inclusionAI/Ring-lite-rl-data huggingface dataset:
Riley has 64 cubes with dimensions $1 \times 1 \times 1$. Each cube has its six faces labelled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a $4 \times 4 \times 4$ cube. Riley determines the total of the numbers on the outside of the $4 \times 4 \times 4$ cube. How many different possibilities are there for this total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Given a trapezoid \( ABCD \) with \( AD \parallel BC \). It turns out that \( \angle ABD = \angle BCD \). Find the length of segment \( BD \) if \( BC = 36 \) and \( AD = 64 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Find the center of gravity of the arc of a circle \( x^{2} + y^{2} = R^{2} \), located in the first quadrant, if the density \( \rho = \rho_{0} x y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{2R}{3}, \frac{2R}{3}\right)}
There are 11 ones, 22 twos, 33 threes, and 44 fours on the blackboard. The following operation is performed: each time, three different numbers are erased, and two more of the fourth type of number (the one that was not erased) are added. For instance, if one 1, one 2, and one 3 are erased, then two more of 4 are added. After several operations, only three numbers remain on the blackboard, and no further operations can be performed. What is the product of the remaining three numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
今有茭草一千八百二十束，欲令撒星形垛之，问底子几何？（撒星形的放法是第一层1束，第二层(1+3)束，第三层(1+3+6)束，···，第n层$( 1 + 3 + 6 + 1 0 + \cdots + \frac { n ( n + 1 ) } { 2 } )$束．本题即n层共1820束，求n.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
若实数 $x, y$ 满足 $x-4 \sqrt{y}=2 \sqrt{x-y}$, 则 $x$ 的取值范围是 \qquad. 请给出区间端点之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
称平面上的 n 个正方形是"三连通的"，如果(1) 所有正方形都全等；(2) 若两个正方形有公共点 P，则 P 是它们每个的顶点；(3) 每个正方形都恰与其他三个正方形有公共点.问：在 2018\leq n\leq3018 中有多少个正整数 n，使得存在 n 个三连通的正方形？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Given a rectangle \(ABCD\). On two sides of the rectangle, different points are chosen: six points on \(AB\) and seven points on \(BC\). How many different triangles can be formed with vertices at the chosen points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231}
给定正整数 $n(n \geqslant 3)$. 对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 $\{1,2, \cdots, n\}$ 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间. 求 $m$ 的最大值。当 $n = 11$ 时，请给出答案。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
If x and y are numbers such that (x + 4)(y - 4) equals some number, the smallest possible value of x^2 + y^2 is 16. What is the number that (x + 4)(y - 4) equals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
抛物线 $y=\frac{1}{2} x^{2}$ 与直线 $l: y=x+4$ 所围成区域中有一个矩形 $A B C D$, 且点 $A, B$ 在抛物线上, 点 $D$ 在直线 $l$ 上, 其中点 $B$ 在 $y$ 轴右侧, 且 $|A B|=2 t(t>0)$.  
当边 $C D$ 在直线 $l$ 上时,求矩形 $A B C D$ 面积的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
On the island of Liars and Knights, a circular arrangement is called correct if everyone standing in the circle can say that among his two neighbors there is a representative of his tribe. One day, 2019 natives formed a correct arrangement in a circle. A liar approached them and said: "Now together we can also form a correct arrangement in a circle." How many knights could there have been in the initial arrangement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1346}
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids: 

$\circ$ Roger's cookies are rectangles: 

$\circ$ Paul's cookies are parallelograms: 

$\circ$ Trisha's cookies are triangles: 

Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Ostap Bender organized an elephant distribution to the population in the city of Fux. 28 union members and 37 non-members arrived, and Ostap distributed the elephants equally among all the union members and equally among all the non-members. It turned out that there is only one way to distribute the elephants (so that all elephants are distributed). What is the maximum number of elephants Ostap Bender could have had? (It is assumed that each attendee received at least one elephant.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2072}
For which natural number \( K \) does the expression \(\frac{n^{2}}{1.001^{n}}\) reach its maximum value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2001}
In the diagram, \(PW\) is parallel to \(QX\). \(S\) and \(T\) lie on \(QX\), and \(U\) and \(V\) are the points of intersection of \(PW\) with \(SR\) and \(TR\), respectively. If \(\angle SUV = 120^\circ\) and \(\angle VTX = 112^\circ\), calculate the measure of \(\angle URV\). Express your answer in degrees (°).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52^{\circ}}
Let \( S(x) \) represent the sum of the digits of the natural number \( x \). Determine the set of \( x \) that satisfy the equation \( x + S(x) + S(S(x)) = 2014 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\varnothing}
Given two subsets $A$ and $B$ of size $k$ in the symmetric group $S_n$, both containing a given integer $j$, how many permutations in $S_n$ take $B$ onto $A$ under the action of $S_n$ induced on the subsets of $[n]$ of size $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k!(n-k)!}
In a chess festival that is held in a school with  $2017$  students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most  $22$  matches. What is the maximum possible number of matches in this event?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43890}
Given a function f(x) defined on the real numbers, satisfying f''(x) < f''(-x) for any x in (0, +inf), and non-zero real numbers a and b such that f(a) - f(b) > f(-b) - f(-a), what is the relationship between a^2 and b^2? Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 < b^2}
Find the number of 9-digit US social security numbers that do not use the digit 8, using exponential generating functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{387420489}
Shaq sees the numbers  $1$  through  $2017$  written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased  $-2, -1, 0$  he would replace them with  $0$ .) If  $M$  is the maximum possible final value remaining on the board, and if m is the minimum, compute  $M - m$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Let  $ABCD$  be a quadrilateral with an inscribed circle, centre  $O$ . Let 
\[AO = 5, BO =6, CO = 7, DO = 8.\] 
If  $M$  and  $N$  are the midpoints of the diagonals  $AC$  and  $BD$ , determine  $\frac{OM}{ON}$  .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35/48}
Find the explicit Dirichlet density of primes $p$ in $\\mathbb{Z}$ that split completely in the field $\\mathbb{Q}(\\sqrt{m})$ and remain inert in the field $\\mathbb{Q}(2^{1/3})$, where $m$ is a non-zero, square-free integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Four friends went mushroom picking in the forest. Upon returning, each pair of them counted how many mushrooms they collected in total. They obtained the numbers $6, 7, 9, 9, 11, 12$. How many mushrooms did each friend collect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 4, 5, 7}
How many different (non-similar) triangles are there whose angles have an integer number of degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2700}
一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率的256倍为_______。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
A solid metal sphere with a radius of 20 has a cylindrical hole with a radius of 12 drilled through its center along a diameter. The remaining part is melted and recast into a new solid sphere. Find the radius of the new sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine all positive integer numbers  $n$  satisfying the following condition:
 the sum of the squares of any  $n$  prime numbers greater than  $3$  is divisible by  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2, 3, 4, 6, 8, 12, 24}
Given that all elements of set A are real numbers, and it satisfies: if $a \in A$, then $\frac{1+a}{1-a} \in A$. If $a = -3$, please write down all elements of set A \_\_\_\_\_\_ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-3, -\frac{1}{2}, \frac{1}{3}, 2\}}
Let \( M \) be a subset of the set \(\{1, 2, \cdots, 25\}\). Any two disjoint subsets of \( M \) must have different sums of their elements. Find the maximum possible sum of the elements in \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123}
Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{876321}
Given an arithmetic sequence where all terms are positive, the sum of the first $n$ terms is $S_{n}$. It is known that $2S_{2}=a_{2}(a_{2}+1)$ and $a_{1}=1$. Find the minimum value of $\frac{2S_{n}+13}{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{33}{4}}
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$.  When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
A height $A H$ is drawn from the vertex $A$ to the base $BC$ of isosceles triangle $ABC$. A point $P$ is marked on the side $A B$ such that $C P = BC$. The segment $C P$ intersects the height $A H$ at point $Q$. It turns out that the area of triangle $BH Q$ is 4 times smaller than the area of triangle $AP Q$. Find the angles of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^
\circ, 75^
\circ, 75^
\circ}
Find the value of $a \in \mathbb{R}$, where $a \neq 1$, such that \[ \lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[ (na+1) + (na+2) + \cdots + (na+n) \right]} = \frac{1}{60}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $p$  be a prime number, and define a sequence by:  $x_i=i$  for  $i=,0,1,2...,p-1$  and  $x_n=x_{n-1}+x_{n-p}$  for  $n \geq p$ 
Find the remainder when  $x_{p^3}$  is divided by  $p$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p-1}
Example 1.7. Find $\int \frac{6 d x}{x(x-1)(x-2)(x-3)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln|x|+3\ln|x-1|-3\ln|x-2|+\ln|x-3|+C}
In a Craps game, calculate the expected number of rolls needed to get outcome 2, given that outcome 2 occurs before outcome 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{7}}
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1925}
Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Calculate the line integral of the vector field \( F(x,y,z) = \left(4x\left(1+\frac{z^2}{8}\right),\frac{\pi}{3}\cos\left(\frac{\pi}{3}y\right),\frac{x^2z}{2}-e^{\frac{z}{3}}\right) \) along the curve \( C \) parametrized by \( \gamma(t) = (e^{t(t-3)}, t, 2t) \). Evaluate \( \int_C F \cdot ds \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 - 3e^2}
Compute the line integral \( \oint_C \left( \frac{-y^3}{(x^2+y^2)^2} \right) dx + \left( \frac{xy^2}{(x^2+y^2)^2} \right) dy \) using Green's theorem, where \( C \) is the boundary of any simply connected region that contains the unit disk centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer $n$ such that $2^n < S^{2007} < 2^{n+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

[asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy]

