inclusionAI/Ring-lite-rl-data huggingface dataset:
In the isosceles triangle \(ABC\), points \(K\) and \(L\) are marked on the equal sides \(AB\) and \(BC\) respectively so that \(AK = KL = LB\) and \(KB = AC\). Find the measure of angle \(ABC\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^\circ}
Since the beginning of the school year, Andrey has been recording his math grades. Upon receiving a new grade (2, 3, 4, or 5), he called it unexpected if, up to that point, it had occurred less frequently than each of the other possible grades. (For example, if he received the grades 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 in order, the first 5 and the second 4 would be unexpected.) Over the entire school year, Andrey received 40 grades - exactly 10 fives, 10 fours, 10 threes, and 10 twos (the order is unknown). Can you definitively determine the number of grades that were unexpected for him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
For all possible non-empty sets $A$ and $B$ such that $A \subseteq B$, how many possible subsets $K$ of $B\setminus A$ are there if $|A|+|B|=8$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
A person draws cards one at a time from a standard deck of 52 cards without replacement, continuing until they draw the 3 of Hearts. What is the expected maximum value of the cards drawn? Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.35}
Given that \( g(f(x)) = x \) and \( g'(x) = \frac{1}{1 + x^2} \), find \( f'(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sec^2 x}
Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school, arriving 20 minutes late. By how many times does the speed of Uncle Vanya's car exceed Nikita's running speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Calculate the expected number of days required for a stack of \( n \) books, initially arranged in order, to return to their original order. Each day, one book is picked uniformly at random and placed on top of the stack. Provide your answer as a function of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Let $V$ be an $n$-dimensional vector space over a field $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Suppose $\psi : V \to V$ is a linear transformation and $A$ is the matrix representing $\psi$ with respect to $\mathcal{B}$. Let $\psi^t : V^{*} \to V^{*}$ be the dual transformation and $B$ be the matrix representing $\psi^t$ with respect to $\mathcal{B}^{*}$. Determine the relationship between the matrices $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B = A^\top}
Given a triangle $ABC$, an inscribed circle touches the sides $AB, AC, BC$ at points $C_1, B_1, A_1$ respectively. Find the radius of the excircle $w$, which touches the side $AB$ at point $D$, the extension of side $BC$ at point $E$, and the extension of side $AC$ at point $G$. It is known that $CE = 6$, the radius of the inscribed circle is $1$, and $CB_1 = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the mass percentage of Ca in Calcium hydroxide
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.09\%}
The set $A=\{x|x\in\mathbb{N}, \text{ and } \frac{4}{2-x}\in\mathbb{Z}\}$ can be represented by listing method as $A= \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0, 1, 3, 4, 6\}}
The sum of the positive numbers \(a, b, c, and d\) does not exceed 4. Find the maximum value of the expression

$$
\sqrt[4]{a^{2}+3 a b}+\sqrt[4]{b^{2}+3 b c}+\sqrt[4]{c^{2}+3 c d}+\sqrt[4]{d^{2}+3 d a}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{2}}
Let \( x \) be a real number satisfying \( x^{2} - \sqrt{6} x + 1 = 0 \). Find the numerical value of \( \left| x^{4} - \frac{1}{x^{4}} \right|.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{3}}
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, with $a_1=1$ and $S_7=28$. Let $b_n=\left\lfloor \log a_n \right\rfloor$, where $\left\lfloor x \right\rfloor$ denotes the greatest integer less than or equal to $x$, such as $\left\lfloor 0.9 \right\rfloor = 0$ and $\left\lfloor \log 99 \right\rfloor = 1$. Find the sum of the first 1000 terms of the sequence $\{b_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1893}
Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, how many arrangements can Mereduth make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let $X_1, X_2, \ldots$ be a sequence of independent uniform random variables on $(0,1)$. Define $N$ as the minimum index $n \geq 2$ such that $X_n < X_{n-1}$. Calculate the expected value $E(N)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Consider an m×n rectangular grid of points in the plane, where m and n are integers greater than 1. Some k of these points are colored red such that no three red points form the vertices of a right-angled triangle with two sides parallel to the sides of the grid. Determine the greatest possible value of k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 2}
Find the length of $BP$, in terms of the width $w$ of a sheet of paper, such that the area of triangle $B'PQ$ is minimized when the bottom right corner is folded over crease $\overline{PQ}$, assuming the sheet extends vertically indefinitely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}w}
Determine the number of distinct ways to arrange 9 black and 9 white rooks on a 6 × 6 chessboard such that no white rook can threaten any black rook. A rook threatens another piece if they share the same row or column, with no other pieces between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Determine the smallest closed set containing $C^1[-1, 1]$ in the space $({\scr{B}}[-1, 1],||\cdot||_\infty)$. Provide your answer as a set description.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C[-1, 1]}
