inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a market where time is measured in milliseconds, and focus on market buy orders only. The market imbalance can be in one of three states: high (H), medium (M), or low (L). The imbalance changes due to incoming market orders or changes in standing orders. If the imbalance is low, it can remain low or increase to medium, each with probability \( \frac{1}{2} \), and no market buy orders arrive. If the imbalance is medium, it can decrease to low with probability \( \frac{1}{3} \) or increase to high. The increase from medium to high can occur due to a new market buy order with probability \( \frac{1}{2} \) or without a new order with probability \( \frac{1}{6} \). If the imbalance decreases from medium to low, no market buy orders arrive. If the imbalance is high, it can decrease to medium or stay high, each with probability \( \frac{1}{2} \), and a market order arrives with probability \( \frac{2}{3} \).

Betty keeps a unit standing sell order in the exchange, and every market buy order executes her order, earning her 1 cent. What is Betty's long-run profit per unit of time?

Hint: Construct a Markov chain with state space including more than just H, M, and L.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{54}}
设实数 $x_{1}, x_{2}, \cdots, x_{22}$ 满足对任意 $1 \leqslant i \leqslant$ 22, 有 $2^{i-1} \leqslant x_{i} \leqslant 2^{i}$ 。 求
$$
\left(x_{1}+x_{2}+\cdots+x_{22}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{22}}\right)
$$
的最大值。原始答案为$(2^{12}-1-\frac{1}{2^{11}})^{2}$，请给出近似整数值。
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Answer: \boxed{16769021}
If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth.
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Answer: \boxed{0.16}
There are three boxes of bottles. Box A contains 2 dozen water bottles, 7/4 dozen orange juice bottles, and half a dozen more apple juice bottles than there are water bottles. Box B has 25% more water bottles and 30% fewer orange juice bottles than Box A. Box C has twice as many water bottles and 1.5 times as many apple juice bottles as Box B, with no orange juice bottles. How many bottles are in all three boxes combined?
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Answer: \boxed{255}
How many bridge hands consist of exactly two 5-card suits, a void, and a remaining suit with a run of 3 cards?
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Answer: \boxed{218640708}
Given \(a > b > c > 0\), find the minimum value of \(2a^{2}+ \dfrac {1}{ab}+ \dfrac {1}{a(a-b)}-10ac+25c^{2}\). Express your answer as a single number.
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Answer: \boxed{4}
In a lake, there is a patch of lily pads. The patch grows at a certain rate. It takes 39 days for the patch to cover the entire lake, and 38 days for the patch to cover half of the lake. What is the growth rate of the patch of lily pads?
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Answer: \boxed{100\%}
Determine the number of homomorphisms from the alternating group $A_5$ to the symmetric group $S_5$. Provide your answer as a single integer.
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Answer: \boxed{121}
A positive integer is called "oddly even" if the sum of its digits is even. Find the sum of the first 2013 oddly even numbers.
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Answer: \boxed{4055187}
Jenna is buying concert tickets for her group of 10 friends. She's able to get three of the tickets from the concert website for $50 each before they sell out. A scalper offers to sell her four more tickets for 250% of the normal price, but Jenna convinces him to give her a $15 off discount that total payment. Finally, two of her friends get discounted tickets through their jobs - one for 60% of the normal price and one for 75% of the normal price. Additionally, there is a 10% service fee on each ticket purchased from the concert website and the scalper. How much do Jenna's friends pay for their tickets in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$766}
Evaluate the limit \( \lim_{n \to \infty} \oint_{\gamma} \frac{\cot(\pi z)}{(z+a)^2} \, dz \), where \( \gamma \) is the rectangular path with vertices at \( n+\frac{1}{2}+ni, -n-\frac{1}{2}+ni, -n-\frac{1}{2} -ni, n+\frac{1}{2}-ni, n+\frac{1}{2}+ni \), and \( a \) is a complex constant. Use the Residue Theorem to determine the behavior of this limit.
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Answer: \boxed{0}
Dragoons take up $1 \times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the $y$-coordinates of their centers are both at most $6$, regardless of any dragoons in between. For example, a dragoon centered at $(4, 5)$ can fire at a dragoon centered at the origin, but a dragoon centered at $(7, 0)$ cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?
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Answer: \boxed{168}
Given the number of minutes studied by Clara and Mira over a period of one week, calculate the average difference in minutes per day in the number of minutes studied by Mira and Clara.
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Answer: \boxed{5}
Given a sequence $\{a_n\}$ that satisfies $a_0= \frac{1}{2}$ and $a_{n+1}=a_n+ \frac{a_n^2}{2018}$ for $n=0,1,2,\ldots$, if $a_k < 1 < a_{k+1}$, what is the value of the positive integer $k$?
