inclusionAI/Ring-lite-rl-data huggingface dataset:
Identify the quotient group $G/H$ where $G = GL_n(\mathbb{R})$ and $H$ is the subgroup of $G$ consisting of all matrices with positive determinant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
Provide an example of a function \( f(x) \) such that the improper integral \( \int_{1}^{\infty} f(x) \, dx \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(x^2)}
Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1121}
Let $S$ be a subset of the natural numbers such that $0 \in S$, and for all $n \in \mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
7. In a race with six runners, $A$ finished between $B$ and $C, B$ finished between $C$ and $D$, and $D$ finished between $E$ and $F$. If each sequence of winners in the race is equally likely to occur, what is the probability that $F$ placed last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{16}}
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$40$}
A figure in the plane has exactly two axes of symmetry. Find the angle between these axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
In a running competition, Jungkook came in the 7th and Minyoung came in the 5th. Taehyung did worse than Minyoung but better than Jungkook. In what place did Taehyung come in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6th}
Determine the number of irreducible polynomials in $GF(2)[x]$ of degree 8 whose roots generate a subfield of $GF(2^8)$ rather than the whole field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the asymptotic growth rate of the function defined by the recurrence relation: \[ F(n) = \begin{cases} \sqrt{3F\left(\frac n3\right)} + n & \mbox{if }\ n>3\\ 2 & \mbox{if }\ n\leq3 \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Determine the total number of automorphisms of the symmetric group $S_6$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
Assuming a Poisson distribution with an unknown rate parameter \(\lambda\) for software failures, where two failures occurred in one week, determine the waiting time without any additional failures needed to declare with 95% confidence that a workaround is successful. A successful workaround will permanently eliminate the problem. Calculate this waiting time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
Consider the Burgers equation \(u_t + uu_x = 0\) with the initial condition given by:
\[
u(x,0) = \phi(x) = \begin{cases} 2, & x \leq \pi/2 \\ \sin x + 1, & \pi/2 \leq x \leq 3\pi/2 \\ 0, & x \geq 3\pi/2 \end{cases}.\]
Determine the breaking time \(t_B\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: $$\lim_{x \rightarrow \infty} \sum_{n=0}^\infty (-1)^{n} \frac{x \lvert x \rvert^{2n}}{2n+1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the limit:
\[ \lim\limits_{n\rightarrow\infty}\sum\limits_{k=2}^n\frac{1-e^{-(b_k-a_{k})}}{1+e^{-(b_k-a_{k})}} \]
where \(0 \leq a_k \leq b_k \leq 1\) and \(0 \leq b_k - a_k \leq \epsilon\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the range of values for $c$ such that the equation $x(x^2-7) = 2x+c$ has exactly two solutions. Express your answer in the form $c_1 < c < c_2$, where $c_1$ and $c_2$ are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10 < c < 10}
Determine the number of possible values for $a$ such that the equation $ax^2 + xy + y^2 + bx + cy + d = 0$ represents a pair of straight lines with slopes $m$ and $m^2$. How many such values of $a$ exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $K$ be a number field and $G_K = \text{Gal}(\overline{K}/K)$ be its absolute Galois group. Consider a discrete $G_K$-module $M$ with a trivial $G_K$-action. For each place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$. The restriction to a decomposition group $G_v$ at $v$ defines a homomorphism $H^1(K, M) \to H^1(K_v, M)$. Define $$F^1(K, M) := \text{Ker}\left(H^1(K, M) \to \prod_v H^1(K_v, M)\right).$$ Determine the structure of $F^1(K, M)$ when the $G_K$-action on $M$ is trivial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the residue of the function \( \frac{z}{e^z - 1} \) at its poles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i k}
Determine the range of values for \( r \) such that the curves \( C_1 = \arg(z) = \frac{\pi}{6} \) and \( C_2 = |z - i2\sqrt{3}| = r \) intersect at exactly two points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 2\sqrt{3})}
Calculate the flux of the vector field \( \mathbf{v} = \mathrm{curl}\, \mathbf{u} \) through the surface \( \Sigma \), where \( \Sigma \) is the portion of the paraboloid \( z = 1 - x^2 - y^2 \) with \( x \geq 0 \) and \( z \geq 0 \). The vector field \( \mathbf{u} \) is given by \( \mathbf{u} = (xy, xz^2, x^2y) \). Use Stokes' theorem to compute the flux.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the linearization of the differential equation \( y''(t) + y(t) e^{0.02y} = 0 \) near \( y = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y''(t) + y(t) = 0}
Let \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) be a \( 2 \times 2 \) matrix over a field \( k \) such that \( A^2 = 0 \). Consider the polynomials \( a^2 + bc, \ c(a+d), \ b(a+d), \ d^2 + bc \) in \( a, b, c, d \). Let \( I \) be the ideal generated by these polynomials. Determine the smallest integer \( r \) such that \( (a+d)^r \in I \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Is 6 the minimum number of bishops needed to guarantee that Player A can always checkmate Player B's knight on an infinite chessboard, assuming both players play optimally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find a basis for the field extension $\mathbb{R}(x^2)(x)$ over $\mathbb{R}(x^2)$. Provide your answer as a set of elements that form a basis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, x\}}
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3001}
Given that \(a, b, c, d\) are positive real numbers such that \(a+b+c+d = 2\), determine the range of values that the expression \(\lambda = (1+a+b)(1+c+d)\) can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 4]}
Let $\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\circ}$ around these lines maps $\mathcal{O}$ onto itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Three non-collinear lattice points  $A,B,C$  lie on the plane  $1+3x+5y+7z=0$ . The minimal possible area of triangle  $ABC$  can be expressed as  $\frac{\sqrt{m}}{n}$  where  $m,n$  are positive integers such that there does not exists a prime  $p$  dividing  $n$  with  $p^2$  dividing  $m$ . Compute  $100m+n$ .