Evaluate the integral \( \int_C \operatorname{Log}\left(1-\frac{1}{z} \right)\,dz \) where \( C \) is the circle \( |z|=2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Determine the degree of the splitting field of the polynomial \((x^{15}-1)(x^{12}-1)\) over the finite field \(\mathbb{F}_7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the complete bipartite graph $G = K_{9,15}$ with $V(G) = 24$ vertices and $E(G) = 135$ edges, determine the minimum number of additional edges needed to make $G$ a Hamiltonian graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For a given \( n \times n \) array, the first row is \( x_1, x_2, x_3, \ldots, x_{n-1}, x_n \). Each successive row is obtained by a cyclic shift of \( r \) places from the previous row. Determine the values of \( r \) for which this construction results in a Latin square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gcd(r, n) = 1}
Two squares, \(ABCD\) and \(ABEF\), are positioned such that the angle between their planes is \(120^{\circ}\). Points \(M\) and \(N\) lie on the diagonals \(AC\) and \(BF\) respectively, with \(AM = FN\). Given that \(AB = 1\), determine the maximum possible value of the distance \(MN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Jerry buys a bottle containing 150 pills. Using a standard 12-hour clock, he notices that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what time will the clock show when he takes the last pill from the bottle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of zeros of the function \( f(z) = 2 + z^4 + e^{iz} \) within the set \( \{z : |z| \leq r, \Im z \geq 0\} \) for any \( r > 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a tetrahedron \(ABCD\), it is given that \(AB \perp BC\), \(BC \perp CD\), and \(AB = BC = CD = 2\sqrt{3}\). The volume of the tetrahedron is 6. Calculate the angle between the skew lines \(AD\) and \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Let $S$ be the set of all positive integer divisors of $100,000.$ Find how many numbers are the product of two distinct elements of $S.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117}
With all angles measured in degrees, the product $\prod\limits_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$ , where $m$ and $n$ are integers greater than $1$. Find $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Determine the number of inequivalent cusps of the modular group \(\Gamma_0(4)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For $p > 0$, determine the range of values for $p$ that ensures the convergence of the integral \( \displaystyle\int_0^{\infty} x^pe^{-x^8\sin^2x}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 3)}
Let $G$ be a finite group with two generators $x$ and $y$ such that $|x|=2$, $|y|=2$, and $|xy|=4$. Determine the isomorphism class of the quotient group $G/Z(G)$. Provide your answer as the name of the group or its standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_2 \times C_2}
Find the non-negative integer value(s) of \(a\) for which the following limit holds true:

\[\lim_{{x \to 1}} \left\{ \frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $A_0=(0,0)$. Distinct points $A_1,A_2,\dots$ lie on the $x$-axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$. For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let \( \phi(n) \) denote the number of positive integers less than or equal to \( n \) which are relatively prime to \( n \). Compute \( \displaystyle \sum_{i=1}^{\phi(2023)} \frac{\gcd(i,\phi(2023))}{\phi(2023)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{385}{34}}
On the bases \(AD\) and \(BC\) of trapezoid \(ABCD\), squares \(ADEF\) and \(BCGH\) are constructed outside the trapezoid. The diagonals of the trapezoid intersect at point \(O\). Find the length of segment \(AD\) if \(BC = 2\), \(GO = 7\), and \(GF = 18\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{22}{7}}
Determine the fractal dimension of a structure formed by repeatedly dividing a $2r \times 2r \times 2r$ cube into smaller cubes, removing the cubes along the axes, and continuing this process for every odd number raised to the power of three.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\log 20}{\log 3}}
Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the walls of the maze. After completing the maze, he noticed that it was possible to reach any unit square from any other unit square in exactly one way, excluding paths that pass through any unit square more than once. What is the side length of the initially drawn large square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
From a large grid paper, cut out 5 connected squares (two squares with only one common vertex are not considered connected), such that the resulting shape can be folded into an open-top cube. How many different shapes can be cut out, considering shapes that are the same after rotation or flipping as the same? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a sequence $\{x^k\}$ that converges superlinearly to a point $x^*$, meaning that \( \lim_{k\to\infty}\frac{||x^{k+1} - x^*||}{||x^k - x^*||^p}=r, \) where $0<p<2$ and $r$ is a constant, determine the value of \( \lim_{k\to\infty}\frac{||x^{k+1} - x^k||}{||x^k - x^*||} \) according to the definition of superlinear convergence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of $a$ for which the equation $a^x = x + 2$ has exactly two real solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the digits of $n$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Problem 1. The angle $\alpha$ is greater than its complementary angle by as many times as it is smaller than its supplementary angle. Calculate the angle $\alpha$.

