inclusionAI/Ring-lite-rl-data huggingface dataset:
Among 100 products, there are 5 defective ones. If 10 products are randomly selected with replacement, let X represent the number of defective products among the 10 selected. What is the distribution of X, including its parameters?

Express your answer in the format of the distribution type and its parameters, e.g., "X ~ B(n, p)".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X ~ B(10, 0.05)}
$已知函数f(x)=$
$\begin{cases}
-x^2+ax+1,& x \leq 1,
ax,& \text{x > 1}.
\end{cases}$
$若f(x)恰有两个极值点，则a的取值范围是\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(0,2)$}
A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{308253}
9.5. At the summit, 2018 politicians gathered. Every two of them were supposed to hold private negotiations. At some point, it was found that among any four, there is one who has already talked to the other three. What is the maximum number of negotiations left to be held?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inward. Calculate the ratio of the shaded area formed between the semicircles and the quarter-circle to the area of a circle with a radius equal to the radius of the quarter circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
John decides to replace all his VHS with DVDs. He can trade in his VHS for different rates depending on the movie genre: 

- $3 for action movies
- $2 for comedy movies
- $1 for drama movies 

He then must buy the DVDs for various prices based on their popularity: 

- $12 for popular movies 
- $8 for moderately popular movies 
- $5 for unpopular movies 

John has the following 100 movies to replace: 

- 20 popular action movies
- 30 moderately popular comedy movies
- 10 unpopular drama movies
- 15 popular comedy movies
- 25 moderately popular action movies

How much does it cost to replace all of his movies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$675}
Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{382}
A train 500 m long can cross an electric pole in 20 seconds. Find the speed of the train.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
There were five teams entered in a competition. Each team consisted of either only boys or only girls. The number of team members was $9, 15, 17, 19$, and $21$. After one team of girls had been knocked out of the competition, the number of girls still competing was three times the number of boys. How many girls were in the team that was eliminated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let \( p \) denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let \( q \) denote the probability that the second shot is also a blank if the cylinder is not spun after the first shot. Find the smallest integer \( N \) such that
\[ 
N \geq \frac{100 p}{q} .
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Let $z$ be a complex number and $k$ a positive integer such that $z^{k}$ is a positive real number other than 1. Let $f(n)$ denote the real part of the complex number $z^{n}$. Assume the parabola $p(n)=an^{2}+bn+c$ intersects $f(n)$ four times, at $n=0,1,2,3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
If the odd function \( f(x) \) is defined on \( (-\infty, 0) \cup (0, +\infty) \) and is increasing on \( (0, +\infty) \), \( f(1) = 0 \). Also, there is a function \( g(\theta) = \sin^2 \theta + m \cos \theta - 2m \), where \( \theta \in \left[0, \frac{\pi}{2}\right] \). Given the set \( M = \{m \mid g(\theta) < 0\} \) and the set \( N = \{m \mid f[g(\theta)] < 0\} \), find the set \( M \cap N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > 4 - 2\sqrt{2}}
It is known that the sum of the first \( n \) terms of a geometric progression consisting of positive numbers is \( S \), and the sum of the reciprocals of the first \( n \) terms of this progression is \( R \). Find the product of the first \( n \) terms of this progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{S}{R}\right)^{\frac{n}{2}}}
There are 300 children in a camp. Everyone has no more than $k-1$ friends. What is the smallest $k$ for which it might be impossible to create some new friendships so that everyone has exactly $k$ friends?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151}
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7$}
A solid cube of side length $2$ is removed from each corner of a larger solid cube of side length $4$. Find the number of edges of the remaining solid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
# Problem 8. (5 points)

