inclusionAI/Ring-lite-rl-data huggingface dataset:
Problem 6.6. Several oranges (not necessarily of equal weight) were picked from a tree. When they were weighed, it turned out that the weight of any three oranges taken together is less than $5 \%$ of the total weight of the remaining oranges. What is the smallest number of oranges that could have been picked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Two brothers raised a flock of sheep. When the price of each sheep (in yuan) exactly equals the number of sheep in the flock, they sold all the sheep and divided the money equally between them: the elder brother first took 10 yuan, then the younger brother took 10 yuan; they continued this way alternately. In the end, the elder brother took 10 yuan first, and the younger brother took less than 10 yuan. At this point, the elder brother gave his straw hat to the younger brother, making the amount of money they each got equal. How much is the straw hat worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Triangle \(ABC\) is isosceles \((AB = BC)\). Segment \(AM\) divides it into two isosceles triangles with bases \(AB\) and \(MC\). Find angle \(B\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^\circ}
Determine the number of triples B needs to name to be sure of identifying the member (X,Y,Z) from the set of all triples (x,y,z) with non-negative integers \( x,y,z < 10 \):
B names a triple (a,b,c) in T. A replies with \( |X + Y - a - b| + |Y + Z - b - c| + |Z + X - c - a| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
It is known that there exists a number \( S \), such that if \( a + b + c + d = S \) and \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = S \) (where \( a, b, c, d \) are different from zero and one), then \(\frac{1}{a - 1} + \frac{1}{b - 1} + \frac{1}{c - 1} + \frac{1}{d - 1} = S \). Find \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
How many ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3456}
A reduction of 30% in the price of oil enables a housewife to obtain 9 kgs more for a certain amount of money. The reduced price per kg is Rs. 60. How much money does the housewife spend on oil at the reduced price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
The number \(21! = 51,090,942,171,709,440,000\) has over \(60,000\) positive integer divisors. One of them is chosen at random. What is the probability that it is odd? Provide your answer as a fraction in simplest form. The answer should be in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Lines parallel to the sides of an equilateral triangle are drawn such that they divide each side into $10$ equal segments, resulting in the triangle being divided into $100$ congruent smaller triangles. Each of these $100$ triangles is referred to as a "cell." Additionally, lines parallel to each side of the original triangle are drawn through each vertex of the original triangle. The cells located between any two adjacent parallel lines form a "stripe." Determine the maximum number of cells that can be selected such that no two selected cells belong to the same stripe.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Solve for $m$ in the equation $\frac{x}{x-3}=2-\frac{m}{3-x}$, such that the equation has a root. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Each boy is friends with 5 girls, and all the girls are friends with different numbers of boys. What is the minimum number of children in this group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Shapovalov A.V.

A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. Now, seven times more soldiers were looking at the back of their neighbor than in the face. How many soldiers are in the row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq  3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and  $$ 2a_k\leq a_{k-1} + a_{k+1} \ \  \ \text{for} \ \  \  k = 2, 3, \cdots , n-1.$$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n+1}{n-1}}
Find the value of $$S=\sum_{r=0}^{n}\binom{n}{r}\sum_{k=2n-2r+1}^{2n+1}\binom{2n+1}{k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(2^{3n}\)}
Determine the limit of the sequence $b_n$ defined by $b_n = \frac{\Gamma \left( {n + 3/2} \right)}{\Gamma \left( {n + 2} \right)\Gamma \left( {1/2} \right)}$, as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and - signs before each of the integers, yielding an expression $+1-2+3-4+5-6+7-8+9-11+12-\cdots-199$. What is the value of the resulting expression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-100}
The extension of the median $A M$ of triangle $A B C$ intersects its circumscribed circle at point $D$. Find $B C$ if $A C = D C = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given that F is the focus of the parabola C: x²=4y, the line l passing through point F intersects with parabola C at two distinct points A and B. The tangents to the parabola C at points A and B are l₁ and l₂, respectively, which intersect at point P. Find the minimum value of |PF| + $\frac {32}{|AB|}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The integers $a$, $b$, and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
The circles \( k_{1} \) and \( k_{2} \) intersect at points \( A \) and \( B \). The line \( t \) is tangent to \( k_{1} \) and \( k_{2} \) at points \( M \) and \( N \), respectively. Given that \( t \perp AM \) and \( MN = 2 AM \), calculate the measure of \( \widehat{NMB} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Given a quadrilateral \(PQRS\) made from two similar right-angled triangles \(PQR\) and \(PRS\), where \(PQ = 3\), \(QR = 4\), and \(\angle PRQ = \angle PSR\), calculate the perimeter of \(PQRS\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given the function $f(x) = \log\_2(x^2 - ax + 3a)$, determine the range of values for $a$ such that the function is increasing on the interval $[2, +\infty)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4 < a \leq 4}
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with magnitudes $|\overrightarrow {a}| = 6\sqrt {3}$ and $|\overrightarrow {b}| = \frac {1}{3}$, and their dot product $\overrightarrow {a} \cdot \overrightarrow {b} = -3$, determine the angle $\theta$ between $\overrightarrow {a}$ and $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
Find the equation of the tangent plane to the surface \( z = 9y^2 - 9x^2 \) at the point \((-1, 4, 135)\). Express the answer in terms of \(x\) and \(y\), for example, \(3x - 4y + 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18x + 72y - 135}
22 football players took part in the football training. They were divided into teams of equal size for each game (11 vs 11). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
In a company, there are 30 employees. The average age of the first group of 10 employees is 24 years, the average age of the second group of 12 employees is 30 years, and the average age of the third group of 7 employees is 35 years. Find the age of the 30th employee, if the average age of all 29 employees without this person is 29 years.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
If $S$ is a set consisting 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}
There are  $2020\times 2020$  squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that square.

