inclusionAI/Ring-lite-rl-data huggingface dataset:
Marco loves trading cards. He has a total of 750 cards on hand, and a third of them are duplicates. Marco decides to trade one-fifth of these duplicates for completely new cards to his friend Josh. After trading with Josh, Marco trades an additional one-third of these new cards with his friend Alex for more distinct cards. Finally, Marco exchanges half of the remaining new cards from the trades with Josh and Alex with his friend Mia. How many distinct new cards is Marco left with after all three trades?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given that the universal set U = ℝ, let set P = {x | x ≥ 3} and M = {x | x < 4}. Calculate the intersection of set P with the complement of set M with respect to the universal set U, denoted as P ∩ (C_U M). Express your answer as a set definition using proper set notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{x | x ≥ 4}}
4. Find the smallest positive number $c$ such that for any integer $n \geqslant 4$, and any set $A \subseteq\{1,2, \cdots, n\}$, if $|A|>c n$, then there exists a function $f: A \rightarrow\{1,-1\}$, satisfying $\left|\sum_{a \in A} f(a) \cdot a\right| \leqslant 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
The domain of $y= \frac{\sqrt{\sin x} + \lg(\cos x)}{\tan x}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2k\pi, \frac{\pi}{2} + 2k\pi), k \in \mathbb{Z}}
Let $A B C$ be a triangle and $\omega$ be its circumcircle. The point $M$ is the midpoint of arc $B C$ not containing $A$ on $\omega$ and $D$ is chosen so that $D M$ is tangent to $\omega$ and is on the same side of $A M$ as $C$. It is given that $A M=A C$ and $\angle D M C=38^{\circ}$. Find the measure of angle $\angle A C B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33^{\circ}}
Consider the function \( f(x) = \frac{1}{(1+2x)^4} \), which has a linear approximation \( L(x) = 1 - 8x \) around \( x = 0 \). Determine the interval of \( x \) values for which the linear approximation is accurate within an error of \( 0.1 \). Express this as an inequality: \( -0.1 \leq f(x) - L(x) \leq 0.1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-0.05 ≤ x ≤ 0.05}
Given the quadratic equation $x(kx+1)-x^{2}+3=0$, find the largest integer value of $k$ for which the equation has real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\frac{AP}{AB} =\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ intersects the circle twice at points $S$ and $T$ . Two more circles $s$ and $t$ are drawn with diameters $SP$ and $ST$ respectively. For any circle $\omega$ let $A(\omega)$ denote the area of the circle. What is $\frac{A(s)+A(t)}{A(a)+A(b)}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{205}{1681}}
In the trapezoid \(ABCD\), the side \(CD\) is equal to the diagonal \(AC\). On the smaller arc \(BC\) of the circumcircle of triangle \(BCD\), a point \(E\) is chosen such that \(CD = CE\). Find the angle \(\angle AEB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
In a random arrangement of 3 red, 2 white, and 2 blue balls, what is the probability of having 2 adjacent white balls or 2 adjacent blue balls? Express your answer as a simplified fraction in the form \(\boxed{\frac{a}{b}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{21}}
The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
How many positive integers $n$ satisfy \[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6$}
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Solve the equation or inequality using your calculator: $\ln (x)=e^{-x}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x \approx 1.3098$}
How many times longer is the staircase to the fourth floor of a building compared to the staircase to the second floor of the same building?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}=3b^{2}+3c^{2}-2\sqrt{3}bc\sin A$, find the value of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
15. Xiaohua wrote a computer program. One minute after the program runs, some bubbles first appear on the computer screen, and at each subsequent integer minute, the same number of new bubbles appear. Half a minute after the 11th appearance of bubbles, one bubble bursts. Thereafter, every minute, more bubbles burst, with the number increasing by 1 each minute (i.e., half a minute after the 12th appearance of bubbles, 2 bubbles burst...). At a certain moment, the total number of burst bubbles is exactly equal to the total number of bubbles that have appeared on the screen, meaning all bubbles have disappeared at this moment. How many bubbles were there at most on the screen at any given time during the entire process of the program running?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1026}
A number, when divided by the sum of a certain number and 445, gives 2 times their difference as quotient and 40 as remainder. The number is 220040. What is the first number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{555}
Given the function $f(x)=f'( \frac {\pi}{6})\sin x+\cos x$, then the value of $f( \frac {\pi}{6})$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Charlie noticed his golden ticket was golden in two ways! In addition to being gold, it was a rectangle whose side lengths had ratio the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$. He then folds the ticket so that two opposite corners (vertices connected by a diagonal) coincide and makes a sharp crease (the ticket folds just as any regular piece of paper would). The area of the resulting shape can be expressed as $a + b \varphi$. What is $\tfrac{b}{a}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The numbers 1 to 2031 are written on a blackboard.
1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.
2. Then select another number on the blackboard, find the absolute value of its difference from the previous absolute value obtained, and erase this number.
