inclusionAI/Ring-lite-rl-data huggingface dataset:
How many diagonals in a regular 32-sided polygon are not parallel to any of its sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
The angle at the apex of the axial section of a cone is $60^{\circ}$. Inside the cone, there are three spheres with a radius of 1. Each sphere is tangent to the other two spheres, the base of the cone, and its lateral surface. Find the radius of the base of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{\sqrt{3}}}
4. (10 points) A sequence of numbers: $8,3,1,4, . \cdots$, starting from the third number, each number is the unit digit of the sum of the two closest preceding numbers. What is the 2011th number? $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a, b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{648}
$a,b,c,x,y,z \in R$ 且满足$( x + y ) c - ( a + b ) z = \sqrt { 6 }$，求下式的最小值.

$$
a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + a x + b y + c z
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Three students write on the blackboard three two-digit square numbers next to each other. In the end, they observe that the resulting 6-digit number is also a perfect square. Find this number!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{166464}
Through two vertices \( A \) and \( B \) of \( \triangle ABC \), perpendicular lines \( AD \) and \( BE \) are drawn on the same side of the plane \( ABC \), forming an equilateral triangle \( CDE \). Given \( AD = 1 \), \( BE = 2 \), and \( DE = 3 \), determine the dihedral angle between planes \( CDE \) and \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arcsin \frac{2}{3}}
Find the number of elements of order 12 in the automorphism group of the cyclic group \( \mathbb{Z}_{1080} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Calculate the total mass of a thin plate with the shape defined by the region bounded by the line $y = x - 1$ and the parabola $y^{2} = 2x + 6$, given that the density of the plate is $ho(x,y) = xy$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
17 Let $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n}$ be real numbers, and $\sum_{i=1}^{n} x_{i}^{2}+\sum_{i=1}^{n-1} x_{i} x_{i+1}=1$, for each given positive integer $k, 1 \leqslant k \leqslant n$, find the maximum value of $\left|x_{k}\right|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{2 k (n+1-k)}{n+1}}}
Find the area of the region intersected by two circles on the xy-plane given by the equations \(x^2 + y^2 = 1\) and \(x^2 + y^2 - 2x = 0\), using a double integral in polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2\pi}{3} - \frac{\sqrt{3}}{2}\)}
For reals  $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$ , let  $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$  for each  $k$ . If  $S_2 = 777$ , compute the least possible value of  $S_3$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
How many terms of the series $\sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!}$ are required to approximate $\int_{0}^{1}e^{-x^2}$ with an error less than $10^{-6}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
若 \triangle ABC 的内角 A,B,C 满足 \sin A=\cos B=\tan C，求 \cos^3A+\cos^2A-\cos A 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $\triangle ABC$ be a right triangle with hypotenuse $AC$. A square is inscribed in the triangle such that points $D,E$ are on $AC$, $F$ is on $BC$, and $G$ is on $AB$. Given that $AG=2$ and $CF=5$, what is the area of $\triangle BFG$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{500}{841}}
Solve the equation $2^x\log(x) + \log(2) = 0$ for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.641}
He wanted to make sure that he is protected from the cold evenings in the forest, so he decided to build a fireplace made of different types of cement. He bought 215 lbs of type-A cement, his son brought 137 lbs of type-B cement, and his neighbor contributed 150 lbs of type-C cement. They mixed these three types of cement together to create type-D cement. During the drying process, type-A cement lost 3% of its weight, type-B cement lost 2% of its weight, and type-C cement lost 5% of its weight. After all the weight losses, they had a certain amount of type-D cement. Find out what percentage of the total original cement they now have in the form of type-D cement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96.67\%}
Suppose $ABCDEF$ is a regular hexagon with area $ 1$, and consider the diagonals $AC$, $BD$, $CE$, $DF$, $EA$ and $FB$. A star is formed by alternating between the vertices of the hexagon and the points of intersection of the diagonals. What is the area of the star?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac23}
