inclusionAI/Ring-lite-rl-data huggingface dataset:
Let  $x_{1}$  be a positive real number and for every integer  $n \geq 1$  let  $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$ . 
If  $x_{5} = 43$ , what is the sum of digits of the largest prime factors of  $x_{6}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
C and D are points on a circle with diameter AB such that \( \angle AQB = 2\angle COD \). The tangents at C and D intersect at point P. Given that the circle has a radius of 1, determine the distance from point P to the center of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\sqrt{3}}{3}}
Consider the sequence of random variables $X_1, X_2, \ldots, X_n, \ldots$ where $X_n \sim \text{Uniform}(0,n)$. Define $Y_n = e^{-X_n}$. Determine whether the sequence $Y_n$ converges in probability, and if it does, find its limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A man nearing retirement wants to invest some money in a fund that compounds annually at a rate of 6 percent, so that in five years, he will have at least $\$100,000$.  How much money must he invest now to make that happen?  (Give your answer to the nearest dollar.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74726}
Let \( g(x) = \frac{2}{x^2 - 8x + 17} \). Determine the difference between the largest and smallest possible values of the parameter \( a \) such that the function is defined for all real numbers \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{\gamma} f(z) \, dz \), where \( f(z) = \pi e^{\pi \bar{z}} \) and \( \gamma \) is the square with vertices \( 0, 1, 1+i, \) and \( i \) in the counterclockwise direction. Determine if the integral is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(e^{\pi} - 1)}
In an undirected, weighted, connected graph with distinct edge weights, if there are multiple minimum spanning trees, is it possible for there to be multiple shortest paths between two vertices? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. Find the number of such different patterns.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine whether the limit \( \lim_{n\rightarrow\infty}\int_0^\infty \frac{\sin(x^n)}{x^n}\,dx \) exists, and if it does, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A positive integer is called \textit{superprime} if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find number of solutions in non-negative reals to the following equations: 
\begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^n }
How many bricks each measuring 20 cm x 10 cm x 8 cm will be needed to build a wall 10 m x 8 m x 24.5 m?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,225,000}
$P$ is the intersection point of diagonals of cyclic quadrilateral $ABCD$. The circumcenters of $\triangle APB$ and $\triangle CPD$ lie on the circumcircle of $ABCD$. If $AC + BD = 18$, find the area of $ABCD$. The original answer is in the form \(\frac{k\sqrt{m}}{n}\), please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
Bogan laid out 10 maggots for her pet beetle. The beetle only ate 1 and Bogan had to throw out the rest. Later that day, she tried feeding again and the beetle ate 3. If Bogan served 20 maggots in total, how many did she attempt to feed the beetle the second time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The non-zero digits \( p, q \) and \( r \) are used to make up the three-digit number \( pqr \), the two-digit number \( qr \), and the one-digit number \( r \). The sum of these numbers is 912. Find the value of \( q \). Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that they form a rectangle $ABEF$. Rectangle $KLMN$ is positioned such that its vertices $K, L, M$, and $N$ lie on segments $CD, DF, FE$, and $EC$, respectively. It is given that rectangles $ABEF$ and $KLMN$ are congruent with $AB = MN$. Find the length of segment $AL$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.8 \times 10^{-37}}
A ball, sliding on a smooth horizontal surface, catches up with a block that was moving on the same surface. The speed of the ball is perpendicular to the face of the block it hits. The mass of the ball is much smaller than the mass of the block. After an elastic collision, the ball slides on the surface in the opposite direction with a speed that is 2 times less than its initial speed.

Find the ratio of the velocities of the ball and the block before the collision.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If Billy rode his bike 17 times, John rode his bike twice as many times, their mother rode her bike 10 times more than John, their cousin Amy rode her bike 3 times more than the square root of the sum of John's and Billy's rides, and their friend Sam rode his bike a total of half their mother's biking frequency minus 5, how many times did they ride their bikes in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{133}
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ Find the remainder when $N$ is divided by $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $f(x)$ be a quadratic polynomial with a positive leading coefficient. Define $g(x) = 1 - \frac{2}{x-5}$ with the domain $x < 5$. For any real number $t < 3$, let $h(t)$ be the minimum of $f(g(x))$ for $t \leq x \leq t+2$. It is given that $h(t) = f(g(t+2))$ when $t < 1$, $h(t) = 6$ for $1 \leq t < 3$, and $h(-1) = 7$. Determine the value of $f(5)$. Assume that continuity cannot be used in the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Find the value of $m$ that minimizes the expression $\binom{m}{k} + \binom{n-m}{k}$, given that $n$ and $k$ are fixed integers and $m = n/2$ if $n$ is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \dfrac{n}{2} \right\rfloor}
Consider the sequence defined by $a_1 = 0$, $a_2 = 1$, and for $n \geq 3$, $a_n = (n-1)(a_{n-1} + a_{n-2})$. Determine the remainder when $a_{2019}$ is divided by 2019.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
There are three bags:
1. Bag A contains 3 green candies and 1 red candy.
2. Bag B contains 2 green candies and 2 red candies.
3. Bag C contains 1 green candy and 3 red candies.

