inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the Lebesgue measure of Luzin's non-Borel set of reals, defined as follows: For $x \in [0,1]$, let \( x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} \) be the continued fraction expansion of $x$. Let $A$ be the set of numbers $x \in [0,1]$ whose corresponding sequence $a_0, a_1, a_2, \cdots$ admits an infinite subsequence $a_{k_1}, a_{k_2}, a_{k_3}, \cdots$ such that for each $i$, $a_{k_{i+1}}$ is divisible by $a_{k_i}$. What is the Lebesgue measure of the set $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the proposition $P$: "For all $x \in \mathbb{R}$, there exists an $m \in \mathbb{R}$ such that $4^{x}-2^{x+1}+m=0$". If the negation of the proposition $P$ is a false proposition, then the range of the real number $m$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \leq 1}
Determine the limit of the sequence \( \frac{3^n+3.75^n+\frac{5^n}{n^5}}{\frac{5^n}{n^5}+n^44^n} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( P \) be a regular polygon with 2006 sides. A diagonal of \( P \) is called good if its endpoints divide the perimeter of \( P \) into two parts, each having an odd number of sides of \( P \). The sides of \( P \) are also called good. Suppose \( P \) has been subdivided into triangles by 2003 diagonals that do not intersect at an interior point of \( P \). Find the maximum number of isosceles triangles that can appear in such a subdivision, where two sides of the isosceles triangles are good.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
Evaluate the limit: $$ \lim_{r\to 0}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 - \cos\left(2r\cos t\right)}{4r^{2}\cos t}\,\mathrm{d}t $$ without explicitly solving the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the residue of the function \( f(z) \log \frac{z-b}{z-a} \) at \( z = a \), where \( f \) is an entire function and \( a, b \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the minimum number of check bits required to create a correctable code with a minimum distance of 4 for a list of 10 key words.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n \in \{1, 2, 4\}$}
Given the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ where $a>b>0$, with left and right foci $F_{1}$ and $F_{2}$, and point $P$ in the first quadrant on the ellipse. Let $PF_{1}$ extend to meet the ellipse at another point $Q$, with $PF_{2}\perp F_{1}F_{2}$ and $|PF_{1}|=2|F_{1}Q|$. Calculate the eccentricity of the ellipse. Express your answer as a simplified fraction or decimal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{5}}
The ancient "Five Elements" theory posits that there are five types of properties in materials: metal, wood, earth, water, and fire. Metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal. If five different properties of materials are arranged in a row, but two materials with overcoming properties are not adjacent, then the number of such arrangements is $\boxed{\text{number}}$ (the result is expressed in numerical value).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let \( T: \mathbb{C}^{11} \rightarrow \mathbb{C}^{11} \) be a linear transformation such that \( \dim \ker T = 4 \), \( \dim \ker T^3 = 9 \), and \( \dim \ker T^4 = 11 \). Find \( \dim \ker T^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2{,}042{,}220}
In a triangle with known sides \(a = 2\), \(b = 3\), and area \(S = \frac{3 \sqrt{15}}{4}\), the median drawn to its third side is less than half of that side.

Find the radius of the circumcircle of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{\sqrt{15}}}
At a certain school, there are 6 subjects offered, and a student can take any combination of them. It is noticed that for any two subjects, there are fewer than 5 students taking both of them and fewer than 5 students taking neither. Determine the maximum possible number of students at the school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the Krull dimension of the ring $B = A[x,y,z]/\langle xy, xz \rangle$, where $A$ is a Noetherian commutative ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim A + 2}
Navi and Ozna are playing a game where Ozna starts first and the two take turns making moves. A positive integer is written on the board. A move consists of either:

1. Subtracting any positive integer at most $2015$ from it, or
2. If the integer on the board is divisible by $2014$, dividing by $2014$.

The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$ such that $a \geq 2015$, ensuring that all numbers of the form $an + b$ will not be the starting integer, where $n$ is any positive integer.

