inclusionAI/Ring-lite-rl-data huggingface dataset:
3. A function $g: \mathbb{N} \rightarrow \mathbb{N}$ satisfies the following:
(a) If $m$ is a proper divisor of $n$, then $g(m)<g(n)$.
(b) If $m$ and $n$ are relatively prime and greater than 1 , then
$$
g(m n)=g(m) g(n)+(n+1) g(m)+(m+1) g(n)+m+n \text {. }
$$

Find the least possible value of $g(2016)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3053}
If the positive solution for the fractional equation $\frac{a}{{x+3}}=\frac{1}{2}$ with respect to $x$ is $a$, then the range of values for $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > \frac{3}{2}}
How many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimeter less than 20?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Find the positive integer $k$ such that the roots of the polynomial $x^3 - 15x^2 + kx - 1105$ are three distinct collinear points in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
In an isosceles trapezoid, one base is \(40 \text{ cm}\) and the other is \(24 \text{ cm}\). The diagonals of this trapezoid are mutually perpendicular. Find its area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Given a sequence ${a_n}$ that satisfies ${a_{n+1}} + {a_n} = (n+1)\cos \frac{n\pi}{2}$ $(n\geqslant 2, n\in \mathbb{N^*})$, and ${S_n}$ is the sum of the first $n$ terms of the sequence. If ${S_{2017}} + m = 1012$ and ${a_1}m > 0$, find the minimum value of $\frac{1}{{a_1}} + \frac{1}{m}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
10. Centipedes and three-headed dragons. In a certain herd of centipedes and three-headed dragons, there are a total of 26 heads and 298 legs. Each centipede has one head. How many legs does a three-headed dragon have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the convex quadrilateral ABCD, it is known that ∠DAB = ∠BCD = 90°, and ∠CDA : ∠ABC = 2 : 1, AD : CB = 1 : √3. Calculate the measure of ∠BDA in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In a tennis tournament, 512 schoolchildren participate. For a win, 1 point is awarded, for a loss, 0 points. Before each round, pairs are drawn from participants with an equal number of points (those who do not have a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Calculate the surface integral of the second kind:

$$
\iint_{S}\left(x y^{2}+z^{2}\right) d y d z+\left(y z^{2}+x^{2}\right) d z d x+\left(z x^{2}+y^{2}\right) d x d y
$$

where \( S \) is the exterior surface of the upper ( \(z \geq 0 \) ) hemisphere \( x^{2}+ y^{2}+z^{2}=a^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi a^4}{20}(8a + 5)}
What is the closed form for a sum involving permutations, given by $nPk + (n-1)Pk + (n-2)Pk + \ldots + kPk$, where $n$ is the total number of items and $k$ is the number of items being permuted? For example, when $n=7$ and $k=2$, the sum is $7P2 + 6P2 + 5P2 + \ldots + 2P2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k!\binom{n+1}{k+1}}
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]
is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{929}
考虑正 60 边形的 60 个顶点，将其中 10 个染红色， 20 个染黄色，其余 30 个染蓝色. 求最小的正整数 $m$ ，使得总能找到不过任何顶点的 $m$ 条直线将平面分成若干区域，满足每个区域中所有点的颜色相同。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. What is the degree measure of $\angle E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
A two-digit number is given in which the first digit is greater than the second. Its digits were rearranged to obtain a new two-digit number. The difference between the first and the second number turned out to be a perfect square. How many possible two-digit numbers satisfy this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
In Mrs. Dawson's rose garden, there are 30 rows of roses. In each row, there are 50 roses. The roses are planted in such a way that, going from left to right:

- 2/5 of the roses are red until a blue rose appears.
- After a blue rose, 1/4 of the remaining roses are white until two yellow roses appear together.
- After the two yellow roses, 3/8 of the remaining roses are purple until three orange roses appear together.
- The rest of the roses in the row are pink.

