inclusionAI/Ring-lite-rl-data huggingface dataset:
When the polynomial \( x^{1951}-1 \) is divided by \( x^{4}+x^{3}+2x^{2}+x+1 \), a quotient and a remainder are obtained. Find the coefficient of \( x^{14} \) in the quotient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Carter has a 14-hour road trip. He wants to stop every certain number of hours to stretch his legs. He also wants to make 2 additional stops for food and 3 additional stops for gas. If each pit stop takes 20 minutes, his road trip will become 18 hours. How often does he want to stop to stretch his legs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If the least common multiple of three consecutive natural numbers is 660, find these three numbers. Express your answer as three specific numbers in ascending order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10, 11, 12}
A circle with radius \( R \) and center at point \( O \) is divided by points \( A, B, C, D, E, F \) into six equal parts. Determine the area of the figure \( COE \), bounded by the arc \( OC \) centered at point \( B \), the arc \( OE \) centered at point \( F \), and the arc \( CE \) centered at point \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi R^2}{6}}
Draw a line on a plane so that it is equidistant from three given points that are not collinear. How many such lines can be drawn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A sample consisting of five observations has an arithmetic mean of 10 and a median of 12. Find the smallest possible range (largest observation minus smallest) for such a sample. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
已知双曲线$C$的方程为${{x}^{2}}-\frac{{{y2}}}{8}=1$，右焦点为$F$，若点$N\left( 0,6 \right)$，$M$是双曲线$C$的左支上一点，则$\Delta FMN$周长的最小值为_____
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6\sqrt{5}+2$}
$ABCD$ is a convex quadrilateral such that $AB<AD$.  The diagonal $\overline{AC}$ bisects $\angle BAD$, and $m\angle ABD=130^\circ$.  Let $E$ be a point on the interior of $\overline{AD}$, and $m\angle BAD=40^\circ$.  Given that $BC=CD=DE$, determine $m\angle ACE$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55^\circ}
Determine the values of $p \in (0, \infty)$ for which the function $\frac{1}{1-z}$ belongs to the Hardy space $H^p(\mathbb{D})$. Here, $H^p(\mathbb{D})$ is defined as the space of analytic functions $f$ on the unit disk $\mathbb{D}$ such that \( \sup_{0 < r < 1} \left( \frac{1}{2 \pi} \int_{0}^{2\pi}|f(re^{i \theta})|^p d \theta \right)^{\frac{1}{p}} < \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$240$}
Misha has accepted a job in the mines and will produce one ore each day. At the market, he can perform the following transactions:
- Buy or sell one ore for $3.
- Buy or sell bundles of three wheat for $12 each.
- Sell one wheat for one ore.

His ultimate goal is to build a city, which requires:
- Three ore
- Two wheat

How many dollars must Misha begin with in order to build a city after three days of working?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Andrew flips a fair coin  $5$  times, and counts the number of heads that appear. Beth flips a fair coin  $6$  times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{170}
A trapezoid \(ABCD\) with \(BC \parallel AD\) is inscribed in a circle with center \(O\). In the same circle, diameter \(CE\) and chord \(BE\) are drawn, intersecting \(AD\) at point \(F\). Point \(H\) is the foot of the perpendicular dropped from point \(F\) to \(CE\). Point \(S\) is the midpoint of segment \(EO\), and point \(M\) is the midpoint of \(BD\). It is known that the radius of the circle is \(R\) and \(CH = \frac{9R}{8}\). Find \(SM\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3R}{2\sqrt{2}}}
Let $\left\{b_{n}\right\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2 \sqrt{1+\sqrt{b_{n-1}}}$. Calculate the sum of $b_{n} 2^{n}$ from $n=1$ to $\infty$. The original answer includes a constant term and a logarithmic term. Please provide the integer part of the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \(\left\{a_{i}\right\}_{i \geq 0}\) be a sequence of real numbers defined by
$$
a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}
$$
for \(n \geq 0\). Determine the largest value for \(a_{0}\) such that \(\left\{a_{i}\right\}_{i \geq 0}\) is bounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \frac{1}{2^{2020}}}
使得 n^2+2023n 为平方数的正整数 n 的最小值是__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{425}
Given that \(MN\) is a moving chord of the circumscribed circle of an equilateral triangle \( \triangle ABC \) with side length \( 2\sqrt{6} \), and \(MN = 4\), \(P\) is a moving point on the sides of \( \triangle ABC \). Find the maximum value of \( \overrightarrow{MP} \cdot \overrightarrow{PN} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
