inclusionAI/Ring-lite-rl-data huggingface dataset:
Find $x$ such that the sum $\sum_{k=1}^{2014} k^k$ is congruent to $x$ modulo $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the asymptotic growth rate of the recurrence relation \(T(n) = 2T\left(\frac{n}{3}\right) + T\left(\frac{n}{4}\right) + n^2\) using the Akra-Bazzi theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^2)}
Let $P(x)$ be a polynomial of degree at most 3 such that $P(x)=\frac{1}{1+x+x^{2}}$ for $x=1,2,3,4$. What is $P(5) ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-3}{91}}
Evaluate the infinite series ratio:

$$\frac{1+\frac{2^2}{2!}+\frac{2^4}{3!}+\frac{2^6}{4!}+\ldots}{1+\frac{1}{2!}+\frac{2}{3!}+\frac{2^2}{4!}+\ldots}$$

Simplify the expression to find the numerical value of the ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^2 - 1}
In the triangular pyramid \(P-ABC\), \(\triangle PAC\) and \(\triangle ABC\) are both equilateral triangles with side length 6. If the dihedral angle \(P-AC-B\) is \(120^\circ\), find the surface area of the circumscribed sphere around \(P-ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84 \pi}
Cary is an engineer in charge of designing an irrigation system for three farmers. Farmer Bob grows 3 acres of corn, 9 acres of cotton, 12 acres of beans, and 4 acres of wheat. Farmer Brenda grows 6 acres of corn, 7 acres of cotton, 14 acres of beans, and 5 acres of wheat. Farmer Bernie grows 2 acres of corn, 12 acres of cotton, and 3 acres of wheat. If corn takes 20 gallons of water an acre, cotton takes 65 gallons of water an acre, beans take three times as much water as corn, and wheat takes 45 gallons of water an acre, what percentage of the total water used will go to Farmer Bob's farm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.32\%}
There are  $n\geq 3$  cities in a country and between any two cities  $A$  and  $B$ , there is either a one way road from  $A$  to  $B$ , or a one way road from  $B$  to  $A$  (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define  $d(A,B)$  to be the minimum number of roads you must go through to go from city  $A$  to  $B$ . Consider all possible ways to build the roads. Find the minimum possible average value of  $d(A,B)$  over all possible ordered pairs of distinct cities in the country.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
At a family gathering, Jane’s mom prepared a dessert platter with a variety of bite-size treats. There were 15 cinnamon swirls, 24 chocolate brownie bites, and 18 fruit tartlets. In total, 8 people including Jane ate an equal number of each dessert item. How many pieces of each dessert item did Jane eat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Write down the natural numbers starting from 1, consecutively until reaching the 198th digit. Find the remainder when this number is divided by 9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Fabián has four cards, each with a different positive integer less than 10 written on it. The numbers are written in different colors, and the following conditions hold:

- The product of the green and yellow numbers is the green number.
- The blue number is the same as the red number.
- The product of the red and blue numbers is a two-digit number written with the green and yellow digits (in this order).