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Answer: \boxed{2018}
Given natural numbers \( M \) and \( N \), each greater than ten, consisting of the same number of digits, and such that \( M = 3N \). To obtain the number \( M \), it is necessary to add 2 to one of the digits of \( N \), and to each of the remaining digits, add an odd digit. Which digit could the number \( N \) end with? Find all possible answers.
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Answer: \boxed{6}
The kid cut out a rectangle from grid paper, with each cell having a side length of 1, and calculated its area and perimeter. Carlson then took the scissors and cut out a square from this rectangle, along the grid lines, that was adjacent to the boundary of the rectangle.

- "My rectangle..." the kid sobbed. "There is something strange about this figure!"
- "Nonsense, do not mention it," Carlson said, waving his hand carelessly. "Here you see, in this figure, the perimeter is the same as the area of the rectangle was, and the area is the same as the perimeter was!"

What size square did Carlson cut out?
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Answer: \boxed{2}
Chuck the llama is tied to the corner of a 2 m by 3 m shed on a 3 m leash. Calculate the area in which Chuck has to play if he can only go around the outside of the shed. Express your answer in square meters (m²), including π.
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Answer: \boxed{7\pi \text{ m}^2}
Let \( p(x) = x^2 - x + 1 \). Let \(\alpha\) be a root of \( p(p(p(p(x)))) \). Find the value of
\[ 
(p(\alpha) - 1) p(\alpha) p(p(\alpha)) p(p(p(\alpha))) 
\]
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Answer: \boxed{-1}
在正 $\triangle A B C$ 的边 $B C, C A, A B$ 上有内分点 $D, E, F$ 将边分成 $3:(n-3)(n>6)$, 线段 $A D, B E, C F$ 相交所成的 $\triangle P Q R(B E$ 交 $A D$ 于 $P$, 交 $F C$ 于 $Q)$ 是 $\triangle A B C$ 的面积的 $\frac{4}{49}$时.求 $n$ 的值.
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Answer: \boxed{8}
已知复数 $z_{1}, z_{2}, z_{3}$ 使得 $\frac{z_{1}}{z_{2}}$ 为纯虚数, $\left|z_{1}\right|=\left|z_{2}\right|=1,\left|z_{1}+z_{2}+z_{3}\right|=1$, 则 $\left|z_{3}\right|$ 的最小值是以 $\sqrt{m}-n$ 的形式给出，请提供m+n的值。
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Answer: \boxed{3}
What is the maximum possible value of $\int_{0}^{1}(f(x))^3dx$ if $f(x)$ is a function such that $-1\leq f(x)\leq 1$ for all $x \in [0,1]$, and $\int_{0}^{1}f(x)dx=0$?
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Answer: \boxed{1/4}
Calculate the mass of the region \( \Omega \) in the first quadrant, bounded by the curves \( x^2 + y^2 = 4 \), \( x^2 + y^2 = 9 \), \( x^2 - y^2 = 1 \), and \( x^2 - y^2 = 4 \). The density at each point \((x,y)\in \Omega\) is equal to the product of the distances of the point to the coordinate axes. Use a change of variables to find the mass of \( \Omega \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{8}}
For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote
    \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying
    \[a_1+a_2+\cdots+a_{10}=20.\]
    Compute the number of positive integer divisors of $N$.


[i]2021 CCA Math Bonanza Individual Round #14[/i]
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Answer: \boxed{462}
A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set?
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Answer: \boxed{39}
Define a finite sequence $\left( s_i \right)_{1\le i\le 2004}$ with initial conditions $s_0 + 2 = s_1 + 1 = s_2 = 2$ and the recurrence relation:
$$ s_n = 1 + s_{n-1} + s_{n-2} - s_{n-3}. $$
Calculate its last element.
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Answer: \boxed{1005006}
Calculate the dimension of the space of divisors on the Jacobian $J$ of a smooth compact complex curve $C_g$ (of genus $g$) that are linearly equivalent to the theta divisor $\Theta$. Specifically, determine $\dim( H^0(J,\mathcal{O}(\Theta)))$. 
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Answer: \boxed{1}
Given two random matrices $A$ and $B$ in $\mathbb{R}^{n\times n}$, determine the probability that the matrix algebra generated by $A$ and $B$ is the full matrix algebra $\mathbb{R}^{n\times n}$.
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Answer: \boxed{1}
Let $X$ and $Y$ be independent random variables, each following a normal distribution $N(1,1)$. Calculate the probability $P(2-X < Y < X)$. Express your answer as a single probability value.