*Proposed by Yannick Yao*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8302}
Given $f_{1}(x)=\sin x+\cos x$, let $f_{2}(x)=f'_{1}(x)$, ..., $f_{n+1}(x)=f'_{n}(x)$, ..., determine the value of $f_{1}\left( \frac{\pi}{3}\right)+f_{2}\left( \frac{\pi}{3}\right)+f_{3}\left( \frac{\pi}{3}\right)+\ldots+f_{2017}\left( \frac{\pi}{3}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1+\sqrt{3}}{2}}
The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5,1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt{5}}
Determine the radius of convergence for the power series $\displaystyle\sum_{n = 0} \dfrac{(n+1)!}{n^2} x^{2n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The digits 1, 2, 3, 4 can be arranged to form twenty-four different four-digit numbers. If these twenty-four numbers are then listed from smallest to largest, what is the position of 3142? Express your answer as an ordinal position (e.g., 1st, 2nd, 3rd, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14th}
Find the sum of all the positive solutions of    
\[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1\]    
If the answer is \(x\pi\), what is the value of \(x\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1080}
Given positive integers \( r \) and \( n \), find the smallest positive integer \( m \) such that for any partition of the set \( S = \{1, 2, \ldots, m\} \) into \( r \) pairwise disjoint subsets \( A_1, A_2, \ldots, A_r \), there exist two positive integers \( a \) and \( b \) in the same subset \( A_i \) (\(1 \leq i \leq r\)) such that \( b < a \leq \frac{n+1}{n} b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n + 1)r}
The sales tax on an article was reduced to 3 1/3%. Due to this change, a person who purchases an article with a market price of Rs. 10,800 saves Rs. 18. What was the original sales tax percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5\%}
A boat travels upstream from port $A$ and encounters a drifting raft at a distance of 4 kilometers from port $A$. The boat continues its journey to port $B$ upstream and immediately turns around, arriving at port $A$ simultaneously with the raft. Given that the current speed is 2 meters/second, and the boat's speed downstream is 8 meters/second, what is the distance between ports $A$ and $B$ in kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
First, select $n$ numbers from the set $ \{1, 2, \cdots, 2020\} $. Then, from these $n$ numbers, choose any two numbers $a$ and $b$, such that $a$ does not divide $b$. Find the maximum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
88. Xiaoming and his sister Xiaomei go to two different schools, which are located to the east and west of their home, respectively. Xiaoming leaves home 6 minutes before his sister Xiaomei starts for her school. Two minutes after Xiaomei leaves, their father, who is at home, realizes that both Xiaoming and Xiaomei have forgotten their homework. So, he sets out on a bike to catch up with Xiaoming and Xiaomei to deliver their homework. It is known that the father's speed is three times that of Xiaoming and twice that of Xiaomei. Throughout the process of delivering the homework, Xiaomei and Xiaoming continue moving forward and do not reach their schools. How many minutes at least will the father need from the time he sets out until he returns home? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Petya wants to place 99 coins on a \(2 \times 100\) board so that no two coins are in adjacent cells (sharing an edge) and each cell contains at most one coin. How many ways are there to place the coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{396}
A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that
$$N(t) \leq at^2 +bt+c$$
for all $t\gt 0$ and all initial points $P_0 .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$a = \pi$}
What is the maximum possible number of sides of smaller convex polygons formed by the diagonals of a convex heptagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the integral \( \int _{ -\infty }^{ +\infty }{ { e }^{ { (ix-b) }^{ 2 } } } \, dx \) for real constants \( x \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
The $600$ students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. Find the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group. 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{10}
Determine the function \( f(x) \) represented by the Riemann sum: \[ \lim\limits_{n \to \infty} \sum\limits_{i=1}^{n} \left(\frac{i^3}{n^4}-\frac{i}{n}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 - x}
Evaluate the Cauchy Principal Value of the integral: \[ \int_{-\infty}^\infty \frac{\cos kx}{x-a} \, dx \] where \( a \) is a real number and \( k > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi \sin(ka)}
Given the differential equation $$0 = -gt - \frac{F}{\alpha}\cdot \log\left( 1 - \frac{\alpha t}{M+m} \right),$$ with $M=5~kg$, $m=10~kg$, $\alpha = 1 kg/s$, and $F = 0.9(M +m)g$, find the second solution for $t$ when $\alpha t \ll M+m$, given that one solution is $t = 0$. What is the second, more important solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{3}}
Line l is tangent to circle O at point A. Let D be the foot of the perpendicular from A to chord BC. The perpendiculars from points B and C to line l are drawn with feet at E and F, respectively. Given that CF = 12 and AD = 9, find the length of BE.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{4}}
Determine the values of \(\beta\) for which the inequality \(\left(\sum_{i=1}^{N}{x_i}\right)^{\beta} \geq \sum_{i=1}^{N}{x_i^{\beta}}\) holds for all positive real numbers \(\{x_i\}_{i=1}^{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
Find the sum of the decimal digits of the number
 \( 5\mathop{\sum }\limits_{{k = 1}}^{{99}}k\left( {k + 1}\right) \left( {{k}^{2} + k + 1}\right) . \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Evaluate the integral $$\int_{0}^{+\infty }\frac{x \log(x)}{(x^{2}+1)^{2}}dx$$ and determine whether it converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Natural numbers \( x, y, z \) are such that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \cdot \operatorname{LCM}(\operatorname{GCD}(x, y), z) = 1400 \).