When a student multiplied the number $66$ by the repeating decimal,  
$$\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$$  
where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. Find the $2$-digit number $\underline{a} \ \underline{b}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Let \( f(x) \) be a polynomial with integer coefficients. Suppose there are four distinct integers \( p, q, r, s \) such that \( f(p) = f(q) = f(r) = f(s) = 5 \). If \( t \) is an integer and \( f(t) > 5 \), what is the smallest possible value of \( f(t) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance of getting them right due to my prior knowledge. What is the minimum number of words I need to learn in order to guarantee at least a 90% score on this test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{534}
Find the closed form for the sum \( \sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k} \), given that \( k = 0, 1, \ldots, n \) and \( n \) is any natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0^{n - k}}
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the area of the region defined by the set \( \{ (x, y) : |x + yt + t^2 | \leq 1 \text{ for all } t \in [0, 1] \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{23}{6}}
Given the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the triple integral \( \int \int \int _{E} \frac{1}{x^2+y^2+z^2 }dV \), where the region \( E \) is defined as \( E=\{(x,y,z) \in \mathbb{R}^3 : z\geq \sqrt{\frac{x^2+y^2}{3}}, 2z\leq x^2+y^2+z^2 \leq 9\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Find a function \( g: J \to \mathbb{R} \) (where \( J \) is an interval) that is Hölder continuous for some \( \alpha > 0 \), but not for any \( \alpha > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = \sqrt{x}}
Find the degree of the field extension \( \mathbb{Q}(i, \sqrt[4]{3}, \sqrt[6]{3}) \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find an imprimitive permutation group acting on a set of 12 elements such that the group has the maximum possible order. Provide the order of this group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1036800}
The "Goldbach Conjecture" states that any even number greater than 2 can be written as the sum of two prime numbers. If we split 10 into the sum of two positive integers, calculate the probability that both addends are prime numbers under the condition that both addends are greater than 2. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{5}}
Determine the smallest size of a rectangular board on which a knight's tour, either open or closed, is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \times 4}
Embed a cube into a dodecahedron in such a way that all eight vertices of the cube are vertices of the dodecahedron. How many solutions does this problem have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If $S$ is a set which consists of $12$ elements, what is the maximum number of pairs $(a,b)$ such that $a, b\in S$ and $\frac{b}{a}$ is a prime number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
20
}
Let \(I\) be the set of points \((x, y)\) in the Cartesian plane such that
\[ x > \left( \frac{y^4}{9} + 2015 \right)^{1/4} \]

Let \(f(r)\) denote the area of the intersection of \(I\) and the disk \(x^2 + y^2 \leq r^2\) of radius \(r > 0\) centered at the origin \((0,0)\). Determine the minimum possible real number \(L\) such that \(f(r) < L r^2\) for all \(r > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
What is the number that, when increased by 20%, equals twice the result of 20 less than the number decreased by 20%? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Determine the value of $n$ for which the subnormal length of the curve $xy^n = a^{n + 1}$ is constant at any point on the curve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the Frobenius number for the arithmetic sequence of integers 100, 105, 110, ..., 200.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Evaluate the limit: $$\lim_{x \to 0} \ln\left(\frac{\cos x}{1+\sin x}\right)^{\frac{1}{x}}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In a group of freshmen, any group of 3 people includes at least 2 who know each other, and any group of 4 people includes at least 2 who do not know each other. What is the maximum number of freshmen in the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$ , where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m+n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
There are  $2024$  cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest  $k$  for which the tourist will always be able to visit at least  $k$  cities? During the route, he can return to the cities he has already visited.

