inclusionAI/Ring-lite-rl-data huggingface dataset:
已知 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围。标准答案格式为$\left(-\infty, \frac{a}{b}\right)$，请给出区间端点a+b的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A \cdot B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1811716}
When the mean, median, and mode of the list
\[10,2,5,2,4,2,x\]
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $n > 1$ be an integer that is divisible by exactly two distinct primes, $p$ and $q$. If $G$ is a group with exactly $n$ elements of order $n$, find the maximum possible number of cyclic subgroups of order $n$ in $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
若直线 \(y = x + m\) 与半圆 \(y = \sqrt{1 - x^2}\) 有2个交点，则 \(m\) 的取值为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1 \leq m < \sqrt { 2 }$}
In trapezoid \(A B C D\), the side \(A B\) is equal to the diagonal \(A C\). On the smaller arc \(A D\) of the circumcircle of triangle \(A B D\), a point \(E\) is selected such that \(A B = A E\). Find the angle \(\angle C E D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
已知抛物线$E:{{y}^{2}}=2px(p>0)$的焦点为$F$，点$A$在$E$上，以$A$为圆心的圆与$y$轴相切，且交$AF$于点$B$，若$\left| AB \right|=2\left| BF \right|$，则圆$A$截线段$AF$的垂直平分线所得弦长为$\sqrt{7}$，则$p=$______．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2$}
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is the length of $AC$? Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $$\sum_{n=1}^{210} nf(n)$$ is divided by $211$.

[i]Proposed by ApraTrip[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
A cylinder has a height of 8 centimeters. If its height increases by 2 centimeters, the surface area increases by 25.12 square centimeters. What is the original surface area of the cylinder in square centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125.6 \text{ square centimeters}}
Given that the function $f(x) = \log\_2 \frac{1+nx}{1-2x}$ defined on the interval $[-m, m]$ is an odd function, and $f(-\frac{1}{4}) \neq f(\frac{1}{4})$, determine the range of $n^m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[
    \sqrt[4]{2}, 
    \sqrt{2}
)}
Calculate the limit of $$ \frac{1}{K} \int_0^K K^{\frac{1}{2x+1}} \textrm{d} x $$ as $K \to +\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given triangle $ABC$, the altitude on side $AB$ is equal to the length of side $AB$. Find the maximum value of $\frac{AC}{BC} + \frac{BC}{AC} + \frac{AB^2}{BC \cdot AC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Given the parabola $C_{1}$: $y^{2}=16x$ and the circle $C_{2}$: $\left(x-4\right)^{2}+y^{2}=1$, where point $M$ has coordinates $\left(8,0\right)$, and points $P$ and $Q$ are moving points on $C_{1}$ and $C_{2}$ respectively, satisfying $|PM|=|PQ|$. Find the range of the x-coordinate of point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{39}{10}, \frac{55}{6}]}
What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of 63.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111888}
Given the function $y=\cos \left( 2x+\dfrac{\pi}{3} \right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\cos 2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
For the symbols , m ” n = (n ^ 2 − m) / x + y for all values of m, n, x, and y. If m = 4, n = 3, x = 2, and y = 5, what is the value of 4 ” 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Brian starts at the point $(1,0)$ in the plane. Every second, he performs one of two moves: he can move from $(a,b)$ to $(a-b,a+b)$ or from $(a,b)$ to $(2a-b,a+2b)$. How many different paths can he take to end up at $(28,-96)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
For a positive integer n not smaller than 2, let n lines divide a plane into at most M_n parts, and let n+2 rays divide a plane into at most N_{n+2} parts. Determine the relationship between M_n and N_{n+2} for all n ≥ 2. Express your answer as a mathematical relationship (e.g., <, >, =).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M_n < N_{n+2}}
Given $ω\in\mathbb{N}^{*}$ and $ω\leqslant 15$, find the number of values of $ω$ such that the function $y=\sin(ωx)$ is not monotonic in the interval $\left[\frac{π}{4}, \frac{π}{3}\right]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If for the real numbers $x, y, z, k$ the following conditions are valid: $x \ne y \ne z \ne x$ and \[ x^3 + y^3 + k(x^2 + y^2) = y^3 + z^3 + k(y^2 + z^2) = z^3 + x^3 + k(z^2 + x^2) = 2008, \] find the product $xyz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004}
Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{000}
In an orchard, there are 200 apples, 150 oranges, and 100 pears. The apples are 50% red and 50% green, the oranges are 40% orange and 60% yellow, and the pears are 30% green and 70% brown. Among these fruits, 40 percent of apples, 25 percent of oranges, and 35 percent of pears are rotten. Further, 70 percent of the rotten apples, 50 percent of the rotten oranges, and 80 percent of the rotten pears have a strong smell. How many green rotten fruits in the orchard did not smell?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Evaluate the exact value of the alternating series $\displaystyle \sum_{n=0}^\infty \frac{(-1)^{n+1}n}{(2n+1)!} \cdot \pi^{2n}$, which is known to converge based on the ratio test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A printing house determines the cost of printing a book by adding the cost of the cover to the cost of each page and rounding the result up to the nearest whole ruble (for example, if the result is 202 rubles and 1 kopek, it is rounded up to 203 rubles). It is known that the cost of a book with 104 pages is 134 rubles, and a book with 192 pages costs 181 rubles. What is the cost of printing the cover, given that it is a whole number of rubles and the cost of a single page is a whole number of kopeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