What is the maximum value that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \) can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider the differential equation \((1+e^x)y'+ye^x=1\). Given that \(y(0)=2\), determine the interval(s) where the solution \(y(x)\) has critical points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0)}
In the figure, quarter arc $AD$ of the circle $x^2+y^2=16$ is given. Points $B$ and $C$ are arbitrary on the arc $AD$ such that $C$ is between $D$ and $B$. Determine the maximum possible area of the pentagon $OABCD$, where $O$ is the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that $\phi = \frac{1 + \sqrt{5}}{2}$, determine the value of $n$ defined as follows:

$$
n = \frac{1}{1} + \frac{1}{1 + \phi} + \frac{1}{1 + \phi + \phi^2} + \frac{1}{1 + \phi + \phi^2 + \phi^3} + \dots
$$

What is the value of $\lfloor2n\rfloor + \lceil2n\rceil$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that point $P$ is a common point of circle $O\_1$ and circle $O\_2$, where circle $O\_1$ is defined as $(x-a)^{2}+(y-b)^{2}=b^{2}+1$, and circle $O\_2$ is defined as $(x-c)^{2}+(y-d)^{2}=d^{2}+1$. If $ac=8$, $\frac {a}{b} = \frac {c}{d}$, then the minimum value of the distance between point $P$ and any point $M$ on line $l$: $3x-4y-25=0$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given the function $$f(x)= \begin{cases} \overset{2^{x},x\geq 3,}{f(x+1),x<3}\end{cases}$$, find the value of $f(\log_2 6)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A point A moves uniformly with velocity $v$ so that the vector $v$ is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity $u<v$. Initially, $v⊥u$, and the points are separated by a distance $l$. What is the time at which the points converge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{vl}{v^2-u^2}\)}
9.3. Three cyclists started simultaneously from one point in the same direction on a circular track. The first one completes the entire track in 5 minutes, the second in 7 minutes, and the third in 9 minutes. What is the least amount of time after which all the cyclists will be at the same point on the track again? The speeds of all cyclists are constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157.5}
There are relatively prime positive integers $s$ and $t$ such that
\[
\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}
\]
Find $s + t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25249}
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8$}
Determine the values of \( p > 0 \) for which the sequence of functions \( f_n(x) = \frac{nx}{1+n^2x^p} \) converges uniformly to the pointwise limit \( f(x) = 0 \) on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Two trains of length 100 m and another length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. From the moment they meet, they will be clear of each other in 18.998480121590273 seconds. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{279.97}
Given the function $f(x)=2x+\sin x$, the solution set of the inequality $f(m^{2})+f(2m-3) < 0$ (where $m\in\mathbb{R}$) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3,1)}
A customer makes three deposits towards the purchase of a certain product. The first deposit is 10% of the total product cost, and it amounts to $130. The second deposit is twice the amount of the first deposit and covers an additional 20% of the total product cost. There is also an ongoing promotional discount of 5% on the total product cost, which the customer hasn't availed yet. If the customer decides to pay off the remaining balance including the discount in a final payment which also has an interest rate of 2%, determine how much more the customer needs to pay for the product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$861.90}
When one ounce of water is added to a mixture of acid and water, the new mixture is $20\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\frac13\%$ acid. The percentage of acid in the original mixture is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$25\%$}
Find the area bounded by the inverse of the function $f(x) = x^3 + 3x + 2$, the x-axis, and the vertical lines $x = -2$ and $x = 6$. You do not need to explicitly find the inverse function to solve this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
A man buys three items - a cycle for Rs. 1,400, a helmet for Rs. 400, and two safety lights costing Rs. 200 each. The store offers a 10% discount on the cycle, 5% discount on the helmet, and the safety lights are sold at the listed price. After purchasing the items, the man pays an additional 5% tax on the total cost. After a few months, he decides to sell the cycle at a 12% loss, the helmet at a 25% profit, and the safety lights at a no profit no loss condition. He also buys a new lock for Rs. 300 and includes it in the sale. If a 3% transaction fee is applied to the total selling price and he decides to round the final selling price down to the nearest whole number, what is the final selling price of all the items combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2215}