Context:
A person has 120 light bulbs, each with a lifetime that follows an independent exponential distribution with a mean of 6 hours. When a bulb fails, it is immediately replaced by a new one, with the replacement time being uniformly distributed over (0, 0.25) hours. Approximate the probability that all bulbs have failed by time 750 hours.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \[N=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}},\] then $N$ equals
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game where the winner is the first player to have 7 or more points and is 2 points ahead of the other player, you currently lead with a score of 6-5. Assuming both players have equal skill and each point is equally likely to be won by either player, what is the probability that you will eventually win the game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the category \(\mathbf{FSet}\) of finite sets as an algebraic theory. What is the category of models for this theory?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum possible value of
\[\sqrt{58 - 42x} + \sqrt{149 - 140 \sqrt{1 - x^2}}\]where $-1 \le x \le 1.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many powers of 2 have only the digits 0, 1, 2, 4, and 8 in their decimal representation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine \( \lim_{t \to \infty} u(1,t) \), where \( u(x,t) \) is the solution to the heat equation \( u_t = u_{xx} \) with the initial condition \( u(x,0)=\frac{e^{2x}-1}{e^{2x}+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In each cell of  $5 \times 5$  table there is one number from  $1$  to  $5$  such that every number occurs exactly once in every row and in every column. Number in one column is *good positioned* if following holds:
 - In every row, every number which is left from *good positoned* number is smaller than him, and every number which is right to him is greater than him, or vice versa.
 -  In every column, every number which is above from *good positoned* number is smaller than him, and every number which is below to him is greater than him, or vice versa.
What is maximal number of good positioned numbers that can occur in this table?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression: $$ \sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) \, dx. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Twelve couples are seated around a circular table such that all of men are seated side by side, and every women are seated to opposite of her husband. In every step, a woman and a man next to her are swapping. What is the least possible number of swapping until all couples are seated side by side?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{q\rightarrow 1^-}\frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}} \), where the Euler function is defined as \( \phi(q)=\prod_{n=1}^{\infty}(1-q^{n}) \) and \( q=\exp(2\pi i \tau) \) with \( |q|<1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $P$  be a point outside a circle  $\Gamma$  centered at point  $O$ , and let  $PA$  and  $PB$  be tangent lines to circle  $\Gamma$ . Let segment  $PO$  intersect circle  $\Gamma$  at  $C$ . A tangent to circle  $\Gamma$  through  $C$  intersects  $PA$  and  $PB$  at points  $E$  and  $F$ , respectively. Given that  $EF=8$  and  $\angle{APB}=60^\circ$ , compute the area of  $\triangle{AOC}$ .

*2020 CCA Math Bonanza Individual Round #6*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Right $\triangle ABC$ has $AB=3$, $BC=4$, and $AC=5$.  Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $\overline{AC}$, $W$ on $\overline{AB}$, and $Z$ on $\overline{BC}$.  What is the side length of the square?

[asy]
pair A,B,C,W,X,Y,Z;
A=(-9,0); B=(0,12); C=(16,0);
W=(12A+25B)/37;
Z =(12C+25B)/37;
X=foot(W,A,C);
Y=foot(Z,A,C);
draw(A--B--C--cycle);
draw(X--W--Z--Y);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,E);
label("$W$",W,NW);
label("$X$",X,S);
label("$Y$",Y,S);
label("$Z$",Z,NE);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral of \( f(z) = \frac{z^2}{e^{\frac{1}{(z+i)^2}}} \) over the contour \(|z|=2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the vector space of functions \( f: \mathbb{R} \to \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{x \to {0_+}} \frac{\ln\Bigl(1+ \cfrac{\ln (x \ln a)}{\ln x}\Bigr)}{\ln a}-\frac{\ln\Bigl(1-\cfrac{\ln (x \ln a)}{\ln x}\Bigr)}{\ln a}\) where \(a > 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[ N = 1111\cdots 11. \] Find the thousandth digit after the decimal point of $\sqrt{N}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若 $z \in \mathbf{C}, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, 请提供 $z$ 的两个解中实部和虚部之和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine which function grows faster as $n$ approaches infinity: $f(n) = n!$ or $g(n) = n^{\log_2 n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of the function \( f(x) = \arctan\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) \) with respect to \( \arccos(x^2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\alpha$ and $\beta$ be nonzero real numbers such that $2 (\cos \alpha-\cos\beta)+\cos\alpha\cos\beta=1$. Find the value of $\tan \left(\frac{\alpha+\beta}{2}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given polynomial functions \( P, Q, R: \mathbb{C} \rightarrow \mathbb{C} \) with maximum degree 2, and complex numbers \( a, b, c \) such that:
\[
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(b) & Q(b) & R(b) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} = 1,
\]
calculate the following sum of determinants:
\[
\begin{vmatrix}
P(1) & Q(1) & R(1) \\
P(b) & Q(b) & R(b) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} +
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(1) & Q(1) & R(1) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} +
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(b) & Q(b) & R(b) \\
P(1) & Q(1) & R(1) \\
\end{vmatrix}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of subsets \( S \) of \(\{1,2, \ldots 6\}\) satisfying the following conditions:
- \( S \) is non-empty.
- No subset of \( S \) has the property that the sum of its elements is 10.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular octahedron [ABCDEF] is positioned such that its vertices lie on the coordinate axes. The face [ABC] is numbered with the number 1. Number the remaining seven faces with the numbers 2 to 8, using a different number for each face. How many different ways can the remaining faces be numbered such that at least three of the faces concurrent at vertex A have odd numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided by 2010?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), consider two planes \( \alpha \) and \( \beta \):
- Plane \( \alpha \) is perpendicular to the line \( A_{1} C_{1} \).
- Plane \( \beta \) is parallel to the line \( C D_{1} \).

