inclusionAI/Ring-lite-rl-data huggingface dataset:
Alice, Bob, and Carol play a game in which each of them chooses a real number between \(0\) and \(1.\) The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between \(0\) and \(1,\) and Bob announces that he will choose his number uniformly at random from all the numbers between \(\frac{1}{2}\) and \(\frac{2}{3}.\) Armed with this information, what number should Carol choose to maximize her chance of winning? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Amelia has 27 unit cubes. She selects one and paints one of its faces. She then randomly glues all 27 cubes together to form a $3 \times 3 \times 3$ cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2/3
}
Consider a sequence of independent random variables $\{X_1, X_2, \ldots, X_N, \ldots\}$, where each $X_i$ is distributed such that $\text{Pr}(X_i = 1) = \text{Pr}(X_i = -1) = \frac{1}{2}$. Define $Y_N$ as the maximum absolute partial sum among the first $N$ terms, given by $Y_N = \max_{1 \leq i \leq N} \left| \sum_{j=1}^{i} X_j \right|$. Determine an upper bound for the limit of the probability $\lim_{N\to\infty}\text{Pr}\left(\frac{Y_N}{N} \leq \alpha\right)$ as $N$ approaches infinity, for $0 < \alpha < 1$. Provide your answer as a single expression or value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a circle with center $O$ and radius $r$, let $AB$ be a chord smaller than the diameter. The radius of the inscribed circle in the smaller circular segment formed by the radii $AO$ and $BO$ is denoted by $\varrho$. Express $AB$ in terms of $r$ and $\varrho$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 r \varrho}{r - \varrho}}
How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given real numbers $a$, $b$, $c$ that satisfy $3x+4y-5\leqslant ax+by+c\leqslant 3x+4y+5$ for any real numbers $x$, $y$, find the minimum value of $a+b-c$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a regular pentagon with vertices numbered from 1 to 5 in a clockwise direction. Initially, two ladybugs are placed at vertices 1 and 3. At each time step, each ladybug moves independently to one of its two adjacent vertices with equal probability. Calculate the expected number of time steps for the two ladybugs to meet at the same vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
4. In space, there are 4 points $A, B, C, D$, satisfying $A B=B C=C D$. If $\angle A B C=\angle B C D=\angle C D A=36^{\circ}$, then the angle formed by line $A C$ and line $B D$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the number of distinct permutations of an $n$-element list such that no two permutations are equivalent under the condition that a permutation matches another if they can be obtained by flipping one another, preserving the relative order of neighboring elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n!}{2}}
Evaluate the integral \( I = \int_{-1}^0 \int_{0}^1 \delta(x-y) \, dx \, dy \), where \( \delta(t) \) is the Dirac delta function defined as the limit of a symmetric Gaussian probability density function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of integer sequences $m_1, m_2, \ldots, m_k$ such that $m_1, m_k \in \{1, 2, \ldots, 12\}$, $m_n \in \{1, 2, \ldots, 6\}$ for $1 < n < k$, and $m_1 + 2m_2 + \ldots + 2m_{k-1} + m_k = 24$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5776}
Let \(g(x)\) be a function defined as:

\[g(x)=\begin{cases} cx(x-1)+d&;x<1\\ x+2&;1\le x\le3\\ rx^2+sx+2&;x>3 \end{cases}\]