Note : We say the square  $(a,b)$  is on the diagonals containing the square  $(c,d)$  when  $|a-c|=|b-d|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
Let  $ABC$  be a triangle inscribed in circle  $\Gamma$ , centered at  $O$  with radius  $333.$  Let  $M$  be the midpoint of  $AB$ ,  $N$  be the midpoint of  $AC$ , and  $D$  be the point where line  $AO$  intersects  $BC$ . Given that lines  $MN$  and  $BO$  concur on  $\Gamma$  and that  $BC = 665$ , find the length of segment  $AD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{444}
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads <tt>002005</tt>, how many miles has the car actually traveled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1462$}
We call a monic polynomial  $P(x) \in \mathbb{Z}[x]$  *square-free mod n* if there <u>dose not</u> exist polynomials  $Q(x),R(x) \in \mathbb{Z}[x]$  with  $Q$  being non-constant and  $P(x) \equiv Q(x)^2 R(x) \mod n$ . Given a prime  $p$  and integer  $m \geq 2$ . Find the number of monic *square-free mod p*  $P(x)$  with degree  $m$  and coeeficients in  $\{0,1,2,3,...,p-1\}$ .

*Proposed by Masud Shafaie*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ p^m - p^{m-1} }
Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\cos x}{a\sin x+b\cos x}\,dx$$ where $a$ and $b$ are positive constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{b\pi+2a\ln\frac{a}{b}}{2(a^2+b^2)}\)}
In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \(a\), there are as many numbers smaller than \(a\) among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockwise. A number that is larger than all 30 numbers closest to it is called "huge." What is the smallest possible number of huge numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The number  $15$  is written on a blackboard. A move consists of erasing the number  $x$  and replacing it with  $x+y$  where  $y$  is a randomly chosen number between  $1$  and  $5$  (inclusive). The game ends when the number on the blackboard exceeds  $51$ . Which number is most likely to be on the blackboard at the end of the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
In an isosceles triangle $ABC$ with base $AC$ and side $AB$ equal to 2, the segment $AD$ is the angle bisector. A tangent $DH$ to the circumcircle of triangle $ADB$ is drawn through point $D$, with point $H$ lying on side $AC$. Find the area of triangle $ABC$ if $CD = \sqrt{2} CH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4\sqrt{2} - 5}}
Find the order of the element 250 in $U(641)$, where $U(n)$ is the collection of positive integers which are coprime to n and forms a group under multiplication modulo n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a plane, several points are marked, each colored blue, yellow, or green. On any segment connecting points of the same color, there are no points of that same color, but there is at least one point of a different color. What is the maximum possible number of all points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the largest real number \(\theta\) less than \(\pi\) (i.e. \(\theta < \pi\)) such that
$$
\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0
$$
and
$$
\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2046\pi}{2047}}
17. From a square with a side length of 20, remove a rectangle with an area of 36, where the two side lengths of the rectangle are integers, and one side of the rectangle is part of one side of the square. The maximum perimeter of the remaining shape is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
Let  $n\ge 1$   be a fixed integer. Calculate the distance  $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$   , where  $p$   runs over polynomials of degree less than  $n$   with real coefficients and  $f$   runs over functions  $f(x)= \sum_{k=n}^{\infty} c_k x^k$    defined on the closed interval  $[0,1]$  , where  $c_k \ge 0$   and  $\sum_{k=n}^{\infty} c_k=1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{-2n+1} }
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north, south, east, or west. He wakes up one morning on the grid with 100 Joules of energy, and hops till he falls asleep with 0 energy. How many different places could he have gone to sleep?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10201}
Let \( A \) and \( B \) be two moving points on the ellipse \( x^{2}+3 y^{2}=1 \), and \( OA \) is perpendicular to \( OB \) (where \( O \) is the origin). Then, the product of the maximum and minimum values of \( |AB| \) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{3}}{3}}
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.