Find the minimum possible angle between planes \( \alpha \) and \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( w = z \tan^{-1}\left(\frac{x}{y}\right) \), calculate the value of \( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of subgroups of order \(p^2\) in the group \(\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Cramer-Rao lower bound for the parameter $\lambda$ of the exponential distribution, $f(x) = \lambda e^{-\lambda x}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through vertex $C$ of square $ABCD$, a line passes, intersecting diagonal $BD$ at point $K$ and the perpendicular bisector of side $AB$ at point $M$ ($M$ lies between $C$ and $K$). Find the angle $\angle DCK$ if $\angle AKB = \angle AMB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit for $0<p<1$:
$$\lim_{n\rightarrow \infty}\frac{(p+1)\sum\limits_{k=1}^n k^p \log k-n^{p+1}\log(n)}{n^{p+1}(p+1)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through the midpoint $D$ of the base of an isosceles triangle, a line is drawn at an angle of $30^{\circ}$ to this base, on which the angle $\angle ACB$ intercepts a segment $EF$. It is known that $ED = 6$ and $FD = 4$. Find the height of the triangle drawn to the base.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Gaussian curvature of a surface with the metric \( ds^2 = du^2 + \cosh^2(u) dv^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a convex quadrilateral ABCD with side lengths AB=8, BC=4, CD=DA=6, express the range of angle A using an inequality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int x^2 \, d\mu \) where \( C \) is the Cantor set and \( \mu \) is the uniform measure on \( C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From a container having pure milk, 30% is replaced by water. This process is repeated five times. At the end of the fifth operation, what is the composition of the mixture in the container?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Linda spent 3 / 4 of her savings on furniture and the rest on a TV and a refrigerator. The TV cost her $210, and the refrigerator cost her 20% more than the TV. Additionally, she got a 7% discount on the furniture and had to pay a 6% tax on the TV. What were her original savings before making these purchases?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two independent random variables $x_i$ and $x_j$ sampled from the same distribution $X$ with finite moments, find the expected value of their product, $\mathbb{E}[x_ix_j]$. Assume $x_i$ and $x_j$ are independent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that a ( a - 6 ) equals some number and b ( b - 6 ) equals the same number, where a ≠ b , the sum of a and b is 6. What is the value of the number that both equations equal to?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \( \int_0^\infty \frac{\ln(1/x)}{1-x^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. Find the remainder when $S(2015)$ is divided by 12.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12, what is the area of the equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-negative integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 45\) given the following conditions: \(x_1 + x_2 > 0\), \(x_2 + x_3 > 0\), \(x_3 + x_4 > 0\), and \(x_4 + x_5 > 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the order of the group generated by all cycles of length $n$. For example, when $n=4$, the set of cycles is: \( G = \{(1234), (1243), (1342), (1324), (1432), (1423)\} \). Determine the order of the group generated by this set.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of distinct real roots that the equation \( \sum_{i=1}^{n}a_{i}b_{i}^{x} + c = 0 \) can have, where \( a_{i}, b_{i}, c \) are real constants, \( b_{i} > 0 \), \( b_{i} \neq b_{j} \) for \( i \neq j \), and \( c \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real solutions to the equation \( x = 99 \sin(\pi x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carrie wants to take a trip to New York. She can get a 20% discount on a flight with Delta Airlines. She can also save 30% off an $1100 flight with United Airlines. She would save $90 by choosing the cheapest flight. What is the original price of the Delta Airlines flight?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a society of 30 people, any two people are either friends or enemies, and everyone has exactly six enemies. How many ways are there to choose three people from the society so that any two of them are either all friends or all enemies?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p, q$ be two-digit numbers, neither of which are divisible by $10$. Let $r$ be the four-digit number formed by placing the digits of $p$ followed by the digits of $q$ in order. As $p$ and $q$ vary, a computer prints $r$ on the screen if $\gcd(p, q) = 1$ and $p + q$ divides $r$. Suppose that the largest number printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{\sin(nx)}{n^\alpha} \) converges, given that \( \alpha > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle. Given that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$, find a simple expression for $d$ in terms of $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). Find the area of triangle \( ABC \) if it is known that \( S_{BQC} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many times must a fair die be rolled so that there is less than a 1% chance that the mean of the scores differs from 3.5 by more than 0.1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Prime Land, there are seven major cities, labeled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e., we take the indices modulo $7$. Al initially starts at city $C_0$.

Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise, he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\frac{m}{1024}$, find $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Little Tiger places chess pieces on the grid points of a 19 × 19 Go board, forming a solid rectangular dot matrix. Then, by adding 45 more chess pieces, he transforms it into a larger solid rectangular dot matrix with one side unchanged. What is the maximum number of chess pieces that Little Tiger originally used?
Please reason step by step, and put your final answer within \boxed{}.


Context:
从一个正 9 边形的 9 个顶点中选 3 个使得它们是一个等腰三角形的三个顶点的方法数是多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABC$  be a scalene triangle whose side lengths are positive integers. It is called *stable* if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the system of equations:
\[ x^2 - y^2 - u^3 + v^2 + 4 = 0 \]
\[ 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \]
find \( \frac{\partial u}{\partial x} \) at the point \((x,y) = (2,-1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \(\sqrt{x} = 1 + \ln(5 + x)\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the minimum possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each face of a cube is painted either  red or blue, each with probability 1/2. The color of each face is determined independently.  What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A right circular cone has a base radius $R$; its height is $m=2R$. At what distance from the vertex must the cone be cut by a plane parallel to the base so that the total surface areas of the resulting parts are equal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
(5 points) $2\lg5 \cdot 2\lg2 + e\ln3 =$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box measuring some inches long by 20 inches wide by 10 inches deep is to be filled entirely with identical cubes. No space is to be left unfilled. The smallest number of cubes that can accomplish this objective is 56. What is the length of the box?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base three representation of $x$ is $12112211122211112222$. The first digit (on the left) of the base nine representation of $x$ is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $Y_1, Y_2, \ldots$ are independent and identically distributed random variables, and $M$ is a random variable independent of the $Y_i$. Define $Z = \sum_{i=1}^M Y_i$. What is the expected value of $Z$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle between two lines whose direction cosines \((l, m, n)\) satisfy the equations:

\[ l + m + n = 0 \]
\[ \frac{mn}{q-r} + \frac{nl}{r-p} + \frac{lm}{p-q} = 0 \]

where \(p\), \(q\), and \(r\) are constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $a$ such that the integral

$$\int \limits^{\infty }_{0} \frac{dx}{a^2 + \left(x - \frac{1}{x}\right)^2} = \frac{\pi}{5050}$$

holds true, given that $a^2 \geq 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
11.5. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, two diagonals of the lateral faces $A_{1} C_{1}$ and $C_{1} D$ are drawn. Find the angle between them, given that the diagonal $A_{1} C_{1}$ is equal in length to one of the edges of the parallelepiped, and the diagonal $C_{1} D$ forms an angle of 30 degrees with this same edge.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在 $\triangle A B C$ 中, $\sin ^{2} A+\sin ^{2} C=2018 \sin ^{2} B$, 则 $\frac{(\tan A+\tan C) \tan ^{2} B}{\tan A+\tan B+\tan C}=$ $\qquad$.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{}.


Context:
On a similar one-way single-lane highway, cars travel at the same speed and obey a modified safety rule: the distance from the back of the car ahead to the front of the car behind is exactly two car lengths for each 20 kilometers per hour of speed or fraction thereof. Assuming each car is 5 meters long and that cars can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye in one hour. Find the quotient when $N$ is divided by 10.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \( \lim_{n \to \infty} -n \mathbb{1}_{Z \geq 1 - \frac{1}{n}} \), where \( Z \) is a random variable on the sample space \( \Omega = (0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{a_{n}\}(a_{n} \gt 0)$, where $S_{n}$ is the sum of its first $n$ terms. If $\frac{1}{3}$, $S_{3}+1$, and $S_{9}$ form a geometric sequence, then the minimum value of $\frac{S_{6}}{S_{3}}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the base of a regular triangular prism is a triangle \( ABC \) with a side length of \( a \). Points \( A_{1} \), \( B_{1} \), and \( C_{1} \) are taken on the lateral edges, and their distances from the base plane are \( \frac{a}{2} \), \( a \), and \( \frac{3a}{2} \) respectively. Find the angle between the planes \( ABC \) and \( A_{1} B_{1} C_{1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{S^2} \frac{dS}{((x-a)^2 +y^2+z^2)^{1/2}} \), where \( a > 1 \) and \( S \) is the unit sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two positive numbers are in a certain ratio. The second number is 20% more than the first. If the ratio is 1 / 6 : 1 / 5, what is the second number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum value of \( n \) such that for every subset \( A \subseteq N_n \) with \(|A| \geq 10\), the inequality \( \max_{\substack{a, b \in A \\ a \neq b}} |a-b| \leq 10 \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
10. Betya and Vasya are playing a game. Betya selects 100 (possibly the same) non-negative numbers $x_{1}, x_{2}, \cdots, x_{100}$, whose sum equals 1. Vasya then divides them into 50 pairs, calculates the product of the two numbers in each pair, and writes the largest of these products on the blackboard. Betya wants the number written on the blackboard to be as large as possible, while Vasya wants it to be as small as possible. What will the number written on the blackboard be under the correct strategy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A convex pentagon  $P=ABCDE$  is inscribed in a circle of radius  $1$ . Find the maximum area of  $P$  subject to the condition that the chords  $AC$  and  $BD$  are perpendicular.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function f(x-1) is an even function and the graph of the function f(x) is symmetric about the point (1,0), and f(x) = ax - 1 for x in [-1,1], calculate f(2022).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The probability of selecting a first-grade product is 0.65, and the probability of selecting a second-grade product is 0.3. Calculate the probability of selecting a defective product. Express your answer as a decimal value between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let point $M(x_{0},2-x_{0})$ be given, and suppose there exists a point $N$ on the circle $O:x^{2}+y^{2}=1$ such that $\angle OMN=30^{\circ}$. Then, the range of the real number $x_{0}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Question 224, Given a positive integer $n(n \geq 2)$, choose $m$ different numbers from $1, 2, \ldots, 3n$. Among these, there must be four pairwise distinct numbers $a, b, c, d$, satisfying $a=b+c+d$. Find the minimum value of $m$.

