Context:
One side of an equilateral triangle of height $24$ lies on line $\ell.$ A circle of radius $12$ is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b} - c\pi,$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c\,?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ordered triples of nonzero integers  $(a, b, c)$  satisfy  $2abc = a + b + c + 4$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hotel has three types of rooms: a triple room, a double room, and a single room, each type having one room available. Three adults and two children come to stay, and the children cannot stay in a room by themselves (they must be accompanied by an adult). Find the total number of different ways to arrange their accommodation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the figure, a geometric shape is formed by stacking 26 identical small cubes. Its bottom layer is composed of $5 \times 4$ small cubes. If its entire outer surface (including the bottom) is painted red, then after this geometric shape is disassembled, the number of small cubes with 3 red faces is $\qquad$ cubes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $R$ be a non-commutative ring and consider a function $f: R \rightarrow R$ such that $f(ab) = f(a)f(b) + f(a)b + af(b)$ for all $a, b \in R$. Determine the expression for $f(x^n)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Convex quadrilateral $A B C D$ has right angles $\angle A$ and $\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that points $P, M$, and $Q$ are collinear. Suppose that $m \angle A B C=160^{\circ}$ and $m \angle Q M C=40^{\circ}$. Find $M P \cdot M Q$, given that $M C=6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\overline{AB}$ and $\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of angle $ACP$? Express your answer as a decimal to the nearest tenth.

[asy]
draw((0,0)--10dir(180),linewidth(2));
draw((0,0)--10dir(67.5),linewidth(2));
draw((0,0)--10dir(0),dashed);

draw(10dir(180)..10dir(90)..10dir(0),linewidth(2));

draw((5,0)+5dir(180)..(5,0)+5dir(-90)..(5,0)+5dir(0),linewidth(2));

dot((0,0));
dot((5,0));

