inclusionAI/Ring-lite-rl-data huggingface dataset:
The graphs of the quadratic functions $y=x^2-2x+2$ and $y=-x^2+ax+b$ ($a>0$, $b>0$) have tangent lines at one of their intersection points that are perpendicular to each other. Find the minimum value of $\frac{1}{a} + \frac{4}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{18}{5}}
一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 $p(0 \leq p \leq 1)$ 的概率消失，有 $\frac{1-p}{3}$的概率保持不变，有 $\frac{1-p}{3}$ 的概率分裂成两个，有 $\frac{1-p}{3}$ 的概率分裂成三个.对所有新产生的生物每天也会发生上述事件.假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 $\frac{1}{2}$ ，则 $p$ 至多为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Rectangular tiles each of size 25 cm by 65 cm must be laid horizontally on a rectangular floor of size 150 cm by a certain width, such that the tiles do not overlap and they are placed with edges jutting against each other on all edges. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is 36. What is the width of the floor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{450}
Let $ABC$ be a triangle with circumcenter $O$ such that $AC=7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{2}}
Determine the number of points among M(1, 1), N(1, 2), P(2, 1), Q(2, 2), and G(2, $\frac{1}{2}$) that satisfy the condition of being a "lucky point," where a "lucky point" is defined as a point that lies on both the graph of a logarithmic function and an exponential function. Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 2:7.$  Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some constants $t$ and $u.$  Enter the ordered pair $(t,u).$

[asy]
unitsize(1 cm);

pair A, B, P;

A = (0,0);
B = (5,1);
P = interp(A,B,2/9);

draw(A--B);

dot("$A$", A, S);
dot("$B$", B, S);
dot("$P$", P, S);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{7}{9}, \frac{2}{9} \right)}
Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Let \( A = \{1, 2, \ldots, 2011\} \). Find the number of functions \( f \) from \( A \) to \( A \) that satisfy \( f(n) \leq n \) for all \( n \) in \( A \) and attain exactly 2010 distinct values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2011} - 2012}
Two opposite edges of a tetrahedron are equal to \( a \), two other opposite edges are equal to \( b \), and the remaining two opposite edges are equal to \( c \). Find the radius of the circumscribed sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4} \sqrt{2 \left(a^2 + b^2 + c^2\right)}}
Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulting fraction can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Determine the number of distinct ellipses that can be formed by selecting three different numbers from -3, -2, -1, 1, 2, 3 to serve as coefficients of an elliptic equation $ax^2+by^2+c=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \( q \) be a 7th root of unity. Calculate \(\frac{q}{1+q^{2}} + \frac{q^{2}}{1+q^{4}} + \frac{q^{3}}{1+q^{6}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Out of 1,500 senior high school students, 70% passed their English exams, 80% passed their Mathematics exams, and 65% passed their Science exams. To get their degree, a student must pass in all three subjects. Assuming that retaking the exams isn't an option, how many students failed to get their degree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{954}
After removing the two jokers from a deck of playing cards, a card is drawn at random. What is the probability of drawing a diamond or an Ace?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{13}}
Determine the limit of the sequence \( a_n = \frac{3 \cdot 5 \cdot 7 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2n} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
将 4 个数 $1,9,8,8$ 写成一行并进行以下操作：对每一对相邻的数都作一次减法, 即用右边的数减左边的数, 然后将所得的数写在两数之间, 算是完成了一次操作. 然后再对这 7 个数所排成的一行进行同样的操作, 如此继续下去, 共操作 100 次, 求最后所得到的一行数的和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{726}
Find the smallest possible non-zero difference $|a-b|$ between two distinct positive integers $a$ and $b$ such that the expressions $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are all integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Determine the number of digits that can replace □ in the multiplication □4×32 to make the product a three-digit number. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the total number of "magic" circles for a given integer \( n \). A "magic" circle is defined as a circle with \( 2^n \) items on its periphery, where each item holds either 0 or 1. For each item, consider the sequence of \( n \) items to the right. The set of these sequences must consist of all possible \( n \)-length combinations of 0 and 1. What is the number of such "magic" circles for a given \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^{n-1} - n}}
Evaluate the limit of the following sum as \(x\) approaches 0 from the right: \[\lim_{x\to 0^{+}}\,\,\sum_{n = 2}^{\infty}{(-1)^{n} \over \ln^{x}(n+1)}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the double integral: $$\int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21600}
Find the values of \( k \) for which the matrix \( A = \begin{bmatrix} 1 & 1 & k \\ 1 & 1 & k \\ 1 & 1 & k \end{bmatrix} \) has three linearly independent eigenvectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \neq -2}
Let $C$ be a code with a distance $d = 2t + u + 1$. Determine the maximum number of errors that $C$ can detect if it is used simultaneously to correct $t$ errors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t + u}
One third of a solution that was 8% sugar by weight was replaced by a second solution, resulting in a solution that was 20% sugar by weight. The second solution was what percent sugar by weight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44\%}
Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is $12$. Given that $EG^2 = p - q\sqrt {r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p + q + r.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
Let a be four years older than b who is a half-sibling to c. If b is twice as old as c, and the total of their ages in months is 720, then how old is b in years, considering that a is three quarters of a year older than his half-sibling d?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.4}
Find $x$ if \[ \frac{1}{\frac{1}{\frac{1}{\frac{1}{x} + \frac{1}{2}} + \frac{1}{\frac{1}{x} + \frac{1}{2}}} + \frac{1}{\frac{1}{\frac{1}{x} + \frac{1}{2}} + \frac{1}{\frac{1}{x} + \frac{1}{2}}}} = \frac{x}{36}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Find the units digit of the decimal expansion of\[(15 + \sqrt{220})^{19} + (15 + \sqrt{220})^{82}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
0