Find the minimum number of starting integers where Navi wins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013}
Let $T: P_k \rightarrow P_k$ be a linear operator defined as $T(p(x)) = p(x+a)$, where $a$ is a constant. If $L$ is a differential operator, find the dimension of the kernel of the operator $L - T$. \( \text{(Express your answer as a single integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{337}
Jan's three-eyed lizard has 3 times more wrinkles than eyes, and a number of spots equal to seven times the square of the number of wrinkles. The lizard's cousin has 2 times more wrinkles than eyes, and a number of spots equal to 5 times the square of the number of wrinkles. How many fewer eyes do both lizards have combined, compared to the combined total of spots and wrinkles on both lizards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{756}
In a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third? Provide your answer in the form of a fraction \frac{k}{m}, and then calculate the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Benny plans to visit one of 8 different zoos each week for the next 13 weeks. Determine the number of ways Benny can schedule his visits such that he visits each zoo at least once and at most three times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61751289600}
Given the function $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{12}}
Convert the following triple integral from Cartesian coordinates to cylindrical or spherical coordinates, choosing the system that simplifies the integral the most:
\[ \int_0^1 \int_0^x \int_0^{ \sqrt{x^2+y^2}} \frac{(x^2+y^2)^{3/2}}{x^2+y^2+z^2}\,dz\,dy\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5\%}
Determine the range of values for \( \alpha \) such that the improper integral \( \int_{0}^{\infty}\frac{\sin x}{x^{\alpha}} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Let \( A_1, A_2, A_3, \ldots, A_n \) be subsets of a set \( S \) with \(|S| = 2019\). The union of any three of the \( A_k \) is equal to \( S \), but the union of any two is not equal to \( S \). Determine the maximum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Determine the magnitude of the vertex angle \( \alpha \) of an isosceles triangle with a given area \( A \) such that the radius \( r \) of the inscribed circle is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
A waiter had 44 customers in his section. If 12 of them left, his remaining customers occupied tables with 8 people at each table and some tables with just 4 people due to social distancing. If 25% of the remaining tables had only 4 people, how many total tables did he have in his section?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the maximum number of digits a natural number can have such that all the digits are different and the number is divisible by each of its digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
24. $[\mathbf{1 4}]$ In how many ways may thirteen beads be placed on a circular necklace if each bead is either blue or yellow and no two yellow beads may be placed in adjacent positions? (Beads of the same color are considered to be identical, and two arrangements are considered to be the same if and only if each can be obtained from the other by rotation).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Find the minimum height of an isosceles right triangle that can contain all non-overlapping circles with radii $1/n$ for $n \in \mathbb{N}$, given that the total area of these circles is $\frac{\pi^3}{6}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{2}}
Point A chases point B at a unit velocity. Point B moves to the right at a unit velocity. Initially, at time $t = 0$, point A is at $(0,1)$ and point B is at $(0,0)$. Determine the limiting distance $||B(t) - A(t)||$ as $t$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the sequence \(\{a_n\}\) satisfying
\[ 
a_{n+1} + (-1)^n a_n = 2n - 1, 
\]
and the sum of the first 2019 terms of the sequence \(\{a_n - n\}\) is 2019, find the value of \(a_{2020}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Jamie accidentally misinterprets the rules of the order of operations, and adds or subtracts before multiplying or dividing. What would be her result for the equation $4 + 3 \times 1 - 2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-7

