inclusionAI/Ring-lite-rl-data huggingface dataset:
Compute the partial derivative \( \frac{\partial\alpha}{\partial x} \) for \( \alpha = x^TAx \), where \( x \in \mathbf{R}^{n \times 1} \) and \( A \in \mathbf{R}^{n \times n} \), using the product rule without expressing \( A \) in terms of its coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(A + A^\top)x}
Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red packets are collected in any order before the fourth click.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum_{z \in S}|z|^2$. For a given $m \in \mathbb{N}$, determine a rough upper bound for the size of $\varphi^{-1}(m)$, the set of all finite subsets of $\mathbb{Z}^d$ with sum of squares equal to $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{O(m^{d/2})}}
Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4096 \sqrt{2}}
Suppose $A$ has $n$ elements, where $n \geq 2$, and $C$ is a 2-configuration of $A$ that is not $m$-separable for any $m<n$. What is (in terms of $n$) the smallest number of elements that $C$ can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
\binom{n}{2}
\]}
Calculate $E[\mu^T\Lambda]$ for a normal-wishart distribution, given the parameters $\mu_0$, $\lambda$, $W$, and $v$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu_0^T v W}
Determine the values of \(a \in \mathbb{R}\) for which the function \(f_n(x, y, z) = \frac{\arctan^a(|x| + |y|)e^z}{n^4 + 4x^2 + y^2}\) is integrable over the domain \(D = \{(x, y, z): 1 < x^2 + y^2 + z < 2\}\) for all \(n \geq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > -2}
Evaluate the definite integral \( \frac{105}{19}\int^{\pi/2}_0 \frac{\sin 8x}{\sin x}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x\le 29$, a subset of $A$ sums to $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
On the xOy coordinate plane, there is a Chinese chess "knight" at the origin (0,0). The "knight" needs to be moved to the point P(1991,1991) using the movement rules of the chess piece. What is the minimum number of moves required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1328}
In a right-angled triangle, the center $O$ of the inscribed circle is such that the half of the hypotenuse closer to $O$ is seen at a right angle from $O$. What is the ratio of the sides of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:4:5}
Calculate \( \frac{\partial T}{\partial x}\left(\frac{\pi}{2}, - \frac{\pi}{2}\right) \) for the function \( T(x,y) = \int_{x-y}^{x+y} \frac{\sin(t)}{t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A 270 meters long bullet train running at a certain speed crosses another bullet train running in opposite direction at the speed of 80 kmph in 9 seconds. The length of the other bullet train is 230.04 meters. What is the speed of the first bullet train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120.016}
In  $\triangle ABC$ , points  $E$  and  $F$  lie on  $\overline{AC}, \overline{AB}$ , respectively. Denote by  $P$  the intersection of  $\overline{BE}$  and  $\overline{CF}$ . Compute the maximum possible area of  $\triangle ABC$  if  $PB = 14$ ,  $PC = 4$ ,  $PE = 7$ ,  $PF = 2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{136}}
A puddle is inhabited by amoebas of three types: red, blue, and yellow. From time to time, any two amoebas of different types can merge into one amoeba of the third type. It is known that in the morning there were 47 red, 40 blue, and 53 yellow amoebas, and in the evening, only one amoeba remained. What color is it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Blue}}
Determine the rate of divergence of the series \( \sum_{m=1}^{\infty}J_{0}(mx) \) as \( x \to 0^{+} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{x}}
How many distinct trees with exactly 7 vertices are there? Here, a tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting \(n\) vertices with \(n-1\) edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces

$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Let proposition $p$: The equation $5x^{2}+my^{2}=1$ represents an ellipse with foci on the $x$-axis; proposition $q$: The equation $(m+1)x^{2}-my^{2}=1$ represents a hyperbola with foci on the $x$-axis. If $p \land q$ is false and $p \lor q$ is true, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,5]}
Determine the number of distinct normal subgroups of a non-abelian group $G$ of order $21$, excluding the trivial subgroup $\{e\}$ and the entire group $G$ itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
设$x$，$y$为非负整数，使得$x+2y$是5的倍数，$x+y$是3的倍数，且$2x+y$≥99，求$7x+5y$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{366}
Let $a < 0$, and $|a| \cdot x \leq a$. Simplify: $|x+1| - |x-3| = \ ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$  Two runners start at point A at the same time and run at the same speed.  The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track.  There is a first time after they begin running when their two positions are collinear with the point A.  At that time each runner will have run a distance of $\frac{m\pi}{n}$ where  $m$  and  $n$ are relatively prime positive integers.  Find  $m + n. $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Determine the number of solutions to the equation \(|\ln|x|\;|=\sin(\pi x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( x \) and \( y \) be positive integers such that \( \frac{100}{151} < \frac{y}{x} < \frac{200}{251} \). What is the minimum value of \( x \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the equation $|2z + 3i| = |z^2|$, where $z = a + ib$, find the range of values for $\|z\|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 3]}
Calculate the functional derivative of the functional \( J(y) = \int_{x_0}^{x_1} \left( y^6(x) + 3 y^2(x) \right) \, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6y^5(x) + 6y(x)}
When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
Evaluate the limit \( \lim_{R\to\infty} \int_{C_R} \frac{e^{-iz}}{z^2} dz \), where \( C_R \) is the semi-circle of radius \( R \) in the lower half-plane, centered at the origin and oriented clockwise, using a direct approach without Jordan's Lemma.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the norm of the operator \( \hat{C}: L^2[0,1] \to L^2[0,1] \) defined by \((\hat{C}f)(x) = \int_0^x f(t) \, dt\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Find the smallest positive period of the function \( y = f(x) \) given that its graph is symmetric about the points \( A\left(a, y_{0}\right) \) and \( B\left(b, y_{0}\right) \) on \( \mathbf{R} \), with no other symmetric points in the interval \( (a, b) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(b - a)}
Let  $L$  be the number formed by  $2022$  digits equal to  $1$ , that is,  $L=1111\dots 111$ .
Compute the sum of the digits of the number  $9L^2+2L$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4044}
Given the function \( f(x) = \frac{x}{\sqrt{1+x^2}} \), find the value of \( f^{(99)}(1) \), where \( f^{(n)}(x) \) denotes the function \( f \) applied \( n \) times to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) where \( C \) is the boundary of the square with vertices \((0,0), (2,1), (1,-2), (3,-1)\) oriented clockwise, and \( \mathbf{F}(x,y) = \langle xy - \sin(x), y^2 + \cos(y) \rangle \). Use Green's theorem to solve this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{2}}
Suppose we have matrices $A_1, A_2, \ldots, A_m$ and $B_1, B_2, \ldots, B_m$ such that $A_i, B_i \in \mathbb{R}^{p_i \times p_{i+1}}$ and $\|A_i - B_i\|_F \leq \varepsilon$ for $1 \leq i \leq m$, where $\|\cdot\|_F$ denotes the Frobenius norm. Determine the upper bound, in terms of $\varepsilon$, for the quantity $\|A_1 \cdot A_2 \cdot \ldots \cdot A_m - B_1 \cdot B_2 \cdot \ldots \cdot B_m\|_\infty$, where $\|\cdot\|_\infty$ is the maximum absolute value of the entries of a matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \varepsilon}
Suppose that the least common multiple of the first $25$ positive integers is equal to $26A7114B4C0$. Find $100 \times A + 10 \times B + C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{740}
Let the polynomial \( p_{k}(x)=\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2} \), where \( k \) is any given positive integer. Find the coefficient of \( x^{2} \) in \( p_{k}(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{1}{3}\left( 4^{2k-1} - 4^{k-1}\right) }
Find the sum a + b + c for the ordered triplet (a, b, c) of real numbers that maximizes the expression \((1 - x)(2 - y)(3 - z)(x + \frac{y}{2} + \frac{z}{3})\) under the conditions \(x < 1\), \(y < 2\), \(z < 3\), and \(x + \frac{y}{2} + \frac{z}{3} > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
The incircle of triangle $A B C$ touches its sides $A B$, $B C$, $C A$ at points $C_{1}$, $A_{1}$, $B_{1}$ respectively. The segment $B B_{1}$ intersects the circle again at point $K$. It is known that $A B = B C = 5$, $A C = 6$. Find $B K$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the random variables $X, X_1, X_2, I_1,$ and $I_2$ which are independent, where $X, X_1,$ and $X_2$ have the same distribution and $I_1, I_2 \sim \text{Bernoulli}\left(\frac{1}{2}\right)$, and the relationship $X = 2 + I_1X_1 + I_2X_2$, determine the correct sign in the quadratic solution for the probability-generating function $M_X(s)$ of $X$ given by:

$$M_X(s) = \frac{2-s^2 \pm \sqrt{1-s^2}}{s^2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-}
If $X_1, X_2 \stackrel{i.i.d}{\sim} N(0,1)$ and $Y = \min(X_1, X_2)$, what is the distribution of $Y^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2_1}
## Task A-4.5.

Given are 100 points arranged in 10 rows and 10 columns. Each point is either red or blue. A point is in a corner if it is at the end of both a row and a column, and on the edge if it is at the end of a row or a column, but not both. Two points are adjacent if they are in the same row or column, with no other points between them. Each pair of adjacent points is connected by a line of red, blue, or green color, such that each red line connects two red points, each blue line connects two blue points, and each green line connects one red and one blue point. It is known that there are 52 red points in total, of which 16 are on the edge and two are in the corners. It is also known that there are a total of 98 green lines.

Determine the total number of blue lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
What is the largest number for which the $3x+1$ conjecture has been verified to be true?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{68}}
A triangle moves in space such that its center of gravity remains stationary, while one of its vertices describes a circle, and the plane of the triangle remains constantly perpendicular to the plane of the circle. What paths are traced by the vertices of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Circles}}
Angelina walked 150 meters from her home to the grocery store at a constant speed. She then walked 200 meters to the gym at double her initial speed. Then, she strolled another 250 meters to the park at half the speed she used from her home to the grocery store. After that, she rushed 300 meters to the library at thrice the speed from the grocery to the gym. She spent 10 seconds less on her way from the grocery to the gym than on her way from home to the grocery, but spent 20 seconds more on her way to the park than on her way to the library. What was Angelina ' s speed , in meters per second , from the park to the library?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Calculate \( \lim_{n\to\infty}\sqrt[n]{|1-z^n|} \) where \( z \in \mathbb{C} \) and \( |z| \neq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{1, |z|\}}
Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then how many students are at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In the school, bananas were to be distributed among students on a 5-day schedule (Monday through Friday). However, varying numbers of students were absent each day, leading to the remaining students receiving extra bananas. Here are the details:

- On Monday, the goal was 3 bananas per student. Due to 370 students being absent, each remaining student received 2 extra bananas.
- On Tuesday, the goal was 4 bananas per student. With 150 students absent, each remaining student got 3 extra bananas.
- On Wednesday, the goal was 2 bananas per student. Given that 490 students were absent, each remaining student received 1 extra banana.
- On Thursday, the goal was 5 bananas per student. When 250 students were absent, each remaining student got 2 extra bananas.
- On Friday, the goal was 1 banana per student. With 340 students absent, each remaining student received 3 extra bananas.

