inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the limit \( \lambda = \lim\limits_{n \to \infty}{n\int_0^{\frac{\pi}{2}}(\sin x)^{2n} dx} \) without using Wallis' integral formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $S$ be a helicoid defined by the parametric equation $\sigma(u, v) = (u \cos v, u \sin v, v)$. Determine the normal curvature in the direction of a general unit tangent vector at the origin $p = (0, 0, 0)$. Provide your answer in terms of the components of the unit tangent vector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2ab}
Given random variables $X_1, \dots, X_n$ and $Y_1, \dots, Y_n$ taking values in $\{0,1\}$, with the conditions $\sum_{i=1}^n X_i = \sum_{i=1}^n Y_i$, $\Pr[X_i=0] \geq \alpha$, and $\Pr[Y_i=0] \geq \beta$, determine the upper bound on $\alpha + \beta$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12
}
Let $q$ be the sum of the expressions $a_1^{-a_2^{a_3^{a_4}}}$ over all permutations $(a_1, a_2, a_3, a_4)$ of $(1,2,3,4).$ Determine $\lfloor q \rfloor.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8
}
Determine the value of $p$ such that the function \( f(x,y) = \frac{1}{(x^2+y^3-2)^p} \) is integrable over the region \( 1 \leq x \leq 2, 1 \leq y \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p < 2}
Determine the maximum number of edges in a directed graph with 7 vertices, 2 connected components, and 4 strongly-connected components.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the limit: $$\lim_{x \to 0} \frac{\ln\left[\dfrac{(1-3x)(1+x)^3}{(1+3x)(1-x)^3}\right]}{x^3}$$ without using L'Hôpital's rule, Taylor series, or differentiation (or integration).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-16}
Let $T$ be the linear transformation defined by $T=\begin{bmatrix}3&1\\1&3\end{bmatrix}$. Determine the area of the region enclosed by $\bigcap_{n\geq1}T^{n}D$, where $D$ is the unit disk in $\mathbb{R^{2}}$. \(\text{Provide your answer as a single numerical value.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi}
Determine the number of positive integers \( n \leq 1988 \) for which the function \( f(n) = n \), given that \( f(1) = 1 \), \( f(3) = 3 \), \( f(2n) = f(n) \), \( f(4n + 1) = 2f(2n + 1) - f(n) \), and \( f(4n + 3) = 3f(2n + 1) - 2f(n) \) for all positive integers \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
Find the function $f(X)$ that minimizes the conditional expected value $E[(Y-f(X))^2|X]$, given that $X$ and $Y$ are random variables with $E(Y) = \mu$ and $E(Y^2) < \infty$. Provide your answer in terms of $X$ and $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E(Y|X)}
In parallelogram $ABCD$, $
abla ABC=105^{\circ}$. Triangle $BMC$ is equilateral, and $
abla CMD=135^{\circ}$. Point $K$ is the midpoint of $AB$. Determine the measure of $
abla BKC$. \( \text{(Express your answer in degrees.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
What is the smallest integer $k$ such that $PSU_3$ is a subgroup of $GL_k(\mathbb{C})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the group of symmetries for the elements of the quaternion group, excluding 1 and -1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_4}
Evaluate the line integral \( \int_\gamma \frac{-y^2 \, dx + 2xy \, dy}{x^2 + y^4} \) along the path \( \gamma: r(t) = (t, 2t^2 - 2) \) for \( -1 \leq t \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. What is the probability that exactly one of the stacks consists of two red boxes and two blue boxes? Express your answer as a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, and find \( m + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{843}
Find the norm of the inverse operator \(A^{-1}\) for the operator \(A\) defined on \(L^2[0,1]\) by \((Ax)(t) = x(t) - \frac{1}{2}\int_{0}^{1}x(s)ds\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $p,q \ge 2$ be such that $p+1$ is a divisor of $q+1$. Let $S$ be a star with $p$ edges. Calculate the Ramsey number $R(S,K_{1,q})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + q}
Suppose \(X_1, \ldots, X_n\) are a random sample from a normal distribution with mean \(\theta\) and variance \(\sigma^2\), where \(\sigma^2\) is fixed and \(\theta > 0\) is a parameter. Find the maximum likelihood estimator of \(\sqrt{\theta}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\overline{X}}}
Given independent and identically distributed (i.i.d.) random variables $Z, \epsilon_1, \epsilon_2 \sim N(0,1)$, where $X = Z + \epsilon_1$ and $Y = Z + \epsilon_2$, find the expected value of $\exp(XY)$. \( \text{E}[\exp(XY)] \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \( X_1, X_2, \ldots, X_{48} \) be a random sample of size 48 from a distribution with probability density function \( f(x) = \frac{1}{x^2} \) for \( 1 < x < \infty \). Approximate the probability that at most 10 of these random variables have values greater than 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3085}