3. Repeat step (2) until all numbers on the blackboard are erased.
What is the maximum final result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2030}
A solid $\Omega$ is bounded by the planes $x=2$, $y=0$, $y=1$, $z=0$ and the cylinder $z^2 = 6x$. The volume density of the material at each point is proportional to the distance from the $XOY$ plane. Find the moment of inertia of the solid $\Omega$ relative to the $z$-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14k}
If $A$ and $B$ are two different points on the parabola $y^{2}=4x$, and the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $D(4,0)$, then the maximum value of $|AB|$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Define a sequence of functions: $f_{1}(x)=f(x)= \dfrac {x}{1-x}$, $f_{2}(x)=f(f_{1}(x))$, $f_{3}(x)=f(f_{2}(x))$,..., $f_{n}(x)=f(f_{n-1}(x))$. Find the coordinates of the intersection point between the graph of the function $y=f_{2017}(x)$ and the curve $y= \dfrac {1}{x-2017}$. Express your answer as an ordered pair (x, y) with exact values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,- \dfrac {1}{2018})}
Evaluate the limit $\lim_{x \to 0} \frac {\sin(xe^x)-\sin(xe^{-x})}{x^3}$ using only asymptotic simplifications and Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A line segment AB has a length of 2, with its endpoints A and B sliding along the x-axis and y-axis, respectively. What geometric figure is formed by the trajectory of the midpoint of line segment AB?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Circle}}
Given two lines $l_{1}: 2x-ay-1=0$ and $l_{2}: ax-y=0$. If $l_{2} \nparallel l_{1}$, then the real number $a=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{2}}
1. (YUG) Find all real numbers $x \in[0,2 \pi]$ such that  $$ 2 \cos x \leq|\sqrt{1+\sin 2 x}-\sqrt{1-\sin 2 x}| \leq \sqrt{2} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi / 4 \leq x \leq 7 \pi / 4}
On the segment \( AB \), point \( C \) is taken, and segments \( AB \) and \( CB \) serve as diameters of circles. Chord \( AM \) touches the smaller circle at point \( D \). Line \( BD \) intersects the larger circle at point \( N \). Given that \( \angle DAB = \boldsymbol{\alpha} \) and \( AB = 2R \), find the area of quadrilateral \( ABMN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R^2 \cos \alpha (1 + \sin \alpha)}
Let \( n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) be two sets of real numbers, satisfying \( \sum_{j=1}^{n} \alpha_{j}^{2}<1 \) and \( \sum_{j=1}^{n} \beta_{j}^{2}<1 \). Define
\[ A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, \, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, \, W=\frac{1}{2}\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2} . \]

Find all real numbers \(\lambda\) such that the equation \( x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0 \) has only real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Define a function $g : \mathbb{N} \rightarrow \mathbb{R}$ such that:
\[
g(x) = \sqrt{4^x + \sqrt{4^{x+1} + \sqrt{4^{x+2} + \cdots}}}
\]
Find the last two digits in the decimal representation of $g(2021)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
The distance from $A$ to $B$ is 999 km. Along the road, there are kilometer markers that indicate the distance to $A$ and to $B$ as follows:

$0|999, 1|998, \ldots, 999|0$.

How many of these markers have signs that contain only two different digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find the integer $n,$ $-180 \le n \le 180,$ such that $\cos n^\circ = \cos 430^\circ.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-70}
Triangle \(\triangle A B C\) has \(A B=21, B C=55\), and \(C A=56\). There are two points \(P\) in the plane of \(\triangle A B C\) for which \(\angle B A P=\angle C A P\) and \(\angle B P C=90^{\circ}\). Find the distance between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{2} \sqrt{409}}
Convert the number 380,180,000,000 to scientific notation. Express your answer in the form a × 10^n, where 1 ≤ a < 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.8018 × 10^11}
To understand the scores of 1252 students in a knowledge competition at a certain school, it was decided to use systematic sampling to draw a sample of 50. The number of individuals randomly eliminated from the population is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Calculate the value of $(1+a)(1+b)(1+c)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
设多项式 \(f(x)\) 的各项系数都是非负实数，且 \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=f^{\prime \prime \prime}(1)=1\)。求 \(f(x)\) 的常数项的最小值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? Express your answer in the form $a\pi + b\sqrt{3}$, where $a$ and $b$ are constants. Please find the value of $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A circle centered at point \( p \) is tangent to the circle given by the equation \((x+1)^{2} + y^{2} = 1\) externally and is tangent to the circle given by the equation \(x^{2} + y^{2} - 2x - 8 = 0\) internally. Determine the coordinates of the point \( p \) that is the farthest distance from the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,0)}
Out of ten disks, three have both sides red, two have both sides blue, and the remaining five have one side blue and the other side red. The ten disks are placed in a hat, one is drawn, and one side of it is observed. How should we guess the color of the other side of the drawn disk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{20}}
A box contains 11 balls, numbered 1, 2, 3, ..., 11. If 6 balls are drawn simultaneously at random, find the probability that the sum of the numbers on the balls drawn is odd. Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{349}
Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3}}{3}}
