the number of ways to express the natural number \( n \) as a sum of natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Determine the minimum number of interior-disjoint almost-squares required to tile an L-shaped region, where the aspect ratio of each almost-square is between 1/2 and 2, and n is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine whether the limit \( \lim_{m,n\to\infty} a_{mn} \) exists for the double sequence given by \[ a_{mn} = \frac{m^2n^2}{m^2+n^2}\left(1-\cos\left(\frac{1}{m}\right)\cos\left(\frac{1}{n}\right)\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
On a manifold with a metric tensor, let \( \Omega \) be the volume form. Is it true that the Hodge star operator applied to \( \Omega \) results in 1, i.e., \( *\Omega = 1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If the unit vectors $e_{1}$ and $e_{2}$ have an angle of $\dfrac{\pi}{3}$ between them, and the vector $a=e_{1}+\lambda e_{2} (\lambda\in\mathbb{R})$, with $|a|= \dfrac{ \sqrt{3}}{2}$, then $\lambda=$ _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
The field shown has been planted uniformly with wheat. [asy]
draw((0,0)--(1/2,sqrt(3)/2)--(3/2,sqrt(3)/2)--(2,0)--(0,0),linewidth(0.8));
label("$60^\circ$",(0.06,0.1),E);
label("$120^\circ$",(1/2-0.05,sqrt(3)/2-0.1),E);
label("$120^\circ$",(3/2+0.05,sqrt(3)/2-0.1),W);
label("$60^\circ$",(2-0.05,0.1),W);
label("100 m",(1,sqrt(3)/2),N);
label("100 m",(1.75,sqrt(3)/4+0.1),E);
[/asy] At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. What is the fraction of the crop that is brought to the longest side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{12}}
Find the number of non-isomorphic groups of order 36 that have a normal subgroup of order 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given $a \gt 2b \gt 0$, then the minimum value of $\frac{{a}^{4}+1}{b(a-2b)}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A tetrahedron has all its faces triangles with sides $13,14,15$. What is its volume?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42 \sqrt{55}}
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665}
In a convex quadrilateral $ABCD$, points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that \( \frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1 \). Another convex quadrilateral $E_1F_1G_1H_1$ is such that $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ respectively, with $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that \( \frac{E_1A}{AH_1} = a \), find \( \frac{F_1C}{CG_1} \) in terms of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{120}{169}}
Given an $8\times 8$ chess board, in how many ways can we select $56$ squares on the board while satisfying both of the following requirements:
(a) All black squares are selected.
(b) Exactly seven squares are selected in each column and in each row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
576
}
Bob is attempting to feed his pets. He feeds his pets as follows:
- No dog receives more food than any other dog
- No cat receives any less food than any other cat
- No hamster receives any less food than any other hamster
- The cats, as a group, receive exactly 1.5 times as much food as a single dog
- The hamsters, as a group, receive exactly 0.25 times as much food as a single dog

If Bob has 9 dogs, 6 cats, and 10 hamsters, approximately what percentage of the food bag does a cat receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.33\%}
Compute the total area of the infinitely many triangles in the figure, where each triangle has a height of \( \frac{6}{7} \). The \( x \)-values for the bases of the triangles, from left to right, are \( \frac{27}{64} \), \( \frac{9}{16} \), \( \frac{3}{4} \), \( 1 \), and so on.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{112}}
Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Given a strict order $(P, \prec)$ and a sequence of distinct elements $a_1, a_2, \ldots, a_n$ from $P$, determine whether there exists a permutation $\sigma$ of $\{1, 2, \ldots, n\}$ such that $a_{\sigma(n)} \not\prec a_{\sigma(m)}$ for all $n < m$. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
On the board we write a series of  $n$  numbers, where  $n \geq 40$ , and each one of them is equal to either  $1$  or  $-1$ , such that the following conditions both hold:

(i) The sum of every  $40$  consecutive numbers is equal to  $0$ .
(ii) The sum of every  $42$  consecutive numbers is not equal to  $0$ .

We denote by  $S_n$  the sum of the  $n$  numbers of the board. Find the maximum possible value of  $S_n$  for all possible values of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate the value of the following expression: \[ \frac{\int_0^1 \frac{dt}{(1-t^4)^{1/2}}}{\int_0^1 \frac{dt}{(1+t^4)^{1/2}}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given real numbers $a$ and $b$ satisfying $b(e^{a}-1)+a=e^{b}-\ln b$, then the range of $b^{2}e^{n}-2e^{b}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-e, +\infty\right)}
Find the real value of $a$ for which the definite integral \( f(a) = \int_{0}^{\infty} \frac{x^a}{2x^6 + 4x^5 + 3x^4 + 5x^3 + 3x^2 + 4x + 2} \, dx \) is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Use the fourth order Runge-Kutta method with step size \( h = 1 \) to approximate \( \sqrt{2} \) from the initial value problem given in the previous question. Provide your answer rounded to five decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.41438}
Find the limit of the series as \( x \) approaches 0: \[ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the limit:
$$\lim_{x\to 0} \frac{\sin(\sin(\sin x)) - x}{x^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
In triangle \(ABC\), the angle bisectors \(BK\) and \(CL\) are drawn. Point \(N\) is marked on segment \(BK\) such that \(LN \parallel AC\) and \(NK = LN\). Determine the measure of angle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Determine how many different ways there are to assign the elements of the set \( M = \{1, 2, 3, 4, 5\} \) into three ordered sets \( A, B,\) and \( C \) such that the following conditions are satisfied: each element must belong to at least one of the sets, the intersection of all three sets is empty, and the intersection of any two sets is not empty. (i.e., \( A \cup B \cup C = M, A \cap B \cap C = \varnothing \), and \( A \cap B \neq \varnothing, B \cap C \neq \varnothing, C \cap A \neq \varnothing \))
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1230}
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$10$}
Let $a=256$. Find the unique real number $x>a^2$ such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ 2^{32} $}
\section*{Problem 3 - 131033}