When $\frac{1}{909}$ is expressed as a decimal, what is the sum of the first 30 digits after the decimal point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Dmitry's parents decided to buy him a laptop for his birthday. They calculated that they could save the required amount in two ways. In the first case, they need to save one-tenth of their salary each month for six months. In the second case, they would save half of one month's salary and then deposit it in a bank for ten months at 3% interest per month (simple interest). In the first case, the saved money will be just enough for the laptop. In the second case, after buying the laptop, there will be leftover money sufficient to buy a computer table for 2875 rubles. What is the mother's salary, if the father's salary is 30% higher? (Provide the answer as an integer, without spaces or units of measurement.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25000}
Find the radius of convergence of the following power series: $$1 + 3z + 2z^2 + 9z^3 +...+ 2^nz^{2n} + 3^{n+1}z^{2n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
For positive integers $a$ and $N$, let $r(a, N) \in\{0,1, \ldots, N-1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \leq 1000000$ for which $r(n, 1000)>r(n, 1001)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499500}
A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that $$ N(T) \leq a T^{2}+b T+c $$ for all $T>0$ and all initial points $P_{0}$. The original answer is in \(\pi\) format, please give the integer part of the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many nonempty subsets of {1, 2, ..., 10} have the property that the sum of their largest element and smallest element is 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Two players are playing a game. One of them thinks of a sequence of integers ($x_{1}, x_{2}, \ldots, x_{n}$), which are single-digit numbers, both positive and negative. The other player is allowed to ask for the value of the sum $a_{1} x_{1}+\ldots+a_{n} x_{n}$, where $(a_{1}, \ldots, a_{n})$ is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The lengths of the altitudes of $\triangle ABC$ are the roots of the polynomial $x^3 - 34x^2 + 360x - 1200.$ Find the area of $\triangle ABC.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given \( n \) new students such that among any 3 students, there are at least 2 students who know each other, and among any 4 students, there are at least 2 students who do not know each other. Determine the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If the expression $\sqrt{3}-a$ is meaningful within the real number range, then the real number $a$ can be _______. (Fill in one number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the function $y=\frac{x-b}{x+2}$ defined on the interval $(a, b+4)$ where $b<-2$, and its range is $(2, +\infty)$, find the value of $a^b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
There are 4 people in the family. If Masha's scholarship is doubled, the total family income will increase by 5%. If the mother's salary is doubled instead, the total family income will increase by 15%. If the father's salary is doubled, the total family income will increase by 25%. By how many percent will the family's income increase if the grandfather's pension is doubled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55\%}
在平面直角坐标系 $x O y$ 中， $F_{1} 、 F_{2}$ 分别是椭圆 $\frac{x^{2}}{2}+y^{2}=1$ 的左、右焦点. 设不经过焦点 $F_{1}$ 的直线 $l$ 与椭圆交于两个不同的点 $A 、 B$, 焦点 $F_{2}$ 到直线 $l$ 的距离为 $d$. 如果直线 $A F_{1} 、 l 、 B F_{1}$ 的斜率依次成等差数列，已知 $d$ 的取值范围形如$(\sqrt{m},n)$，求$m+n$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{500}}
Find the largest positive integer $n < 30$ such that $\frac{1}{2}(n^8 + 3n^4 - 4)$ is not divisible by the square of any prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A baseball team has averaged 15 hits per game over their first 5 games. There are 11 players on the team. Their best player has some total hits. If the other players continue their average over the next 6 games, each player will average 6 hits across the 6 games in total. How many total hits does the best player have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find the value of \(m + n + p\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Mark does a gig every other day for 2 weeks. For each gig, he plays 3 songs. 2 of the songs are 5 minutes long and the last song has a certain relationship to the length of the first two songs. He played for 280 minutes. What is the ratio of the length of the last song to the length of the first two songs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
The Ebbinghaus forgetting curve describes the law of human brain forgetting new things. Based on this, a research team found that after learning course $A$, 20% of the memorized content is forgotten every week. In order to ensure that the memorized content does not fall below $\frac{1}{12}$, the content needs to be reviewed after $n$ ($n\in N$) weeks. Find the value of $n$. ($\lg 3\approx 0.477$, $\lg 2\approx 0.3$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1$, 前 $n$ 项和为 $S_{n}, S_{n+1}=4 a_{n}+2$, 求 $a_{2013}$ 的值。原始答案为$x \times y^{z}$的形式，其中$y$不能是2的倍数，请给出$x + y + z$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5033}
Dan is holding one end of a 26 inch long piece of light string that has a heavy bead on it with each hand. If he starts with his hands together at the start and leaves his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
On the surface of a cube, there is a closed broken line with eight segments, whose vertices coincide with the vertices of the cube.

