inclusionAI/Ring-lite-rl-data huggingface dataset:
the h.c.f. of two numbers is 120 and the other three factors of their l.c.m. are 13, 17 and 23. The larger of the two numbers is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26520}
For given natural numbers \( k_{0}<k_{1}<k_{2} \), determine the minimum number of roots in the interval \([0, 2\pi)\) that the equation

$$
\sin \left(k_{0} x\right) + A_{1} \cdot \sin \left(k_{1} x\right) + A_{2} \cdot \sin \left(k_{2} x\right) = 0
$$

can have, where \( A_{1}, A_{2} \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 k_0}
You have two blackboards, $A$ and $B$. You need to write some integers on them, where each integer is greater than or equal to $2$ and less than or equal to $20$. The condition is that each number on blackboard $A$ must be co-prime with each number on blackboard $B$. Determine the maximum possible value of the product of the number of integers written on $A$ and the number of integers written on $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
A certain number with a sum of digits equal to 2021 was divided by 7 and resulted in a number composed exclusively of the digit 7. How many digits 7 can this number contain? If there are multiple answers, provide their sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
A teacher grades students' tests by using a certain method. Student B answers each of the 100 questions on her test and receives a score of 73. She answered 91 questions correctly. What is the ratio of the number of incorrect responses subtracted to the number of correct responses in the grading method?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$. If $PQ$ is a focal chord and the equation of the line $PQ$ is $y=mx+c$, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{p+q}}
Each half of this figure is composed of $3$ red triangles, $5$ blue triangles, and $8$ white triangles. When the upper half is folded down over the centerline, $2$ pairs of red triangles coincide, as do $3$ pairs of blue triangles. There are $2$ red-white pairs. How many white pairs coincide? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If \(i^2 = -1\), then find the sum 
\[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \]
\[ + \cdots + i^{40}\cos{3645^\circ} \]
The original answer is in the form \(\frac{k}{m}(\sqrt{2})(a - bi)\), please calculate the value of a + b + k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
7. On the side $B C$ of triangle $A B C$, a point $K$ is marked.

It is known that $\angle B+\angle C=\angle A K B, \quad A K=5, \quad B K=6, \quad K C=2$.

Find the area of the circle inscribed in triangle $A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21}{11}\pi}
In triangle \( ABC \), the midpoints of sides \( AC \) and \( BC \) are marked as points \( M \) and \( N \) respectively. The angle \( MAN \) is \( 15^{\circ} \), and the angle \( BAN \) is \( 45^{\circ} \).

Find the angle \( ABM \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^\circ}
Given $\triangle ABC$, with $\sin A : \sin B : \sin C = 1 : 1 : \sqrt{2}$, the degree measure of the largest interior angle of this triangle is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
已知四面体 PABC，点 A_1 在 \triangle PBC 内，满足 \triangle A_1BP, \triangle A_1CP, \triangle A_1BC 的面积之比为 3:2:1. G 在线段 AA_1 上，直线 PG 交平面 ABC 于点 M，且 \df{AG}{GA_1}=\df{PG}{GM}，则四面体 PABC 与 A_1AMB 的体积之比为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In trapezoid \(ABCD\), the bases \(AD = 12\) and \(BC = 8\) are given. A point \(M\) is chosen on the extension of side \(BC\) such that \(CM = 2.4\).

In what ratio does the line \(AM\) divide the area of trapezoid \(ABCD\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
Calculate $$\int^{1}_{-1}\frac{e^\frac{-1}{x}dx}{x^2(1+e^\frac{-2}{x})}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\pi - 2\arctan e\)}
For any integer \( n (n \geqslant 4) \), consider the set \(\{1, 2, \cdots, n\}\) with \( m \) subsets \( A_{1}, A_{2}, \cdots, A_{m} \), where \( A_{1} \) has one element, \( A_{2} \) has two elements, ..., \( A_{m} \) has \( m \) elements, and no subset is contained within another subset. Find the maximum value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-2}
Triangle  $ABC$  is a triangle with side lengths  $13$ ,  $14$ , and  $15$ . A point  $Q$  is chosen uniformly at random in the interior of  $\triangle{ABC}$ . Choose a random ray (of uniformly random direction) with endpoint  $Q$  and let it intersect the perimeter of  $\triangle{ABC}$  at  $P$ . What is the expected value of  $QP^2$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{84}{\pi}}
设 $A=\{1,2,3, \cdots, 4 n+2\}, M=\{2 n+1,4 n+3,6 n+5\}$. 对 $A$ 的任意非空子集 $B$, 当 $B$ 的任意两个数之和不属于 $M$ 时, 称 $B$ 为 $M$-自由集。如果 $A=A_1 \cup A_2, A_1 \cap A_2=$ $\varnothing$, 且 $A_1, A_2$ 均为 $M$-自由集, 那么称有序对 $\left(A_1, A_2\right)$ 为 $A$ 的一个 $M$ - 划分. 试求 $A$ 的所有 $M-$ 划分的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{n+1}$}
For which values of the parameter \( a \), the equation