}
In how many ways can the sequence $1,2,3,4,5$ be arranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32
}
Determine the order of growth of the entire function defined by the infinite product \( \prod_{n=0}^{\infty} \left(1 - \frac{z}{(n+1)!}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the Riemann-Stieltjes integral \( \int_{0}^{2}{\alpha\left(\dfrac{x}{\sqrt{2}}+1\right) \mathrm{d}\alpha(x)} \), where \( \alpha(x) = 3[x] \) and \([x]\) is the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
A soft drink can is $h$ centimeters tall and has a radius of $r$ cm. The cost of material in the can is $.0005$ cents per cm$^2$ and the soda itself costs $.001$ cents per cm$^3$. The cans are currently $9$ cm tall and have a radius of $4$ cm. Use calculus to estimate the effect on costs of increasing the radius by $0.4$ cm and decreasing the height by $0.7$ cm. Round your answer to $3$ decimals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.068}
$p, q, r$ are real numbers satisfying the following equations:  
\[ \frac{(p+q)(q+r)(r+p)}{pqr} = 24 \]  
\[ \frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10. \]  
Given that \( \frac{p}{q} + \frac{q}{r} + \frac{r}{p} \) can be expressed in the form \( \frac{m}{n} \), where \( m, n \) are relatively prime positive integers, compute \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Evaluate the limit: \[ \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1 + x^2 + x^4 + \cdots + x^{2k}} \, \text{dx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Determine the index of $K^{*n}$ in $K^*$ for an infinite extension field $K/\mathbb{Q}_p$ where $p\nmid n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
1. Today's date is written as: 22.11.2015. Name the last past date that is written with the same set of digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.12.2012}
If the curve $y=(x-a)\ln x$ has two tangent lines passing through the origin, then the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -e^2)}
Find the maximum likelihood estimate (MLE) of $m$ in the linear model $y = mx + e$, where $m$, $x$, and $e$ are independent and identically distributed as $N(0,1)$, given a single observation $(x, y)$. Assume $x = 1$ and $y = 3$. Provide the analytical expression for the MLE of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the function $f\left(x\right)=\cos 2x+\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\left(x\right)$, then $\cos (x_{1}+x_{2})=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Evaluate the infinite sum: $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\sum_{n=0}^\infty \frac{1}{k2^n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
We say a non-negative integer $n$“contains” another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ contains $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not contain a multiple of $7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$999993$

}
Find a function \( \mu(x,y) \) such that the differential equation \( \mu(x,y)(3x^2y+2xy+y^3)dx + \mu(x,y)(x^2+y^2)dy = 0 \) is exact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{3x}}
Find the angle at the apex of a regular quadrilateral pyramid if it is equal to the angle between a lateral edge and the base plane of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \left( \frac{\sqrt{5}-1}{2} \right)}
Evaluate the following double integral, where $D$ is a domain in $\mathbb{R}^2$ that is bounded, closed, and symmetric with respect to the line $y=-x$:

$$
I:=\iint_{D} \frac{x^2+y^2}{1+x^4+y^4} \mathrm{d}x \mathrm{d}y
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the vertex of $\mathcal{P}$ to all four intersection points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{9653} + \sqrt{5} + \sqrt{5220} + \sqrt{68}}
A rancher has 500 head of cattle. He was about to sell them all for $300,000 when 350 of them fell sick and died. Because of the sickness, his customers lost confidence in his cattle, forcing him to lower his price by $250 per head. How much money would the devastated farmer lose if he sold the remaining cattle at the lowered price compared to the amount he would have made by selling them at the original price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$247,500}
For positive integers $n$, let the numbers $c(n)$ be determined by the rules:  
- $c(1) = 1$  
- $c(2n) = c(n)$  
- $c(2n+1) = (-1)^n c(n)$  