---

The translation maintains the original format and line breaks as requested.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Zaslavsky A.A.

The number of the current Olympiad (70) is formed by the last digits of the year of its holding, written in reverse order. How many more times will such a situation occur in this millennium?

#
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( x, y \) be nonnegative integers such that \( x + 2y \) is a multiple of 5, \( x + y \) is a multiple of 3, and \( 2x + y \geq 99 \). Find the minimum possible value of \( 7x + 5y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can 4 purple balls and 4 green balls be placed into a $4 \times 4$ grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four points \( A, O, B, O' \) are aligned in this order on a line. Let \( C \) be the circle centered at \( O \) with radius 2015, and \( C' \) be the circle centered at \( O' \) with radius 2016. Suppose that \( A \) and \( B \) are the intersection points of two common tangents to the two circles. Calculate \( AB \) given that \( AB \) is an integer \( < 10^7 \) and that \( AO \) and \( AO' \) are integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}{cccccc}&A&B&B&C&B\ +&B&C&A&D&A\ \hline &D&B&D&D&D\end{array}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the vertex of the obtuse angle \( A \) of triangle \( ABC \), the height \( AD \) is dropped. A circle with center \( D \) and radius \( DA \) is drawn, which intersects the sides \( AB \) and \( AC \) again at points \( M \) and \( N \) respectively. Find \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limiting velocity of a spherical object increasing in size due to accumulation of material, given that the rate of increase in radius is proportional to the radius and neglecting air resistance.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(W_t)_{t\ge0}$ be a Standard Brownian Motion. Compute the conditional probability $$P\left(\inf_{t\in [1,2]} W_t < 0 ~ \middle| ~ W_1 >0,~ W_2 >0\right).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形 $A B C D$ 所在平面上一点, 满足
$$
\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\overrightarrow{0}
$$
设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$ 的面积, 则 $\frac{t}{s}=$ $\qquad$.
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{}.


Context:
Given the sequence $\left\{a_{n}\right\}$ with the partial sum $S_{n}=2 a_{n}-1$ for $n=1,2,\cdots$, and the sequence $\left\{b_{n}\right\}$ that satisfies $b_{1}=3$ and $b_{k+1}=a_{k}+b_{k}$ for $k=1,2,\cdots$, find the partial sum of the sequence $\left\{b_{n}\right\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p,$ $q,$ $r,$ $x,$ $y,$ and $z$ be positive real numbers such that $p + q + r = 2$ and $x + y + z = 1$. Find the maximum value of:
\[\frac{1}{p + q} + \frac{1}{p + r} + \frac{1}{q + r} + \frac{1}{x + y} + \frac{1}{x + z} + \frac{1}{y + z}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In quadrilateral ABCD, AB = √2, BC = √3, CD = 1, ∠ABC = 75°, ∠BCD = 120°. Find the measure of ∠CDA. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train of a certain length running at the speed of 54 km/hr takes 16.13204276991174 seconds to cross a bridge 132 m in length. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the closed figure formed by the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(n)$ be the sum of the squares of the digits of the positive integer $n$ (in decimal notation). For example, $f(123) = 1^2 + 2^2 + 3^2$. Define $f_1(n) = f(n)$, and $f_{k+1}(n) = f[f_k(n)]$ (where $k=1,2,3,\ldots$). Calculate the value of $f_{2007}(2007)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $G$ is a group with exactly $12$ elements of order $4$. How many elements of order $2$ does $G$ have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\Gamma\) denote the circumcircle of triangle \(ABC\). Point \(D\) is on \(\overline{AB}\) such that \(\overline{CD}\) bisects \(\angle ACB\). Points \(P\) and \(Q\) are on \(\Gamma\) such that \(\overline{PQ}\) passes through \(D\) and is perpendicular to \(\overline{CD}\). Compute \(PQ\), given that \(BC = 20\), \(CA = 80\), and \(AB = 65\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can a committee of three people be formed if the members are to be chosen from four married couples?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? (For example, Raashan and Ted may each decide to give $1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0$, Sylvia will have $2$, and Ted will have $1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $1$ to, and the holdings will be the same at the end of the second round.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box contains some tablets of medicine A and 14 tablets of medicine B. The least number of tablets that should be taken from the box to ensure that at least two tablets of each kind are among the extracted is 16. How many tablets of medicine A are in the box?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Mrs. Evans' freshmen economics class, 40 students are enrolled. Some students answered question 1 correctly, 29 answered question 2 correctly, and 10 did not take the test. If 29 students answered both questions correctly, how many students answered question 1 correctly?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the smallest positive integer $m$ such that
\[
r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tim came second in math. When his mother asked him how much he had scored, he answered that he got the sum of the first 'n' even numbers, where 'n' is a square number. His mother immediately worked out the answer. What is the value of 'n'?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_{n}\}$ with $a_{1}=111$, $a_{2}=217$, and for $n\geqslant 3$, $a_{n}=a_{n-2}-\frac{n-p}{a_{n-1}}$, find the value of the constant $p$ when the maximum number of terms $m$ is $220$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the sum \( \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n} \) as a rational number. The original answer is in the form of \( \frac{m}{n} \). Please provide the value of \( m + n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which $n$ can values of the variables $x_{1}, x_{2}, \ldots, x_{n}$ be chosen from the set $\{+1, -1\}$ such that the equation $x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + \ldots + x_{n-1}x_{n} + x_{n}x_{1}=0$ is satisfied?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A Battleship game board measures $10 \times 10$. What is the maximum number of $1 \times 4$ ships that can be placed on it?
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. In an isosceles trapezoid $A B C D$ with lateral sides $A B$ and $C D$, the lengths of which are 10, perpendiculars $B H$ and $D K$ are drawn from vertices $B$ and $D$ to the diagonal $A C$. It is known that the bases of the perpendiculars lie on segment $A C$ and $A H: A K: A C=5: 14: 15$. Find the area of trapezoid $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive reals \( p \) and \( q \), define the remainder when \( p \) is divided by \( q \) as the smallest nonnegative real \( r \) such that \( \frac{p - r}{q} \) is an integer. For an ordered pair \( (a, b) \) of positive integers, let \( r_1 \) and \( r_2 \) be the remainder when \( a \sqrt{2} + b \sqrt{3} \) is divided by \( \sqrt{2} \) and \( \sqrt{3} \) respectively. Find the number of pairs \( (a, b) \) such that \( a, b \leq 20 \) and \( r_1 + r_2 = \sqrt{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, $\angle C=50^{\circ}$. With point $C$ as the center, draw an arc with a suitable length as the radius, intersecting $AC$ at point $M$ and $BC$ at point $N$. With $M$ and $N$ as centers, draw arcs with lengths greater than $\frac{1}{2}MN$ as the radius. These two arcs intersect at point $D$. Connect $CD$. Find the degree of $\angle ACD$. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set \( S = \{1, 2, \ldots, 30\} \). There are 10 subsets, \( A_1, A_2, \ldots, A_{10} \), each containing 3 elements, where any two subsets \( A_i \) and \( A_j \) have at least one element in common. Define \( n_i \) as the number of subsets \( A_j \) to which element \( i \) belongs. Determine the smallest possible value of \( \max (n_i) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle with radius $r$ is tangent to sides $AB, AD$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. The area of the rectangle, in terms of $r$, is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\{1,2, \ldots, k\}$ is a multiple of 11 can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function \( u(x, y) = x^{2} + \frac{81}{x^{2}} - 2xy + \frac{18}{x} \sqrt{2 - y^{2}} \) where \( x \) and \( y \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sides $a, b, c, d$ of a trapezoid, how long is the line segment that joins the midpoints of the parallel sides?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M=\{x|0 < x < 4\}$ and $N=\{x|\frac{1}{3}\leqslant x\leqslant 5\}$. Find the intersection $M \cap N$ and express your answer as a set definition in the format $\{x|\text{condition}\}$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\{x|\frac{1}{3}\leqslant x < 4\}}


