inclusionAI/Ring-lite-rl-data huggingface dataset:
At a temperature of $13^{\circ}$, a 41% sugar solution in water becomes saturated. How many grams of sugar will not dissolve in water (and will precipitate) if (at a temperature of $15^{\circ}$) you thoroughly mix a glass of water (220 g) with 280 g of sugar?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
In the plane Cartesian coordinate system \( xOy \), a moving line \( l \) is tangent to the parabola \( \Gamma: y^{2} = 4x \), and intersects the hyperbola \( \Omega: x^{2} - y^{2} = 1 \) at one point on each of its branches, left and right, labeled \( A \) and \( B \). Find the minimum area of \(\triangle AOB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{5}}
Find the limit of the integral \( \int_0^{+\infty}\frac{\mathrm{d}x}{(1+x^2)^{1/n}} \) as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the sets $A=\{x|(x-3)(x+1)\geqslant 0\}$ and $B=\left\{ x \mid x < - \frac{4}{5}\right\}$, calculate the intersection $A\cap B$. Express your answer as a set definition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{x|x\leqslant -1\}}
Around the vertices of an equilateral triangle with a side length of 4 units, arcs of circles with radii of 1 and 5 units were drawn as shown in the figure.

What is the width of the narrowest strip between two parallel lines that can contain the resulting plane figure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \text{ units}}
Find the annual yield on a 4-year zero-coupon bond given the following information: You have two 4-year annual-coupon bonds, each with a face value and redemption value of 8000. The first bond has a coupon rate of 7% and a price of 7908.57, while the second bond has a coupon rate of 8% and a price of 8101.55.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.10\%}
Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of $k$ : \[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
\boxed{\frac{(k^2 - 4)^2}{4k(k^2 + 4)}}
\]}
For  $ 0<a<1$ , let  $ S(a)$  is the area of the figure bounded by three curves  $ y\equal{}e^x,\ y\equal{}e^{\frac{1\plus{}a}{1\minus{}a}x}$  and  $ y\equal{}e^{2\minus{}x}$ .
Find  $ \lim_{a\rightarrow 0} \frac{S(a)}{a}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ -2 }
Evaluate the integral \( \oint_{|z| = 1} \frac{z+\bar{z}}{(2z-i)^2} \, dz \) using the Residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi i}{2}}
将一个四棱锥的每一个顶点染色，要求同一条棱上的两个顶点异色．如果只有5种颜色可供选用，一共有多少种不同的染色方法？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$ ? Here $[x]$ is the greatest integer $\leqslant x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches, including the top, bottom, and sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Ranu starts with one standard die on a table.  At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step.  After 2013 such steps, let  $D$  be the number of dice on the table.  What is the expected value (average value) of  $6^D$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10071}
Find the smallest positive integer $k$ such that 
\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] 
for some ordered triple of positive integers $(a,b,c)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1297}
What is the molecular weight of 3 moles of Na2SO4·10H2O (sodium sulfate decahydrate)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{966.75}
What is the cube root of a certain number if the result is 0.06000000000000001?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.06000000000000001}
A train passes a platform in 40 seconds. The same train passes a man standing on the platform in 20 seconds. The speed of the train is some km/hr, and the length of the platform is 300.024 meters. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.00432}
A cake has a shape of triangle with sides  $19,20$  and  $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't  stick out of the plate. What is the minimal diameter of the plate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
A robot is standing on the bottom left vertex  $(0,0)$  of a  $5\times5$  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  $0\leq a,b\leq5$ . 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}
Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1.  Length $r$ is found to be $32$ inches.  After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches.  How high is the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
$P$ 是 $\triangle A B C$ 内一点, 引线段 $A P D, B P E$ 和 $C P F$, 使 $D$ 在 $B C$ 上, $E$ 在 $A C$ 上, $F$ 在 $A B$上. 已知 $A P=6, B P=9, P D=6, P E=3, C F=20$, 求 $\triangle A B C$ 的面积.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac {3}{16}$}
A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find \(\frac{v_f}{v_0}\). The answer should be in the form \(\frac{k}{\sqrt{m}}\). Please calculate the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? Express your answer in the form $k\sqrt{m}$, and find the value of $k + m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{less}}