满足方程 \df{1}{k_1}+\df{1}{k_2}+\cdots+\df{1}{k_n}=1，k_1k_2\cdots k_n\leq 200 的所有有序正整数组 (n,k_1,\cdots,k_n) 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The number represented by a point on the number line that is a distance of $\sqrt{5}$ from the origin is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{5}}
If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have a positive integer solution \( (x, y, z) \), then \( n \) is called a "good number." How many good numbers are there that do not exceed 2,019?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1346}
Using an electric stove with a power of $P=500 \mathrm{W}$, a certain amount of water is heated. When the electric stove is turned on for $t_{1}=1$ minute, the water temperature increases by $\Delta T=2^{\circ} \mathrm{C}$, and after turning off the heater, the temperature decreases to the initial value in $t_{2}=2$ minutes. Determine the mass of the heated water, assuming the thermal power losses are constant. The specific heat capacity of water is $c_{B}=4200$ J/kg$\cdot{ }^{\circ} \mathrm{C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.38}
Given the function $$f(x)=x+\int_0^1t(x+t)f(t){\rm d}t$$, find the value of the integral $$\eta=\int_0^1f(x){\rm d}x$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{42}{23}}
Let $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively. When $x < 0$, $f'(x)g(x) + f(x)g'(x) > 0$, and $g(-3) = 0$, then the solution set of the inequality $f(x)g(x) < 0$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -3) \cup (0, 3)}
If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A robot vacuum cleaner called "Orthogonal" decides to clean the coordinate plane. It starts from point $O$, located at the origin of the coordinates, and travels a distance $X_{1}$ along a straight line. At the point where it arrives, the vacuum cleaner turns and describes a circle centered at the origin. After completing the circle, it continues, without changing direction, traveling a distance $X_{2}$ along a straight line. It then turns sideways towards the origin to describe another circle centered at the origin and, after completing it, travels a distance $X_{3}$. The vacuum cleaner continues this pattern. It is known that the values $X_{1}, X_{2}, X_{3}, \ldots$ follow a certain distribution with a mean $a$ and variance $d$. Find the expected value of the area of the circle described by the $n$-th circle the vacuum cleaner travels along.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi n \left( d + a^2 \right)}
If Leo gains 12 pounds, he will weigh 70% more than his sister Kendra. On the other hand, Leo's brother Mark weighs 20% less than the combined weight of Leo and Kendra. Currently, their combined weight (Leo, Kendra, and Mark) is 305 pounds. What is Leo's current weight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102.24}
10.1. Find the sum of all such integers $a \in[0 ; 400]$, for each of which the equation $x^{4}-6 x^{2}+4=\sin \frac{\pi a}{200}-2\left[x^{2}\right]$ has exactly six roots. Here the standard notation is used: $[t]$ - the integer part of the number $t$ (the greatest integer not exceeding $t$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60100}
The monotonic decreasing interval of the function $f\left(x\right)={\left( \frac{1}{3}\right)}^{-{x}^{2}-4x+3}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2]}
Junior has 34 rabbits. On Monday, he bought 8 toys for his rabbits. On Tuesday, he bought three times as many toys as he did on Monday. On Wednesday, he bought double the number of toys he bought on Tuesday. On Thursday, he bought the same amount of toys as he did on Monday. On Friday, he bought five times as many toys as he did on Monday. On Saturday, he bought half as many toys as he did on Wednesday. If he split all the toys evenly among the rabbits, how many toys would each rabbit have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the function $f(x) = -tx^2 + 2x + 1$ (where $t < 0$, and $t$ is a constant), for any two different $x_1$, $x_2$, when $x_1$, $x_2 \in [-2, 2]$, it always holds that $|f(x_1) - f(x_2)| \leq k|x_1 - x_2|$ (where $k$ is a constant, $k \in \mathbb{R}$). Then, the range of values for the real number $k$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-4t + 2, +\infty)}
Find the mass percentage of Na in NaClO
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30.89\%}
In the acute-angled triangle \(ABC\), the angle bisector \(BL\) is drawn. The circumcircle of triangle \(ABL\) intersects the side \(BC\) at point \(D\). It turns out that point \(S\), symmetric to point \(C\) with respect to line \(DL\), lies on side \(AB\) and does not coincide with its endpoints. What values can \(\angle ABC\) take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