Context:
Given the lines $l_{1}$: $\left(m+3\right)x+5y=5-3m$ and $l_{2}$: $2x+\left(m+6\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, then the value of $m$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains, 431 meters and 365 meters in length respectively, are running in opposite directions, one at the rate of 120 km/h and the other at the rate of 105 km/h. In what time will they be completely clear of each other from the moment they meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the graph of the inverse proportion function $y=\frac{2}{x}$ passes through points $A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$, $C(x_{3}, y_{3})$, and $x_{1} \lt x_{2} \lt 0 \lt x_{3}$, then the relationship between $y_{1}$, $y_{2}$, $y_{3}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \( I \) is the center of the inscribed circle of triangle \( ABC \), point \( O \) is the center of the escribed circle that touches side \( AC \), and line segments \( AC \) and \( OI \) intersect at point \( K \).

It is known that \( OI = 50, IK = 18, \) and \( AK = 24 \). Find the length of the angle bisector of angle \( B \) in triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Farm tax is levied on the 60% of the cultivated land. The tax department collected a certain amount through the farm tax from the village of Mr. Willam. Mr. Willam paid only $500 as farm tax. The percentage of total land of Mr. Willam over the total taxable land of the village is 20.833333333333332%. How much money did the tax department collect from the village?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limiting value of the following expression as $\beta$ approaches infinity:

$$\lim_{\beta\to+\infty}\frac{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}(t-1)dt}{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}dt}$$

where $I_0(x)$ represents the modified Bessel function of the first kind.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer tasks receive fewer candies, possibly receiving none. Determine the smallest value of $N$ for which this distribution is possible, regardless of the number of tasks in the Olympiad and the students' successes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
四个半径为 1 的球两两相切，则它们的外切正四面体的棱长为多少？答案的形式为k(1+\sqrt{m})，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive numbers $x$, $y$, and $z$ that satisfy the equation $x^2 + y^2 + z^2 = 2xyz + 1$, find the maximum value of the expression:
$$P = (x - 2yz)(y - 2zx)(z - 2xy).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of all $6$-digit natural numbers such that the sum of their digits is $10$ and each of the digits $0, 1, 2, 3$ occurs at least once in them.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Patrick has been saving money to buy a bicycle that costs $150. He saved half the price, but then lent $50 to his friend Alice at a 5% annual interest rate and $30 to his friend Bob at a 7% annual interest rate. If Alice repays the loan after 8 months and Bob after 6 months, how much money does Patrick have in total, including the interest accrued on both loans?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ali turned 8 years old this year and is now 3 years older than Yusaf. Umar is twice Yusaf's age, and Aisha is half a year younger than the average age of Ali and Umar. If Aisha reached a milestone age last month, how old is Aisha?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the price of a book is first decreased by 40% and then increased by 35%, what will be the net change in the price?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An integer $n$ is chosen uniformly at random from the set $\{1,2,3, \ldots, 2023!\}$. Compute the probability that $$\operatorname{gcd}\left(n^{n}+50, n+1\right)=1$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $\sin\left(\theta + \frac{\pi}{4}\right) = \frac{1}{4}$ with $\theta \in \left(-\frac{3\pi}{2}, -\pi\right)$, find the value of $\cos\left(\theta + \frac{7\pi}{12}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Calculate the maximum slope of the line $OQ$. Express your answer as a single numerical value, including the square root symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On an island, there live knights, liars, and yes-men; each knows who is who among them. In a row, 2018 island inhabitants were asked the question: "Are there more knights than liars on the island?" Each inhabitant answered either "Yes" or "No" in turn such that everyone else could hear their response. Knights always tell the truth, liars always lie. Each yes-man answered according to the majority of responses given before their turn, or gave either response if there was a tie. It turned out that exactly 1009 answered "Yes." What is the maximum number of yes-men that could be among the island inhabitants?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remaining purple shirts are also hanging consecutively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the right pan of a balance scale lies a weight with a mass of 11111 g. The scale operator successively places weights on the pans, the first of which has a mass of 1 g, and each subsequent one is twice as heavy as the previous one. At some point, the scales are balanced. On which pan is the 16 g weight placed?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{The 16-gram weight is on the left pan}