A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. 

- If the first candy is green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag.  
- If the first candy is red, the child randomly selects a second candy from the same bag as the first candy.  

If the probability that the second candy is green is given by the fraction $\frac{m}{n}$ in lowest terms, find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{217}
Calculate the number of selfish subsets of the set \( \{1, 2, \ldots, n\} \), where a subset is defined as selfish if it contains its own cardinality and all its other elements are greater than its cardinality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_n}
Find all positive integer solutions \((x, m, n)\) for the equation \( x^{m} = 2^{2n+1} + 2^{n} + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(23, 2, 4)}
Determine whether the set of numbers $x > 1$ for which the set $\{\sin(x^n) \mid n \in \mathbb{N}\}$ is dense in $[-1, 1]$ is meager or has cardinality $2^{\aleph_0}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
On their way driving to Los Angeles, Benjamin and James see lots of billboard ads. They spot two types of billboards: electronic and traditional. Curious, Benjamin starts to count only the electronic billboards they see within specific periods each hour. In the first hour, he notes the billboards during a 25-minute period and counts 15 electronic billboards. They stop for a 15-minute break within this first hour. In the second hour, he only counts during a 45-minute period and sees 31 electronic billboards. In the third hour, during their 35-minute lunch break, Benjamin counts 10 electronic billboards from the restaurant. After resuming their journey, he counts for a 20-minute period and sees 12 electronic billboards. James asks Benjamin the average number of electronic billboards they've seen per hour, considering the varying counting periods and breaks they took during the trip. What should Benjamin tell him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.64}
For which values of \( n \) is it possible to divide any given angle into \( n \) equal parts using only a compass and a straight edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
Let \( T:V\rightarrow W \) be a linear map of inner product spaces with \( T^\ast \) the dual map. Calculate the function \( f(\lambda) = \operatorname{tr}e^{-\lambda T^\ast T} - \operatorname{tr}e^{-\lambda TT^\ast} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim V - \dim W}
In the given figure, the line segment \( KM_1 \) intersects the extension of \( AB \) at point \( N \). Determine the measure of angle \( \angle DNA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Is there a numeral system where numbers can have negative values without using a negative sign? If so, determine the base of such a system.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Two numbers between $0$ and $1$ on a number line are to be chosen at random. What is the probability that the second number chosen will exceed the first number chosen by a distance greater than $\frac 14$ unit on the number line? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{32}}
In how many ways can 4 distinct beads be placed on a bracelet that has a knot in it?  (Beads cannot slide past the knot.  Two ways are considered the same if one is a reflection of the other, since we can flip the bracelet over.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $X_t=B_t-tB_1$, where $B_t$ represents standard Brownian motion for $0\leq t\leq 1$. For $0\leq s<t\leq 1$, find $$\mathbb{E}(X_s|X_t=1).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{s}{t}\)}
Given a list of numbers from 1 to 1000, there are 499,500 unique pairs. If you draw 8 numbers at once, what is the minimum number of tries needed to pick all unique pairs, allowing for repetitions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17875}
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given a geometric surface and three points A, B, and C on it, with known geodesic distances AB and AC, determine if the geodesic distance BC satisfies the triangle inequality: AC ≤ AB + BC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{AC \leq AB + BC}
An urn contains 4 blue and 4 red marbles. Initially, a marble is drawn (without looking) and removed from the urn. Then, a marble is drawn from the urn, its color recorded, and put back in the urn. This process is repeated 1000 times. Let event D be that between 450 and 550 blue marbles are drawn, and let event E be that between 379 and 479 blue marbles are drawn. Which event is more likely, D or E?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E}
Let \( f \in \mathcal{S}(\mathbb{R}^3) \) be a real function. Consider the integral \[ I_f = \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{f(x)f(y)}{|x-y|^2} \, dy \, dx. \] Determine whether it is true that \( I_f \geq 0 \) for a generic function \( f \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_f \geq 0}
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p=3,q=7,r=5,s=2}
In triangle $A B C$, points $M$ and $N$ are the midpoints of $A B$ and $A C$, respectively, and points $P$ and $Q$ trisect $B C$. Given that $A, M, N, P$, and $Q$ lie on a circle and $B C=1$, compute the area of triangle $A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{7}}{12}}
Let $T  =  TNFTPP$. As $n$ ranges over the integers, the expression $n^4  -  898n^2  +  T  -  2160$ evaluates to just one prime number. Find this prime.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1801
}
The area of a rectangle remains unchanged when it is made $2\frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2\frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is:

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20
}
Consider a square matrix $A$ with real entries and a generalized eigenvector $v$ of rank $r$ for the eigenvalue $\lambda$, where $v$ is non-null. It is known that $v$ is also a generalized eigenvector of $A^n$ with rank $s \leq r$. If $\lambda \neq 0$, determine whether $s = r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s = r}
Determine if the function \( f(x) = x^2 \) is Riemann-Stieltjes integrable with respect to \( g(x) = 3x \) on the interval \([0,1]\), given that \( g(1) = 4 \). If it is integrable, find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. Find the number of heavy-tailed permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine whether the limit \( \lim_{x \to \infty} \ln\left(\frac{x}{m}\right) \sum_{p \le x} \frac{1}{\ln p} \) converges, where \( p \) denotes all primes and \( m \) is the largest prime less than \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If $f(x) = x^2+6x^2+12x+6$, solve the equation $f(f(f(x))) = 0.$


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ x = -2 + \sqrt[27]{2} \]}
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and$$S(P(n)) = P(S(n)).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The only solution is \( P(x) = x \).}
Solve the partial differential equation \( \frac{\partial z}{\partial x} \cdot \frac{\partial z}{\partial y} = 1 \) with the initial condition \( z(0,0) = 0 \). Determine the value of \( z(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+$ $f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{2}}
Determine the structure of the group \( G = \mathbb{Z} \times \mathbb{Z} / H \) where \( H = \langle (4, -3) \rangle \), and decide if it is torsion-free.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Given a geometric surface representing a real-world topography, such as a mountain range, and three points A, B, and C on this surface, where the geodesic distances $AB$ and $AC$ are known, determine if the geodesic distance $BC$ necessarily satisfies the triangle inequality, i.e., is $AC \leq AB + BC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{AC \leq AB + BC}
A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
75%}
Determine whether the limit \( \lim _{t \to \infty} t \cdot \int_t^\infty \frac {\sin^2 (u)}{u^2} \, \mathrm{d}u \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
4. In the Olympionov family, it is a tradition to especially celebrate the day when a person turns as many years old as the sum of the digits of their birth year. Kolya Olympionov had such a celebration in 2013, and Tolya Olympionov had one in 2014. Who is older and by how many years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the cardinality of the $\sigma$-algebra generated by the sets $X_i := \{(x_1, x_2, ..., x_n): x_i < \frac{1}{2}\}$ for $i = 1, 2, ..., n$ in the space $X = [0, 1]^n$. Is the cardinality of this $\sigma$-algebra $2^n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
Determine whether the result $x(t) = Ce^t$ holds for solutions $x \in \mathcal{D}'(\mathbb{R})$ of the differential equation $x' = x$ in the theory of distributions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x(t) = Ce^t}
Let \(\mathcal{F}\) be the family of holomorphic functions \(f\) on the open unit disk such that \(\Re f > 0\) for all \(z\) and \(f(0) = 1\). Compute \(\alpha = \sup\{|f'(0)| : f \in \mathcal{F}\}\). Determine whether or not the supremum \(\alpha\) is attained.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine if the limit \( \lim_{x \to x_0} f(x) \) exists, where \( f(x) = \begin{cases} h(x) & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} \) and \( h(x) \) is a continuous function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Is the genus of $CP^N$ equal to 0 for all $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$,

$$a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.$$

What is $|a_{2009}|$?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0
}
## Zadatak B-4.6.

Zadane su elipsa s jednadžbom $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ i hiperbola kojoj su žarišta u tjemenima elipse na velikoj osi, a tjemena u žarištima elipse. Kolika je površina šesterokuta kojemu su vrhovi tjemena elipse na maloj osi i sjecišta zadanih krivulja?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12}{7}(\sqrt{10}+\sqrt{35})}
NT3 Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
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}
Assume that the time between emissions from a radioactive source is independent and exponentially distributed with rate one. Each emission is detected by a Geiger counter with probability \( p \). Compute the density of the distribution of the time between detections of particles and identify the distribution by name.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p e^{-pt}}
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?

(Walther Janous)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{257}
Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
In a Euclidean manifold $\mathbb{R}^n$, given two tangent vectors at a point $p$, $v_p = v^i \frac{\partial}{\partial x^i}_p$ and $w_p = w^j \frac{\partial}{\partial x^j}_p$, define the dot product between $v_p$ and $w_p$ and provide its mathematical expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{v^i w^i}
A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. What fraction of the equilateral triangle's area is covered by the infinite series of squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 - \sqrt{3}}{2}}
用1～4这四个数字分别组成两个两位数，使这两个两位数的乘积最大。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41 \times 32 = 1312}