Determine these four numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8, 1, 9, 9}
Evaluate the integral $$\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \text{ inches}}
A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{5}{16}$}
We dilute one liter of $a\%$ alcohol with water to $b\%$. A portion of this mixture is set aside, and the remaining mixture is further diluted until it becomes $c\%$. How much water was used in total, given that the amount of the resulting $c\%$ mixture is equal to the amount of the set-aside $b\%$ mixture?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2a}{b + c} - 1\)}
Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
What is the molecular weight of 8 moles of Al2O3
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{815.68}
Determine the values of \( \alpha \in \mathbb{R} \) for which the improper integral \( \int_{0}^{+\infty}\frac{1}{(\arctan x)^\alpha} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
What is the mass percentage of Ca in Calcium hydroxide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.09\%}
The sum of the squares of the projections of the sides of an equilateral triangle onto any axis is constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2} a^2}
Compute the line integral \( \int_C \vec{F} \cdot d\vec{s} \) where \( \vec{F} = A_0(x\hat{y} - y\hat{x}) \), along a circular path, counterclockwise, about the origin with radius 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi A_0}
In the Cartesian coordinate plane $xOy$, two circles $C_1$ and $C_2$ are defined as follows: $C_1: x^2+(y-t)^2=4$ and $C_2: (x-2)^2+y^2=14$. If there exists a point $P$ on circle $C_1$ such that a tangent line to circle $C_2$ passes through $P$ at point $Q$ and $PO = \sqrt{2}PQ$, determine the range of values for the real number $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\sqrt{3}, 4\sqrt{3}}
An aluminum and a copper part have the same volume. The density of aluminum is \(\rho_{A} = 2700 \, \mathrm{kg/m}^{3}\), and the density of copper is \(\rho_{M} = 8900 \, \mathrm{kg/m}^{3}\). Find the mass of the copper part if it is known that the masses of the parts differ by \(\Delta m = 60 \, \mathrm{g}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 + \ldots + 10^{2013}$. If one writes down the number $S$, how often does the digit $5$ occur in the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4022}
Let $a$ and $b$ be positive real numbers with $a \ge b$. Let $\rho$ be the maximum possible value of $\frac{a}{b}$ for which the system of equations  
$$a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2$$  
has a solution in $(x,y)$ satisfying $0 \le x < a$ and $0 \le y < b$. Then $\rho^2$ can be expressed as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given triangle  $ ABC$ . Point  $ O$  is the center of the excircle touching the side  $ BC$ . Point  $ O_1$  is the reflection of  $ O$  in  $ BC$ . Determine angle  $ A$  if  $ O_1$  lies on the circumcircle of  $ ABC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given the digits 1, 2, 3, 4, 5, 6 to form a six-digit number (without repeating any digit), requiring that any two adjacent digits have different parity, and 1 and 2 are adjacent, determine the number of such six-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find all prime numbers  $p$  for which the number  $p^2+11$  has less than  $11$  divisors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A man takes $37 to eat pizza in a restaurant. The restaurant offers a discount that increases by multiples of 4 for every additional pizza eaten. For 1 pizza there is 0% off, for 2 pizzas there is 4% off, for 3 pizzas there is 8% off. The pizza costs a certain amount without discount. The man eats 3 pizzas. What is the cost of the pizza without discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$13.41}
A runner, two cyclists, and a motorcyclist are moving with constant speeds on a circular highway. The runner and one of the cyclists are moving in one direction, while the motorcyclist and the other cyclist are moving in the opposite direction. The runner meets the second cyclist every 12 minutes, the first cyclist overtakes the runner every 20 minutes, and the motorcyclist overtakes the second cyclist every 5 minutes. Determine how often the motorcyclist meets the first cyclist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Sasha discovered that there are exactly \( n \) working buttons with digits on the calculator. It turned out that any natural number from 1 to 99,999,999 can either be entered using only the working buttons or obtained as the sum of two natural numbers, each of which can be entered using only the working buttons. What is the smallest \( n \) for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the positive value for the slope of an asymptote of the hyperbola described by the equation:
\[
\sqrt{(x-2)^2 + (y-3)^2} - \sqrt{(x-2)^2 + (y+1)^2} = 4
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \( PQ \) of the larger circle is a tangent to both unshaded circles. The length of \( PQ \) is 6 units. What is the area, in square units, of the shaded region?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9\pi}{2}}
设双曲线 $C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$. 若 $C_{2}$ 的短轴长与 $C_{1}$ 的实轴长的比值等于 $C_{2}$ 的离心率，求 $C_{1}$ 在 $C_{2}$ 的一条准线上截得线段的长。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A seller has weights of 1, 2, 4, 8, 16, and 32 grams (one of each) and a balance scale. A 25 gram candy and three of the weights were placed on the first pan, while the remaining three weights were placed on the second pan, such that the scale balanced. Indicate the weights of all three weights on the second pan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 8, 32}
记数集 A=\{a_1,a_2,\cdots,a_n\} 中所有元素的算术平均值为 P(A). 若 B 是 A 的非空子集，且 P(B)=P(A)，则称 B 是 A 的一个"均衡子集". 求数集 M=\{1,2,3,4,5,6,7,8,9\} 的均衡子集的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
If \( a = \log 25 \) and \( b = \log 49 \), compute
\[
5^{a/b} + 7^{b/a}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the signature of the quadratic form \( f(x, y) = xy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1)}
Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\overline{B C}$ and $\overline{D A}$. Compute $P Q^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{116}{35}}
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
设 V 是 10 维列向量构成的内积空间，其中对 v=(v_1,v_2,\cdots,v_{10})^T, w=(w_1,w_2,\cdots,w_{10})^T，v,w 的内积 \langle v,w\rangle=\ds{i=1}{10}v_iw_i.对 u\in V，定义 V 上的线性变换 P_u 如下：P_u:V\rightarrow V,\quad x\mapsto x-\df{2\langle x,u\rangle u}{\langle u,u\rangle}.给定 v,w\in V，满足0<\langle v,w\rangle<\sqrt{\langle v,v\rangle\langle w,w\rangle}.记 Q=P_v\circ P_w，则所有满足 P\circ Q=Q\circ P 的线性变换 P:V\rightarrow V 构成的线性空间的维数是__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the area is $x\pi$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let  $ABCD$  be a square with side length  $1$ . How many points  $P$  inside the square (not on its sides) have the property that the square can be cut into  $10$  triangles of equal area such that all of them have  $P$  as a vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given that $11^{-1} \equiv 3 \pmod{31}$, find $20^{-1} \pmod{31}$. Provide the result as a residue modulo 31 (a number between 0 and 30, inclusive).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
任意＿个自然数，就可以保证其中必有四个数的和是4的倍数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $n$ be the smallest positive integer with exactly 2015 positive factors. What is the sum of the (not necessarily distinct) prime factors of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
Given two fuzzy topological spaces $(X, \tau_1)$ and $(Y, \tau_2)$, where $X=\{a\}$, $Y=\{x, y\}$, $\tau_1 =\{0_X, 1_X, \{(a, 0.3)\}\}$, $\tau_2 =\{0_Y, 1_Y, \{(x, 0.2), (y, 0.2)\}\}$, and a fuzzy function $f:X \rightarrow Y$ such that $f(a)=x$, find the inverse of the fuzzy set $\{(x, 0.5), (y, 0.3)\}$ in $X$. What is the inverse fuzzy set in $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{(a, 0.5)\}}
Evaluate the following integral considering the parity of the integrand:
$$\hat g(n) = \frac{1}{2\pi}\int_0^{2\pi} xe^{-inx} dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
By writing 6 different numbers, none of which is the number 1, in ascending order and multiplying them, Olya obtained the result 135135. Write down the numbers that Olya multiplied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 5, 7, 9, 11, 13}
$O K R A$ is a trapezoid with $O K$ parallel to $R A$. If $O K=12$ and $R A$ is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to $O K$, through the intersection of the diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{353}
How many of the integers between 25 and a certain number are even if the total number of even integers is 10?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let $M$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_5x_1$ where $x_1, x_2, \cdots, x_5$ is a permutation of $(1, 2, 3, 4, 5)$. Let $N$ be the number of permutations for which this maximum is attained. Evaluate $M + N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
Determine the points in \(\mathbb{R}^3\) where the distribution \(D = \langle X_1, X_2 \rangle\) has integral manifolds, given the vector fields: \[X_1 = \frac{\partial}{\partial x} + z \frac{\partial}{\partial y}, \quad X_2 = \frac{\partial}{\partial y} + z \frac{\partial}{\partial z}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z = 0}