Context:
Given the sequence $$a_1=\sqrt{2}, \quad a_{n+1}=\sqrt{2}\left(a_n+\frac {1}{a_n}\right),$$ compute $$\lim_{n\to \infty}\frac{a_n}{(\sqrt{2})^n}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the line $y=2x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0, b \gt 0\right)$ do not have any common points, then the range of eccentricity of the hyperbola is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the inequality $2x-m\leqslant 3$ holds, a sufficient and necessary condition is $-5 \lt x \lt 4$. The range of real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
11. A Trip for Milk. Anton went to a dairy store. He had no money, but he had empty bottles - six one-liter bottles (worth 20 kopecks each) and six half-liter bottles (worth 15 kopecks each). In the store, there was milk sold by the liter for 22 kopecks. What is the maximum amount of milk he could bring home? He had no other containers besides the empty bottles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a function $g$ satisfies $g(x) = g(x+3)$ for all $x \in \mathbb{R}$, and it is given that $\int_2^4 g(x) dx = 5$ and $\int_{-3}^{-1} g(x) dx = -3$, calculate the value of $\int_1^{10} g(x) dx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem: Given $f \in C^{(1)} [0,1]$, with $f(0) = a$ and $\int_{0}^{1} f(x) \, dx = 0$, determine the minimum value of $\int_{0}^{1} (f'(x))^2 \, dx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that there are 5 points in the plane no three of which are collinear, draw all the segments whose vertices are these points, and determine the minimum number of new points made by the intersection of the drawn segments.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if
\[k = \sum_{i=2}^n a_i F_i,\]
where $a_i \in \{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i \ge 2$). This representation is said to be \textit{minimal} if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many non-trivial ring homomorphisms are there from the ring $Z_{20}$ to the ring $Z_{44}$, considering them as rings modulo 20 and 44, respectively?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 5 volunteers to be divided into 3 groups to serve at three different venues for the Youth Olympic Games. Two of the groups will have 2 people each, and one group will have 1 person. How many different allocation schemes are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the value of
$$3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
The answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a freight yard, there are 100 shipping containers. Each container carries 30 crates. Each crate is filled with 15 boxes, and each box is filled with 10 washing machines. There are 6 workers, who work simultaneously to remove a certain number of washing machines from each box. If each worker removes 4 washing machines from each box they manage, how many washing machines were removed in total by all workers? Note that all workers remove the same number of washing machines from each box.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 a,b>0，满足：关于 x 的方程 \sqrt{|x|}+\sqrt{|x+a|}=b 恰有三个不同的实数解 x_1, x_2, x_3，且 x_1<x_2<x_3=b，则 a+b 的值为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ordered pairs of real numbers  $(x, y)$  are there such that  $x^2+y^2 = 200$  and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tank has 3 inlet pipes A, B, and C with flow rates of 8 liters/min, 12 liters/min, and 6 liters/min respectively. The tank also has 2 outlet pipes, X and Y, with flow rates of 20 liters/min and 15 liters/min respectively. Inlet pipe A and outlet pipe X are open for the entire time, while inlet pipes B and C are open for the first 3 hours and 4 hours, respectively. Outlet pipe Y is open for the last 5 hours. At the end of 7 hours, the tank is full. Find the capacity of the tank.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > b > 0)$, a circle with the origin as its center and the semi-focal distance as its radius intersects the right branch of the hyperbola at points $A$ and $B$, and $\triangle F_{1}AB$ is an equilateral triangle, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bug starts at a vertex of a grid made of equilateral triangles with a side length of $1$. At each step, the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. Find the probability that after $5$ moves, the bug will never have been more than $1$ unit away from the starting position. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer \( n \geq 1 \), we define the recursive relation given by
\[ a_{n+1} = \frac{1}{1 + a_{n}}. \]

Suppose that \( a_{1} = a_{2012} \). Find the sum of the squares of all possible values of \( a_{1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive digit $D$ and positive integer $k$, we use the symbol $D_{(k)}$ to represent the positive integer having exactly $k$ digits, each of which is equal to $D$. For example, $2_{(1)}=2$ and $3_{(4)}=3333$. There are $N$ quadruples $(P, Q, R, k)$ with $P, Q$ and $R$ positive digits, $k$ a positive integer with $k \leq 2018$, and $P_{(2k)}-Q_{(k)}=\left(R_{(k)}\right)^{2}$. What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
James can do a farmer's walk with 300 pounds per hand for 20 meters. He can lift a certain percentage more if he only goes 10 meters. In addition, lifting straps give him another 20%. If he increases his 20-meter distance without straps by 50 pounds and everything else increases in proportion, he can move 546 pounds with lifting straps for 10 meters. What is the percentage increase in weight he can lift if he only goes 10 meters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bank serves a million clients, and Ostap Bender knows the list of these clients. Each client has a unique six-digit PIN code. On each move, Ostap Bender can choose a client he has not yet chosen and peek at the digits of the code in any $N$ positions (he can choose different positions for different clients). Ostap wants to find out the PIN code of millionaire Koreiko. What is the smallest number, $N$, that guarantees he can do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define the infinite products:
\[ A = \prod_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \]
If \(\frac{A}{B} = \frac{m}{n}\) where \(m,n\) are relatively prime positive integers, determine \(100m+n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pad of paper comes with 60 sheets. Evelyn uses a pad of paper for writing notes at work every week. She usually works from Tuesday to Thursday, but takes Monday and Friday off. However, during her workdays, she alternates her paper usage. On the first working day of the week, she uses 2 sheets, on the second day, she uses 4 sheets, and on the third day, she uses 6 sheets.

Considering that Evelyn works for 48 weeks a year, but she takes 8 additional days off for holidays or sick days, how many sheets of paper does she use on average per day she works?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the solid generated by rotating the region bounded by the curves $$y=e^x,\>\>\>\>\>y=\cos(\pi x^2/2),\>\>\>\>\>x=0,\>\>\>\>\>x=1$$ about the $y$-axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
40. (5 points)
Form a ten-digit number without repeating digits using numbers $0 \sim 9$, where the number can be divided by 11. There are $\qquad$ such numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive integers $n$ such that
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) > 0.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular tetrahedron with 2 unit edges has two of its vertices on the circumference of the base and the top of a right circular cylinder. What is the volume of the cylinder?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integers from 1 to 2001 have a digit sum that is divisible by 5?
Please reason step by step, and put your final answer within \boxed{}.


Context:
7th Australian 1986 Problem A2 What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let a 1 = 3, a 2 = 3 3 , and a n+1 = 3 a n . Similarly, b 1 = 100, b 2 = 100 100 etc. What is the smallest n for which b n > a 100 ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are many $n$-fold coverings (For instance $\Bbb R/n\Bbb Z$ as $S^1$ cover) but I can't find one with trivial automorphism group. What is the smallest $n$ for which there is a covering with trivial automorphism group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the correlation function of the stationary random function $X(t)$, given its spectral density $s_{x}(\omega)=s_{0}$ in the interval $-\omega_{0} \leq \omega \leq \omega_{0}$, and outside this interval $s_{x}(\omega)=0$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \frac{2s_0 \sin(\omega_0 \tau)}{\tau} }


Context:
Given the sets \( A = \{(x, y) \mid ax + y = 1, x, y \in \mathbb{Z}\} \), \( B = \{(x, y) \mid x + ay = 1, x, y \in \mathbb{Z}\} \), and \( C = \{(x, y) \mid x^2 + y^2 = 1\} \), find the value of \( a \) when \( (A \cup B) \cap C \) is a set with four elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangular prism $M-ABCD$ with a square base and $MA = MD$, $MA \perp AB$, if the area of $\triangle AMD$ is 1, find the radius of the largest sphere that can be placed within the prism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x,$ $y,$ $z,$ $v,$ $w$ be positive real numbers such that $x^2 + y^2 + z^2 + v^2 + w^2 = 2016.$  Let $M$ be the maximum value of
\[xz + 2yz + 3zv + 7zw,\]and let $x_M,$ $y_M$, $z_M,$ $v_M,$ $w_M$ be the values of $x,$ $y,$ $z,$ $v,$ $w,$ respectively, that produce the maximum value of $M.$  Find $M + x_M + y_M + z_M + v_M + w_M.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
a person travels from p to q a speed of 80 km / hr and returns by increasing his speed by 10 % . what is his average speed for both the trips ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A ball inside a rectangular container of width 7 and height 12 is launched from the lower-left vertex of the container. It first strikes the right side of the container after traveling a distance of $\sqrt{53}$ (and strikes no other sides between its launch and its impact with the right side). How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mrs. Hilt measured the distance from her desk to the water fountain. It was 30 feet. If she goes to the water fountain 4 times today , Mrs. Hilt walked _____ feet on her trips to the fountain .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an obtuse triangle \(ABC\), a point \(L\) is chosen on the side \(AB\), which is equal to 14, such that it is equidistant from the lines \(AC\) and \(BC\). On the segment \(AL\), there is a point \(K\) that is equidistant from the vertices \(A\) and \(B\). Find the sine of angle \(ACB\) if \(KL = 1\) and \(\angle CAB = 45^\circ\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
在复平面上,复数 $z_{1}, 1+\mathrm{i}, 1+a \mathrm{i}(a \in \mathbf{R})$ 对应的点分别是 $A, B, C$, 点 $A$ 在线段 $B C$ 上运动, 复数 $z_{2}$ 满足 $\left|z_{2}\right|=1$. 若复数 $z_{1}+z_{2}$ 对应的点组成的图形的面积是 $\pi+4$ ，请给出 $a$ 可能的取值的和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let a, b, c, and d represent the current ages of four individuals. In five years, the total age of a and b will be 10 years more than the total age of b and c at that time. Currently, the total age of c and d is 12 years less than the total age of a and d.