*Proposed by Lewis Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Find the sum of all four-digit numbers in which the digits $0, 4, 5, 9$ are absent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6479352}
Consider a Poisson process with rate $λ$ and let $L$ be the time of the last arrival in the interval $[0, t]$, with $L = 0$ if there was no arrival. Compute $E(t − L)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\lambda} (1 - e^{-\lambda t})}
试求满足下列条件的大于 $5$ 的最小奇数 $a$: 存在正整数 $m_{1} 、 n_{1} 、 m_{2} 、 n_{2}$ ，使得
$a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2},$
 且 $m_{1}-n_{1}=m_{2}-n_{2} .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{261}
For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499500}
Determine the number of roots of the equation:

$$
\left(21 x - 11 + \frac{\sin x}{100}\right) \cdot \sin(6 \arcsin x) \cdot \sqrt{(\pi - 6x)(\pi + x)} = 0
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given $sinα+cos({α-\frac{π}{6}})=\frac{{\sqrt{3}}}{3}$, then the value of $sin({α+\frac{{7π}}{6}})$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
Given the equation $2^a+2^b=3^c+3^d$, find the maximum number of integers $a,b,c,d$ that can be negative. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The lengths of the three sides of a triangle, \( a, b, c \) satisfy the equation \( \frac{c^{2}}{a+b}+\frac{a^{2}}{b+c}=b \). What is the measure of the angle \( B \) opposite the side of length \( b \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Let $p(x)$ be a polynomial of degree strictly less than $100$ such that it does not have $(x^3-x)$ as a factor. If \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \) for some polynomials $f(x)$ and $g(x)$, then find the smallest possible degree of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
3. Find $g(2022)$, if for any real $x, y$ the equality holds

$$
g(x-y)=2022(g(x)+g(y))-2021 x y .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2043231}
A club charges $30 to enter. James buys 3 rounds for his 10 friends. He also buys 8 drinks for himself throughout the night. Cocktails cost $10 each while non-alcoholic drinks cost $5 each. There is a 20% discount on cocktails if you order three or more. He purchases 7 cocktails and the remaining drinks are non-alcoholic. James also decides to eat some food, so he orders a gourmet burger which costs $20 and a side of fries for $8. He leaves a 20% tip on the food and a 15% tip on the non-discounted value of the drinks. How much did he spend for the night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$308.35}
Bus tickets have numbers ranging from 000001 to 999999. A number is considered lucky if the first three digits are odd and different from each other, and the last three digits are even, with 7 and 8 not standing next to each other. How many different lucky numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7200}
在 $\triangle A B C$ 中, $A=90^{\circ}, A B=1, A C=\sqrt{3}$, 点 $P$ 满足 $\frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}$。如果答案是 $x^\circ$，求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
$\sin 75^{\circ}\cos 30^{\circ}-\sin 15^{\circ}\sin 150^{\circ}=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{2}}{2}}
$A(x)=\sum_{m=1}^{\infty} \sum_{n=0}^{\infty} \frac{m!\cdot x^{m} \cdot n!}{(m+n+1)!}$, 则 $\left[100 A\left(\frac{2}{3}\right)\right]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109}
For the hyperbola $x^{2}- \frac {y^{2}}{m}=1$, find the sufficient but not necessary condition for the eccentricity to be greater than $\sqrt {2}$. Express your answer as an inequality involving $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > 2}
In triangle \( ABC \), angle \( B \) is right. The midpoint \( M \) is marked on side \( BC \), and there is a point \( K \) on the hypotenuse such that \( AB = AK \) and \(\angle BKM = 45^{\circ}\). Additionally, there are points \( N \) and \( L \) on sides \( AB \) and \( AC \) respectively, such that \( BC = CL \) and \(\angle BLN = 45^{\circ}\). In what ratio does point \( N \) divide the side \( AB \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that $$4\left(a_{n-1}^{2}+a_{n}^{2}\right)=10 a_{n-1} a_{n}-9$$ What is the largest positive integer less than $a_{8}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{335}
In a right triangle, one of the acute angles $\alpha$ satisfies
\[\tan \frac{\alpha}{2} = \frac{1}{\sqrt[3]{2}}.\]Let $\theta$ be the angle between the median and the angle bisector drawn from this acute angle.  Find $\tan \theta.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Ahmed and Emily are having a contest to see who can get the best grade in the class. There have been 9 assignments and Ahmed has a 91 in the class. Emily has a 92. The final assignment is worth the same amount as all the other assignments. Ahmed needs a minimum grade of 100 to beat Emily if all grades are whole numbers. What grade did Emily get on the final assignment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
Find the limit of $ \sqrt[n^2]{a_n} $ given that $ a_2>0,\;a_3>0 ,\; a_{n+2}a_n=(1+1/n)^n a_{n+1}^2, n\ge2 $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1225}
Let \(\{a, b, c, d\}\) be a subset of \(\{1, 2, \ldots, 17\}\). If 17 divides \(a - b + c - d\), then \(\{a, b, c, d\}\) is called a "good subset." Find the number of good subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{476}
How many ways are there to remove an  $11\times11$  square from a  $2011\times2011$  square so that the remaining part can be tiled with dominoes ( $1\times 2$  rectangles)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2,002,001}
How many numbers can be formed with the digits from 1 to 4, where all the digits are different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Find the number of ordered triples of integers $(a,b,c)$ such that $$a^2 + b^2 + c^2  - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2$$ and $|a|, |b|, |c| \le 2021.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14152}
In triangle \( \triangle ABC \), \( AB = 2\sqrt{3} \), \( AC = 2 \), and the height from \( BC \) is \( \sqrt{3} \). What is the length of \( BC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}