In triangle $ABC$, $AB=13$ and $BC=15$. A point $D$ is taken on side $AC$ such that $AD=5$ and $CD=9$. The bisector of the angle adjacent to angle $A$ intersects line $BD$ at point $E$. Find $DE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Piercarlo chooses \( n \) integers from 1 to 1000 inclusive. None of his integers is prime, and no two of them share a factor greater than 1. What is the greatest possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
The equations
\[60x^4 + ax^3 + bx^2 + cx + 20 = 0\]and
\[20x^5 + dx^4 + ex^3 + fx^2 + gx + 60 = 0\]have a common rational root $r$ which is not an integer, and which is positive.  What is $r?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \frac{8}{5}a_n + \frac{6}{5}\sqrt {4^n - a_n^2}$ for $n\geq 0$. Find the greatest integer less than or equal to $a_{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{983}
What is the logical negation of the proposition p: ∃x ∈ ℕ, x^3 < 3^x? Express your answer as a logical statement using quantifiers and predicates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∀x ∈ ℕ, x^3 ≥ 3^x}
For any positive integer  $n$ , let  $D_n$  denote the set of all positive divisors of  $n$ , and let  $f_i(n)$  denote the size of the set   $$ F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \} $$  where  $i = 1, 2$ .
Determine the smallest positive integer  $m$  such that  $2f_1(m) - f_2(m) = 2017$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cdot 5^{2016}}
Given the function $y=\lg ( \frac {1}{x}-1)$, its domain is $A$. If the inequality $\frac {9x}{2-2x}-m^{2}x-2mx > -2$ always holds for any $x\in A$, then the range of the positive real number $m$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac {3 \sqrt {6}-2}{2})}
Arrange the following numbers in ascending order and determine the position of 12211 in the sorted list: 121, 112, 1112, 12112, 11122, 12211, 21211, 12121, 11221. Express your answer as a numerical position (e.g., 1st, 2nd, 3rd, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \(M\) be the intersection of the diagonals \(AC\) and \(BD\) of cyclic quadrilateral \(ABCD\). If \(|AB|=5\), \(|CD|=3\), and \(m(\widehat{AMB}) = 60^\circ\), find the circumradius of the quadrilateral. If the circumradius is expressed in the form \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers, what is the value of \(a + b + c\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given any number a from the set {1, 2, 3, ..., 99, 100} and any number b from the same set, calculate the probability that the last digit of 3^a + 7^b is 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{16}}
A rectangular tiled patio is composed of 160 square tiles. The rectangular patio will be rearranged so that there will be fewer columns of tiles and 4 more rows of tiles. After the change in layout, the patio will still have 160 tiles, and it will still be rectangular. There were 10 rows in the tile patio before the change in layout. How many fewer columns of tiles will there be after the change in layout?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the asymptotic order of $\sum_{k=0}^n \binom{n}{n-k}\binom{n+m}{k}$, where $m$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(2^{2n})}
During the Nanchang Flower Expo, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that each of the exhibition areas A and B should have one person, and the remaining two exhibition areas should have two people each. Additionally, Xiao Li and Xiao Wang cannot be in the same group. Calculate the total number of different arrangement plans possible under these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
Given an isosceles triangle \( ABC \), where \( BC = a \) and \( AB = AC = b \). A triangle \( ADC \) is constructed such that \( AD = DC = a \) on the external side of \( AC \). Let \( CM \) and \( CN \) be the angle bisectors in triangles \( ABC \) and \( ADC \) respectively. Find the radius of the circumcircle of triangle \( CMN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ab}{a+b}}
Rhona wrote down a list of nine multiples of ten: $10, 20, 30, 40, 50, 60, 70, 80, 90$. She then deleted some of these multiples so that the product of the remaining multiples was a square number. What is the least number of multiples that she could have deleted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the number of eight-digit numbers for which the product of the digits is equal to 16875. The answer should be an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1120}
for any integer k > 1 , the term “ length of an integer ” refers to the number of positive prime factors , not necessarily distinct , whose product is equal to k . for example , if k = 24 , the length of k is equal to 4 , since 24 = 2 × 2 × 2 × 3 . if x and y are positive integers such that x > 1 , y > 1 , and x + 3 y < 1000 , what is the maximum possible sum of the length of x and the length of y ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A cargo ship moving from Shanghai to Vancouver navigates for 21 days before reaching port. Customs and regulatory processes in Vancouver last 4 days. Finally, moving the cargo from the port to your rural warehouse takes some time and it always arrives on the seventh day. How many days ago should the ship have departed if your warehouse is expecting the shipment 2 days from today?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
If $a$ is a root of $x^3-x-1 = 0$, compute the value of $$a^{10 }+ 2a^8 -a^7 - 3a^6 - 3a^5 + 4a^4 + 2a^3 - 4a^4 - 6a - 17.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-15}
Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given vectors \(\vec{a} = \{1,2\}\) and \(\vec{b} = \{-3,2\}\), find the real number \(k\) such that \(k \vec{a} + \vec{b}\) is in the same or opposite direction as \(\vec{a} - 3 \vec{b}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
In quadrilateral \( \square ABCD \), point \( M \) lies on diagonal \( BD \) with \( MD = 3BM \). Let \( AM \) intersect \( BC \) at point \( N \). Find the value of \( \frac{S_{\triangle MND}}{S_{\square ABCD}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
A certain number, when divided by 195, leaves a remainder of 79, and when divided by 273, leaves a remainder of 109. What is the remainder when the same number is divided by 39?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $P(x)=(x-1)(x-2)(x-3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree 3 such that $P(Q(x))=P(x) \cdot R(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
There are 21 nonzero numbers. For each pair of these numbers, their sum and product are calculated. It turns out that half of all the sums are positive and half are negative. What is the maximum possible number of positive products?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given the function $f(x)=2022^{x-3}+(x-3)^{3}-2022^{3-x}+2x$, the solution set of the inequality $f(x^{2}-4)+f(2-2x)\leqslant 12$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2,4]}
A company that ships boxes to a total of 12 distribution centers uses color and shape coding to identify each center. If either a single color, a pair of two different colors, a single shape, or a pair of two different shapes is chosen to represent each center, and if each center is uniquely represented by that choice of one color, two colors, one shape, or two shapes, what is the minimum number of colors and shapes needed for the coding? (Assume that the order of the colors in a pair does not matter, but the order of the shapes in a pair does matter. Additionally, specific color combinations should be used only with specific shapes.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is $\text{CCAMT}$? Here are examples of possible sequences:

- $C, CA, CAM, CCAM, CCAMT$.
- $A, AT, CAT, CAMT, CCAMT$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given the set $A=\left\{ x|{x^2}-x-2\leqslant 0 \right\}$ and the set $B=\left\{ x|1 < x\leqslant 3 \right\}$, find ${C_R}A\cup B=\_\_\_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,-1)\cup(1,+\infty)}
For real numbers $x$, let  
$$P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)$$
where $i = \sqrt{-1}$. Find the number of values of $x$ with $0\leq x<2\pi$ for which  
$$P(x)=0.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In $\triangle ABC$, $\angle A = 60^\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
The price of electricity went up by 25%. John's old computer used a certain amount of watts and his new computer uses 50% more. If the old price of electricity was 12 cents per kilowatt-hour, his computer costs $9 to run for 50 hours. How many watts did John's old computer use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1500}
The product \(\left(\frac{1}{2^3-1}+\frac{1}{2}\right)\left(\frac{1}{3^3-1}+\frac{1}{2}\right)\left(\frac{1}{4^3-1}+\frac{1}{2}\right)\cdots\left(\frac{1}{100^3-1}+\frac{1}{2}\right)\) can be written as \(\frac{r}{s2^t}\) where \(r\), \(s\), and \(t\) are positive integers and \(r\) and \(s\) are odd and relatively prime. Find \(r+s+t\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5990}
Find the largest positive integer  $n$  such that the following is true:
There exists  $n$  distinct positive integers  $x_1,~x_2,\dots,x_n$  such that whatever the numbers  $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$  are, not all null, the number  $n^3$  do not divide  $\sum_{k=1}^n a_kx_k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In a group of 100 cars, some cars do not have air conditioning. If at least 41 cars have racing stripes, the greatest number of cars that could have air conditioning but not racing stripes is 59. How many cars do not have air conditioning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Given differentiable functions $z$ and $w$ of $x$ and $y$ that satisfy the equations:

$$xw^3 + yz^2 + z^3 = -1$$
$$zw^3 - xz^3 + y^2w = 1$$

Find the partial derivative of $z$ with respect to $x$, $\frac{\partial z}{\partial x}$, at the point $\left(x, y, z, w\right) = \left(1, -1, -1, 1\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{4}}
Given the function $f(x)=\frac{x}{ax+b}(a≠0)$, and its graph passes through the point $(-4,4)$, and is symmetric about the line $y=-x$, find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Bob knows that Alice has 2021 secret positive integers \( x_{1}, \ldots, x_{2021} \) that are pairwise relatively prime. Bob would like to figure out Alice's integers. He is allowed to choose a set \( S \subseteq\{1,2, \ldots, 2021\} \) and ask her for the product of \( x_{i} \) over \( i \in S \). Alice must answer each of Bob's queries truthfully, and Bob may use Alice's previous answers to decide his next query. Compute the minimum number of queries Bob needs to guarantee that he can figure out each of Alice's integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{457064}
Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) (where \( a, b > 0 \)), with left and right foci denoted as \( F_{1} \) and \( F_{2} \) respectively, and given that the hyperbola intersects with the circle \( x^{2}+y^{2}=r^{2} \) (where \( r > 0 \)) at a point \( P \). If the maximum value of \( \frac{\left| P F_{1} \right| + \left| P F_{2} \right|}{r} \) is \( 4 \sqrt{2} \), determine the eccentricity of the hyperbola \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{2}}
James' keyboard is designed for both English and Russian languages, and it also has customizable function keys. In addition to the 26 English alphabet keys, there are 33 Russian alphabet keys, 8 customizable function keys, 5 accent marks, and 3 special characters. If the keyboard is missing 3/8 of the English consonants, four English vowels, 1/4 of the Russian consonants, three Russian vowels, 1/3 of the customizable function keys, 2/5 of the accent marks, and half of the special characters, how many keys are missing in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Mike and Harry play a game on an 8x8 board. Mike chooses \( k \) squares and writes an \( M \) in each of them. Harry then chooses \( k+1 \) squares and writes an \( H \) in each of them. After Harry is done, Mike wins if there is a sequence of letters forming " \( H M M \) " or " \( M M H \) ", when read either horizontally or vertically, and Harry wins otherwise. Determine the smallest value of \( k \) for which Mike has a winning strategy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$, and area $S_{\triangle ABC} = \frac{a^2 + b^2 - c^2}{4}$, find angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^{\circ}}
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the Cartesian coordinate system, given the set of points $I=\{(x, y) \mid x$ and $y$ are integers, and $0 \leq x \leq 5,0 \leq y \leq 5\}$, find the number of distinct squares that can be formed with vertices from the set $I$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
A car goes uphill at the rate of 30 km an hour and downhill at the rate of 50 km an hour. After 15 hours, it has covered a certain distance. It went downhill and uphill for 5 hours respectively. What is the total distance covered by the car?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}