\[
3^{x^{2}+6 a x+9 a^{2}} = a x^{2}+6 a^{2} x+9 a^{3}+a^{2}-4 a+4
\]

has exactly one solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the generating function for the sequence 1,1,1,1,1,1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x^6-1}{x-1}}
Round down the largest four-digit number you can make by playing four 7, 9, 3, and 6 cards only once, to the hundredth place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9700}
Red spends his Sunday at home with the following schedule: 30 minutes on the computer, 20 minutes listening to music, 35 minutes cooking (including 3 minutes for rice washing), 15 minutes tidying up the room, and 25 minutes practicing the piano. Among these activities, how many minutes are needed for the most reasonable arrangement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是以$\frac{m}{n}$的形式表示，请给出$m+n$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{289}
Calculate the following expression:
$$2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))))))$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4094}
设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点,已知 $B F=4, B D=2, B E=5, \angle B A D=\angle A C D, \angle B A F=\angle C A E$, 求 $B C$ 的长。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{217}
Consider all functions $f: \mathbb{N} \to \mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $\mathbb{N}$. Determine the least possible value of $f(1998)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
A semicircular sponge with a diameter of $20 \text{ cm}$ is used to wipe a corner of a room's floor such that the ends of the diameter continuously touch the two walls forming a right angle. What area does the sponge wipe?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \pi \text{ cm}^2}
If $x > 0$, $y > 0$, and $\frac{1}{2x+y} + \frac{4}{x+y} = 2$, find the minimum value of $7x + 5y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7 + 2\sqrt{6}}
Let $\{a_n\}$ be a geometric sequence, $\{b_n\}$ be an arithmetic sequence, and $b_1=0$, $c_n=a_n+b_n$. If $\{c_n\}$ is $1$, $1$, $2$, $\ldots$, find the sum of the first $10$ terms of the sequence $\{c_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{978}
Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}}
marts income is 60 percent more than tims income and tims income is 50 percent less than juans income. additionally, katies income is 20 percent more than marts income, and jasons income is 30 percent less than katies income. what percentage of jasons income is juans income
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148.81\%}
一个 10\times10 的数表初始每个位置的数字均为 1，现在做有限次操作，每次把某行或某列的所有数换成相反数. 称非负整数 N 是好的，如果最终数表中 -1 的个数恰为 N. 则好的 N 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Determine the number of isolated subgroups of a value group with rank $n$, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
In order to investigate the number of students from various classes participating in the extracurricular calligraphy group at a school, 5 classes were randomly selected, and the number of students from each class participating in the group was taken as sample data. It is known that the sample mean is 7, the sample variance is 4, and the sample data are all different from each other. Find the maximum value among the sample data.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the dimension of the irreducible highest weight module for the simple Lie algebra of type $G_2$ using the Weyl dimension formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $ABC$ be a triangle with $AB = 2$ and $AC = 3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2 = \frac{a + b\sqrt{c}}{d}$ for positive integers $a, b, c$, and $d$, where $\gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Compute $1000a + 100b + 10c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270382}
Among the four-digit numbers, the number of four-digit numbers that have exactly 2 digits repeated is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3888}
Evaluate the following limit:

$$\lim_{n \to \infty} \frac{1}{n}\left((m+1)(m+2) \cdots (m+n)\right)^{\frac{1}{n}}$$

