inclusionAI/Ring-lite-rl-data huggingface dataset:
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number  $k$  such that there may be  $k$  equal numbers between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the general method to find the number of lines that are both tangent and normal to a given curve, and how many such lines exist for the curve $y^2=x^4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a function $f$ for which \[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\] holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\ldots,f(999)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{177}
有1000盏灯和1000个开关，每个开关同时控制着所有号码是它的倍数的灯，开始所有灯都是亮着的，现在拉2,3,5号三个开关，那么亮着的灯有＿个．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499}
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
An editor is working on a new translation of a classic novel. The novel has a total of 480 pages. Each day, the editor translates an average of 12 pages. However, she also spends time reviewing her translations, which takes an additional 3 pages worth of time each day. If she works 5 days a week, how many weeks will it take her to complete the translation and review of the entire novel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In summation notation, what is the expected value of the squared difference between the sample mean and each individual observation, assuming the observations are iid normally distributed with $N(\mu , \sigma ^2)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)\sigma^2}
In triangle \(ABC\), medians \(AL\) and \(BM\) intersect at point \(K\). Vertex \(C\) lies on the circle that passes through points \(K\), \(L\), and \(M\). The length of side \(AB\) is \(a\). Find the length of median \(CN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a \sqrt{3}}{2}}
Evaluate $$\int \ln(2x+3)\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(x+\frac{3}{2}\right)\ln(2x+3) - x + C}
计算下列实变函数定积分．

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } + 1 } { x ^ { 6 } + 1 } d x ,$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ $\pi/2$}
The current value of a piece of machinery is $40,000. Its value depreciates at a rate of 5% per annum. Additionally, there are annual maintenance costs of $2000 that also increase by 3% every year due to inflation. Calculate the total net worth of the machine after 2 years considering both depreciation and maintenance costs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$32,040}
Points \( M \) and \( K \) are marked on the sides of angle \( ABC \) such that angles \( \angle BMC \) and \( \angle BKA \) are equal, \( BM = BK \), \( AB = 15 \), \( BK = 8 \), and \( CM = 9 \).

Find the perimeter of triangle \( COK \), where \( O \) is the intersection point of lines \( AK \) and \( CM \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Five line segments are drawn inside a rectangle, creating a decagon (10-sided polygon). Find the sum of the six marked angles. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1080^\circ}
A hotel has one triple room, one double room, and one single room available. Three adults and two children come to stay, and the children cannot stay in a room by themselves (they must be accompanied by an adult). If all three rooms are occupied, there are \_\_\_\_\_\_\_\_\_ different arrangements for their accommodation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数就是第二个函数，而第三个函数的图像与第二个函数的图像关于直线 $x+y=0$ 对称。求第三个函数的表达式。原始答案的形式为 $y = k\varphi(mx)$，请给出 $k + m$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
On a table, there are three cones standing on their bases, touching each other. The heights of the cones are the same, and the radii of their bases are \(2r\), \(3r\), and \(10r\). A ball with a radius of 2 is placed on the table, touching all three cones. It turns out that the center of the ball is equidistant from all points of contact with the cones. Find \(r\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: 
(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$; 
(b) $f\left(t_0\right)=1 / 2$; 
(c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$; 
(d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$.
Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given \( k \geq 1 \). Find the maximum real number \( \lambda \) such that for any positive real numbers \( x_i > 0 \) \((i=1,2,\ldots,n)\) satisfying \( \sum_{i=1}^{n} \frac{1}{x_{i}} = 2016 \), the following inequality holds:
$$
\lambda \sum_{i=1}^{n} \frac{x_{i}}{1+x_{i}} \leq \left[\sum_{i=1}^{n} \frac{1}{x_{i}^{k}\left(1+x_{i}\right)}\right]\left(\sum_{i=1}^{n} x_{i}^{k}\right).
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
The letters L, K, R, F, O and the digits 1, 7, 8, 9 are "cycled" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In a hockey championship, teams receive a certain number of points when they win and 1 point when they tie. In one season, a team won a championship with 60 points. They won 12 games more than they tied. How many points does a team receive when they win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the minimum number of acute-angled triangles that must be arranged without gaps and overlaps to form an isosceles triangle with a $120^\circ$ vertex angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. 