Let R be the difference in years between the individuals a and c. Also, let L(a) and L(c) represent the linear functions of the ages of a and c after five years, respectively:

L(a) = a + 5
L(c) = c + 5

Given this information, find the value of R - the difference in years between individuals a and c in the current year.
Please reason step by step, and put your final answer within \boxed{}.


Context:
一枚不均匀的硬币，若随机抛掷它两次均得到正面的概率为 \frac{1}{2} ，则随机抛掷它两次得到正面、反面各一次的概率的近似整数值是多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
There exists a polynomial $P$ of degree 5 with the following property: if $z$ is a complex number such that $z^{5}+2004 z=1$, then $P(z^{2})=0$. Calculate the quotient $P(1) / P(-1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle is tangent to the extensions of two sides \( AB \) and \( AD \) of square \( ABCD \) with side length \( 6 + 2 \sqrt{5} \) cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 36^{\circ} \), and it is known that \( \sin 18^{\circ} = \frac{\sqrt{5} - 1}{4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = 2 \) and \( BC = 5 \), then \( BX \) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f\left(x\right)=2a^{x}-ex^{2}+18$, where $a \gt 0$ and $a\neq 1$. If $f\left(x\right)$ has two critical points $x_{1}$ and $x_{2}$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Isabel wants to save 40 files onto disks, each with a capacity of 1.44 MB. 5 of the files take up 0.95 MB each, 15 files take up 0.65 MB each, and the remaining 20 files each take up 0.45 MB. Calculate the smallest number of disks needed to store all 40 files.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six positive numbers, not exceeding 3, satisfy the equations \(a + b + c + d = 6\) and \(e + f = 2\). What is the minimum value of the expression

$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the given acute triangle ABC, the incenter I and circumcenter O intersect the line BC at point X. Let M be the midpoint of the arc BC (not containing point A) of the circumcircle of triangle ABC. If the points A, O, M, and X are concyclic, calculate the measure of angle BAC in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an equilateral triangle \( ABC \) with side length 1, points \( D \) and \( E \) are taken on sides \( AB \) and \( AC \) respectively such that when the triangle is folded along line segment \( DE \), vertex \( A \) falls exactly on side \( BC \). The minimum length of \( AD \) is \( \qquad \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The rabbit and the turtle are traveling from point $A$ to the forest amusement park. The rabbit starts at 9:00 AM and jumps 40 meters in one minute, playing for 2 minutes after every 3 minutes of jumping. The turtle starts at 6:40 AM and crawls 10 meters per minute without any breaks. If the turtle arrives at the amusement park 15 seconds earlier than the rabbit, how many meters is it from point $A$ to the forest amusement park?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the square root of 4. Express your answer in the form ±y, where y is a positive number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square \(ABCD\) with side length 1, determine the maximum value of \(PA \cdot PB \cdot PC \cdot PD\) where point \(P\) lies inside or on the boundary of the square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rowena is able to mow 2/5 of a lawn in 18 minutes. If she began the job at 10:00 a.m., and mowed at this same constant rate, what time did she finish mowing the entire lawn? Express your answer in 12-hour time format (e.g., 10:45 a.m.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A young man bought a gift priced at $21 but costing $18 from Mr. Wang's store using a $100 bill. Mr. Wang gave the young man $79 in change but later had to return the $100 bill to a neighbor because it was counterfeit. Calculate Mr. Wang's total financial loss in this transaction. Express your answer in dollars.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A number is chosen uniformly at random from the set of all positive integers with at least two digits, none of which are repeated. Find the probability that the number is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. Gwendoline
repeatedly rolls the two dice and records the product of the two values until one of the values she records
appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jack received 10 emails and 12 letters in the morning. He then received 3 emails and some letters in the afternoon. Jack received 7 more emails in the morning than in the afternoon. How many letters did Jack receive in the afternoon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four students are to complete a 100-meter relay race, with the condition that the distance run by each person cannot exceed three times the distance run by any other student. If the distance run by a certain student is x meters, find the range of values for x. Express your answer as an inequality in the format a ≤ x ≤ b.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alex is an entrepreneur who measures success by the financial growth of their business and the assets they accumulate. Last year, Alex's business made a profit of $120,000. This year, Alex set a goal to increase the profit by 25%. Additionally, Alex plans to invest $50,000 in a new project, expecting a 10% return on this investment by the end of the year. What is the total expected financial gain for Alex this year from both the increased profit and the investment return?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知抛物线 $x^{2}=4 y$ 上的两个动点 $A\left(x_{1}, y_{1}\right) 、 B\left(x_{2}, y_{2}\right)\left(y_{1}+y_{2}=2, y_{1} \neq y_{2}\right)$ 。若线段 $A B$ 的垂直平分线与 $y$ 轴交于点 $C$, 则 $\triangle A B C$ 面积的最大值为某个复杂分数形式，标准答案格式为$\frac{a\sqrt{b} }{c}$，则a+b+c的值为多少。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Emma has saved $1,200 in her bank account, which earns a 3% compound annual interest. On January 1st, she withdrew 8% of her balance to buy a new pair of shoes. The day after, she deposited 25% of the amount she withdrew. On February 5th, Emma received her paycheck and deposited 150% of the amount she spent on shoes. On March 16th, one week after her birthday, she withdrew 5% of her current balance to contribute to a friend's birthday gift. Her bank compounds interest monthly.