If \(g(x)\) is continuous for all \(x \in \mathbb{R}\) except \(x=1\), \(|g(x)|\) is differentiable everywhere, \(g'(x)\) is continuous at \(x=3\), and \(|c+r+d+s|=m\), find the value of \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
一个盒子里有无限可数个球，编号分别为 0,1,2,3,\cdots. 现在从中随机取球，每次取出编号为 k 的球的概率为 \df{e^{-1}}{k!}. 如果连续有放回地取 6 个球，记这 6 个球编号之和为 4 的概率为 p. 则 [10|\ln p|]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given $n$ zigzag lines, where each zigzag line intersects every other zigzag line exactly twice and no three lines intersect at the same point, determine into how many regions they divide the plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 + 1}
Farmer James invents a new currency such that for every positive integer $n \leq 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be *nice* if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Determine the number of functors from the category of diagrams with exact rows in an abelian category to the category of six-term exact sequences in the abelian category, where the functors act identically on objects and morphisms except for possibly associating a different connecting morphism to each diagram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The product of four consecutive positive odd numbers ends in 9. Find the two penultimate digits of this product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{09}
What is the range of values of \( m \) for which the equation \( m(x^2 + y^2 + 2y + 1) = (x - 2y + 3)^2 \) represents an ellipse in a Cartesian coordinate system?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5,+\infty)}
Calculate the integral \( \int_{-1}^1 |z| \, dz \) where the integration contour is the upper semi-circle with unit radius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the limit of the sum of binomial coefficients given by:
\[ S_{n} = \sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the smallest integer $n$ such that a square $ABCD$ can be decomposed into $n$ non-overlapping triangles, each having only acute angles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the dimension of the vector space of alternating n-linear functions on the set of n x n matrices over a field F.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the exterior $2$-form $\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$ on $\mathbb{R}^n$ using the vectors $A = (1, 1, 1, \dots, 1)$ and $B = (-1, 1, -1, \dots, (-1)^n)$. Compute the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + (-1)^n}
We have some cards that have the same look, but at the back of some of them is written $0$ and for the others $1$.(We can\u0027t see the back of a card so we can\u0027t know what\u0027s the number on it\u0027s back). we have a machine. we give it two cards and it gives us the product of the numbers on the back of the cards. if we have $m$ cards with $0$ on their back and $n$ cards with $1$ on their back, at least how many times we must use the machine to be sure that we get the number $1$? (15 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + 1}
A group of 10 children are taken to an amusement park. They all try a variety of rides and attractions with the following costs:

Ferris wheel: $5 per child (6 children)
Roller coaster: $7 per child (4 children)
Merry-go-round: $3 per child (everyone)
Bumper cars: $4 per child (7 children)
Haunted house: $6 per child (5 children)
Log flume: $8 per child (3 children)

During their visit, they also bought some snacks:

3 cones of ice cream: $8 per cone (4 children)
1 hot dog: $6 each (5 children)
1 slice of pizza: $4 each (3 children)
1 large pretzel: $5 each (2 children)
1 cotton candy: $3 each (6 children)
1 soda: $2 each (7 children)

How much did they spend altogether?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$286}
The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.86\%}
Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
What is the quantity equivalent to '2% of 1'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{100}}
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 100$ such that $i^x+i^y$ is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1850}
Given the ellipse \( C: \frac{x^{2}}{3} + y^{2} = 1 \) with the upper vertex as \( A \), a line \( l \) that does not pass through \( A \) intersects the ellipse \( C \) at points \( P \) and \( Q \). Additionally, \( A P \perp A Q \). Find the maximum area of triangle \( \triangle A P Q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}}
Given vectors $\vec{a}=(m,4)$ and $\vec{b}=(3,-2)$, and $\vec{a}$ is not parallel to $\vec{b}$, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
x = 3
}
Consider the bounded solution $u(x,t)$ of the heat equation \(\frac{\partial u}{\partial t} -\frac{\partial^2 u}{\partial x^2} = 0\), where \(u(x, 0) = \frac{e^{2x} - 1}{e^{2x} + 1}\). Determine \(\lim_{t \to \infty} u(1, t)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
8.4. On a chessboard, the centers of some cells are marked in such a way that no triangle with marked vertices is a right triangle. What is the maximum number of points that could have been marked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
What is the minimum number of elementary row or column transformations needed to find the inverse of a given matrix using Gauss-type methods?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
How many distinct garlands can be formed by arranging all $6n$ flowers of one kind and $3$ flowers of another kind in a circular fashion, considering that the garland is considered the same if it can be rotated or reflected to match another arrangement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n^2 + 3n + 1}
6. In the Cartesian coordinate system $x O y$, the point set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly selected from $K$, then the probability that the distance between each pair of these three points is no more than 2 is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{14}}
Let $x_1 = \frac{1}{20}$, $x_2 = \frac{1}{13}$, and \[ x_{n+2} = \frac{2x_nx_{n+1}(x_n + x_{n+1})}{x_n^2 + x_{n+1}^2} \] for all integers $n \geq 1$. Evaluate $\sum_{n=1}^\infty \left( \frac{1}{x_n + x_{n+1}} \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Consider a random walk on the set of natural numbers \(\mathbb{N}\), starting at 0. The transition probabilities are given by:

\[ p(0,1)=0.75, \quad p(n,n-1)=0.5, \quad p(n,n+1)=0.5 \quad \text{for } n>0. \]

Calculate the expected time \(\mathbb{E}[T_{50}]\) for the random walk to reach the value 50.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7550}{3}}
Determine the number of orderings of the totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}[\zeta]$, where $\zeta$ is a primitive $p$-th root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p-1}{2}}
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Find the positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
A function \(f\left(a \frac{b}{c}\right)\) for a simplified mixed fraction \(a \frac{b}{c}\) returns \(\frac{a + b}{c}\). For instance, \(f\left(2 \frac{5}{7}\right) = 1\) and \(f\left(\frac{4}{5}\right) = \frac{4}{5}\). Find the sum of the three smallest positive rational \(x\) where \(f(x) = \frac{2}{9}\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
In a $4 \times 4$ table, we place $k$ blocks such that:

1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we remove a block, there is at least one cell that is not covered.

Find the maximum value of $k$.

Note: The blocks can overlap.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that:
[LIST]
[*] There are no two numbers written in the first row that are equal to each other.[/*]
[*]  The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*]
[*] The two numbers written in each column are different and they add up to a rational number.[/*]
[/LIST]
Determine the maximum quantity of irrational numbers that can be in the chessboard.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032}
How many strings of length $5$ formed from the digits $0$, $1$, $2$, $3$, $4$ are there such that for each $j \in \{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies this condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1296$}
设数列$\{ a _ { n } \}$的通项公式为$a _ { n } = n ^ { 3 } - n , n \in N ^ { * }$，将该数列中个位数字为0的项，按从小到大的顺序排列构成数列$\{ b _ { n } \}$，则$b _ { 2 0 1 8 }$被7除所得的余数为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the maximal number of real intersections of the unbounded components of two plane cubic curves, $C_1$ and $D_1$, in the projective plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
If $a,b,c,d$ are distinct positive real numbers such that $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $ac=bd$, find the minimum value of $$\dfrac{a}{c}+\dfrac{b}{d}+\dfrac{c}{a}+\dfrac{d}{b}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
From the set $\{1,2,3,…,10\}$, select any three different numbers so that these three numbers form a geometric sequence. There are \_\_\_\_\_\_ such geometric sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \( x \) and \( y \) be acute angles such that \( \sin y = 2005 \cos (x + y) \sin x \). Find the greatest possible value of \( \tan y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2005 \sqrt{2006}}{4012}}
Determine the maximum order of an element in the group $SL_2(\mathbb{Z} / p \mathbb{Z})$ for an odd prime $p$, given that the characteristic polynomial $P_A(x) = x^2 - \text{tr}(A)x + 1$ has no roots in $\mathbb{Z} / p \mathbb{Z}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p+1}
Find the range of the function $y=2^{-x^{2}+2x}$. Express your answer as an interval, specifying the endpoints and whether they are included or excluded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,2]}
Find the angle between the vector \((2,2,1,1)\) and the linear subspace spanned by the vectors \((3,4,-4,-1)\) and \((0,1,-1,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
## Problem Statement

Find the derivative.

