inclusionAI/Ring-lite-rl-data huggingface dataset:
Let  $ABC$  be a triangle with  $\angle  A = 90^o, \angle B = 60^o$  and  $BC = 1$  cm. Draw outside of  $\vartriangle ABC$  three equilateral triangles  $ABD,ACE$  and  $BCF$ . Determine the area of  $\vartriangle DEF$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9\sqrt{3}}{16}}
Solve the equation: \(2^x + x = 4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.386}
There are 99 identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a 9 by 11 rectangle, what is the expected value of the number of full circles formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{16}}
Let  $\Omega$  be a unit circle and  $A$  be a point on  $\Omega$ . An angle  $0 < \theta < 180^\circ$  is chosen uniformly at random, and  $\Omega$  is rotated  $\theta$  degrees clockwise about  $A$ . What is the expected area swept by this rotation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Consider a triangle \( ABC \) with angle \( A = \theta \) where \( \theta \) is in the range \( (0, \pi) \), and side \( BC = 1 \). Determine the number of incongruent triangles for which the points \( A \), the centroid of the triangle, the midpoint of \( AB \), and the midpoint of \( AC \) are concyclic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For two lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel, determine the value of $a$ that satisfies this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
On a circle of radius 12 with center at point \( O \), there are points \( A \) and \( B \). Lines \( AC \) and \( BC \) are tangent to this circle. Another circle with center at point \( M \) is inscribed in triangle \( ABC \) and touches side \( AC \) at point \( K \) and side \( BC \) at point \( H \). The distance from point \( M \) to line \( KH \) is 3. Find \( \angle AOB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$
[i](K. Ivanov )[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Evaluate the integral \(\int_{S^2} \frac{dS}{\sqrt{(x-a)^2 + y^2 + z^2}}\) over the unit sphere \(S^2\), given that \(a > 1\), using only multivariable calculus techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{a}}
Given that $f(x)$ is an odd function defined on $(−∞,0)∪(0,+∞)$, and $f(x+ \frac{5}{2})f(x)=1$, if $f(-1) > 1$ and $f(2016)= \frac{a+3}{a-3}$, then the range of $a$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,3)}
Find the residue at infinity for the function \( \frac{e^{z^2}}{z^{2n+1}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{n!}}
Find the large number from the below question. The difference of two numbers is 24672. On dividing the larger number by the smaller, we get 13 as quotient and 257 as remainder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26706}
A bus stop has two independent Poisson arrival processes: buses arrive according to a Poisson process with rate $\mu$, and passengers arrive according to a Poisson process with rate $\lambda$. Assuming that all passengers waiting at the stop board the bus when it arrives, find the probability that there are exactly $k$ passengers on the third bus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(\lambda/\mu)^k e^{-\lambda/\mu}}{k!}}
In a math competition, there are 6 questions, each worth 7 points, with scores ranging from 0 to 7. The total score of a participant is the product of their scores on the 6 questions. If two participants have the same product score, their ranking is determined by the sum of their scores. If these are also the same, the participants are tied. There are 262,144 participants with no ties. What is the total score of the participant ranked 117,649th?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum number of tokens that must be placed in the cells of a $99 \times 99$ grid such that every $4 \times 4$ subgrid contains at least eight tokens.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4901}
Consider all the permutations of the 26 English alphabets that start with the letter 'z'. In how many of these permutations is the number of letters between 'z' and 'y' less than the number of letters between 'y' and 'x'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144 \times 23!}
Determine the values of \( \alpha \) for which the generalized Gaussian probability density \( p_{\alpha}(t) \propto \exp (- |t|^\alpha) \) is infinitely divisible, given that \( 0<\alpha<\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2]}
Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222480}
Given that \(2 \cdot 50N\) is an integer and its representation in base \(b\) is 777, find the smallest positive integer \(b\) such that \(N\) is a fourth power of an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Among all the line segments connecting a point outside a line to every point on the line, the shortest is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{the perpendicular segment}}
Smith, Brown, and Robinson each bought shares with their wives. Mary bought 50 more shares than Mr. Brown, and Mr. Robinson bought 120 more than Jane. Each man paid as many shillings per share as he bought shares, and each wife as many pence per share as she bought shares. Each man spent one guinea more than his wife. Determine Eliza's surname given these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Robinson}
Given that the terminal side of angle $\alpha$ passes through the fixed point $P$ on the function $y=\log _{a}(x-3)+2$, find the value of $\sin 2\alpha+\cos 2\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{5}}
In Figure 1, \( P, B \) and \( C \) are points on a circle with center \( O \) and diameter \( BC \). If \( A, B, C, D \) are collinear such that \( AB = BC = CD \), \( \alpha = \angle APB \) and \( \beta = \angle CPD \), find the value of \( (\tan \alpha)(\tan \beta) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Evaluate the limit: \[ \lim_{n\to\infty} \frac{\sqrt[n]{(n+1)!}}{n+1} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
3 candidates in an election and received 1136 , 7636 and 10628 votes respectively . what % of the total votes did the winning candidate gotin that election ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.78\%}
2. Let $ABCD$ be a convex quadrilateral such that

$$
\measuredangle ABC=90^{\circ}, \measuredangle DAB=60^{\circ} \text{ and } \measuredangle BCD=120^{\circ}.