How many roses at Mrs. Dawson's rose garden are pink, considering the above color distribution pattern?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
已知数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=2, a_{n+1}=2^{a_{n}}(n \in$ $\left.\mathbf{N}^{*}\right)$ ，其中数列 $\left\{b_{n}\right\}: b_{1}=5, b_{n+1}=5^{b_{n}}\left(n \in \mathbf{N}^{*}\right)$ ，则满足 $b_{m}>a_{25}$ 的最小正整数 $m$ 的值是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In different historical periods, the conversion between "jin" and "liang" was different. The idiom "ban jin ba liang" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in today's terms. Currently, 10 liang equals 1 jin, with each jin equivalent to 500 grams today. There is a batch of medicine, with part weighed using the ancient system and the other part using the current system, and it was found that the sum of the number of "jin" is 5 and the sum of the number of "liang" is 68. How many grams is this batch of medicine in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2800}
Find the sum of the real part and the imaginary part of the complex number $\frac{i}{1+i}-\frac{1}{2i}$. Express your answer as a single fraction or integer in the form $\boxed{[answer]}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
If $a_n=2n^2+\lambda n+3$ where $\lambda$ is a real constant and $n\in\mathbb{N}^*$, and the sequence $\{a_n\}$ is monotonically increasing, then the range of real $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda > -6}
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.2}
Given the quadratic equation $x^2 + px + q = 0$, where $p$ and $q$ are positive numbers, and the roots of this equation differ by $2$, find the value of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{q+1}}
Simplify $rac{1}{2+rac{2}{3}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{8}}
Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$. Given $AB = 21$, $CD = 7$, $AD = 13$, and the area of trapezoid $[ABCD] = 168$. A line parallel to $AB$ through $P$ intersects the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at points $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let  $n$  be a positive integer,  $n \ge 2$ , and put  $\theta=\frac{2\pi}{n}$ . Define points  $P_k=(k,0)$  in the *xy*-plane, for  $k=1,2,\dots,n$ . Let  $R_k$  be the map that rotates the plane counterclockwise by the angle  $\theta$  about the point  $P_k$ . Let  $R$  denote the map obtained by applying in order,  $R_1$ , then  $R_2$ , ..., then  $R_n$ . For an arbitrary point  $(x,y)$ , find and simplify the coordinates of  $R(x,y)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x+n, y)}
The vertices of  $\triangle ABC$  are labeled in counter-clockwise order, and its sides have lengths  $CA = 2022$ ,  $AB = 2023$ , and  $BC = 2024$ .  Rotate  $B$   $90^\circ$  counter-clockwise about  $A$  to get a point  $B'$ .  Let  $D$  be the orthogonal projection of  $B'$  unto line  $AC$ , and let  $M$  be the midpoint of line segment  $BB'$ .  Then ray  $BM$  intersects the circumcircle of  $\triangle CDM$  at a point  $N \neq M$ .  Compute  $MN$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Given that $e$ is the base of the natural logarithm, if for any $x \in [0,1]$, there exists a unique $y \in [-1,1]$ such that $x + y^2e^y - a = 0$ holds, then the range of the real number $a$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 + \frac{1}{e}, e]}
(4) Given a regular tetrahedron $P-ABC$ with the side length of the equilateral triangle base being 1, and the center $O$ of its circumscribed sphere satisfies $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\overrightarrow{0}$, then the volume of this regular tetrahedron is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
Given \( a, b \in \mathbf{R} \), the circle \( C_{1}: x^{2}+y^{2}-2 x+4 y-b^{2}+5=0 \) and the circle \( C_{2}: x^{2}+y^{2}-2(a-6) x-2 a y +2 a^{2}-12 a+27=0 \) intersect at two distinct points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \), and \( \frac{y_{1}+y_{2}}{x_{1}+x_{2}}+\frac{x_{1}-x_{2}}{y_{1}-y_{2}} = 0 \). Find the value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Solve in positive integers the following equation:

$$ \left[ \sqrt{1} \right] + \left[ \sqrt{2} \right] + \left[ \sqrt{3} \right] + \ldots + \left[ \sqrt{x^2 - 2} \right] + \left[ \sqrt{x^2 - 1} \right] = 125, $$

where $[a]$ denotes the integer part of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A positive integer is considered **detestable** if the sum of its digits is a multiple of $11$. Determine how many positive integers below $10,000$ are detestable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{908}
When $0.73\overline{864}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{737910}
Find the smallest possible value of $x+y$ where $x, y \geq 1$ and $x$ and $y$ are integers that satisfy $x^{2}-29y^{2}=1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11621}
In the Generic Math Tournament,  $99$  people participate. One of the participants, Alfred, scores 16th in Algebra, 30th in Combinatorics, and 23rd in Geometry (and does not tie with anyone). The overall ranking is computed by adding the scores from all three tests. Given this information, let  $B$  be the best ranking that Alfred could have achieved, and let  $W$  be the worst ranking that he could have achieved. Compute  $100B+W$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167}
Nine lines parallel to the base of a triangle divide the other sides each into  $10$  equal segments and the area into  $10$  distinct parts. If the area of the largest of these parts is  $38$, calculate the area of the original triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Let $P$ be a point not on line $XZ$ and $Q$ a point on line $XZ$ such that $PQ \perp XZ.$ Meanwhile, $R$ is a point on line $PZ$ such that $SR \perp PZ.$ If $SR = 5,$ $PQ = 6,$ and $XZ = 7,$ then what is the length of $PZ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.4}
Let  $ABC$  be a non-equilateral triangle with integer sides. Let  $D$  and  $E$  be respectively the mid-points of  $BC$  and  $CA$  ; let  $G$  be the centroid of  $\Delta{ABC}$ . Suppose,  $D$ ,  $C$ ,  $E$ ,  $G$  are concyclic. Find the least possible perimeter of  $\Delta{ABC}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Danny is sitting on a rectangular box. The area of the front face of the box is half the area of the top face, and the area of the top face is some multiple of the area of the side face. The volume of the box is 192 and the area of the side face of the box is approximately 32. What is the ratio of the area of the top face to the side face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:2}
a、b为正整数，满足$\frac { 1 } { a } - \frac { 1 } { b } = \frac { 1 } { 2 0 1 8 }$，则所有正整数对（a，b）的个数为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then among the remaining students, there will be twice as many liars as those who tell the truth." How many liars were at the party?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151}
3. There are 207 different cards with numbers $1,2,3,2^{2}, 3^{2}, \ldots, 2^{103}, 3^{103}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer divisible by 6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{267903}
In quadrilateral \( \square ABCD \), \( AB = 4 \), \( BC = 2 \), and \( \angle DAB = 60^\circ \). A point \( P \) on side \( BC \) and a point \( Q \) on the extension of side \( DC \) satisfy \( S_{\triangle BPQ} = \frac{\sqrt{3}}{32} \). Find the minimum value of \( \overrightarrow{AP} \cdot \overrightarrow{BQ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{97}{16}}
Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide on a specific time. Each of the four group members signs on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
-1200° is an angle in the    quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{third}
A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) The original answer is in the form \( k\sqrt{m} - n \). Please give the value of \( k + m + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Let \( n \) (\( \geqslant 3 \)) be a positive integer, and \( M \) be an \( n \)-element set. Find the largest positive integer \( k \) that satisfies the following condition:

There exists a family \( \psi \) of \( k \) three-element subsets of \( M \) such that the intersection of any two elements in \( \psi \) (note: the elements of \( \psi \) are all three-element sets) is nonempty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n-1}{2}}
There are 100 participants in an international StarCraft championship. The tournament is single-elimination, meaning in each match, two players compete, the loser is eliminated, and the winner continues. Find the maximum possible number of participants who won exactly two matches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
A unit square \(ABCD\) and a circle \(\Gamma\) have the following property: if \(P\) is a point in the plane not contained in the interior of \(\Gamma\), then \(\min (\angle APB, \angle BPC, \angle CPD, \angle DPA) \leq 60^\circ\). The minimum possible area of \(\Gamma\) can be expressed as \(\frac{a \pi}{b}\) for relatively prime positive integers \(a\) and \(b\). Compute \(100a+b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
30. In each of the following 6-digit positive integers: $555555,555333,818811$, 300388, every digit in the number appears at least twice. Find the number of such 6-digit positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11754}
The population of a certain bacteria is known to double every 10 hours. Assuming exponential growth, determine the time that it would take for the bacteria to triple in number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.85 \text{ hours}}
Given a parabola \(C\) with the center of ellipse \(E\) as its focus, the parabola \(C\) passes through the two foci of the ellipse \(E\), and intersects the ellipse \(E\) at exactly three points. Find the eccentricity of the ellipse \(E\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{5}}{5}}
Let $k>0$ and define $$A=-\sqrt{2} \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})}$$ and $$\gamma(k)=\frac{(k+1)A-3kA+2A^3}{(k-A^2)^{\frac{3}{2}}}.$$ Where $\Gamma$ is the Gamma function. What is $$\lim_{k \to 0^{+}} \gamma(k)?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12 square units, find the area of the equilateral triangle. Express your answer in square units as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A flea is jumping on the vertices of square \(ABCD\), starting from vertex \(A\). With each jump, it moves to an adjacent vertex with a probability of \(\frac{1}{2}\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be the last one visited.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, find the value of $y$. Express your answer as a decimal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67.5}
38. In a tournament where each pair of teams played each other twice, 4 teams participated. For each win, two points were awarded, for a draw - one, and for a loss - 0. The team that finished in last place scored 5 points. How many points did the team that finished in first place score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many distinct trees with exactly 7 vertices exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Each pair  $(x, y)$  of nonnegative integers is assigned number  $f(x, y)$  according the conditions: $f(0, 0) = 0$ ; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$ , $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$  for  $x, y \ge 0$ .
Let  $n$  be a fixed nonnegative integer and let  $a$ ,  $b$  be nonnegative integers such that  $f(a, b) = n$ . Decide how many numbers satisfy the equation  $f(a, x) + f(b, x) = n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two projects simultaneously, team A is assigned to project $A$, and teams B and C work together on project $B$. After a few days, team C joins team A to complete project $A$. How many days did team C and team B work together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The middle school sold 6 more than two times the number of fair tickets as it did tickets to the baseball game. If 25 fair tickets were sold, how many baseball game tickets did the school sell?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Given that \(\alpha, \beta \in\left(0, \frac{\pi}{2}\right)\), and
\[
\tan ^{2}(\alpha + \beta) + \cot ^{2}(\alpha + \beta) = 3\alpha - \alpha^{2} - \frac{1}{4}.
\]
Determine \(\beta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi - 6}{4}}
If the function \( f(x) = \frac{(\sqrt{1008} x + \sqrt{1009})^{2} + \sin 2018 x}{2016 x^{2} + 2018} \) has a maximum value of \( M \) and a minimum value of \( m \), then \( M + m \) equals \_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that \( 2^{2013} < 5^{867} < 2^{2014} \), how many pairs of integers \((m, n)\) satisfy: 
\[ 5^n < 2^m < 2^{m+2} < 5^{n+1} \]
where \( 1 \leq m \leq 2012 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{279}
An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, find the maximum number of students that can be seated for an exam.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}