Consider a four-sided pyramid with a square base. The vertices of this base are the points \(A, B, C\), and \(D\). The apex of the pyramid is \(S\). All eight edges have the length \(a\).

\(E\) and \(F\) are the midpoints of the edges \(S B\) and \(S C\), respectively. A plane through the points \(A, E, F\), and \(D\) divides the pyramid into two sub-bodies.

Calculate the ratio of the volumes of these two sub-bodies!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:5}
Solve the inequality \(\sqrt{x^{2}+x-56}-\sqrt{x^{2}+25 x+136}<8 \sqrt{\frac{x-7}{x+8}}\) and find the sum of its integer solutions within the interval \([-25, 25]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285}
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
Ant Amelia starts on the number line at \(0\) and crawls in the following manner. For \(n=1,2,3,\) Amelia chooses a time duration \(t_n\) and an increment \(x_n\) independently and uniformly at random from the interval \((0,1).\) During the \(n\)th step of the process, Amelia moves \(x_n\) units in the positive direction, using up \(t_n\) minutes. If the total elapsed time has exceeded \(1\) minute during the \(n\)th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most \(3\) steps in all. What is the probability that Amelia’s position when she stops will be greater than \(1\)? The answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Vasya wrote down 11 natural numbers in one notebook. Petya wrote down the greatest common divisors of each pair of numbers recorded in Vasya's notebook in another notebook. It turned out that every number written in one of the two notebooks is also in the other notebook. What is the maximum number of distinct numbers that could have been written in Vasya's notebook?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
We define: \( a @ b = a \times (a+1) \times \ldots \times (a+b-1) \). Given \( x @ y @ 2 = 420 \), determine \( y @ x = (\quad) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
find the value of 18396 x 9999 = m ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183941604}
Find the joint density function $g(u, v)$ for the transformed random variables $U = \exp(X) + 2\exp(Y)$ and $V = 2\exp(X^2) + \exp(Y^2)$, where $X$ and $Y$ are independent Exponential random variables with mean 1. Then, evaluate $g(3e, 3e)$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6e^4}}
In a group of $n$ people, each person initially knows $k$ unique pieces of gossip. What is the minimum number of phone calls required for everyone in the group to know all the gossip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 4}
Find the degree and a basis for the field extension \( \mathbb{Q}(\sqrt{2},\sqrt[3]{3}) \) over \( \mathbb{Q}(\sqrt{2}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine whether the following limit exists: \[\lim_{m \to \infty} \left(\frac{N}{m} \right)^{\frac{N}{m-N}}-\left(\frac{N}{m} \right)^{\frac{m}{m-N}}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the probability density function (pdf) \( f(x) = 10x^9 \) for \( 0 < x < 1 \), and assuming independence, let \( Y = \max(X_1, X_2, \ldots, X_8) \). Find the cumulative distribution function (CDF) for \( Y \) and calculate the probability \( P(0.9999 < Y < 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.007968}
Determine the smallest positive integer  $n \ge 3$  for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where  $A$  denotes the result when the numbers  $2^{10}$ ,  $2^{20}$ ,  $\dots$ ,  $2^{10n}$  are written in decimal notation and concatenated (for example, if  $n=2$  we have  $A = 10241048576$ ).

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
How many ways are there to color every integer either red or blue such that \(n\) and \(n+7\) are the same color for all integers \(n\), and there does not exist an integer \(k\) such that \(k, k+1\), and \(2k\) are all the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the probability that it takes exactly $n$ random numbers uniformly distributed in $(0,1)$ to exceed $1$ when summed together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{n-1}{n!}\)}
Two chords, measuring 9 and 17, are drawn from a single point on a circle. Find the radius of the circle if the distance between the midpoints of these chords is 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{85}{8}}
Consider a sequence of one-to-one conformal mappings $\{f_n\}$, each mapping the open unit disk $\mathbb{D}$ onto the right half-plane $\mathbb{A} = \{z \in \mathbb{C} : \mathfrak{R} z > 0\}$. Assume that $f_n$ converges uniformly to a function $f$ on compact subsets of $\mathbb{D}$, but $f$ is not one-to-one. Determine the real part of $f(0)$. \( \mathfrak{R} f(0) = \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}