The graph of the function $y=\sin(\omega x+ \frac {5\pi}{6})$ where $0<\omega<\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \frac {1}{2})$ and $( \frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \( \frac{48}{\sqrt{K}} \). What is \( K \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{36}{\pi^3}}
Solve the inequality: 
\[ 8.57 \log_{3} \log_{4} \left(\frac{4x - 1}{x + 1}\right) - \log_{\frac{1}{3}} \log_{\frac{1}{4}} \left(\frac{x + 1}{4x - 1}\right) < 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{2}{3}, \infty \right)}
在正四棱雉 $P-ABCD$ 中，$M、N$ 分别为 $PA、PB$ 的中点，且侧面与底面所成二面角的正切为 $\sqrt{2}$。求异面直线 $DM$ 与 $AN$ 所成角的余弦值。答案的格式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In each cell of a $5 \times 5$ table, a natural number is written with invisible ink. It is known that the sum of all the numbers is 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In the geometric sequence {a_n}, a_3a_7 = 64. Find the value of a_5. Express your answer as a numerical value, including the possibility of a negative value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm8}
For what value of $k \in \mathbb{R}$ does the stochastic process $$Y_t = (W_t + t)e^{-W_t + kt}, \quad t \geq 0$$ have zero drift, where $W_t$ is a Wiener process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
This morning, farmer Rita's workers collected 30,000 gallons of milk and chilled it in a storage tank. They then spent the next 4 hours pumping milk at a rate of 2,880 gallons/hour from the storage tank into a milk tanker. For the following 7 hours, the workers added more milk into the storage tank at varying rates each hour. In the first hour, they added 1,200 gallons, and every subsequent hour they increased the amount of milk produced by 200 gallons. How many gallons of milk were left in the storage tank at the end of this period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31,080}
Given a set  $A$  which contains  $n$  elements. For any two distinct subsets  $A_{1}$ ,  $A_{2}$  of the given set  $A$ , we fix the number of elements of  $A_1 \cap A_2$ . Find the sum of all the numbers obtained in the described way.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \left( 2^{2n-3} - 2^{n-2} \right)}
The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is the value of $DE+EF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? Provide your answer in the form of \frac{k}{m}, and calculate the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The true discount on a bill of Rs. 74500 due 1 year hence at 15% per annum is Rs. 11175. What is the banker's discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11175}
Given that the complex number $z$ and $(z+2)^{2}-8i$ are both purely imaginary, then $z=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2i}
A soccer team won 45 percent of its first 36 games in a particular season, and at least 75 percent of its remaining games. At the end of the season, the winning percentage of the team was 62 percent. The team played an even number of games, and they did not tie any of their games. What was the total number of games that the team played in that season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Find the most general entire functions of the form \( f(z) = u(x) + iv(y) \), where \( u \) and \( v \) depend only on one real variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = az + b}
Determine the characteristic lines for the partial differential equation: $$yu_x + xu_y = u^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - y^2 = C}
Find the number of non-archimedean extensions of the absolute value of \(\mathbb{Q}\) for the field \(\mathbb{Q}(\sqrt[n]{p})\), where \(p\) is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Positive real numbers \( x \) and \( y \) satisfy the following conditions: There exist \( a \in[0, x] \) and \( b \in[0, y] \) such that \( a^{2} + y^{2} = 2 \), \( b^{2} + x^{2} = 1 \), and \( ax + by = 1 \). What is the maximum value of \( x + y \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Arrange all the zeros of the function $f(x)=\sin x$ ($x > 0$) in ascending order to form the sequence $\{a_n\}$. The sequence $\{b_n\}$ satisfies $b_n=3^n \cdot a_n$. Then, the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(2n-1)\cdot 3^{n+1}+3}{4}\pi}
On the side $AB$ of the convex quadrilateral $ABCD$, a point $M$ is chosen such that $\angle AMD = \angle ADB$ and $\angle ACM = \angle ABC$. The tripled square of the ratio of the distance from point $A$ to the line $CD$ to the distance from point $C$ to the line $AD$ is $2$, and $CD = 20$. Find the radius of the inscribed circle of triangle $ACD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{10}-2\sqrt{15}}
