inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the smallest integer \( n \) such that for a random sample \( X_1, \ldots, X_n \) from a normal distribution with mean \( \mu = 0 \) and variance \( \sigma^2 = 2 \), the probability \( P(\min(X_1^2, \ldots, X_n^2) \leq 0.002) \geq 0.8 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the harmonic conjugate of the function \( u(z) = \ln(|z|) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arg(z)}
On a $2020 \times 2020$ chessboard, what is the maximum number of warriors you can place such that no two warriors attack each other? A warrior can move either 3 steps forward and 1 step sideways in any direction, or 2 steps forward and 2 steps sideways in any direction. Determine the maximum number of warriors that can be placed on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1020100}
What is a sufficient condition on the matrix $A$ using the $\ell^2$-norm that guarantees convergence of the sequence $\{x_k\}$ to $0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|A\|_2 < 1}
Determine the residue of the function \( g(z) = \frac{\tan(z)\tanh(z)}{z^5} \) at \( z = 0 \) without using Cauchy's residue formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_{0}^{2\pi} \frac{\sin^4(4x)}{\sin^2(x)} \, dx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins. 

At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."

At the end of the first round, Lele says: "The number of your gold coins is 6 times the number of my gold coins now."

At the end of the second round, Huanhuan says: "The number of my gold coins is 5 times the number of your gold coins now."

Find the minimum number of gold coins Huanhuan had at the beginning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
14.Strike out a hundred digits from the number $12345678910111213 \ldots 5960 \quad$ so that the resulting number is the largest possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99999785960}
Evaluate the double infinite summation: \[ \sum_{i=0}^\infty \sum_{j=0}^\infty (3 + i + j) \cdot \left( \frac{1}{3} \right)^{(1+i)} \cdot \left( \frac{2}{3} \right)^{(1+j)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{2}}
A $4 \times 4$ square piece of paper is cut into two identical pieces along its diagonal. The resulting triangular pieces of paper are each cut into two identical pieces. Each of the four resulting pieces is cut into two identical pieces. Each of the eight new resulting pieces is finally cut into two identical pieces. What is the length of the longest edge of one of these final sixteen pieces of paper?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine all real numbers $a$ for which the system of equations
$$3x^2 + 2y^2 + 2z^2 = a$$
$$4x^2 + 4y^2 + 5z^2 = 1 - a$$
has at least one solution $(x, y, z)$ in the real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[2/7, 3/7]$

}
Let \( O \) be the circumcenter of a triangle \( {ABC} \) , \( P \) be the midpoint of \( {AO} \) , and \( Q \) be the midpoint of \( {BC} \) . If \( \angle {ABC} = 4\angle {OPQ} \)
 and \( \angle {ACB} = 6\angle {OPQ} \) , compute \( \angle {OPQ} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$12^\circ$

}
Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. The treasure was buried in a single cell of an $M \times N$ ($2 \le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure. 

For each detector, you can set it up to scan a specific subgrid $[a,b] \times [c,d]$ with $1 \le a \le b \le 2$ and $1 \le c \le d \le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. 

You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. What is the minimum $Q$ required to guarantee to determine the location of the Blackboard’s legendary treasure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Identify the only symmetric group that possesses a nontrivial outer automorphism group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_6}
A bear is preparing to hibernate for the winter and needs to gain 1200 pounds. At the end of summer, the bear feasts on berries, small woodland animals, and insects. During autumn, it devours acorns, salmon, and honey. It gained a fifth of the weight it needed from berries during summer and one-tenth of the weight it needed from insects. During autumn, it gained twice the amount it gained from berries from acorns, and three times the amount it gained from insects from honey. Salmon made up a quarter of the remaining weight it had needed to gain. How many pounds did it gain eating small animals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For how many positive integers  $n$ ,  $1\leq n\leq 2008$ , can the set \[\{1,2,3,\ldots,4n\}\] be divided into  $n$  disjoint  $4$ -element subsets such that every one of the  $n$  subsets contains the element which is the arithmetic mean of all the elements in that subset?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004}
Find the number of ways to choose two nonempty subsets $X$ and $Y$ of $\{1, 2, \ldots , 2001\}$, such that
$|Y| = 1001$ and the smallest element of $Y$ is equal to the largest element of $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2000}}
What is the difference between the local values of a certain digit in the number 58408 if the difference is 7992?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7992}
Let $X=\ell_1$ and $Y=\ell_2$. Define an operator $B$ as follows:

$$B(x_1,x_2,\ldots)=(x_2,x_3-x_2,x_4-x_3,\ldots)$$

Determine whether $B$ is a bounded linear operator from $X$ to $Y$ and find $\|B\|$ if it is bounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle relate to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}:2\sqrt{2}:3}
求最大的正整数 n，使得存在 8 个整数 x_1, x_2, x_3, x_4 和 y_1, y_2, y_3, y_4，满足：\{0,1,\cdots,n\}\subseteq\{|x_i-x_j|\ |\ 1\leq i<j\leq 4\}\cup\{|y_i-y_j|\ |\ 1\leq i<j\leq 4\}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a Poisson process $\lbrace N(t)\rbrace_{t\geq 0}$ with intensity $\lambda = 3$, calculate the conditional probability:

$$P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{125}}
George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among the other three. If a player has two hot potatoes on his turn, he only passes one. If George goes first, and play proceeds clockwise, what is the probability that Travis has a hot potato after each player takes one turn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{27}}
Given the bivariate probability density function for two independent random variables $X$ and $Y$: \( f(x,y) = 12x^2y^3 \) for \( 0 < x < 1 \) and \( 0 < y < 1 \), and \( f(x,y) = 0 \) otherwise, find the probability that the sample mean of a random sample of size \( n = 50 \) for the random variable $X$ is greater than 0.7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9661}
Syam invested Rs. 1800 in 9% stock A and Rs. 2300 in 7% stock B. His overall earnings from stock A and stock B were Rs. 120 and Rs. 160 respectively. If he later decided to invest Rs. 2500 in 6% stock C but lost track of his overall earnings, find out the rate at which the combined investment in stocks A, B, and C would be quoted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.52\%}
It takes Mary $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$40\%$}
In isosceles triangle \( \triangle ABC \), \( AC = BC \) and \( \angle ACB = 40^\circ \). A circle with diameter \( BC \) intersects \( AC \) and \( AB \) at points \( D \) and \( E \) respectively. The diagonals of quadrilateral \( BCDE \) intersect at point \( F \). Find the measure of \( \angle BFC \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110^\circ}
Consider a symmetric random walk starting at 0. Let $T_a$ and $T_{-b}$ denote the first times the walk visits $a$ and $-b$, respectively, where $a, b > 0$. Find $\mathsf{E}(\min(T_a, T_{-b}))$. Use the fact that $P(T_a < T_{-b})$ can be computed by conditioning on the first step.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab}
The first term of a geometric sequence is 6, the sum of the first $n$ terms is $\frac{45}{4}$, and the sum of the reciprocals of these terms is $\frac{5}{2}$. What is this geometric sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6, 3, \frac{3}{2}, \frac{3}{4}}
Bob knows that Alice has 2021 secret positive integers $x_{1}, \ldots, x_{2021}$ that are pairwise relatively prime. Bob would like to figure out Alice's integers. He is allowed to choose a set $S \subseteq\{1,2, \ldots, 2021\}$ and ask her for the product of $x_{i}$ over $i \in S$. Alice must answer each of Bob's queries truthfully, and Bob may use Alice's previous answers to decide his next query. Compute the minimum number of queries Bob needs to guarantee that he can figure out each of Alice's integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the mass percentage of Ca in Calcium carbonate
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40.04\%}
The left and right foci of the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ are $F_1$ and $F_2$, respectively. Points $A$ and $B$ are on ellipse $C$, and $\overrightarrow{AF_1} = 3\overrightarrow{F_1B}$, $\angle BAF_2 = 90^{\circ}$. The eccentricity of ellipse $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{2}}{2}}
Find the sum of all natural numbers $n$ such that the function $g(x) = 100|e^{x+1} - 1| - \sum_{k=1}^n |e^{x^k + 1} - 1|$ is differentiable over the entire real line $\mathbb{R}$. Here, $n \in \mathbb{N}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 21 }
Given the sequence \((a_{n})\) defined by the recurrence relation \(a_{n+1}=\frac{1}{2}(a_{n}+\frac{3}{a_{n}})\), which converges to \(\sqrt{3}\), determine the rate of convergence of the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{quadratic}
3. (4 points) In a parallelogram with sides 4 and 7, the bisectors of the four interior angles are drawn. Find the ratio of the area of the quadrilateral formed by the intersection of the bisectors to the area of the parallelogram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{56}}
某校高一新生共有784人，每班分配56人．方法是：将每人的入学成绩从高分到低分依次编号（成绩相同的学生按姓氏笔画的顺序），然后按S形顺序编班．例如：若有8个班，按编号1至8号分别编在1至8班，9至16号分别编在8至1班，17至24号分别编在1至8班，···．该校高一新生中编号为300（每号只对应1人）的同学编在___班.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the smallest integer $k$ such that a $k$-vector, whose entries can be set to 0 or 1, can be used to communicate any integer between 1 and 1000 to a friend. The vector is subject to a transformation $f_i$, which sets the $i$-th component to 1 while leaving other components unchanged, and the value of $i$ is unknown. What is the smallest $k$ that ensures your friend can always understand the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Misha made a homemade darts game during the summer at the dacha. A circular board is divided into several sectors by concentric circles, and darts can be thrown at it. Points are awarded based on the number written in the sector, as shown in the diagram.