Sides of a triangle form an arithmetic sequence with common difference  $2$ , and its area is  $6 \text{ cm }^2$ . Find its
sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{6} - 2, 2\sqrt{6}, 2\sqrt{6} + 2}
For which values of \(\alpha \geq 1\) can Hephaestus guarantee victory in a finite number of turns, regardless of Poseidon's initial choice of flooded cells, in the game described?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The sides of  $\triangle ABC$  measure 11,20, and 21 units. We fold it along  $PQ,QR,RP$  where  $P,Q,R$  are the midpoints of its sides until  $A,B,C$  coincide. What is the volume of the resulting tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given that in the expansion of $(1-2x)^n$, only the coefficient of the third term is the largest, find the sum of the coefficients of the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Arrange for five students, namely A, B, C, D, and E, to participate in the volunteer activities at the Shanghai World Expo, with each person undertaking one of the following tasks: translation, tour guiding, etiquette, or driving. Each task must have at least one participant. Students A, B, and C cannot drive but can undertake the other three tasks, while students D and E are capable of handling all four tasks. Calculate the number of different arrangement possibilities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1 , the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
椭圆 $\frac{x^{2}}{4}+\frac{y^{2}}{3}=\lambda, F$ 为左焦点, $A, B$ 为椭圆上两点且 $|F A|=5,|F B|=8$, 求直线 $A B$ 的斜率 $k$ 的范围。请给出该范围上下限绝对值之和的近似整数值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance m with probability $\frac{1}{2^{m}}$. What is the probability that Flora will eventually land at 10?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac12$}
In a sequence of numbers, a term is called "golden" if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, \ldots, 2021$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
Consider the curve $C: y^2z = x^3 - xz^2$ in the projective plane $\mathbb{P}^2$ and the point $P = (0:0:1)$. Define the rational map $\phi: \mathbb{P}^2 \dashrightarrow \mathbb{P}^1$ by $\phi(x:y:z) = (x:y)$. Determine the value of $\psi(P)$, where $\psi = \phi|_C: C \to \mathbb{P}^1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0:1)}
Determine the order of growth of the complex polynomial \( p(z) = a_nz^n + \ldots + a_1z + a_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Bingo has three dogs. On average, they walk a total of 100 miles a week. The first dog walks an average of 2 miles a day with a constant speed of 3 miles per hour. The second dog's daily walks depend on the last digit of the day of the month: 1 mile if it is odd, and 3 miles if it is even, with a varying speed between 4 to 6 miles per hour. The third dog has rest days on weekends and its daily walks are impacted by seasonal weather changes (assume 3 miles during summer, 2 miles during spring and fall, and 1 mile during winter). Calculate the average daily distance the third dog walks in summer, considering a 30-day month, and taking into account its rest days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.2}
If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given \(\frac{1 - |\cos x|}{1 + |\cos x|} = \sin x\), find \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k\pi, 2k\pi + \frac{\pi}{2} (k \in \mathbb{Z})}
Of the goose eggs laid at a certain pond, 1/4 hatched and 4/5 of the geese that hatched from those eggs survived the first month. Of the geese that survived the first month, 3/5 did not survive the first year. If no more than one goose hatched from each egg, how many geese survived the first year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
把一个半径为1.1的圆形纸片放到直角坐标平面上，纸片最少要盖住＿个整点.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The volume of a parallelepiped is \( V \). Find the volume of the polyhedron whose vertices are the centers of the faces of this parallelepiped.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6} V}
Points \(P\) and \(Q\) are two distinct points in the \(xy\)-plane. Find the number of different places in the \(xy\)-plane where a third point, \(R\), can be placed so that \(PQ = QR = PR\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is the power of the triple. Compute the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7290}
The distance between points A and B is 1000 meters. Person A departs from point A and takes 1 hour to reach point B. Person B departs from point B, 20 minutes after person A leaves, and takes 40 minutes to reach point A. How many meters from point A do person A and person B meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}