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with  $3$  buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially  $3$  doors are closed and  $3$  mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Two congruent pentagons are each formed by removing a right-angled isosceles triangle from a square of side-length 1. The two pentagons are then fitted together as shown. Calculate the length of the perimeter of the octagon formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Points $A, B, C,$ and $D$ are sequentially located on a circle. It is known that the degree measures of the smaller arcs $AB, BC, CD,$ and $AD$ have a ratio of $1: 3: 5: 6$. Find the angles of the quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96^\circ, 132^\circ, 84^\circ, 48^\circ}
Determine the length of  $BC$  in an acute triangle  $ABC$  with  $\angle ABC = 45^{\circ}$ ,  $OG = 1$  and  $OG \parallel BC$ . (As usual  $O$  is the circumcenter and  $G$  is the centroid.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$  Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$  Find the greatest integer that does not exceed $1000r.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{777}
Find the angles of a triangle, given that the distance between the center of the circumscribed circle and the orthocenter is half of the longest side and equal to the shortest side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^{\circ}, 60^{\circ}, 30^{\circ}}
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the maximal possible finite number of roots of the equation   $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where  $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$  are distinct reals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
A train is moving at a speed of 132 km/hour. The length of the train is 110 meters. How long will it take to cross a railway platform of a certain length if it takes 7.499400047996161 seconds to cross it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
A circle centered at \(A\) with a radius of 1 and a circle centered at \(B\) with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is in the simplest fractional form \(\frac{k}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
How can we calculate the following sum efficiently?
$$\sum_{k=0}^{38\,204\,629\,939\,869} \frac{\binom{38\,204\,629\,939\,869}{k}}{\binom{76\,409\,259\,879\,737}{k}}\,.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.5}
When a polynomial is divided by \(3x^3 - 4x^2 + 5x - 6\), what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 1, 2}
Let the conjugate of complex number $z$ be $\overline{z}$. If $z+\overline{z}=4$ and $z\cdot \overline{z}=8$, find the value of $\dfrac{\overline{z}}{z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm i}
The polynomial  $P$  is a quadratic with integer coefficients.  For every positive integer  $n$ , the integers  $P(n)$  and  $P(P(n))$  are relatively prime to  $n$ .  If  $P(3) = 89$ , what is the value of  $P(10)$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{859}
The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least  $\sqrt{2}$ . It is known that a regular triangle with side lenght  $3$  cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 5}
How can we determine the number of distinct 3-cycles in $A_4$ that can be formed from the product of two non-disjoint transpositions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
设函数 $\varphi_{1}(x), \varphi_{2}(x)$ 并足 $\varphi_{i}^{\prime \prime}(x)-\ln (x+1) \varphi_{i}^{\prime}(x)+e^{x} \varphi_{i}(x)=0, i=1,2, x>-1$. 定义 $w(x)=\varphi_{1}^{\prime}(x) \varphi_{2}(x)-\varphi_{1}(x) \varphi_{2}^{\prime}(x)$. 若 $w(0)=1$, 则 $[w(2)]=$ $\qquad$  .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
三位数（100,101，···，999）共900个．在卡片上打印这些三位数，每张卡片打印一个三位数，有的卡片所印的，倒过来看仍为三位数，如198倒过来看是861（1倒过来看仍视为1）；有的卡片则不然，如531倒过来看是189．因此有些卡片可以一卡两用，有的不能.于是，至多可少打印＿张卡片.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Determine $x^2+y^2+z^2+w^2$ if 
$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$ 
$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$ 
$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$ 
$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$   satisfy the given conditions
 $ a)$   $ f(0,0)\equal{}1$  ,  $ f(0,1)\equal{}1$  , 
 $ b)$   $ \forall k \notin \left\{0,1\right\}$    $ f(0,k)\equal{}0$  and
 $ c)$   $ \forall n \geq 1$  and  $ k$  ,  $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$ 

find the sum  $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2008}}
Find the largest positive  $p$  ( $p>1$ ) such, that  $\forall a,b,c\in[\frac1p,p]$  the following inequality takes place
\[9(ab+bc+ca)(a^2+b^2+c^2)\geq(a+b+c)^4\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4 + 3\sqrt{2}}}
What is the molecular weight of 9 moles of Carbon tetrachloride?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1384.29}
Each vertex of a tetrahedron has edges of lengths $5$, $\sqrt{41}$, and $\sqrt{34}$ meeting. What are the lengths of the segments connecting the midpoints of the opposite edges of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 5}
A frame for three square photos has an equal width all around (see the diagram). The perimeter of one photo opening is 60 cm, and the total perimeter of the entire frame is 180 cm. What is the width of the frame?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \text{ cm}}