长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 满足的条件是 \frac{\sqrt{k}}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $\sin C + \sin (B-A) = \sin 2A$, $\sin C= \frac{\sqrt{3}}{3}$, and $a-b=3- \sqrt{6}$, find the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{2}}{2}}
Paco had some salty cookies and 34 sweet cookies. He ate 15 sweet cookies and 56 salty cookies. After that, Paco had 19 sweet cookies left. How many salty cookies did Paco have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面，其球心距为 13 。若作一平面与这二球面相切，且与圆柱面相交成一栯圆，则这个粨圆的长轴长与短轴长之和是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
The product of two positive integers plus their sum equals 119. The integers are relatively prime, and each is less than 25. What is the sum of the two integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The "Hua Luogeng" Golden Cup Junior Math Invitational Contest was first held in 1986, the second in 1988, and the third in 1991, and has subsequently been held every 2 years. The sum of the digits of the year of the first "Hua Cup" is: \( A_1 = 1 + 9 + 8 + 6 = 24 \). The sum of the digits of the years of the first two "Hua Cup" contests is: \( A_2 = 1 + 9 + 8 + 6 + 1 + 9 + 8 + 8 = 50 \). Find the sum of the digits of the years of the first 50 "Hua Cup" contests, \( A_{50} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{629}
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15^\circ}
If 0.5% of a equals 95 paise, b is three times the amount of a minus 50, and c is the difference between a and b squared, what is the combined value of a, b, and c in rupees, considering that a and c must be positive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109610}
设正六边形 $A B C D E F$ 的边长为 1. 则 $(\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
In triangle \(ABC\), side \(BC\) is equal to \(a\), and the radius of the inscribed circle is \(r\).

Find the radii of two equal circles that touch each other, if one of them touches sides \(BC\) and \(BA\), and the other touches sides \(BC\) and \(CA\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ar}{a + 2r}}
What is the negation of the proposition '$\forall x > 1$, $x^{2}+1\geqslant 0$'? Express your answer as a logical statement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\exists x > 1$, $x^{2}+1 < 0$}
设$n=1(mod4)$且$n>1$, $P = ( a _ { 1 } , a _ { 2 } , \cdots , a _ { n } )$是$(1，2,⋯,n)$的任意排列，$k_p$表示使不等式$a _ { 1 } + a _ { 2 } + \cdots + a _ { k } < a _ { k + 1 } + a _ { k + 2 } + \cdots + a _ { n }$成立的最大下标$k$.试对一切可能的不同排列$P$，求对应的$k_p$的和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { n ! } { 2 } \cdot ( n - 1 ) $}
Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h, they will meet 16 km away from point C. What was the original speed of car A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find a recursive formula for the integral $I_n = \int_0^2 x^n(4 - x^2)^{1/2} dx$, where $n \geq 2$, using a suitable method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4(n-1)}{n+2} I_{n-2}}
The  $42$  points  $P_1,P_2,\ldots,P_{42}$  lie on a straight line, in that order, so that the distance between  $P_n$  and  $P_{n+1}$  is  $\frac{1}{n}$  for all  $1\leq n\leq41$ . What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{861}
For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote:
$$f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).$$
When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying:
$$a_1+a_2+\cdots+a_{10}=20.$$
Compute the number of positive integer divisors of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}
Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 p (0\leq p\leq 1) 的概率消失，有 \df{1-p}{3} 的概率保持不变，有 \df{1-p}{3} 的概率分裂成两个，有 \df{1-p}{3} 的概率分裂成三个. 对所有新产生的生物每天也会发生上述事件. 假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 \df{1}{2}，则 p 至多为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Two workers started paving paths in a park simultaneously from point $A$. They pave with constant speeds: the first one on the section $A-B-C$, and the second one on the section $A-D-E-F-C$. They finished their work at the same time, spending 9 hours on it. It is known that the second worker works 1.2 times faster than the first. How many minutes did the second worker spend paving the section $DE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45 \text{ minutes}}
Evaluate the integral \( \int \frac{\cos 2x}{\cos^2 x \sin^2 x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\cot x - \tan x + C}
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, find the probability that this number is divisible by $11$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occupied given that all planets are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{981}
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $a_3^2 \neq a_1a_9$. Find the value of $\frac {a_{3}}{a_{6}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
The number of years of service of the 8 employees in a production department are some values. The range in the number of years of service of the 8 employees is 14. If the second lowest number of years of service is 10, what is the lowest number of years of service?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given positive numbers $a$ and $b$ that satisfy $5 - 3a \leq b \leq 4 - a$ and $\ln{b} \geq a$, find the range of values for $\frac{b}{a}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[e, 7]}
For each positive integer n, let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of $n$ for which $f(n) \le 300$.
Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109}
How many five-character license plates consist of a consonant, followed by a vowel, followed by a consonant, a digit, and then a special character from the set {$, #, @}? (For this problem, consider Y as both a consonant and a vowel.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79,380}