}
Use the Central Limit Theorem to approximate the 0.95-quantile \( q_{0.05}(\mathcal{A}_n) \) for the sample size \( n=100 \), given that \( q_{0.05}(\mathcal{A}_1)=3.84 \). Assume \( \mathcal{A}_1 \sim \chi^2_1 \) and \( n \cdot \mathcal{A}_n \sim \chi^2_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.23}
There are $5$ pots of different chrysanthemums, among which $2$ pots are white, $2$ pots are yellow, and $1$ pot is red. They need to be arranged in a row such that the red chrysanthemum is in the middle, the white chrysanthemums are not adjacent, and the yellow chrysanthemums are not adjacent. How many different arrangements are there? (Answer with a number.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Calculate the volume of the region bounded by the surfaces \( z = (x^2 + y^2)^2 \) and \( z = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Determine the domain of the function \( \ln(x)^{\ln(x)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Determine the number of elements of order 10 in the group \( C_{15} \times C_{12} \times C_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the expected area of the projection of a unit cube onto a plane under isotropic random direction of projection using a different approach than Arnold's solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
The area of triangle $ABC$ is 1. Let $A_1, B_1, C_1$ be the midpoints of the sides $BC, CA, AB$ respectively. Points $K, L, M$ are taken on the segments $AB_1, CA_1, BC_1$ respectively. What is the minimum area of the common part of triangles $KLM$ and $A_1B_1C_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
A one-foot stick is marked in 1/4 and 1/5 portions. How many total markings will there be, including the end points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( G \) be a finite group and \( g, h \in G - \{1\} \) such that \( g^{-1}hg = h^2 \). Given that \( o(g) = 5 \) and \( o(h) \) is an odd integer, find the order of \( h \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
设 a_n=\left(\df{1+q}{1-q}\right)\cdot\left(\df{1+q^2}{1-q^2}\right)^{\frac{1}{2}}\cdot\left(\df{1+q^4}{1-q^4}\right)^{\frac{1}{4}}\cdot\cdots\cdot\left(\df{1+q^{2^n}}{1-q^{2^n}}\right)^{\frac{1}{2^n}}，其中 q=\df{11}{13}. 记 a=\displaystyle\lim_{n\rightarrow+\infty}a_n，则 [a]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
某个国家有 2023 个岛和 2022 座桥, 任意一座桥连接两个不同的岛, 任意两个岛之间至多有一座桥相连, 且可以从任意一个岛通过若干座桥到达其他任何岛. 若三个岛中的某个岛与另两个岛都有桥连接, 则称这三个岛组成"岛群", 已知任两个"岛群"中都有相同的岛, 那么恰有一座桥的岛最少有多少个?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
Given point $A$ is a fixed point on the circle $O: x^{2}+y^{2}=4$, and point $B$ is a moving point on the circle $O$. If it satisfies $| \overrightarrow{AO}+ \overrightarrow{BO}|=| \overrightarrow{AO}- \overrightarrow{BO}|$, then $\overrightarrow{AO} \cdot \overrightarrow{AB}=$ \_\_\_\_\_\_ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
On the segment $OA$ of length $L$ on the number line $Ox$, two points, $B(x)$ and $C(y)$, are randomly placed. Find the probability that from the three resulting segments a triangle can be formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
Given that triangle \( \triangle ABC \) satisfies \( \frac{\angle A}{3} = \frac{\angle B}{4} = \frac{\angle C}{5} \), find the value of \( \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{|BC|^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 + \sqrt{3}}{4}}
In each cell of a $5 \times 5$ grid, a natural number is written with invisible ink. It is known that the sum of all the numbers equals 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Compute the surface integral \( \iint_S F \cdot dS \) where \( F = \langle y, z, -x \rangle \) and \( S \) is the portion of the paraboloid \( z = x^2 + y^2 \) that lies below the plane \( z = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
An infinite sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the recurrence $$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$ for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\cdots+a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of non-zero ring homomorphisms from the ring \( \mathbb{Z}_{30} \) to the ring \( \mathbb{Z}_{42} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the minimum number of factors that must be removed from the product of all integers from 1 to 99, denoted as \( 99! = 1 \cdot 2 \cdot \ldots \cdot 99 \), so that the product of the remaining factors ends in the digit 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
How many elements of the set of all $2 \times 2$ matrices with entries in the field with 7 elements are similar to the matrix $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
The teacher secretly thought of a three-digit number $S$. Students $A$, $B$, $C$, and $D$ tried to guess the number by suggesting $541$, $837$, $291$, and $846$ respectively. The teacher informed them that each of them got exactly one digit correct, and it was in the correct position. What is the number $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{247}
A rhombus \(ABCD\) and a triangle \(ABC\), containing its longer diagonal, are inscribed with circles. Find the ratio of the radii of these circles, if the acute angle of the rhombus is \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \cos \frac{\alpha}{2}}