Misha threw 8 darts three times. In the second attempt, he scored twice as many points as he did in the first attempt, and in the third attempt, he scored 1.5 times more than in the second attempt. How many points did he score in the second attempt?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Let \( a^3 - a - 1 = 0 \). Find the exact value of the expression 
\[
\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Jimmy finds that 8 bananas weigh the same as 6 oranges. If Jimmy has 40 oranges, and the weight of 4 oranges equals the weight of 5 apples, how many apples and bananas would Jimmy need to equal the weight of his 40 oranges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the coefficient of $x^{80}$ in the power series $$(1+x+x^{2}+x^{3}+x^{4}+	ext{...})(x^{2}+x^{4}+x^{6}+x^{8}+	ext{...})(1+x^{3}+x^{5}).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{115}
Let  $a/b$  be the probability that a randomly chosen positive divisor of  $12^{2007}$  is also a divisor of  $12^{2000}$ , where  $a$  and  $b$  are relatively prime positive integers.  Find the remainder when  $a+b$  is divided by  $2007$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
There are 55 points marked on a plane: the vertices of a regular 54-gon and its center. Determine the number of ways to color a set of three marked points in red such that the colored points form the vertices of a regular triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Given a point $P$ and a circle $\mathcal{C}$ in the plane, the distance from $P$ to $\mathcal{C}$ is defined as the minimum length $PM$ between $P$ and a point $M$ on the circle $\mathcal{C}$. For example, if $P$ lies on the circle, then the distance from $P$ to $\mathcal{C}$ is zero, and if $P$ is the center of the circle, then the distance from $P$ to $\mathcal{C}$ is equal to the radius of $\mathcal{C}$.

Given four non-cyclic points $A, B, C, D$, what is the maximum number of circles that pass at equal distance from these four points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $a,b,c$  be positive integers such that  $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$  are  $7$  distinct primes. The sum of two of  $a,b,c$  is  $800$ . If  $d$  be the difference of the largest prime and the least prime among those  $7$  primes, find the maximum value of  $d$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1594}
A cylindrical container has a fixed inner volume \(V\) mm\(^3\), a 2 mm thick solid wall, and an open top. The bottom is a solid circular disc with the same thickness and radius as the outer radius of the container. The volume of the material used to construct the container is minimized when the inner radius of the container is 10 mm. What is the value of \(\frac{V}{250\pi}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit: $$L = \lim_{x \to \infty}x^2\left(x^2 - x \cdot \sqrt{x^2 + 6} + 3\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\left(0 \lt m \lt 1\right)$, point $A\left(0,2\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\frac{4}{3}$, then $m=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{9}}
Determine the genus of the curve \( y^{2}+y=\frac{\alpha}{x^{2}+x} \) over a finite field with even characteristic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Maria is a single mother who decided to renovate her kitchen. She hired a contractor who promised to complete the renovation for $8,000. Unfortunately, the contractor turned out to be fraudulent and disappeared after receiving a $2,500 deposit from Maria, leaving the work unfinished. Maria later found another reliable contractor who agreed to finish the renovation for $6,000. How much more money will Maria need to pay to finally complete her kitchen renovation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Find a closed-form expression for the series \( \sum_{n=0}^\infty\frac{(1/2)_n}{n!}(H_n-H_{n-1/2}) \), where \( H_z \) is the harmonic number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Please calculate the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{471}
Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
To which number can the expression

$$
\frac{(a+b)(b+c)(a+c)}{a b c}
$$

be equal if the numbers \( a \), \( b \), and \( c \) satisfy the condition of problem 84?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the value of \(a\) such that the quadrilateral \(ABCD\) is cyclic, where the line \(x + 2y + a = 0\) intersects the circle \(x^2 + y^2 = 4\) at points \(A\) and \(B\), and the line \(12x - 6y - 41 = 0\) intersects the circle \(x^2 + y^2 - 4x - 2y + 1 = 0\) at points \(C\) and \(D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many positive integers less than or equal to 100 have a prime factor that is greater than 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Determine the number of terms required to approximate the integral \( \int_0^{0.5}\frac{1-e^{-x}}{x}dx \) with an error less than \( 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Bob’s Rice ID number has six digits, each a number from $1$ to $9$, and any digit can be used any number of times. The ID number satifies the following property: the first two digits is a number divisible by $2$, the first three digits is a number divisible by $3$, etc. so that the ID number itself is divisible by $6$. One ID number that satisfies this condition is $123252$. How many different possibilities are there for Bob’s ID number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
Evaluate the expression: \[ \sum_{k=0}^{10} {30+k \choose k}{20-k \choose 10} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{51}{10}}
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. 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{2}
Given n balls of different colors arranged in a row, determine the expected number of swaps needed to rearrange them such that they satisfy a condition defined by a function f. Assume that at least one arrangement exists that satisfies the condition, and a function g provides all possible arrangements. What is the expected number of swaps required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - H_n}
Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4060}
Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).

Note: The factorials do not have to be distinct.  For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Among all pairs of real numbers(x,y)such that
 \( \sin \sin x = \sin \sin y \) with \( - {10\pi } \leq  x,y \leq  {10\pi } \) , Oleg randomly
 selected a pair(X,Y). Compute the probability that \( X = Y \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{20}$}
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}