What is the balance in her bank account on April 1st, considering her transactions and interest earned on her account?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. On the right branch of the hyperbola $C$, there exists a point $P$ such that the inscribed circle of $\triangle PF\_1F\_2$ has a radius of $a$. The center of the circle is denoted as $M$, the centroid of $\triangle PF\_1F\_2$ is $G$, and $MG \parallel F\_1F\_2$. Determine the eccentricity of the hyperbola $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all positive integers $n$ with the following properties:

- $n$ is not divisible by any primes larger than $10$.
- For some positive integer $k$, the positive divisors of $n$ are ordered as:
  \[1 = d_1 < d_2 < d_3 < \cdots < d_{2k} = n.\]
- The divisors of $n$ satisfy the condition:
  \[d_1 + d_2 + \cdots + d_k = 3k.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The natural number  $a_n$  is obtained by writing together and ordered, in decimal notation , all natural numbers between  $1$  and  $n$ . So we have for example that  $a_1 = 1$ , $a_2 = 12$ ,  $a_3 = 123$ ,  $. . .$  ,  $a_{11} = 1234567891011$ ,   $...$  . Find all values of  $n$  for which  $a_n$  is not divisible by  $3$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain fraction of the people in the room are under the age of 21, and exactly 5/12 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, and there are 36 people in the room under the age of 21, what is the fraction of people under the age of 21?
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Suppose that $F(8) = \frac{43}{256}$, and $F(10)$ can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The distance from the top of the can on the floor to the top of the bottle on the table is $150 \textrm{ cm}$. The distance from the top of the bottle on the floor to the top of the can on the table is $110 \textrm{ cm}$. Find the height of the table. Express your answer in centimeters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( z \in \mathbf{C} \). If the equation with respect to \( x \):
$$
4 x^{2}-8 z x+4 i+3=0
$$
has real roots, then the minimum value of \( |z| \) is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A person walks at 4 km/h and misses the bus by 10 minutes. If they walk at a different speed, they reach 5 minutes before the arrival of the bus. The distance they walk to reach the bus stand is 5 km. What is their walking speed when they reach 5 minutes before the bus arrives?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For what the smallest \(\$ n \$\) is it true that for any set \(\mathrm{A}\) of \(2007\) sets, there exists a set \(\mathrm{B}\) of \( n \) sets such that every set in \(\mathrm{A}\) is the intersection of two different sets in \(\mathrm{B}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\det \mathbf{A} = -1,$ then find $\det (\mathbf{7A}).$
Please reason step by step, and put your final answer within \boxed{}.


Context:
用 $0,1,2,3,4,5$ 这六个数字组成没有重复数字的四位偶数, 并将这些偶数从小到大排列起来, 则第 99 个数是 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the area of the total surface of the cone to the area of its lateral surface, if the side of the base of the pyramid is 4, and the angle between the height of the pyramid and the plane of the lateral face is $30^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A person's salary was reduced by 14% as part of a company-wide pay cut. Additionally, upon receiving their reduced salary, they must pay 28% as income tax. By what percent should their post-tax, reduced salary be raised so as to bring it at par with their original pre-tax salary?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the expansion of $((x+a)^{2}(2x- \frac {1}{x})^{5})$ does not contain the $x^{3}$ term, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
FemtoPravis is walking on an $8 \times 8$ chessboard that wraps around at its edges (so squares on the left edge of the chessboard are adjacent to squares on the right edge, and similarly for the top and bottom edges). Each femtosecond, FemtoPravis moves in one of the four diagonal directions uniformly at random. After 2012 femtoseconds, what is the probability that FemtoPravis is at his original location?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a two-dimensional space, the one-dimensional measure of a circle (circumference $l=2\pi r$) and its two-dimensional measure (area $S=\pi r^{2}$); in a three-dimensional space, the two-dimensional measure of a sphere (surface area $S=4\pi r^{2}$) and its three-dimensional measure (volume $V= \dfrac {4}{3}\pi r^{3}$); in a four-dimensional space, the three-dimensional measure of a "hypersphere" ($V=8\pi r^{3}$), then it is conjectured that its four-dimensional measure $W=$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves

$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Terry's daily income is $24 for 6 hours of work, while Jordan's daily income is $30 for 8 hours of work. Terry works 7 days a week, and Jordan works 6 days a week. Their standard workweek is defined as 40 hours, and both of them earn an overtime pay rate of 1.5 times their hourly rate for any hours worked beyond the standard workweek. Terry and Jordan have different tax brackets based on their gross weekly income: Terry falls into the 12% tax bracket, while Jordan falls into the 15% tax bracket. Calculate the difference between their weekly net incomes after considering overtime pay rates and tax deductions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $\int_{-\infty}^{\infty} \frac{x^{-1/2}}{(1+x)^2} \, dx$ using residue calculus or real methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
**Problem Statement:**

One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is \(1000\) km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained?

*(Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit $\lim\limits_{n\to\infty} V_{n}$ for all $n \in \mathbb{N}$, where $V_{n} = \left(\sum\limits_{k=0}^{2n}\sqrt{k}\right) - (2n+1)\sqrt{n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a "convex number". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that it is a "convex number"?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using l'Hopital's rule: $$\lim_{x\to0}\left(\csc^2x-\frac{1}{x^2}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For reals $x \ge 3$, let $f(x)$ denote the function \[ f(x) = \frac{-x + x\sqrt{4x-3}}{2}. \]
Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[ a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3}{a_i^2 + a_ia_{i+1} + a_{i+1}^2}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $[x]$ denote the greatest integer less than or equal to $x$. In the set $\left\{\left[\frac{1^{2}}{2003}\right],\left[\frac{2^{2}}{2003}\right], \cdots,\left[\frac{2003^{2}}{2003}\right]\right\}$, how many distinct elements are there? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square has vertices \( P, Q, R, S \) labelled clockwise. An equilateral triangle is constructed with vertices \( P, T, R \) labelled clockwise. What is the size of angle \( \angle RQT \) in degrees?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many $n=2,3,4,\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0) \) with its left focus at point \( F \). A tangent to the ellipse at a point \( A \) intersects the \( y \)-axis at point \( Q \). Given that \( \angle Q F O = 45^{\circ} \) and \( \angle Q F A = 30^{\circ} \), determine the eccentricity \( e \) of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \sum_{k=1}^\infty \frac{k^2-2}{(k+2)!} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $|x|\leqslant \frac{\pi}{4}$, find the minimum value of the function $f(x)=\cos^{2}x+\sin x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ (where $a>0$, $b>0$) with eccentricity $e=2$, and a line through a point $M$ on the hyperbola intersects the hyperbola at points $A$ and $B$, with slopes $k_1$ and $k_2$ respectively. If the line $AB$ passes through the origin, calculate the value of $k_1 \cdot k_2$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$  is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$. Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$, find the value of $p_{2}+2p_{1}-3p_{0}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
[ Combinations and Permutations ] $[$ Graph Theory (other) ]