A die is thrown repeatedly until the sequence "5 followed by 3" appears. What is the expected number of throws required for this sequence to occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Evaluate the integral: $$\int_{-2}^2 (x^4+4x+\cos(x))\cdot \arctan\left(\frac{x}{2}\right)dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi - 16}
Calculate the mass of a half disc of radius $R$ centered at the origin $(0,0)$, with $y>0$, given that the density is \( \rho = \frac{\rho_oy}{\sqrt{x^2+y^2}} \) for \( y>0 \), where \( \rho_o \) is a constant with units of \( \text{kg.m}^{-3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\rho_o R^2}
Given the functions \( D(\rho) = g'(\rho)e^{ia\rho} - g'(-\rho)e^{-ia\rho} \) and \( S(\rho) = g'(\rho)e^{ia\rho} + g'(-\rho)e^{-ia\rho} \), determine if the ratio \( \frac{D(\rho)}{S(\rho)} \) is independent of the function \( g'(\rho) \) when \( g'(\rho) \neq g'(-\rho) \). If it is independent, find the ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-i \cot(a\rho)}
Determine the condition under which the right cosets $Nu$ and $Nv$ are equal, given that $u, v \in G$ for a group $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{uv^{-1} \in N}
Let $X_t, t \geqslant 0,$ be a Brownian motion. Define the stopping times $T_a := \inf \{t \mid X_t = a\}$. Calculate the probability $\mathbb{P}\{T_{2}< T_{-1} < T_{3}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
A stick of length 1 is randomly cut into 3 pieces. What is the expected length of the shortest piece?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{9}}
Evaluate the limit: $$\lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{\frac{1}{x^2}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Evaluate the improper integral: \[ I = \int_{0}^{\infty} \frac{x^3 \ln x}{(x^4+1)^3} \,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{32}}
In a box containing 91 balls, $m$ of them are red and the rest are blue. A sample of 19 balls is drawn, resulting in 7 red balls and 12 blue balls. Estimate the number of red balls, $m$, using the method of maximum likelihood.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Consider a triangular tower with 11 rows, where each square on the bottom row contains either a 0 or a 1. Each square above the bottom row is the sum of the two squares directly below it. Determine the number of ways to fill the bottom row such that the number in the top square is a multiple of 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{640}
Evaluate the expression \((\boldsymbol{x'}\cdot\nabla)^2\boldsymbol{x}\), where \(\boldsymbol{x'},\boldsymbol{x}\in \mathbb{R}^3\) and \(\boldsymbol{x'}\) is independent of \(\boldsymbol{x}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\boldsymbol{0}}
Determine the indicial equation for the differential equation: $$ x^2y'' + x(1+x+x^2)y' + x(2-x)y = 0 $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^2 = 0}
Suppose $f(x)$ is a function such that $f^{(2)}(b), f^{(3)}(b), \ldots, f^{(n-1)}(b)$ are all zero. If $f^{(n)}(x)$ is continuous and non-zero at $x=b$, find the value of the limit:

$$\lim_{k \rightarrow 0} (\varphi_n-1)$$

where:

$$f (b+k)=f(b)+kf'(b)+\frac{k^2}{2!}f''(b)+\ldots+\frac{k^{n-1}}{(n-1)!}f^{n-1}(b+\varphi_{n-1} k)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{n-1}{n}}
Find the area of the region enclosed by the ellipse given by the equation \(5x^2 + 6xy + 2y^2 + 7x + 6y + 6 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Find all \( n \in  \mathbb{N} \) so that 7 divides \( {5}^{n} + 1 \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n \equiv 3 \pmod{6}$}
Determine the maximum number of connected closed sets \(A_1, A_2, A_3, \ldots\) that can be drawn in \(\mathbb{R}^2\) such that for each pair \(i, j\), the interiors \(\operatorname{int}(A_i)\) and \(\operatorname{int}(A_j)\) are disjoint, but the intersection \(A_i \cap A_j\) is non-empty for all \(i \neq j\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a classroom, there are 10 girls and 5 boys. Each of the 15 students plays against every other student exactly once, with each game having a single winner. Each participant has a 0.5 probability of winning any game. Let X be the total number of wins by the boys. What is the minimal value of Var(X), considering possible dependencies between the game results?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \(X_1, X_2, \ldots\) be a sequence of independent random variables, each uniformly distributed on \([0, e]\). Define \(R_n = \prod_{k=1}^n X_k\). Determine the probability that the sequence \((R_n)\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the limit as \( a \to \infty \) of \( a^3 \) times the area of the region \( S \) bounded by the parabola \( y = x^2 \), the line \( y = a^2 \), and the lines \( x = a \) and \( x = a + \frac{1}{2a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Provide an example of an irreducible polynomial \(g\) over \(\mathbb{Q}\) such that the extension \((\mathbb{Q}[x]/(g))/\mathbb{Q}\) is not Galois.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 - 2}