Find the smallest possible number of independent deputies after the vacation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the minimum of the following function defined in the interval $45^{\circ}<x<90^{\circ}$:

$$
y=\tan x+\frac{\tan x}{\sin \left(2 x-90^{\circ}\right)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
What is the volume of a regular quadrangular pyramid, given its base area Q and lateral area S?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{6} \sqrt{Q(S^{2}-Q^{2})}\)}
I was given the following exercise. Let $X$ be a random variable that takes non-negative natural number values such that $E[X]=1,E[X^2]=2,E[X^3]=5$. Find the minimum value of $P(X=0)$ using the Taylor expansion of the probability generating function at $z=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
How many distinct trees with exactly 7 vertices are there? 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. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Compute
\[\tan 5^\circ + \tan 25^\circ + \tan 45^\circ + \dots + \tan 165^\circ.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that the function y=f(x) is an even function, when x>0, f(x)=ln x. If M=f(-π), N=f(e), K=f(2), what is the relationship among M, N, and K? Express your answer as an inequality using the symbols >, <, and =.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M > N > K}
11 将集合$\{ 2 ^ { x } + 2 ^ { y } + 2 ^ { z } \vert x , y , z \in N ,$ $x$&lt;$y$&lt;$z$}中的数从小到大排列，第100个是（用数字作答）．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{577}
Several students are competing in a series of three races. A student earns 5 points for winning a race, 3 points for finishing second, and 1 point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
How many ways can a schedule of 4 mathematics courses - algebra, geometry, number theory, and calculus - be created in an 8-period day if exactly one pair of these courses can be taken in consecutive periods, and the other courses must not be consecutive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
For every integer n ≥ 3, the function g(n) is defined as the product of all the odd integers from 1 to n, inclusive. The difference between g(106) and g(some number) is 105. What is that some number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
There are two values of \(a\) for which the equation \(4x^2 + ax + 8x + 9 = 0\) has only one solution for \(x\). The discriminant of the quadratic equation can be expressed in the form \(m + n = 0\). Please find the sum of these values of \(m\) and \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-16}
Given that \( m \) and \( n \) are two distinct positive integers and the last four digits of \( 2019^{m} \) and \( 2019^{n} \) are the same, find the minimum value of \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{502}
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x \rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1973}
Let $f(x)=-4^{x}+2^{x+1}-1$, $g(x)=\log_{10} (ax^{2}-4x+1)$. If for any $x_{1}\in \mathbb{R}$, there exists $x_{2}\in \mathbb{R}$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 4]}
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3+\sqrt{5}}{2}}
On a $10 \times 10$ chessboard, several knights are placed. In any $2 \times 2$ square, there is at least one knight. What is the smallest number of cells these knights can threaten? (Note: A knight does not threaten the square on which it stands, but it does threaten the squares on which other knights are standing.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Amongst the seven numbers \( 3624, 36024, 360924, 3609924, 36099924, 360999924, \) and \( 3609999924 \), there are \( n \) of them that are divisible by 38. Find the value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n \ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6423}
Let $a=\sqrt{17}$ and $b=i \sqrt{19}$, where $i=\sqrt{-1}$. Find the maximum possible value of the ratio $|a-z| /|b-z|$ over all complex numbers $z$ of magnitude 1 (i.e. over the unit circle $|z|=1$ ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
7. A digital clock displays hours, minutes, and seconds using two digits each (e.g., 10:09:18). Between 05:00:00 and 22:59:59 on the same day, the probability that all six digits on the clock are different is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{135}}
Determine the value of $s$ for which the function $v(x, y) = x^4 − sxy^3 + 10xy − 10x$ satisfies Laplace's Equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the sum of two prime numbers is $102$ and one of the prime numbers is greater than $30$, calculate the product of these two prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2201}
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
将函数 $y=\sqrt{4+6 x-x^{2}}-2(x \in[0, 6])$ 的图象绕坐标原点逆时针方向旋转角 $\theta(0 \leqslant \theta \leqslant \alpha)$，得到曲线 $C$ 。若对于每一个旋转角 $\theta$ ，曲线 $C$ 都是一个函数的图象，则 $\alpha$ 的最大值为多少？答案的形式为$\arctan \frac{a}{b}$，请给出$a+b$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 150 people in the family. What is the smallest fraction of the original plot of land that any member of the family could have received?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2\cdot3^{49}}}