where $m$ is a positive constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{e}\)}
In a rectangular solid with edge lengths of 3, 8, and 9, if a cylindrical hole is drilled through the top surface without changing its surface area, then the radius of the hole is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( f(x) = \cos (\cos (\cos (\cos (\cos (\cos (\cos (\cos x))))))) \), and suppose that the number \( a \) satisfies the equation \( a = \cos a \). Express \( f'(a) \) as a polynomial in \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 - a^2)^4}
For which smallest $n$ does there exist a convex $n$-gon such that the sines of all its angles are equal, but the lengths of all its sides are different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline {AB}$, $\overline{CD}$, and $\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac {15}{32}\sqrt{3}$}
What is the base of the numeral system in which the decimal number 1728 is represented as 3362?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the smallest positive integer $n$ such that for every integer $m$, with $0<m<1993$, there exists an integer $k$ for which
$$
\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3987}
The area of a rectangle divided by a diagonal is 576 square centimeters (cm2). If the width of the rectangle is 32 centimeters (cm), what is the height in centimeters (cm)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n - m + 1 }
Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers $n$, $k$, and $l$, where $n! = 1 \cdot 2 \cdots n$. In your answer, indicate 0 if there are no solutions, specify the value of $n$ if there is only one solution, or provide the sum of all $n$ values if there are multiple solutions. Recall that a solution is a triple $(n, k, l)$; if they differ in at least one component, they are considered different solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let $A$ be the set of all sequences of 0's and 1's with length 4. What is the minimal number of sequences that can be chosen, such that any arbitrary sequence from $A$ differs in at most one position from one of the chosen sequences?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $\left(x_{1}, x_{2}, \cdots, x_{20}\right)$ be a permutation of $(1,2, \cdots, 20)$ that satisfies $\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right)=620$. How many such permutations are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(10!)^2}
In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
If the sum of the coefficients of the expansion of $(x+y)^{3}(2x-y+a)^{5}$ is $256$, find the coefficient of the term which contains the letter $x$ and in which $x$ is raised to the power of $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given $\sin (α- \frac{π}{3})= \frac{3}{5} $, where $a∈(\frac{π}{4}, \frac{π}{2})$, find $\tan α=\_\_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \frac{48+25 \sqrt{3}}{11}}
Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Paulo wants to get a burger meal that costs $6. Aside from that, he also wants to get a soda which costs 1/3 as much as the burger meal. While on the counter, Jeremy asked Paulo to get him 2 of each item Paulo is going to get. Stephanie also wants to order, and she is going to get 3 burger meals and 1 soda. There is a 10% discount on all burger meal purchases during the Happy Hour promo when buying more than 2 items. In addition, a 5% sales tax is applied to the total order cost before any discounts. How much will they be charged for their orders combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$42.42}
John had a stock of 900 books in his bookshop. He sold 75 books on Monday, 50 books on Tuesday, 64 books on Wednesday, 78 books on Thursday, and 135 books on Friday. However, he also had daily customer returns and discounts. On Monday, 5 customers returned books, and he provided a 10% discount to 20 customers. On Tuesday, 10 customers returned books, and he provided a 15% discount to 15 customers. On Wednesday, 3 customers returned books, and he provided a 5% discount to 10 customers. On Thursday, 7 customers returned books, and he provided a 20% discount to 5 customers. On Friday, 15 customers returned books, and he provided a 25% discount to 25 customers. What percentage of the books were not sold by the end of the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59.78\%}
Find the smallest volume bounded by the coordinate planes and  by a tangent plane to the ellipsoid  $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2} abc}
1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94605}
Several identical boxes weigh a total of 10 tons, with each box weighing no more than 1 ton. What is the minimum number of 3-ton trucks needed to transport all this cargo in one trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the smallest positive integer $k$ such that there exist positive integers $M, O > 1$ satisfying:
\[
 (O \cdot M \cdot O)^k = (O \cdot M) \cdot \underbrace{(N \cdot O \cdot M) \cdot (N \cdot O \cdot M) \cdot \ldots \cdot (N \cdot O \cdot M)}_{2016 \text{\ (N \cdot O \cdot M)s}},
\]
where $N = O^M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2823}
对于一个由至少两个正整数构成的集合，若每个元素均与其他至少一个元素不互质，则称该集合为"芳香集".设 P(n)=n^2+n+1. 求最小的正整数 b，使得存在非负整数 a，满足集合 \{P(a+1),P(a+2),\cdots,P(a+b)\} 是芳香集.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A tailor trims 6 feet from opposite edges of a square piece of cloth, and some feet from the other two edges. If 120 square feet of cloth remain, and the length of a side of the original piece of cloth was 22 feet, how many feet did the tailor trim from the other two edges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find $$\lim_{x\to\infty}\frac{\sqrt{1-\cos^3(1/x)}\cdot(3^{1/x}-5^{-1/x})} {\log_2( 1+x^{-2} + x^{-3})}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{\sqrt{6}}{2}\ln(2)\ln(15)$$}
A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their non-negative difference. When just one number remains, it is compared to \(x\). If it has the same parity A wins, otherwise B wins. Find the probability that A wins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Find the Taylor series expansion of $\sqrt{\frac{1-x}{1+2x}}$ around $x=0$ up to the term in $x^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{3x}{2} + \frac{15x^2}{8}}
Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A host standing at the golden ratio point of the stage appears more natural and appropriate. If the length of stage AB is 20 meters and the host currently stands at point A, then they should walk at least ____ meters to reach the ideal position. (Round the result to 0.01 meters)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.64}
Next year's 'leap day', 29th February 2024, may be written in 'ddmmyy' format as the number 290224. What is the sum of all the two-digit factors of 290224?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{397}
There are  $63$  houses at the distance of  $1, 2, 3, . . . , 63 \text{ km}$  from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants,  $63$  elfs named  $E_1, E_2, . . . , E_{63}$  , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf  $E_n$   takes  $n$  minutes to travel  $1 \text{ km}$  for each  $n = 1,2,...,63$  , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Suppose point $P$ is on the parabola $y^2=x$, and point $Q$ is on the circle $(x-3)^2+y^2=1$, then the minimum value of $|PQ|$ is $\_\_\_\_\_\_\_\_\}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{11}}{2}-1}
In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\angle LMN$ is $60^{\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{3}}{3}}