$$

Let the diagonals $AC$ and $BD$ intersect at point $M$, such that $\overline{BM}=1$ and $\overline{MD}=2$. Calculate the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
Given that the first term of an arithmetic sequence is $31$, and the terms from the $16$-th term onwards are less than $1$, find the range of the common difference $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-\frac{15}{7}, -2\right)}
Compute the Riemann-Stieltjes integral \( \int_{-1}^{1} (x^2 + e^x) \, d(\text{Sgn}(x)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle \(ABC\), it is known that \(AB = BC\) and \(AC = 4 \sqrt{3}\). The radius of the inscribed circle is 3. Line \(AE\) intersects the altitude \(BD\) at point \(E\), and the inscribed circle at points \(M\) and \(N\) (with \(M\) being between \(A\) and \(E\)). Given that \(ED = 2\), find \(EN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{33}}{2}}
Using the digits 1, 2, 3, 4, and 9 once each, form the smallest possible even five-digit number. What is the digit in the tens place of this number? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Petya bought a "Constructor" set, which contained 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this statement by forming triangles with the sticks. The sticks are arranged in the constructor in ascending order of length. What is the minimum number of checks (in the worst case) that Petya needs to perform to prove or disprove the statement in the instructions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many positive integers  $a$  with  $a\le 154$  are there such that the coefficient of  $x^a$  in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?

*Author: Ray Li*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
If five points are given on a plane, considering all possible triples of these points, 30 angles can be formed. Let the smallest of these angles be denoted as $\alpha$. Find the maximum value of $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^{\circ}}
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0, 1, 2, \ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the die. If the die lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Given that $17^{-1} \equiv 31 \pmod{53}$, find $36^{-1} \pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
设 $M=\{1,2, \cdots, 1995\}, A$ 是 $M$ 的子集, 且满足条件: 当 $x \in$ $A$ 时, $15 x \bar{\in} A$. 求 $|A|$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$|A|$ 的最大值为 1870 .}
For the NEMO, Kevin needs to compute the product
		\[
		9 \times 99 \times 999 \times \cdots \times 999999999.
		\]
		Kevin takes exactly  $ab$  seconds to multiply an  $a$ -digit integer by a  $b$ -digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{870}
Evaluate the limit as \(x\) approaches 0 from the left: 
\[\lim_{x\to0^-}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
Find the radius of convergence for the series \( f(x) = \sum_{n=1}^\infty \left(1+\frac{(-1)^n}{n}\right)^{n^2} \cdot \frac{(2x+1)^n}{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2e}}
In the field \(\mathbb{F}_2\) of order 2, consider the ideal \(I\) in \(\mathbb{F}_2[x]\) generated by the polynomials \(x^5 + x^2 + x + 1\) and \(x^7 + x^5 + x^3 + x\). How many polynomials of degree 3 are elements of \(I\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Some language has only three letters -  $A, B$  and  $C$ . A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all  $A$ . What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^40}
Xiaohong drives to a place to attend a meeting. 30 minutes after departure, due to traffic congestion, there was a delay of 25 minutes. In order to arrive at the meeting venue on time, Xiaohong increased her speed by 25%. Eventually, she arrived at the meeting venue on time. It took Xiaohong a total of     minutes from departure to the meeting venue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Given a regular quadrilateral pyramid S-ABCD, points P and Q are the midpoints of SB and SD respectively. The plane passing through points A, P, and Q divides S-ABCD into two parts with volumes V1 and V2 such that V1 < V2. What is V2/V1 equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( S \) be a subset of \(\{1, 2, \cdots, 1989\}\), and the difference between any two numbers in \( S \) is not equal to 4 or 7. What is the maximum number of elements in \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{905}
Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB=20$ and $CD=14$, compute the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
140
}
Alex is a practical-minded economics student who loves to track his spending and investments. He decides to start investing in a small business project and wants to calculate his potential profits. Alex invests $500 in the project, which promises a 10% return monthly. He also incurs a monthly cost of $50 for project-related expenses. After four months, Alex wants to know his total profit. Calculate Alex's total profit after four months, considering both the returns on his investment and the monthly expenses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.05}
In triangle \(ABC\), \(\angle CAB = 75^\circ\) and \(\angle ABC = 45^\circ\). Point \(K\) is taken on side \(CA\), and point \(M\) on side \(CB\), such that \(CK:KA = 3:1\).