The function $f(x)$ satisfies
\[f(x) - 3 f\left(\frac{1}{x}\right) = 3^x\]
for all $x \neq 0$. Find $f(2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{9 + 3\sqrt{3}}{8}}
In the expansion of $( \sqrt {x}+ \frac {1}{x})^{n}$, the sum of coefficients of all quadratic terms is $64$. What is the constant term in the expansion?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $N = \prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{624}
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
On the hypotenuse \( AB \) of a right triangle \( ABC \), square \( ABDE \) is constructed externally with \( AC=2 \) and \( BC=5 \). In what ratio does the angle bisector of angle \( C \) divide side \( DE \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:5}
Determine the minimum number of isoclinic rotations required to transform an arbitrary 2-plane in \( \mathbb{R}^4 \), passing through the origin, into any of the Euclidean orthonormal 2-planes in a standard Cartesian 4D coordinate system.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $y=f(x)$ be a function defined on $\mathbb{R}$ with a period of $1$. If $g(x) = f(x) - 2x$ has a range of $[-2, 6]$ on the interval $[2, 3]$, find the range of $g(x)$ on $[-2017, 2017]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-4030, 4044]}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{de} f$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
There are 2015 coins on the table. Two players play the following game: they take turns; on their turn, the first player can take any odd number of coins from 1 to 99, while the second player can take any even number of coins from 2 to 100. The player who cannot make a move loses. How many coins should player number 1 take on their first move to guarantee a win with an unchanged strategy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
A particle starts from rest at $t=0$, $x=0$, and comes to rest again at $x=1$, $t=1$. Let the instantaneous acceleration be $a(t)$. Determine the largest constant $c$ such that $\sup_{t \in [0,1]} |a(t)| \geq c$ is guaranteed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T$ be a tree with exactly two vertices of degree $7$ and exactly $20$ vertices of degree $100$. What is the minimum possible number of vertices in a tree $T$ that satisfies those restrictions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1994}
Given the quadratic function f(x) = x^2 - ax + 1, find the necessary and sufficient condition on 'a' for the inequality f(x) < 0 to have a solution with respect to x. Express your answer as an inequality involving 'a'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|a| > 2}
A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus $3$ stops behind. After how many minutes will Zia board the bus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$17$}
the average of 6 integers is 63 , and none of these integers is greater than 100 . if the average of 3 of the integers is 65 , what is the least possible value of one of the other two integers ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Let $g_n:[0,1] \to \mathbb{R}$ be defined by \( g_n(x) = \frac{nx}{1+n^2 x^q} \) for \( q > 0 \). Determine the values of \( q \) for which the sequence \( g_n \) converges uniformly to its point-wise limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Find a function \( g: \mathbb{N} \to \mathbb{P}(\mathbb{N}) \) such that the set \( A = \{ n \in \mathbb{N} \mid n \notin g(n) \} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(n) = \mathbb{N}}
Let $f(x)=\mathrm e^{-x}\sin(x)g(x)$, where $g:\mathbb{R^+}\to (-M, M)$ for some positive real $M$. Determine the number of solutions to the equation $|f(n_1)|+|f(n_2)|+\cdots+|f(n_k)|=1$, where all $n_i$ are natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Consider i.i.d. random variables $X_{1}, X_{2}, \ldots$, such that $X_{i}$ has a uniform distribution over the interval $(0, 1)$. Find the limit:
\[\lim_{n \to \infty} P\left(\sum_{i=1}^{n} X_i \leq 2 \sum_{i=1}^n{X_{i}^2}\right).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the line integral \( \iint_{S_1\cup S_2} \text{curl}F\cdot dS \) for the surfaces \( S_1=\{(x,y,z):x^2+y^2=1, 0\le z \le 1\} \) and \( S_2=\{(x,y,z): x^2+y^2 +(z-1)^2 =1, z\ge 1\} \), where \( F(x,y,z)=(zx+z^2y+x, z^2yx+y, z^4x^2) \). Apply Stokes' theorem to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the definite integral: $$\int_{0}^{\infty}\frac{4x\ln (x)}{x^4+4x^2+1}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \( \lim_{x\to 0^+}\sum_{n=2020}^{\infty}\frac{(-1)^n}{\log^x(n)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of distinct locations where a treasure could be found, given the following conditions: You are at the origin $(0,0)$, and the treasure lies within a $100$ meter radius. Person A claims to have found the treasure by walking $108$ meters from the origin, with their distance calculated as $|x| + |y|$. Person B claims to have found the treasure by walking $105$ meters from the origin, with their distance calculated as $\frac{|x|}{\sqrt{3}} + \max\left\{\frac{|x|}{\sqrt{3}}, |y|\right\}$. How many distinct locations could the treasure be located?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles needed to divide a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}