*Proposed by Bogdan Rublov*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Determine the number of connected components of the set \(T=\{(a,b,c)\in\mathbb{R}^3: a^2b^2c^2=4\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知圆锥面 x^2+y^2=\df{z^2}{3}，记沿该圆锥面从 P(-\sqrt{3},3,6) 到 Q(\sqrt{3},0,3) 的曲线长度的最小值为 I，则 [10I]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Evaluate the definite integral \( \int_0^\pi \sin(\sin(x))\sin(x)\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi J_1(1)}
Given complex numbers \( z_{1}, z_{2}, \ldots, z_{n} \) such that \( |z_{1}| = |z_{2}| = \cdots = |z_{n}| = r \neq 0 \), and \( T_{s} \) is the sum of all possible products of \( s \) out of these \( n \) numbers, find \( \left|\frac{T_{s}}{T_{n-s}}\right| \) in terms of \( r \), assuming \( T_{n-s} \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^{2s - n}}
Given that \( AE \) and \( BD \) are straight lines that intersect at \( C \), \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the integral \( \int_C \omega \) where \( \omega = \frac{-y \, dx + (x-1) \, dy}{(x-1)^2 + y^2} \) and \( C \) is the cardioid given by \( r = 1 + \cos \varphi \), positively oriented.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
When Xiaoming was doing math problems, he found that $\sqrt{1-\frac{1}{2}}=\sqrt{\frac{1}{2}}$; $\sqrt{2-\frac{2}{5}}=2×\sqrt{\frac{2}{5}}$; $\sqrt{3-\frac{3}{{10}}}=3×\sqrt{\frac{3}{{10}}}$; $\sqrt{4-\frac{4}{{17}}}=4×\sqrt{\frac{4}{{17}}}$; $\ldots$; following this pattern, if $\sqrt{a-\frac{8}{b}}=a•\sqrt{\frac{8}{b}}(a$,$b$ are positive integers), then $a+b=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
For positive real numbers $p$ and $q$, define the remainder when dividing $p$ by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainders when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_1 + r_2 = \sqrt{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the smallest three-digit number with the following property: if you append a number that is 1 greater to its right, the resulting six-digit number will be a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
A checker moves from the bottom-left corner of a $100 \times 100$ board to the top-right corner, with each step moving one square to the right or one square up. Let $a$ be the number of paths where exactly 70 steps are taken below the diagonal that runs from the bottom-left to the top-right, and $b$ be the number of paths where exactly 110 such steps are taken. Determine which is greater: $a$ or $b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b}
Calculate the sum of the alternating series \(\sum_{n=0}^\infty (-1)^n\binom{1/2}{n}\frac{x^{4n+1}}{4n+1}\) such that the error is less than or equal to \(\frac{0.4^9}{72}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.398976}
Find an example of an $S$-composite number from the set $S = \{1, 5, 9, \ldots\}$, which can be expressed as a product of $S$-primes in more than one way. Here, $S$-primes are numbers in the set with only $1$ and themselves as factors, and $S$-composites are the rest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{441}
Determine the values of $n$ for which the expression \( \frac{\sum_{k=0}^{n}k^{2}}{n} \) is a perfect square, excluding cases where $n$ is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular.