What is the minimum number of segments of this broken line that can coincide with the edges of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ \text{segments}}
Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady's $2010$th step, considering the procedure implies hexadecimal (base 16) operations rather than quinary (base 5)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a square with side length \(a\), on each side of the square, a trapezoid is constructed outside such that the top bases and the lateral sides of these trapezoids form a regular dodecagon. Compute its area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3a^2}{2}}
Let \(AA_1B_1B\) be the axial section of a cylinder, and let \(C\) be a point on the circumference of the base. Given \(AA_1 = 1\), \(AB = 4\), and \(\angle BAC = 60^\circ\), find the dihedral angle between plane \(A_1CB_1\) and the base of the cylinder \(AB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
A $20 \times 20 \times 20$ block is cut up into 8000 non-overlapping unit cubes, and a number is assigned to each cube. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Find the sum of all numbers of the cubes outside these slices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{333}
Given a sequence $\left\{a_{n}\right\}$ with $k$ terms $(k\in N^{*})$, where $a_{1}=2$, define vectors $\overrightarrow {c_{n}}=(a_{n}, a_{n+1})$ and $\overrightarrow {d_{n}}=(n, n+1)$ for $n=1,2,3,\ldots, k-1$. If $|\overrightarrow {c_{n}}|=|\overrightarrow {d_{n}}|$, calculate the number of sequences $\left\{a_{n}\right\}$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{k-1}}
Let $\triangle X Y Z$ be a right triangle with $\angle X Y Z=90^{\circ}$. Suppose there exists an infinite sequence of equilateral triangles $X_{0} Y_{0} T_{0}, X_{1} Y_{1} T_{1}, \ldots$ such that $X_{0}=X, Y_{0}=Y, X_{i}$ lies on the segment $X Z$ for all $i \geq 0, Y_{i}$ lies on the segment $Y Z$ for all $i \geq 0, X_{i} Y_{i}$ is perpendicular to $Y Z$ for all $i \geq 0, T_{i}$ and $Y$ are separated by line $X Z$ for all $i \geq 0$, and $X_{i}$ lies on segment $Y_{i-1} T_{i-1}$ for $i \geq 1$. Let $\mathcal{P}$ denote the union of the equilateral triangles. If the area of $\mathcal{P}$ is equal to the area of $X Y Z$, find $\frac{X Y}{Y Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
a and b run around a circular track. a beats b by one round or some minutes. In this race, they had completed 4 rounds. If the race was only of one round, a's time over the course would be 7.5 minutes. How many minutes does a beat b by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
已知正方体 $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ 的棱长为 1, 点 $M, N$ 分别为线段 $A B^{\prime}, A C$ 上的动点, 点 $T$在屏幕 $B C C^{\prime} B^{\prime}$ 内，求 $|M T|+|N T|$ 的最小值。原始答案为 $\frac{a\sqrt{b}}{c}$ 格式，请给出 $a + b+c$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知正整数数列 $a, b, c, d$ 严格递增, $d \leq 101$, 且 $101 \mid a+b+c+d$, 则这样的数组 $(a, b, c, d)$ 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40425}
In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $E$ is on $A_{1} D_{1}$, point $F$ is on $C D$, and $A_{1} E = 2 E D_{1}$, $D F = 2 F C$. Find the volume of the triangular prism $B-F E C_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{27}}
There are 2016 spheres in space, none of which are identical. Some of the spheres are red, while the others are green. Each point of contact between a red sphere and a green sphere is painted blue. Find the maximum possible number of blue points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1016064}
61. (5) In a line, numbers $1,2, \ldots, n$ are written in some order. A pair of numbers is called a pit if these numbers are adjacent, or there are only numbers between them that are smaller than each of them. What is the maximum number of pits? (A number can be part of several pits.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n-3}
Given the function $f\left(x\right)=ax^{2}+bx+c$, $x\in \left[-2a-5,1\right]$ is an even function, then $a+2b=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the midline of an isosceles trapezoid with height $h$, given that the lateral side is seen from the center of the circumscribed circle at an angle of $120^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{h \sqrt{3}}{3}}
In an examination, 30% of total students failed in Hindi, 45% failed in English, 25% failed in Math, and 40% failed in Science. Additionally, 12% in both Hindi and English, 15% in both Hindi and Math, 18% in both Hindi and Science, 20% in both English and Math, 22% in both English and Science, and 24% in both Math and Science. Finally, 10% failed in all four subjects. Calculate the percentage of students who passed in all of the subjects.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61\%}
Let $a$ and $b$ be natural numbers, and $a = 1999b$. The sum of the greatest common divisor and the least common multiple of $a$ and $b$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000b}
The center of the city is a rectangle of \(5 \times 8\) km, consisting of 40 blocks of size \(1 \times 1\) km, with the boundaries being streets that form 54 intersections. What is the minimum number of police officers needed at intersections such that each intersection can be reached by a police officer driving no more than 2 km on the city streets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute the value of the following expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046}