$$
y=x \ln (\sqrt{1-x}+\sqrt{1+x})+\frac{1}{2}(\arcsin x-x)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln(\sqrt{1-x}+\sqrt{1+x})}
Find the smallest positive integer \( k \) such that the sum of the coefficients \( a_0 + a_1 + a_2 + \cdots + a_{k-1} \) in the expansion of \((1+x)(1+2x)(1+3x) \cdots (1+kx)\) is divisible by 2005.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{401}
将有 2023 个顶点的完全图的每条边染色，使得有公共点的任意两条边颜色不同. 定义每个点所连出的边的颜色为一个颜色集，求不同颜色集个数的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Every second Saturday and all Sundays are holidays. In a month of 30 days beginning on a Saturday, how many working days will be there if all Sundays are not holidays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
On the extension of side $AC$ of triangle $ABC$, a point $T$ is marked beyond point $A$ such that $\angle BAC = 2 \angle BTC$. Find the area of triangle $ABC$, given that $AB = AC$, $BT = 42$, and $AT = 29$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Find the sum of the coefficients of $x^{20}$ and $x^{21}$ in the power series expansion of $\frac{1}{(1-x^3)^4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Among the non-empty subsets of the set \( A = \{1, 2, \cdots, 10\} \), how many subsets have the sum of their elements being a multiple of 10?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
A <b>palindrome</b> is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $\boxed{1:01}$, $\boxed{4:44}$, and $\boxed{12:21}$. How many times during a $12$-hour period will be palindromes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$57$}
Evaluate the following limit using L'Hopital's Rule and Leibniz Integral Rule:

$$\lim_{a \to 1} \frac{1}{1-a} \int _{a} ^{1} \frac{\ln(x)}{x-1}\ dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the cardinality of the set $F$ of all entire functions $f$ such that $f(x) = |x^k|$ for all $x \in (-1, 1)$, where $k$ is an odd integer $k \geq 1$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two cars start from the opposite places of a main road, 113 km apart. The first car runs for 25 km and takes a right turn and then runs 15 km. It then turns left and then runs for another 25 km and then takes the direction back to reach the main road. In the meantime, due to a minor breakdown, the other car has run only 35 km along the main road. What would be the distance between the two cars at this point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Solve the equation (for positive $x$)
$$x^x=\frac{1}{\sqrt2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$x = \frac{1}{2}$}
Let \( \{X_t \mid t\in\mathbb{N}\} \) be an iid sequence of normal random variables on a probability space \((\Omega,\mathcal{F},P)\). Define \(A(\omega) = \{X_t(\omega) \mid t\in\mathbb{N}\}\) for all \(\omega\in\Omega\) and let \(\overline{A(\omega)}\) be its closure. Let \(\ell\) be the Lebesgue measure. Determine the probability \(P(\{\omega \mid \ell(\overline{A(\omega)}) = 0\})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How can the centroid of 10 points be constructed using a fixed-opening compass in only 38 steps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Determine the asymptotic behavior of the sum \( \sum_{n=1}^N \Omega(n) \) as \( N \to \infty \), where \( \Omega(n) \) is the sum of the exponents in the prime factorization of a positive integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N \log \log N}
Let $a, b, c$ and $d$ be rational numbers. If it is known that the equation $x^4 + ax^3 + bx^2 + cx + d = 0$ has $4$ real roots, two of which are $\sqrt2$ and $\sqrt{2008}$. The value of $a + b + c + d$ is ...
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2006}
Let  $k$  be a positive integer and  $P$  a point in the plane. We wish to draw lines, none passing through  $P$ , in such a way that any ray starting from  $P$  intersects at least  $k$  of these lines. Determine the smallest number of lines needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k+1}
You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let $\theta$ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is $\lfloor 100 \cdot \theta \rfloor$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Tony and Maria are training for a race by running all the way up and down a 700 m long ski slope. They each run up the slope at different constant speeds. Coming down the slope, each runs at double his or her uphill speed. Maria reaches the top first, and immediately starts running back down, meeting Tony 70 m from the top. When Maria reaches the bottom, how far behind is Tony? Express your answer in meters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Given the function $f(x)= \sqrt {-3x^{2}+ax}- \frac {a}{x}(a > 0).$ If there exists $x\_0$, such that $f(x\_0) \geqslant 0$ holds, then the minimum value of $a$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \sqrt {3}}