If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{\sqrt{29}}{2}$}
In a row of 30 people, each person is either a knight who always tells the truth or a liar who always lies. They are numbered from 1 to 30. Each person with an odd number claims: "All people with numbers greater than mine are liars," and each person with an even number claims: "All people with numbers less than mine are liars." Determine the possible number of liars. If there are multiple solutions, list them in ascending order separated by semicolons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Suppose all solutions \( u(x,t) \) to the partial differential equation \( au_x + bu_t = 0 \) satisfy the condition \( u(1,2) = u(3,6) \). Determine the value of \( \frac{b}{a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Solve the equation:
\[ (x + 1)^{63} + (x + 1)^{62}(x−1) + (x + 1)^{61}(x−1)^{2} + \ldots + (x−1)^{63} = 0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the flux of the vector field \( \mathbf{F}(x, y, z) = 6x\mathbf{i} + 6y\mathbf{j} + 2\mathbf{k} \) across the surface \( G \), where \( G \) is the surface cut from the bottom of the paraboloid \( z = x^2 + y^2 \) by the plane \( z = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\pi}
Find the non-constant polynomial $f(x)$ such that $f(g(x)) = g(f(x))$, where $g(x) = 3x^4 - 1$. Express $f(x)$ in its simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3x^4 - 1}
Let $A$ and $B$ be subsets of a set $M$ and let $S_0=\{A,B\}$. For $i\geq 0$, define $S_{i+1}$ inductively to be the collection of subsets $X$ of $M$ that are of the form $C\cup D, C\cap D,$ or $M-C$ (the complement of $C$ in $M$), where $C,D \in S_i$. Let $S=\cup^\infty_{i=0} S_i$. What is the largest possible number of elements of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the limit: \[ \lim_{n\to\infty} \prod_{k=1}^\infty \left(\frac{p_n}{p_k}\right)^{\frac{p_n}{n(p_n+p_k)}} \] where \( p_n \) is the \( n \)th prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the maximum number of triangles that can be formed using up to 25 nodes and up to 40 edges, allowing for overlapping triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Evaluate the integral \( \left| \oint_{|z| = R} \frac{z^2}{\log z}\,dz \right| \) for \( R > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the values of \( k \) for which the equation \( x^3 - 3x^2 + 6x + k = 0 \) has three real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Let $X$ be an exponentially distributed random variable and $V$ be a uniformly distributed random variable on $\{-1,+1\}$, both independent of each other. Define $Y = XV$. Calculate the conditional probability $P(|Y| > b \mid Y < 0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-b}}
Aubree saw 20 beavers and 40 chipmunks by a tree when going to school. While coming back from school, she realized the number of beavers had changed and the number of chipmunks had decreased by 10. She saw a total of 130 animals that day. What happened to the number of beavers when she came back from school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the number of odd numbers in the $n$-th row of Pascal's triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
Define a function $g$ from the positive integers to the positive integers with the following properties:
(i) $g$ is increasing.
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Compute all possible values of $g(88).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7744}
Point $P$ is located in the second quadrant, with a distance of $2$ to the $x$-axis and a distance of $3$ to the $y$-axis. Determine the coordinates of point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3, 2)}
What is the range of values for \( m \) that makes the equation \( m\left(x^{2}+y^{2}+2 y+1\right)=(x-2 y+3)^{2} \) represent an ellipse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5,+\infty)}
The diagonals of a quadrilateral are both equal to \(a\), and the sum of its midsegments (the segments connecting the midpoints of opposite sides) is \(b\). Calculate the area of the quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b^2 - a^2}{2}}
Question 3 Suppose the number $a$ has the property: for any four real numbers $x_{1}, x_{2}, x_{3}, x_{4}$, there always exist integers $k_{1}, k_{2}, k_{3},$ $k_{4}$, such that
$$
\sum_{1 \leqslant i<j \leqslant 4}\left(\left(x_{j}-k_{i}\right)-\left(x_{i}-k_{j}\right)\right)^{2} \leqslant a .
$$

Find the minimum value of such an $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
某社区组织了一次公益讲座.向社区居民普及垃圾分类知识.为了解讲座效果，随机抽取10位社区居民，让他们在讲座前和讲座后分别回答一份垃圾分类知识问卷.这10位社区居民的讲座前和讲座后答卷的正确率如下表：

| 编号/准确率 | 1号  | 2号  | 3号  | 4号  | 5号  | 6号  | 7号  | 8号  | 9号  | 10号 |
|-------------|------|------|------|------|------|------|------|------|------|------|
| 讲座前      | 65%  | 60%  | 70%  | 100% | 65%  | 75%  | 90%  | 85%  | 80%  | 60%  |
| 讲座后      | 90%  | 85%  | 80%  | 95%  | 85%  | 85%  | 95%  | 100% | 85%  | 90%  |

$从正确率不低于90\%的垃圾分类知识答卷中随机抽取3份，记随机变量x为抽中讲座前答卷的份数，求随机变量x的数学期望；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{6}{7}$}