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find the value of $m+n+p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
the length of each side of square a is increased by 150 percent to make square b . if the length of the side of square b is increased by 60 percent to make square c , by what percent is the area of square c greater than the sum of the areas of squares a and b ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120.69\%}
In a trapezoid with bases $x$ and 5, express the distance between the midpoints of the diagonals as a function of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{|x - 5|}{2}}
What is the maximum number of handshakes that could have occurred in a group of 20 people, given that no matter which three people are chosen from the group, there will always be at least two who have not shaken hands?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A triangle has sides of length 888, 925, and $x>0$. Find the value of $x$ that minimizes the area of the circle circumscribed about the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{259}
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, the sum of the digits of $n$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$15$}
A satellite of mass 'm' is revolving around Earth in a circular orbit of radius \( r_0 \) with speed \( v_0 \). At a certain point in its orbit, the direction of motion of the satellite is suddenly changed by an angle \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \), while maintaining the same speed. As a result, the satellite enters an elliptical orbit around Earth. Calculate the ratio of the speed at perigee to the speed at apogee of the new elliptical orbit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Suppose $g: [-1,1] \to [-1,1]$ is a bijective function defined by $g(x)=px^3+qx+r$. Determine the possible values of $g(-1)$. Provide your answer as a set of values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1, 1\}}
Example 6 The adjacent terms $a_{n}, a_{n-1}$ of the sequence $\left\{a_{n}\right\}$ are the two roots of the equation $x^{2}-c_{n} x+\left(\frac{1}{3}\right)^{n}=0$, and $a_{1}=2$. Find the sum of the first $2 n$ terms of the sequence $\left\{c_{n}\right\}$, $S_{2 n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}[1-(\frac{1}{3})^{n}]}
A servant is sent to a spring with two vessels of 7 pints and 11 pints capacity. She needs to bring back exactly 2 pints of water.

What is the minimum number of operations required in this case? An "operation" is defined as filling a vessel, emptying a vessel, or pouring water from one vessel to another.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Meghal is playing a game with 2016 rounds $1,2, \cdots, 2016$. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2 \pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1019088}
A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/15}
In Figure 1, \(\angle A = 60^\circ\), \(\angle B = \angle D = 90^\circ\), \(BC = 2\), \(CD = 3\), and \(AB = x\). Find the value of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{\sqrt{3}}}
If $\sin \alpha = 3 \sin \left( \alpha + \frac{\pi}{6} \right)$, find the value of $\tan \left( \alpha + \frac{\pi}{12} \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{3} - 4}
Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ such that $N \subset M$. Let $P_M$ and $P_N$ be the orthogonal projections onto $M$ and $N$, respectively. Determine the range of the orthogonal projection $P_M - P_N$. \( \text{Provide your answer as a subspace of } H. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M \ominus N}
Let \( H \) be a Hilbert space with a complete orthonormal sequence \((i_k)_{k=1}^\infty\). Suppose there exist complex numbers \(a_k \in \mathbb{C}\) for \(k \in \mathbb{N}\), and a bounded linear operator \(T: H \rightarrow H\) such that \(T(i_k) = a_ki_k\), with the sequence \((a_k)\) being bounded. Determine the norm of \(T\), denoted by \(||T||\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sup_{k \in \mathbb{N}} |a_k|}
Simplify the following sum:

$$\sum_{k=0}^{10}{20\choose k}(1/4)^k (3/4)^{20-k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum number of squares that must be colored in a 65x65 grid so that in any configuration of four cells forming an "L" shape, at least one of the squares is colored.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1409}
Determine the number of positive integer solutions to the equation \(a^2 + b^2 = p_1p_2p_3\), where \(p_1, p_2,\) and \(p_3\) are distinct primes, each congruent to 1 modulo 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 80 cable cars traveling in a circular route between ski resorts A and B, all moving in the same direction. The distance between two adjacent cable cars is the same. Xiaoming, who is sitting in one of the cable cars, encounters an opposing cable car every 15 seconds. How long does it take to travel from A to B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
When $x \in (0,1)$, the graph of the function $f(x)=e^{x}-1$ is not below the graph of the function $g(x)=x^{2}-ax$. Determine the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2-e,+\infty)}
Let $p,$ $q,$ $r$ be positive real numbers. Find the smallest possible value of
\[4p^3 + 6q^3 + 24r^3 + \frac{8}{3pqr}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the line integral \( \int_{c} (3a(x^2+y^2)-y^3)dx+3xy(2a-y)dy \) where \( c \) is the upper part of the circle defined by \( y=a+\sqrt{a^2-x^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6a^4}
Evaluate the integral \( \int_{0}^1 \int_{0}^{1} \int_{x^2}^{1}  12xze^{zy^2} \, dy \, dx \, dz \) using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3(e - 2)}
A hot air balloon with 200 balloons is blowing up. After about half an hour, while at an altitude of 1500 feet, 1/5 of the total number of balloons in the hot air balloon have blown up. During the following hour, the balloon ascends to an altitude of 3500 feet, causing 30% of the remaining balloons to blow up due to pressure changes. Meanwhile, the wind speed increases, creating a variation in the rate at which balloons blow up. Additionally, balloons inflated at different times show varying durability. Out of the balloons still intact, 10% have double durability and are less likely to blow up in the same conditions. At the end of the second hour, twice the number of balloons that had already blown up, excluding the more durable ones, also blow up, as the wind speed decreases. How many balloons in the hot air balloon remain intact?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}