Determine all positive integers \( k \) for which there exist positive integers \( r \) and \( s \) that satisfy the equation
$$
\left(k^{2}-6k+11\right)^{r-1}=(2k-7)^{s}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 3, 4, 8}
What is the minimum number of planes determined by  $6$  points in space which are not all coplanar, and among which no three are collinear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Does there exist an integer \( n \) such that \( 21n \equiv 1 \mod 74 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what level will the water in the first container settle if the tap is opened? The density of water is 1000 kg/$\mathrm{m}^3$, and the density of oil is 700 kg/$\mathrm{m}^3$. Neglect the volume of the connecting tube. Give the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Let $n$ be the answer to this problem. Given $n>0$, find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Let  $\alpha$  be a root of  $x^6-x-1$ , and call two polynomials  $p$  and  $q$  with integer coefficients  $\textit{equivalent}$  if  $p(\alpha)\equiv q(\alpha)\pmod3$ .  It is known that every such polynomial is equivalent to exactly one of  $0,1,x,x^2,\ldots,x^{727}$ .  Find the largest integer  $n<728$  for which there exists a polynomial  $p$  such that  $p^3-p-x^n$  is equivalent to  $0$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{727}
最多能取多少个两两不等的正整数，使得其中任意三个数之和都为素数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For an integer $n \geq 3$, define the sequence $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ by the prime factorization of $n!$ as $n! = p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}$, where the primes $p_{1}<p_{2}<\ldots <p_{k}$ correspond to the respective exponents $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$. Determine all integers $n \geq 3$ for which $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ form a geometric sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 6, 10}
Five integers form an arithmetic sequence. Whether we take the sum of the cubes of the first four terms, or the sum of the cubes of the last four terms, in both cases, we obtain 16 times the square of the sum of the considered terms. Determine the numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 16, 32, 48, 64}
Suppose that in the previous problem, the person looking at the portrait answered the question like this: "In my family, I grew up alone, completely alone. Nonetheless, the son of the one in the portrait is the son of my father (you heard correctly, it's the son!)."

Whose portrait is this person looking at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{father}
Some friends went hiking for 3.5 hours. They traveled 21 kilometers in that time. Birgit was faster than the average time by a certain amount. If Birgit kept the same pace, it would take her 48 minutes to go 8 kilometers. How much faster was Birgit than the average time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the limit of the function

$$
\lim _{x \rightarrow 1}\left(\frac{2 x-1}{x}\right)^{\ln (3+2 x) / \ln (2-x)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
Let $Q=[b_{ij}] $ be a $5\times 5$ matrix over $\mathbb C$ such that $b_{ij}=2$ for $i\ne j$ and otherwise $b_{ii}=c+2$ where $c>0$. Let $\gamma,\delta$ be the largest and the smallest eigenvalue of $Q$. If $\gamma-\delta=6$, then what is the trace of $Q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
If \(\frac{a}{b} = 5\), \(\frac{b}{c} = \frac{1}{4}\), and \(\frac{c^2}{d} = 16\), then what is \(\frac{d}{a}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{25}}
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 4. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is four times the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider the sequence \(\left\{a_{n}\right\}\) defined by \(a_{0}=\frac{1}{2}\) and \(a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2023}\) for \(n = 0, 1, \ldots\). Find the integer \(k\) such that \(a_{k} < 1 < a_{k+1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
A car travels 200 kilometers in 4 hours, and a train travels 210 kilometers in 3 hours. Which one has a higher speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{train}
If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\pm 2ax + a^2$}
sophia finished 23 of a book . she calculated that she finished 90 more pages than she has yet to read . how long is her book ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Evaluate the improper integral \( \int_1^\infty \frac{\ln x}{x} \, dx \) by determining its convergence or divergence as the upper limit approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
$PS$  is a line segment of length  $4$  and  $O$  is the midpoint of  $PS$ . A semicircular arc is drawn with  $PS$  as diameter. Let  $X$  be the midpoint of this arc.  $Q$  and  $R$  are points on the arc  $PXS$  such that  $QR$  is parallel to  $PS$  and the semicircular arc drawn with  $QR$  as diameter is tangent to  $PS$ . What is the area of the region  $QXROQ$  bounded by the two semicircular arcs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi - 2}