How many natural numbers (ie. 1,2,3,...) equal three times the sum of the digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1}
Let $K$ be the field with exactly $7$ elements. Let $\mathscr{M}$ be the set of all $2 \times 2$ matrices with entries in $K$. How many elements of $\mathscr{M}$ are similar to the matrix \( \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Let \( f:\mathbb{R}\to \mathbb{R} \) be a continuous and convex function such that \( f(0) \leq 0 \). Define the set \( \mathcal{D} = \{x \mid f(x) \leq 0\} \cap \{x \mid x \geq 0\} \). If \( \mathcal{D} \) is bounded, determine the set \( \mathcal{D} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, M]}
What is the fifth smallest positive integer with at least $5$ distinct prime divisors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4830}
Evaluate the limit: $$\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^{n} \left(\frac{k}{n}\right)^{1/k} - n}{\ln(n^2 + n)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Given a random sample $X_1, X_2, \cdots, X_n$ from a distribution with the probability density function $$f(x;\theta)=\frac{1}{\theta}x^{(1-\theta)/\theta}\, , 0<x<\infty\, , 0<\theta<\infty,$$ find the asymptotic variance of the maximum likelihood estimator $\hat\theta$ using the Rao–Cramér lower bound.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\theta^2}{n}}
Compute \(892^{(7918)(7909)} \mod (7919 \cdot 7907)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61031734}
Determine the number of distinct ways to color the 12 edges of a cube using the colors red, blue, and yellow, such that exactly 3 edges are red, 3 edges are blue, and 6 edges are yellow.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{784}
Simplify the expression \((V' A V)^{-1}\) in terms of the matrices \(A^{-1}, V,\) and \(V'\), given that \(V \in \Re^{n \times k}\) with \(n > k\), \(V'V = I\), and \(\mathrm{rank}(V) = \text{rank}(V' A V) = k\). The matrix \(A\) is \(n \times n\) and its inverse \(A^{-1}\) is known.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(V' A V)^{-1} = V' A^{-1} V}
Determine whether the sequence \( \frac{2^n - 3n^3}{1-4^n} \) converges or diverges using the ratio test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two identical sticks, each with the same length and mass density, are used in an experiment. One stick is broken at a uniformly random point, and the other stick is supported at a uniformly random point. The pieces of the broken stick are placed on the ends of the supported stick. What is the probability that the shorter piece of the broken stick will tip downwards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Given a circular list of coins, all initially showing Tails, determine the minimum number of moves required to make all coins show Heads. In each move, flipping the coin at position $i$ also flips the coins at positions $i-1$ and $i+1$. What is the minimum number of moves needed to achieve this for a list of $N$ coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{N}{\gcd(N, 3)}}
Determine the limit of the sequence defined by the recurrence relation: \[ a_{n + 1} = \int_{0}^{a_n} \left(1 + \frac{1}{4} \cos^{2n + 1} t\right) dt, \] where the initial term $a_0$ is in the interval $(0, 2\pi)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find all solutions to the equation \(x^x = \frac{1}{256}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Given the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ on the interval $(0, +\infty)$, determine the value of $m$ that makes it a decreasing function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-1"]}
Determine the number of zero divisors in the ring \( \frac{\mathbb{Z}_3[x]}{(x^4 + 2)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Evaluate the integral: \[ \int_{0}^{1} \frac{x^{a - 1}}{(1 - x)^{a + 1}} e^{\frac{x}{x - 1}} \, dx \] where \( a \in \mathbb{R} \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Gamma(a)}
$\text{What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$215$
}
Let $f : \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ be a function satisfying $f(1)=1$ and \[f\Biggl(f(x+1) + f\Bigl( x + f(x)\Bigr)\Biggr) = x+2.\] Determine the value of $f(10)$. \(\boxed{\text{Your answer here}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the integral of the function \( f(x, y, z) = x + y + z \) over the portion of the plane \( 2x + 2y + z = 8 \) that lies in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Find the infinite sum of the alternating series:
\[\sum_{n=0}^{\infty} \frac{(-1)^n \, (5 \pi)^{2n+1}}{6^{2n+1} \, (2n+1)!}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $N$ be midpoint of the side $AB$ of a triangle $ABC$ with $\angle A$ greater than $\angle B$. Let $D$ be a point on the ray $AC$ such that $CD=BC$ and $P$ be a point on the ray $DN$ which lies on the same side of $BC$ as $A$ and satisfies the condition $\angle PBC =\angle A$. The lines $PC$ and $AB$ intersect at $E$, and the lines $BC$ and $DP$ intersect at $T$. Determine the value of $\frac{BC}{TC} - \frac{EA}{EB}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In how many ways can 10 distinct toys be distributed among 4 children such that each child receives at least one toy and the oldest child receives the most toys?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140580}
Find a cluster point of the sequence \( x_{n} = (\pi \cdot n) \mod 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the surface integral \( \int_S F \cdot n \, dS \) where \( F(x,y,z) = (x^2 + \sin y^2, y^2 + \sin z^2, z + \sin (xy)) \) and \( S \) is the surface of the bounded solid defined by the surfaces \( z = x^2 + y^2 + 1 \) and \( z = 2(x^2 + y^2) \), with the normal vector oriented outward from the solid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
On one of the sides of triangle $ABC$, there is a point $P$. Draw a line through $P$ that cuts off a triangle similar to triangle $ABC$ from the original triangle $ABC$. Determine the number of possible solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine whether the poles of the function \( f(z) = \frac{z}{(1-e^z) \sin z} \) at \( z = 2\pi i k \), where \( k \in \mathbb{Z} \) and \( k \neq 0 \), are of order 1 or 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the asymptotic upper bound for the number of $m$-step walks in the given recursion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(2^m)}
What is the name of a lattice where for every subset \( S \) of the lattice, both the infimum and supremum of \( S \) are elements of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{finite chain}
Find the value of the following limit: \[ \lim\limits_{n \rightarrow \infty}  \sum\limits_{k=1}^n \arctan\left(\frac{1}{\sqrt{k^2+1}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine all real numbers $k$ such that the limit \( \lim_{n\to\infty}\frac{\tan n}{n^k}=0 \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k > 1}
Compute the limit \( \lim_{n \to \infty}\int_{0}^{\pi} \frac{\sin(x)}{1 + 3\cos^{2}(nx)}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set of all countable subsets of the power set of the natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
The angle of inclination of the line $x-y-1=0$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\dfrac{\\pi}{4}"]}
Given a convex quadrilateral \(ABCD\) with sides \(AB=70\) units, \(BC=100\) units, \(CD=35\) units, and \(DA=75\) units, and the projection of \(BC\) onto line \(AD\) is 96 units, find the distance \(MN\) where \(M\) and \(N\) are the intersection points of the angle bisectors at \(A, B\) and \(C, D\) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\sqrt{2}}
What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-11$}
Calculate the limit of the sum \(\sum_{k=1}^n \frac{k^3}{k^3+n^4}\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
The following list shows every number for which more than half of its digits are digits $2$, in increasing order:
$$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$If the $n$th term in the list is $2022$, what is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
31
}
Let $x$ be a uniformly distributed variable across the interval [0, 0.1], where $x$ represents the probability of a particular event occurring during a trial. Given that 528174 trials occur and the event does not occur in any of these trials, find the smallest real number $y$ such that $x < y$ with at least 95% certainty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.00000567}
Let \( A = K[x,y,z] \). Find the radical of the ideal \( I = (xyz, xy+xz+yz, x+y+z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x, y, z)}
Determine the cardinality of the set $A$, where a function $h$ belongs to $A$ if and only if there exists a natural number $k$ such that for all $i \geq k$, $f(i) = h(i)$, given a function $f: \mathbb{N} \rightarrow \{0,1\}$. Assume you cannot use cardinality arithmetic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Determine the natural number \( a \) such that the following equality holds for a given function \( f \) and a positive integer \( n \):
\[ \sum_{k=0}^n f(a+k) = 16(2^n - 1) \]
where \( f \) is a function satisfying the relation \( f(x + y) = f(x) \cdot f(y) \) for all natural numbers \( x \) and \( y \), and \( f(1) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}