翻译完成，但请注意，原文本中的数学问题已经用英文表述。如果您需要解题帮助，请告知。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The base of the pyramid \( T A B C D \) is the trapezoid \( A B C D \) where \( B C \parallel A D \). The distances from points \( A \) and \( B \) to the plane \( T C D \) are \( r_1 \) and \( r_2 \) respectively. The area of triangle \( T C D \) is \( S \). Find the volume of the pyramid \( T A B C D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{S \left( r_1 + r_2 \right)}{3}}
Li Jiang, a student, buys a piece of sportswear at a sports counter in a mall and receives a 100 yuan voucher. This voucher can be used to purchase one of three different types of sports socks, priced at 18 yuan, 30 yuan, and 39 yuan respectively. It is required that the entire amount of the voucher must be used in one go, and any remaining balance cannot be exchanged for cash. Li Jiang does not want to add any additional cash and wishes to utilize the voucher with an efficiency greater than 95%. Calculate the number of different ways to make a selection that fulfills these conditions. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the values of \( s \) for which the integral \[ \int\limits_{e_k}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln(x) \cdot \ln^2(x) \cdot \ln^3(x) \cdots \ln^k(x)^s} \] converges, where \( \ln^k(x) \) denotes the logarithm applied \( k \) times and \( e_k = e^{e^{e^{\ldots e}}} \) is the \( k \)-th iterated exponential.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s > 1}
Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such that $2^{k_2} \mid N$ and $3^{k_3} \mid N$ respectively. Find $k_2 + k_3.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2030612}
Using the letters $ A$, $ M$, $ O$, $ S$, and $ U$, we can form $ 120$ five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $ USAMO$ occupies position
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
115
}
已知$n$个正整数$x_{1}, x_{2}, \cdots, x_{n}$的和为2016。若这$n$个数既可分为和相等的32个组，又可分为和相等的63个组，求$n$的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
Determine the number of elements of order 4 in the symmetric group $S_6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Find the smallest positive integer solution to the indeterminate equation \(195y = 221x + 65\) using the Indian method of kuttaka.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $U \subset W \subset V$ be three linear spaces with dimensions 3, 6, and 10, respectively. Define $E \subset L(V,V)$ as the set of linear transformations $f: V \rightarrow V$ such that $f(U) \subset U$ and $f(W) \subset W$. Determine the dimension of $E$. \( \dim E \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
How many cyclic subgroups of order $p$ does ${\bf Z}_{p^n}\oplus{\bf Z}_p$ have, where $p$ is a prime number and $n \geq 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 1}
Evaluate the integral: $$\int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the limit \( \lim_{n \to \infty} \int_{[0,n]} \left(\frac{n+x}{n+2x}\right)^n \, d\lambda \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the degree of the field extension $[\mathbb{Q}(\beta + 1):\mathbb{Q}((\beta + 1)^2 - 4)]$, where $\beta$ is a root of the polynomial $q(x) = x^2 + 3x - 3$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the minimum value of the function \( f(x) = \tan^2 x - 4 \tan x - 8 \cot x + 4 \cot^2 x + 5 \) on the interval \( \left( \frac{\pi}{2}, \pi \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 + 8\sqrt{2}}
Find the values of $q$ for which the curve $y = x^p \sin \frac{\pi}{x}$, where $0 < x \leq 1$, has finite length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q > 1}
How many seven-digit numbers are divisible by 11 and have a sum of their digits equal to 59?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
What is the automorphism group of the symmetric group of degree three, $Sym(3)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Evaluate the improper integral: $$\int_0^\infty \frac{\cos{kt}}{\pi}\,\mathrm{d}k.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(t)}
Calculate the difference between the number of acyclic digraphs and the number of acyclic transitive digraphs on 5 labeled vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25050}
Find the slope of the curve formed by the intersection of the surface \( z = x^3 + xy^2 \) and the plane \( 2x - 2y = 1 \) at the point \( (1, \frac{1}{2}) \). Use the parametrization \( x = 1 + t \) and \( y = \frac{1}{2} + t \), and calculate the derivative with respect to \( t \) at \( t = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{4}}
Given the generating function \( P(x) = \sum_{n=0}^{\infty} p_n x^n = 1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + 11x^6 + \cdots \), which represents the partition generating function, and \( Q(x) = \sum_{n=0}^{\infty} q_n x^n \), where \( q_n \) is the number of partitions of \( n \) containing no 1s, find the polynomial \( \frac{Q(x)}{P(x)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - x}
Circle $\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA = 13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB + AC$ given that $BC = 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Determine the smallest number of digits $N$, consisting of ones and twos, arranged in a circle such that every possible four-digit number composed only of the digits 1 and 2 can be represented as a sequence of consecutive digits. What is this smallest value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the limit: $$\lim_{x \to 0^-} \frac{5x - \sqrt{x}}{\sqrt{e^{2x} - 1}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\(\frac{1}{\sqrt{2}}\)}
Evaluate the sum: $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k+1)^2 \Gamma (n+1) }{\Gamma (2 k+n+2)}((\psi ^{(0)}(n+1)-\psi ^{(0)}(2 k+n+2)) (\psi ^{(0)}(k+1)-\psi ^{(0)}(2 k+n+2))-\psi ^{(1)}(2 k+n+2)).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Sarah is driving to the airport. After driving at 15 miles per hour for one hour, she realizes that if she continues at that same average rate she will be an hour late for her flight. She then travels at a faster speed for the rest of the trip, and arrives 30 minutes before her flight departs. She drove 45 miles in all. What was her speed for the rest of the trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is the smallest possible value of $b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
3 Let $n$ be a given positive integer, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, let $m$ be the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$. Find the maximum value of $m$ under the condition $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{12}{(n-1) n(n+1)}}}