Find the value of \( \sum_{n=1}^{2013} c(n) c(n+2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11111111100000000}
Let \( f \in \mathcal{C}[0, \infty) \) be such that \( \lim\limits_{x \to \infty} f(x) \in \mathbb{R} \) and let \( a > 0 \). Compute \[ \lim_{n \to \infty} \int_0^a f(nx)\,dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \cdot \lim_{x \to \infty} f(x)}
In the store "Everything for School," three types of chalk packs are sold: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the number of packs of excellent chalk increased by 80%, and the number of regular chalk packs decreased by no more than 10 packs. How many total packs of chalk were in the store initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{260}
Find the minimal value of \( \alpha_3 \) such that the function \( u(x,y,z) = x^\alpha(1-x)y(1-y)z(1-z) \) belongs to the Sobolev space \( H^3(\Omega) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{2}}
Let \( f \) and \( g \) be continuous functions over the reals such that \( g(x) = \int_0^x f(y)(y-x) \, dy \) for all \( x \), and \( g \) is three times continuously differentiable. What is the greatest integer \( n \) such that \( f \) is \( n \) times continuously differentiable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate the surface area of the portion of the cone defined by \( z^2 = x^2 + y^2 \), where \( z \geq 0 \), that is contained within the cylinder \( y^2 + z^2 \leq 81 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81\pi}
For any subset \( S \subseteq \{1, 2, \ldots, 15\} \), a number \( n \) is called an "anchor" for \( S \) if \( n \) and \( n+|S| \) are both members of \( S \), where \( |S| \) denotes the number of members of \( S \). Find the average number of anchors over all possible subsets \( S \subseteq \{1, 2, \ldots, 15\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13/8}
108 singers participate in an art festival. Arrange them into performances with 4 singers in each performance such that among any 8 singers, any two singers perform together the same number of times. What is the minimum number of performances needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2889}
Evaluate the following limit:
$$\lim_{n\rightarrow\infty}\frac{\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)^2}{\prod_{k=1}^{2n}\left(1-\frac{1}{2k}\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\{a_n\}$ be a geometric sequence and $\{b_n\}$ be an arithmetic sequence with $b_1 = 0$. Define $\{c_n\} = \{a_n + b_n\}$, and suppose that $\{c_n\}$ is the sequence 1, 1, 2, .... Find the sum of the first 10 terms of $\{c_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{978}
The tetrahedron $ABCDE$ has congruent side edges and a volume of one unit. Its base is the rectangle $ABCD$. The tetrahedron is intersected by a plane passing through the edge $BC$, creating a cross-section that intersects the side edges $EA$ and $ED$ at points $A_1$ and $D_1$, respectively. The lines $BA_1$ and $CD_1$ intersect at point $M$. What is the volume of the tetrahedron $EMA_1D_1$ if $EM = 2AB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{9}}
Evaluate the limit: \[ \lim_{x\to 0^+} \int_0^\infty \frac{e^{-xt}}{\sqrt{t^2 +t}} \, dt \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( \int_0^{2\pi}\frac{\sin^2\left(\frac{N+1}{2}x\right)}{2\pi(N+1)\sin^2(x/2)} \, dx \) for any \(N\in\mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of lines of symmetry in a regular octahedron such that a rotation of at most $180^{\circ}$ around these lines maps the octahedron onto itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Two concentric circles have radii of 1 and 2 units, respectively. What is the minimum possible area of a cyclic quadrilateral inscribed in the larger circle that also contains the smaller circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3}}
Given a positive integer $k$, determine the condition on the positive integer $m$ such that there exists exactly one positive integer $n$ satisfying the divisibility condition \( n^m \mid 5^{n^k} + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq k + 2}
Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
Given $a>0$, $b>0$, $c>2$, and $a+b=2$, find the minimum value of $$\frac {ac}{b}+ \frac {c}{ab}- \frac {c}{2}+ \frac { \sqrt {5}}{c-2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {10}+ \sqrt {5}}
Evaluate $\iiint_E z\,dV,$ where $E$ lies above the paraboloid $z=x^2+y^2$ and below the plane $z=2y.$ In cylindrical coordinates the paraboloid is given by $z=r^2$ and the plane by $z=2r\sin\theta$ and they intersect in the circle $r=2\sin\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5π/6}
Compute the Lebesgue integral \( \int_{[0,\pi/2]} f \, d\mu \) where the function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases}
\sin(x) & \text{if } \cos(x) \text{ is rational} \\
\sin^2(x) & \text{if } \cos(x) \text{ is irrational}
\end{cases}
\]

Evaluate the integral and provide the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}