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Answer: \boxed{\dfrac{1}{4}}
A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide the answer to the nearest percent.
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Answer: \boxed{97}
On a rectangular table of size $x$ cm $\times 80$ cm, identical sheets of paper of size 5 cm $\times 8$ cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter above and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length of $x$ in centimeters?
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Answer: \boxed{77 \text{ cm}}
What is the lowest prime number that is thirteen more than a cube?
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Answer: \boxed{229}
Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use?
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Answer: \boxed{10}
At the beginning of the school year, Jinshi Middle School has 17 classrooms. Teacher Dong Helong takes 17 keys to open the doors, knowing that each key can only open one door, but not knowing which key matches which door. What is the maximum number of attempts he needs to make to open all 17 locked doors?
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Answer: \boxed{136}
Find the smallest upper bound for the expression:
\[ A(N) = \max_{a_1, a_2, \ldots, a_N \in [0,1]} \sum_{n=1}^{N} a_n^{1/2} \prod_{s=1}^{n-1} (1-a_s)^{1/2} \]
where an empty product is defined to be 1.
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Answer: \boxed{\sqrt{N}}
Find the mass percentage of O in Dinitrogen pentoxide
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Answer: \boxed{74.06\%}
In an isosceles triangle, the base \(a\) and the angle \(\alpha\) at the base are given. Find the length of the median drawn to the lateral side.
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Answer: \boxed{\frac{a}{4}\sqrt{9 + \tan^2 \alpha}}
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{2\sin A - \sin C}{\sin C} = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{{a}^{2} + {c}^{2} - {b}^{2}}$, then the maximum value of $\sin ^{2}A + \sin ^{2}C$ is ______.
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Answer: \boxed{\frac{3}{2}}
It is known that \( A \) is the largest number that is the product of several natural numbers whose sum is 2011.

What is the largest power of three that divides \( A \)?
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Answer: \boxed{3^{669}}
A “palindromic integer” is an integer that remains the same when its digits are reversed. There are 200 palindromic integers of a certain digit length that are both even and greater than 500000. What is the digit length of these palindromic integers?
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Answer: \boxed{6}
A pharmaceutical company needs to optimize the cultivation temperature for a certain medicinal liquid through bioassay. The experimental range is set from 29℃ to 63℃, with an accuracy requirement of ±1℃. When using the fractional method for optimization, the minimum number of experiments required to ensure the best cultivation temperature is found is ______.
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Answer: \boxed{7}
Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is:
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Answer: \boxed{1}
Find an upper bound for the series \( \sum_{i=1}^{n}\left(\frac{n}{i}\right)^{i} \).
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Answer: \boxed{n e^{n/e}}
Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle{CDP}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
546
}
What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen can attack at least one other queen?
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Answer: \boxed{16}
A set of composite numbers from the set $\{1,2,3,4, \ldots, 2016\}$ is called good if any two numbers in this set do not have common divisors (other than 1). What is the maximum number of numbers that a good set can have?
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Answer: \boxed{14}
Given a parabola C: y²=x and a point M (1, -1) on it. Points A and B are two moving points on the parabola C, and $\overrightarrow {MA}\cdot \overrightarrow {MB}$=0. The maximum value of the distance from point M to line AB is _______.
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Answer: \boxed{\sqrt {5}}
A tank has two outlet pipes - outlet pipe A can empty the tank when it is full in 5 hours, and outlet pipe B can empty the tank when it is full in 8 hours. If the inlet pipe is kept open, letting water in at a rate of 4 litres/min, and both outlet pipes A and B are open simultaneously, the combined system would take 3 hours longer to empty the tank. Find the capacity of the tank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200}
A square $ABCD$ has side length $5$. Let $E$ be a point $20$ units from point $D$. For every point $F$ inside the square, rotate $E$ about $F$ $120$ degrees counterclockwise. Determine the area of the region formed by all of the image points of the rotation.
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Answer: \boxed{75}
Let $a_1, a_2, \cdots, a_{n}$ be real numbers such that $a_1^2 + a_2^2 + \cdots + a_{n}^2 = 1$. Determine the maximum value of the expression $$\large |a_1 - a_3| + |a_3 - a_5| + \cdots + |a_{2n-1} - a_{2n+1}| + |a_{2n+1} - a_1|$$ where $n \in \mathbb{N}$ and $n \geq 2$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
In the Cartesian coordinate system, consider the ellipse \(C: \frac{x^{2}}{27}+\frac{y^{2}}{18}=1\) with its right focus \(F\). A line \(l\) passing through \(F\) intersects the ellipse \(C\) at points \(A\) and \(B\). Determine if there exists a fixed point \(P\) on the \(x\)-axis such that the dot product \(\overrightarrow{PA} \cdot \overrightarrow{PB}\) remains constant as the line \(l\) rotates about \(F\).