Find \(KM:AB\) if this ratio is less than \(3/4\), and if the line \(MK\) cuts off a triangle from triangle \(ABC\) that is similar to it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 (\sqrt{3} - 1)}{4}}
Given two permutations \(\sigma = (13624)(587)(9)\) and \(\tau=(15862)(394)(7)\) in \(S_9\), determine the number of permutations \(\alpha\) such that \(\alpha\sigma\alpha^{-1} = \tau\). How many such \(\alpha\) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let \( n \) be a positive integer not exceeding 1996. If there exists a \( \theta \) such that \( (\sin \theta + i \cos \theta)^{n} = \sin \theta + i \cos n \theta \), find the number of possible values for \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["499"]}
Evaluate the integral \( \int_{-1}^{1} \frac{x^{1/3} + 2}{\sqrt{|1 - x^{1/3}|} - \sqrt{x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the minimal cardinality of a quotient ring $R/I$, where $I$ is a proper ideal of the ring $R = \mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$. Assume that the polynomial $x^2 - x + 5$ is irreducible over both $\mathbb{F}_2$ and $\mathbb{F}_3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that $n$ and $k$ are positive integers with $n > k$, and real numbers $a_{1}, a_{2}, \cdots, a_{n} \in (k-1, k)$. Let positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the condition that for any $k$-element subset $I$ of $\{1,2,\cdots, n\}$, the inequality $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$ holds. Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_1 a_2 \cdots a_n}
Let \(f(z)\) be a cubic polynomial with distinct roots \(r\), \(s\), and \(t\). If \(g(z)=f(cz)\) for some complex number \(c\), and \(g(z)\) also has distinct roots, find the number of possible values of \(|c|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A store has three types of boxes containing marbles in large, medium, and small sizes, respectively holding 13, 11, and 7 marbles. If someone wants to buy 20 marbles, it can be done without opening the boxes (1 large box plus 1 small box). However, if someone wants to buy 23 marbles, a box must be opened. Find the smallest number such that any purchase of marbles exceeding this number can always be done without opening any boxes. What is this smallest number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Let \( 0 < c < 1 \) and \( n \) a positive integer. Alice and Bob are playing a game. Bob writes \( n \)
 integers on the board, not all equal. On a player's turn, they erase two numbers from the
 board and write their arithmetic mean instead. Alice starts and performs at most \( {cn} \)
 moves. After her, Bob makes moves until there are only two numbers left on the board.
 Alice wins if these two numbers are different, and otherwise, Bob wins.
 For which values of \( c \) does Alice win for all large enough \( n \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( c \geq  \frac{1}{2} \)}
7.5. Miron and Varya are playing the following game. There is a pile of 32 stones on the table. The players take turns, with Miron starting first. On their turn, a player divides any pile that has more than one stone into several equal piles. The player who cannot make a move (when there is exactly one stone in each pile before their turn) wins. Who will win if both players play optimally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Varya}
What is the maximum number of cells in an 8x8 chessboard that can be cut by a single straight line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let \( A = \{1, 2, \cdots, 2004\} \) and \( f: A \rightarrow A \) be a bijection satisfying \( f^{[2004]}(x) = f(x) \), where \( f^{[2004]}(x) \) denotes applying \( f \) 2004 times to \( x \). How many such functions \( f \) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2004 \cdot 2002!}
In how many ways can 8 people be seated in a row of chairs if three of those people, Alice, Bob, and Charlie, refuse to sit next to each other in any order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36000}
Determine the number of Boolean functions \( F(w, x, y, z) \) that satisfy the condition \( F(\bar{w}, x, \bar{y}, z) = F(w, \bar{x}, y, \bar{z}) = F(w, x, \bar{y}, \bar{z}) \) for all Boolean variables \( w, x, y, \) and \( z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Solve the equation: \[ \frac{10}{x+10}+\frac{10 \cdot 9}{(x+10)(x+9)}+\frac{10 \cdot 9 \cdot 8}{(x+10)(x+9)(x+8)}+\cdots+ \frac{10 \cdot 9 \ldots 2 \cdot 1}{(x+10)(x+9) \ldots(x+1)}=11 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{11}}
Given an isosceles triangle $ABC$ with base $AC$. Points $K$ and $N$ are marked on the side $BC$ (with $K$ between $B$ and $N$). It is found that $KN = AN$ and $\angle BAK = \angle NAC$. Find $\angle BAN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
In the game "set," consider all possible four-digit numbers that can be formed using the digits 1, 2, and 3, with each digit used exactly once in each number. A triplet of numbers forms a set if, for each digit position, either all three numbers have the same digit, or all three numbers have different digits. How many distinct sets can be formed in this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1080}
The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Find the expectation of the posterior distribution given by:
\[ p_{\theta|x}(\theta) = \frac{\theta^x(1-\theta)^x\cdot6\theta(1-\theta)}{\int_0^1\theta^x\cdot6\theta(1-\theta)d\theta} \] where \( X \) is a random variable from the Bernoulli distribution with parameter \( \theta \), and the prior density \( p(\theta) = 6\theta(1 - \theta) \) for \( \theta \in (0, 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the independent coefficient in the inverse of the polynomial $2x^2+2$ in the ring $\mathbb{Z}_3[x]/(x^3+2x^2+2)$. Provide your answer as an integer in $\mathbb{Z}_3$. \(\boxed{\phantom{0}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
David and I each have a sheet of computer paper, mine evenly spaced by $19$ parallel lines into $20$ sections, and his evenly spaced by $29$ parallel lines into $30$ sections. If our two sheets are overlayed, how many pairs of lines are perfectly incident?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}