The 25th question: In country A with 2011 cities, there is a unique bidirectional flight route operated by a certain airline between any city in A and any city in country B, which also has 2011 cities. There are no flight routes between cities within the same country, and at most 19 different airlines operate these routes. Find the maximum integer $\mathrm{k}$, such that no matter how the routes are arranged, there are always $\mathrm{k}$ cities that are connected (one can travel from one city to another, not necessarily directly), and all the routes are operated by a fixed airline.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{212}
What is the angle between the minute and hour hands at exactly 3:05?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.5}
Find the minimum value of the function
$$
f(x)=\sqrt{15-12 \cos x}+\sqrt{4-2 \sqrt{3} \sin x}+\sqrt{7-4 \sqrt{3} \sin x}+\sqrt{10-4 \sqrt{3} \sin x-6 \cos x}.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A dealer offers a certain cash discount and still makes a profit of 36% 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 cash discount percentage offered by the dealer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60\%}
Solve for $x$ in real numbers, where $a$ and $b$ are real, in the equation:

$$\left| a^2 - 2a - b^2 - 4b - x \right| + \left| a^2 - 2a - b^2 - 4b - 3x + 2 \right| + \left| a^2 - 2a + b^2 + 4b + 2x \right| + (a^2 - 2a + b^2 + 4b + 18|x - 2|) + 11x = 20.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Three partners, P, Q, and R invest in a business with the investment ratio of 3:4:5, respectively. The ratio of their profits at the end of the business period is 9:16:25. If partner P invested the money for 4 months and partner R invested for 10 months, find out for how long did partner Q invest the money in the business?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Convex quadrilateral ABCD is given, and lines are extended from points A, B, C, and D to points B', C', D', and A' respectively, such that AB = BB' = 6, BC = CC' = 7, CD = DD' = 8, and DA = AA' = 9. If the area of quadrilateral ABCD is 10, find the area of quadrilateral A'B'C'D'. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the norm of the linear operator $T:C[0,1]\to C[0,1]$ defined by $Tf(t)=f(t)+\int_0^t(1-2s)f(s)ds$, where the norm is the supremum norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
The volume of a regular pyramid is $V$. A plane is passed through the center of a sphere inscribed in the pyramid, parallel to its base. Find the volume of the pyramid that is cut off from the given pyramid by this plane, if the dihedral angle at the base is $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{V}{8 \cos ^{6} \frac{\alpha}{2}}}
Define the minimum real  $C$  where for any reals  $0 = a_0 < a_{1} < \dots < a_{1000}$  then   $$ \min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000}) $$   holds. Find  $\lfloor 100C \rfloor.$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
Calculate the integral $$\int_{0}^{\pi/2} \ln(a^2\sin^2{x} + \cos^2{x})dx$$ for $a > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi\ln\left(\frac{a+1}{2}\right)}
How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Points $A$ and $B$ are located on a circular alley, connected by a straight highway segment 4 km long, which is the diameter of the circular alley. A pedestrian leaves home at point $A$ and walks along the alley at 6 km/h. After 1 hour, he realizes he forgot his keys and asks a neighbor, who is a cyclist, to bring them to him. The cyclist travels at 15 km/h on the highway and 20 km/h on the alley. What is the minimum time in which the pedestrian can receive the keys, assuming he can walk towards the cyclist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{13}}
An employee arrives at a bus stop at 8:10. Two buses are scheduled to depart from the stop. The first bus departs at 8:00, 8:20, and 8:40 with probabilities of 1/4, 1/2, and 1/4 respectively. The second bus departs at 9:00, 9:20, and 9:40 with probabilities of 1/4, 1/2, and 1/4 respectively. The departure times of the two buses are independent of each other. What is the probability that the employee will wait for more than 50 minutes? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{16}}
Let's determine the radius of the sphere that touches three faces of a regular tetrahedron with unit edge length, as well as the three sides of the fourth face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{8}}
What is the smallest integer n such that any subset of {1, 2, 3,..., 20} of size n must contain two numbers that differ by 8? Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}