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Answer: \boxed{(4, 0)}
Find the limit of the series as \( n \) approaches infinity:
\[ x(n) = \sum_{k=1}^n \frac{k^2}{n^3+k^2} \]
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Answer: \boxed{\dfrac{1}{3}}
Let $G$ be a group and $H$ be a subgroup of finite index $n$ in $G$. Determine if there exists a normal subgroup $N$ of $G$ contained in $H$ such that the index $[G:N]$ is finite. If such a subgroup exists, provide an upper bound for $[G:N]$. Provide your answer as a specific number or expression.
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Answer: \boxed{n!}
Evaluate the integral \( I = \int_{-\infty}^{\infty} \frac{\sin(ax)}{x(\pi^2-a^2x^2)}\,dx \) for \( a > 0 \).
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Answer: \boxed{\dfrac{2}{\pi}}
In a round-robin tournament with four football teams (each pair of teams plays one match), the winner of each match earns 3 points, the loser earns 0 points, and in the event of a draw, both teams earn 1 point. After the tournament, it was found that no team won all their matches, and all four teams had different scores. Determine the minimum possible number of draws in all matches. Express your answer as a single integer.
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Answer: \boxed{1}
Evaluate the limit: \( \lim_{n\to\infty} \frac{1}{n}\int_{0}^n \frac{dx}{5+4\cos(x)} \).
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Answer: \boxed{\dfrac{1}{3}}
Calculate the variance of the random variable $Y$, which represents the number of draws needed to replace all black balls with white balls in a bag initially containing 2 white balls and 2 black balls. In each draw, a white ball is returned to the bag, while a black ball is replaced by a white ball. Find $\operatorname{var}(Y)$. \( \boxed{?} \)
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Answer: \boxed{14}
$A$ standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Answer: \boxed{187}
Calculate the expected sojourn time for cars that decide to take gas at a station where cars arrive according to a Poisson process with rate \( \frac{1}{3} \). An arriving car finding \( n \) cars at the station leaves with probability \( q_n = \frac{n}{4} \) and joins the queue with probability \( 1 - q_n \), for \( n = 0, 1, 2, 3, 4 \). Cars are served in order of arrival, and the service time is \( \text{Exponential}(\frac{1}{3}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{384}{71}}
Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$.
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Answer: \boxed{p^3 + p^2 - p}
In the  $ x \minus{} y$  plane, find the area of the region bounded by the parameterized curve as follows.
 $ \left\{\begin{array}{ll} x \equal{} \cos 2t & \quad 
y \equal{} t\sin t & \quad \end{array} \right.\ (0\leq t\leq 2\pi)$ 
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Answer: \boxed{\frac{32}{9}}
A metal sphere with a radius of $\sqrt[z]{16}$ is melted and recast into a cone whose lateral surface area is three times the area of its base. Find the height of the cone.
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Answer: \boxed{8}
Given that $a^{-1}ba = b^2$ and $|a| = 3$ in a group, find the order of the element $b$, where $b \neq e$ and $b$ is not of order 2.
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Answer: \boxed{7}
Compute the following expression:
\[ 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4)))))))) \]
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Answer: \boxed{1398100}
Determine the rate of divergence of the series \( \sum_{j=0}^{\infty} \frac{1}{\sqrt{j}} \) as \( n \to \infty \). Express the rate of divergence of the partial sums \( S_n \) in the form \( O(n^{\alpha}) \) for some \( \alpha > 0 \).
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Answer: \boxed{O(n^{1/2})}
In the diagram, $ABCDEFG$ is a regular heptagon (a $7$-sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Answer: \boxed{727}
Determine the greatest positive integer \( x \) such that the expression \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square.
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Answer: \boxed{571}
Determine the minimum number of symbols required to open a lock with 4 symbols (A, B, C, D) and a 3-symbol password, given that the lock does not "rest" after each attempt. Repeated symbols are allowed.
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Answer: \boxed{66}
Evaluate the limit: $$ \lim_{n \to \infty} \left( \int_1^{1+\frac{1}{n}} \left\{ [(x+1)\ln x +1] e^{x (e^x \ln x +1)} + n \right\} \, dx \right)^n $$
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Answer: \boxed{e^e}
Sandy buys an old scooter for $800 and spends $200 on its repairs. In addition, she pays $50 for a new set of tires and $100 for a paint job. The scooter's value also depreciates by 10% per year. If Sandy sells the scooter for $1,200 after 2 years, what is the gain percent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.35\%}