设a，b，c是互不相等的实数，则$( a - b ) ^ { 2 }$.$( b - c ) ^ { 2 }$. $( c - a ) ^ { 2 }$ 中不超过$\frac { 1 } { 2 } ( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } )$的最少有几个？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. What is the minimum distance the ray could travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10 \sqrt{3}}{9}}
In the figure, there are two identical regular hexagons inside a parallelogram. Calculate the fraction of the area of the parallelogram that the shaded portion occupies. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3),$ and the other two vertices have integer coordinates. How many rectangles satisfy these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Jeff makes 100 donuts a day for 365 days in a year. Jeff indulges in his creations, eating 4% of donuts each day. His friends Chris, Sam, and Emily drop by every month, consuming 15, 20, and 25 donuts respectively each time. He also donates 10% of his monthly donuts to a local charity and gives his neighbor 6% of his donuts at the end of the year. If 50 donuts can fit into one large box, how many such boxes can Jeff fill with his remaining donuts at the end of the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{570}
A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have any other vertices. How many triangles result from this triangulation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1998}
将前 12 个正整数构成的集合 $M=\{1,2, \cdots, 12\}$ 中的元素分成 $M_{1}, M_{2}, M_{3}, M_{4}$ 四个三元子集, 使得每个三元子集中的三数都满足: 其中一数等于另外两数之和, 试求不同的分法的种数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8$}
Given the number 2550, calculate the sum of its prime factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{11}}
We roll a fair die consecutively until the sum of the numbers obtained, \( S \), exceeds 100. What is the most probable value of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$342$}
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A certain amount of money is divided amongst a, b, and c so that a may get 2/3 as much as b and c together. B gets a certain fraction as much as a and c together. The share of a is $80, and the total amount of money is $200. What is the ratio of b's share to the combined share of a and c?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3}
Given the vector 
$$
\boldsymbol{a}_{k}=\left(\cos \frac{k \pi}{6}, \sin \frac{k \pi}{6}+\cos \frac{k \pi}{6}\right),
$$
where $k=0,1,\cdots,12$, find the value of $\sum_{k=0}^{11} \boldsymbol{a}_{k} \cdot \boldsymbol{a}_{k+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \sqrt{3}}
Given a mixture of AlCl3, Al2(SO4)3, and Al(OH)3, determine the mass percentage of Al in the entire mixture if the mass of each compound is 50 g, 70 g, and 40 g, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.87\%}
Given the function $y=f(x)$, let $A(x\_1,y\_1)$ and $B(x\_2,y\_2)$ be two distinct points on its graph with corresponding tangent line slopes $k\_A$ and $k\_B$. Define $K(A,B)=\frac{|{K}\_{A}-{K}\_{B}|}{|AB|}$ (where $|AB|$ is the length of the line segment $AB$) as the "approximate curvature" of the curve $y=f(x)$ between points $A$ and $B$. For the curve $y=\frac{1}{x}$, let $A\left(a,\frac{1}{a}\right)$ and $B\left(\frac{1}{a},a\right)$ with $a>0$ and $a\neq 1$. If $m\cdot K(A,B)>1$ holds true, find the range of values for the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{\sqrt{2}}{2},+\infty)}
Determine the least real number $k$ such that the inequality
\[
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \geq 4\left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
\]
holds for all real numbers $a, b, c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a function $y = f(x)$ defined on $\mathbb{R}$ that is an even function satisfying $f(2 + x) = f(2 - x)$, with at least one zero in the interval $(0, 4)$ and $f(0) = 0$, determine the minimum number of zeros the function $y = f(x)$ has in the interval $(-8, 10]$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference.  Find their sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
A car starts from point \( O \) and moves along a straight road with a constant speed \( v \). At the same time, a person on a bicycle, located at a distance \( a \) from point \( O \) and \( b \) from the road, begins to ride with the intention of delivering a letter to the car driver. What is the minimum speed the cyclist must maintain to achieve this goal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{v b}{a}}
In the following right-angled triangle ABC, ∠C = BC = 1 and DEF is an arc of a circle with center A. Suppose the shaded areas BDE and CEF are equal and AD = x/√π. Calculate the value of x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
According to the Gregorian calendar, a total of 97 leap days must be inserted over 400 consecutive years. After how many years will there be a 1-day discrepancy between the Gregorian calendar and the "exact" calendar if a year has a length of 365 days, 5 hours, 48 minutes, and 46 seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3323}
A person crosses a 2,500 m long street in 8 minutes. What is their speed in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.65}
从 $[0,10]$ 中任取一个数 $x$, 从 $[0,6]$ 中任取一个数 $y$, 计算使得
$$
|x-5|+|y-3| \leqslant 4
$$
的概率。原始答案的形式为\frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}