Find the actual total number of students in the school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{925}
Given that $α$ is an angle in the second quadrant and $P(x, \sqrt {5})$ is a point on its terminal side with $\cos α= \frac{ \sqrt {2}}{4}x$, find the value of $\sin α$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{10}}{4}}
Given the sets \( A = \{ x \mid x^2 - 3x + 2 \leq 0 \} \) and \( B = \left\{ x \left\lvert\, \frac{1}{x-3} < a \right. \right\} \), if \( A \subseteq B \), then the range of values for the real number \( a \) is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{1}{2}, +\infty\right)}
Which is larger: $Ackermann(G_{64}, G_{64})$ or $G_{G_{64}}$? Compare the two expressions and determine which one represents a larger number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G_{G_{64}}}
There are 2 boys and 3 girls standing in a row. If boy A does not stand at either end, and among the 3 girls, exactly 2 girls stand next to each other, then the number of different arrangements is \_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
For  $n > 1$ , let  $a_n$  be the number of zeroes that  $n!$  ends with when written in base  $n$ . Find the maximum value of  $\frac{a_n}{n}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
The length of the upper base of a trapezoid is $L$, and the length of its midline is $m$. The length of the segment connecting the midpoints of the two diagonals is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m - L}
Find the splitting field of the polynomial \( f(x)g(x) \) over \( \mathbb{Z}_2 \), where \( f(x) = x^3 + x + 1 \) and \( g(x) = x^3 + x^2 + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_8}
On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the total length of intervals for which the inequality \( f(a) \leq 5 \) holds, where \( f(a) \) is the number of distinct solutions to the equation:

\[ \sin \frac{a \pi x}{x^{2}+1}+\cos \frac{\pi\left(x^{2}+4 a x+1\right)}{4 x^{2}+4}=\sqrt{2-\sqrt{2}} \]

Round your answer to two decimal places. If there are no solutions, write 0; if the length is infinite, write 9999.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.50}
Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-kx-4=0$, and ${x}_{1}^{2}+{x}_{2}^{2}+x_{1}x_{2}=6$, find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\sqrt{2}}
Given a partial order \( \langle P, \preceq \rangle \) where \( P = \{ \langle k, a_k \rangle : k = 1, \ldots, n^2 + 1 \} \) and \( \langle k, a_k \rangle \preceq \langle \ell, a_\ell \rangle \) if and only if \( k \le \ell \) and \( a_k \le a_\ell \), determine what can be concluded about the sequence \( a_1, a_2, \ldots, a_{n^2+1} \) given that each antichain in \( P \) has length at most \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
If the unfolded diagram of a cone is a sector with a central angle of $\frac{2\pi}{3}$ and a radius of $2$, then the surface area of this cone is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16\pi}{9}}
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.  
Rule $1$: If the integer is less than $10$, multiply it by $9$.  
Rule $2$: If the integer is even and greater than 9, divide it by $2$.   
Rule $3$: If the integer is odd and greater than $9$, subtract $5$ from it.   
$A$ sample sequence: $23, 18, 9, 81, 76, \cdots$.   
Find the value of the $98^\text{th}$ term of the sequence that begins $98, 49, \cdots$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot Y moves clockwise, and robots J and B move counterclockwise. After 12 seconds, robot J reaches point B. After another 9 seconds, when robot J first catches up with robot B, it also meets robot Y for the first time. How many seconds after B first reaches point A will Y first reach point B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4242}
A group of some adults and 70 children go for trekking. There is meal for either 70 adults or 90 children and if 21 adults have their meal, the total number of children that can be catered with the remaining food is approximately 63. How many adults went for trekking?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Calculate the area of the figure bounded by the lines

$$
y = x^3, \quad y = -1, \quad y = -3, \quad \text{and} \quad x = 0.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}(3\sqrt[3]{3} - 1)}
If \((2x + 4)^{2n} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}\) \((n \in \mathbf{N}^{*})\), then the remainder when \(a_2 + a_4 + \cdots + a_{2n}\) is divided by 3 is \(\quad\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the least number of moves it takes a knight to get from one corner of an $n \times n$ chessboard, where $n \geqslant 4$, to the diagonally opposite corner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2\\cdot\\lfloor\\frac{n+1}{3}\\rfloor"]}
A kite is inscribed in a circle with center  $O$  and radius  $60$ . The diagonals of the kite meet at a point  $P$ , and  $OP$  is an integer. The minimum possible area of the kite can be expressed in the form  $a\sqrt{b}$ , where  $a$  and  $b$  are positive integers and  $b$  is squarefree. Find  $a+b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{239}
Given that the odd function $f(x)$ defined on $\mathbb{R}$ is monotonically increasing on $(0,+\infty)$, and $f(-1)=2$, determine the solution set of the inequality $f(x-1)+2\leqslant 0$ on $(0,+\infty)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2]}