In the parliament, there are 30 deputies. Any two of them are either friends or enemies, and each one is friends with exactly six others. Any three deputies form a committee. Find the total number of committees in which all three members are pairwise friends or all three are pairwise enemies.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in the bucket?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the function $f(x)=2a^{x}-b^{x}$, where $b\geqslant 2a > 0$. Find the interval where the root of $f(x)$ is located. Express your answer using standard interval notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设集合 $M \subseteq\{1,2, \cdots, 2011\}$ 满足：在 $M$ 的任意三个元素中都可以找到两个元素 $a 、 b$ ，使得 $a \mid b$ 或 $b \mid a$ 。 求 $|M|$ 的最大值（ $|M|$表示集合 $M$ 的元素个数)。
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $f(x) = x^{x^{x^x}}$ , find the last two digits of $f(17) + f(18) + f(19) + f(20)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points \( A_1 \), \( B_1 \), and \( C_1 \) are taken on the sides \( BC \), \( CA \), and \( AB \) of triangle \( ABC \) respectively, such that the radii of the circles inscribed in triangles \( A_1BC_1 \), \( AB_1C_1 \), and \( A_1B_1C \) are all equal and are denoted by \( r \). The radius of the circle inscribed in triangle \( A_1B_1C_1 \) is \( r_1 \). Find the radius of the circle inscribed in triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$, we have $xy\notin A$. Determine the maximum possible size of $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The orthogonal projection of an isosceles right triangle onto plane $\alpha$ is an equilateral triangle. Find the angle formed by the hypotenuse of the given triangle with plane $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are two pieces of wood, one rectangular and one cylindrical, with the same height and a volume ratio of 1:1. If the rectangular wood is processed into the largest possible cylinder and the cylindrical wood is processed into the largest possible rectangular prism, what is the ratio of the volume of the cylinder to the volume of the rectangular prism? $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Over all real numbers \( x \) and \( y \), find the minimum possible value of
\[ (xy)^2 + (x + 7)^2 + (2y + 7)^2. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a regular tetrahedron with side length $12$. Select points $E$, $F$, $G$, $H$ on $AC$, $BC$, $BD$, and $AD$, respectively, such that $AE = BF = BG = AH = 3$. Compute the area of quadrilateral $EFGH$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle \( \triangle ABC \), point \( E \) is an interior point, and \( BE \) extended intersects \( AC \) at point \( D \). Given \( \angle DCE = 10^\circ \), \( \angle DBC = 30^\circ \), \( \angle ECB = 20^\circ \), and \( \angle ABD = 40^\circ \), find \( \angle BAE \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The vertices of a rhombus are located on the sides of a parallelogram, and the sides of the rhombus are parallel to the diagonals of the parallelogram. Find the ratio of the areas of the rhombus and the parallelogram if the ratio of the diagonals of the parallelogram is \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A marksman has a probability of 0.8 for hitting the target each time he shoots. Using random simulation to estimate the probability that the marksman hits the target at least 3 times in 4 shots: First, generate a random number between 0 and 9 from the calculator, where 0 and 1 represent missing the target, and 2 to 9 represent hitting the target. Group every 4 random numbers to represent the results of 4 shots. After random simulation, the following 20 groups of random numbers are obtained:

75270293 7140 9857 0347 4373 8636 6947 1417 4698

0371 6233 2616 8045 6011 3661 9597 7424 7610 4281

Based on this, estimate the probability that the marksman hits the target at least 3 times in 4 shots is ________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A souvenir went through two price increases, from the original price of $76.8$ yuan to the current price of $120$ yuan. The average percentage increase for each price increase is ______ $\%$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
One antifreeze solution is 10% alcohol. Another antifreeze solution is 18% alcohol. To create 20 liters of antifreeze solution with a certain percentage of alcohol, 7.5 liters of each antifreeze solution should be combined. What is the percentage of alcohol in the resulting antifreeze solution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bear A and Bear B agreed to climb Mount Baiyun along the same route during the weekend. They started their journey from the foot of the mountain at the same time, and descended immediately after reaching the peak. Each of their descending speeds is twice their respective ascending speeds, and Bear A is faster than Bear B. After 2 hours of travel, Bear A and Bear B met 1600 meters away from the peak. When Bear B reached the peak, Bear A was exactly halfway down the mountain. How many hours did it take Bear A to travel from the start to return to the foot of the mountain?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\dfrac {π}{2} < β < α < \dfrac {3}{4}π$, $\cos (α+β)=- \dfrac {3}{5}$, and $\sin (α-β)= \dfrac {5}{13}$, find $\cos 2β$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of all ordered pairs $(x, y)$ of integer solutions to the equation:
\[ 6x^2 + y^2 + 6x = 3xy + 6y + x^2y. \]
$S$ contains a unique ordered pair $(a, b)$ with a maximal value of $b$. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
(4 points) The function $f(x) = 1 + \log_a |x + 1|$, ($a > 0$ and $a \neq 1$) passes through a fixed point, which is ___.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. In a $2 \times 2019$ grid, each cell is filled with a real number, where the 2019 real numbers in the top row are all different, and the bottom row is also filled with these 2019 real numbers, but in a different order from the top row. In each of the 2019 columns, two different real numbers are written, and the sum of the two numbers in each column is a rational number. How many irrational numbers can appear at most in the first row?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the limit of the area of the Koch Snowflake at n iterations as n approaches infinity, given by the following expression: $$\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}} \cdot \frac{s^2 \sqrt{3}}{4}$$ where $s$ represents the side length of the initial equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let
$$
\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots-\frac{1}{1}}}}=\frac{m}{n}
$$
where \(m\) and \(n\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \(m^2 + mn - n^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point is randomly thrown onto the segment [12, 17] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2} + k - 90)x^{2} + (3k - 8)x + 2 = 0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression $\log_{10} 60 + \log_{10} 80 - \log_{10} 15$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a $p \times q$ matrix with characteristic polynomial $(x-c)^r(x-d)^s$, where $c$ and $d$ are distinct complex numbers. Determine the dimension of the complex vector space $W$ of all $q \times p$ matrices $Y$ such that $XY=YX$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2025 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parking lot, there are eight parking spaces numbered from 1 to 8. Now, 2 trucks and 2 cars need to park at the same time. Each parking space can accommodate at most one vehicle. If vehicles of the same type must be parked in adjacent spaces, there are a total of $\boxed{\text{number}}$ parking arrangements. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_1, a_2, \ldots, a_n$ be a permutation of $1, 2, \ldots, n$. The number of elements that are to the left of $a_i$ and smaller than $a_i$ is called the inversion number of $a_i$ ($i=1,2,\ldots,n$). For example, in the permutation $6, 4, 5, 3, 2, 1$, the inversion number of $5$ is $1$, and the inversion number of $3$ is $0$. Find the number of different permutations of the numbers $1, 2, 3, 4, 5, 6, 7, 8$ such that the inversion number of $8$ is $2$, the inversion number of $7$ is $3$, and the inversion number of $5$ is $3$. (Express the result as a number.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider integers with the sum of their digits divisible by 7. What is the greatest difference between two consecutive such integers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( E \) be a point moving inside the square \( ABCD \). Given that the minimum value of the sum of distances from \( E \) to the points \( A \), \( B \), and \( C \) is \( \sqrt{2} + \sqrt{6} \), find the side length of this square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
直线 $y=k x+1$ 与双曲线 $x^{2}-y^{2}=1$ 的左支交于 $A 、 B$ 两点, 直线 $l$ 经过点 $(-2,0)$ 和 $A B$的中点, 求直线 $l$ 在 $y$ 轴的截距 $b$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant starts at the origin of the Cartesian coordinate plane. Each minute, it moves randomly one unit in one of the directions: up, down, left, or right, with all four directions being equally likely. Its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a cube. The midpoint of edge $B^{\prime} C^{\prime}$ is $E$, and the midpoint of edge $C^{\prime} D^{\prime}$ is $F$. The plane $A E F$ divides the cube into two parts. Determine the ratio of the volumes of the two parts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You built a cuboid-shaped pond with a length of 18 meters (m), a width of 10 meters (m), and a height of 2 meters (m). If you are going to paint the inner side of the pond, how many square meters (㎡) of the pond could be painted?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to     .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence of numbers with only even digits in their decimal representation, determine the $2014^\text{th}$ number in the sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ with the right focus and a line inclined at a $45^{\circ}$ angle has two intersections with the right branch of the hyperbola, find the range of the eccentricity $e$ of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.