label("A",10dir(180),W);
label("B",10dir(0),E);
label("C",(0,0),SW);
label("D",5dir(0),NE);
label("P",10dir(67.5),NE);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$  and  $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and  }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in  $S$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A secret facility is a rectangle measuring $200 \times 300$ meters. There is one guard at each of the four corners of the facility. An intruder approaches the perimeter of the facility from the outside, and all the guards run towards the intruder by the shortest routes along the outer perimeter (while the intruder remains stationary). Three guards cover a total distance of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can we find all the numbers $m$ such that $\sigma(m)=24$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $f$ is an analytic function in the region $|z|<2$ and satisfies $|f(z)|\le 2$ for all $z$ in this region, with the condition $f(1)=0$. Determine the best possible bound for $f(1/4)$. Provide your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of monic polynomials in $x$ of degree 6 all of whose roots are members of the set $\\{ -1, 0, 1\\}$.  Let $P$ be the sum of the polynomials in $S$.  What is the coefficient of $x^4$ in $P(x)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n \) such that in any \( n \times n \) grid, where each cell is colored either white or black, there always exists a rectangle whose vertices are the centers of four cells of the same color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral of the vector field \( F(x,y) = (\sin(x)\ln(x) + y^2) \mathbf{a}_x + (\cos(y)e^y - x^2) \mathbf{a}_y \) along the borders of the region bounded by the circumferences: \( C_1 \) with radius 1 centered at \((1,0)\) and \( C_2 \) with radius 2 centered at \((2,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
John is half times younger than his father, who is 4 years older than John's mother. John's grandmother is three times older than John. John's aunt is 5 years younger than twice the age of John's mother, and 10 years younger than his father's brother. If John's father is 40 years old, find the age difference between John, his mother, his aunt, and his grandmother.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many of the first 1000 positive integers can be expressed in the form
$\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$,
 where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $|\vec{a}|=5$, $|\vec{b}|=3$, and $\vec{a} \cdot \vec{b}=-9$, find the projection of $\vec{a}$ onto $\vec{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that
$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$
What is the maximum value of $P(a)?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are eliminated. The remaining team wins. How many different possible competition sequences can occur?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y = (α - 1)x^{-4α - 2}$, find the value of the real number $α$ if the function is a power function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below. $2x_1+x_2+x_3+x_4+x_5=6$ $x_1+2x_2+x_3+x_4+x_5=12$ $x_1+x_2+2x_3+x_4+x_5=24$ $x_1+x_2+x_3+2x_4+x_5=48$ $x_1+x_2+x_3+x_4+2x_5=96$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( {A}_{1}{B}_{1}{C}_{1} \) be a triangle with \( {A}_{1}{B}_{1} = {16},{B}_{1}{C}_{1} = {14} \) ,and \( {C}_{1}{A}_{1} = 10\)
 Given a positive integer \( i \) and a triangle \( {A}_{i}{B}_{i}{C}_{i} \) with circumcenter \( {O}_{i} \) ,define
 triangle \( {A}_{i + 1}{B}_{i + 1}{C}_{i + 1} \) in the following way:
 (a) \( {A}_{i + 1} \) is on side \( {B}_{i}{C}_{i} \) such that \( {C}_{i}{A}_{i + 1} = 2{B}_{i}{A}_{i + 1} \) .
 (b) \( {B}_{i + 1} \neq  {C}_{i} \) is the intersection of line \( {A}_{i}{C}_{i} \) with the circumcircle of
 \( {O}_{i}{A}_{i + 1}{C}_{i} \) .
 (c) \( {C}_{i + 1} \neq  {B}_{i} \) is the intersection of line \( {A}_{i}{B}_{i} \) with the circumcircle of
 \( {O}_{i}{A}_{i + 1}{B}_{i} \) .
 Find
 \( {\left( \mathop{\sum }\limits_{{i = 1}}^{\infty }\left\lbrack  {A}_{i}{B}_{i}{C}_{i}\right\rbrack  \right) }^{2} \) .
 Note: \( \left\lbrack  K\right\rbrack \) denotes the area of \( K \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( \lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. The real quartic  $P x^{4}+U x^{3}+M x^{2}+A x+C$  has four different positive real roots. Find the square of the smallest real number  $z$  for which the expression  $M^{2}-2 U A+z P C$  is always positive, regardless of what the roots of the quartic are.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a$, $b$, and $c$ are the three side lengths of $\triangle ABC$, and they satisfy the relationship $\sqrt{{c}^{2}{-a}^{2}{-b}^{2}}+|a-b|=0$, then the shape of $\triangle ABC$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real numbers \(x\) that satisfy the equation: \[ \left(|x^2-12x+20|^{\log_{10}{x^2}}\right)^{-1+\log_{10}{x}}=|x^2-12x+20|^{1+\log_{10}{\frac{1}{x}}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Paul's grades last semester were very bad. To encourage him, Paul's dad promised him some money for every B+ he gets in his next scorecard and twice that amount for every A. If Paul obtains at least two A+ he'll double the previous rewards and would receive a flat $15 for every A+ obtained (with no extra bonus). If there are 10 courses with grades in Paul's scorecard, the maximum amount he could receive from his dad is $190. How much money does Paul's dad promise him for every B+?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls High School Math Meet:

How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009) have exactly 256 odd entries?

What is the solution to Wendy's second problem?
Please reason step by step, and put your final answer within \boxed{}.


Context:
One rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005^{th}$ digit that you write?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mrs. Jones is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only has enough orange juice to fill one third of the last glass. What fraction of a glass of orange juice does she need to pour from the 3 full glasses into the last glass so that all glasses have an equal amount of orange juice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What geometric solid has the property that its front view, side view, and top view are all the same? Provide the name of the shape as your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The value of an item is $C$ dollars. After a loss of $x \%$ of the selling price, it is sold for $100. Then, it is sold again with a profit of $x \%$ of the new selling price $S$. If the difference between $S$ and $C$ is $1 \frac{1}{9}$ dollars, what is the value of $x$? Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $S=\{1,2, \cdots, 2005\}$ .若 $S$ 中任意 $n$个两两互质的数组成的集合中都至少有一个质数，试求 $n$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the series: \[ \sum^{10}_{i=1} i \left( \frac{1^2}{1+i} + \frac{2^2}{2+i} + \cdots + \frac{10^2}{10+i} \right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
热气球的探测器显示，从热气球A看一栋高楼顶部B处的仰角为30°，看这栋高楼底部C处的俯角为60°，若热气球与高楼的水平距离为120m，则这栋高楼有多高？（结果精确到0.1，$\sqrt{2} \approx 1.414$，$\sqrt{3} \approx 1.732$）
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the error for the numerical integration method given by: $$\int_a^b \left( f'(x) \right)^2\ dx \approx h\cdot \sum_{i=0}^{n-1} \left(\frac{f(x_{i+1}) - f(x_i)}{h}\right)^2$$ where $h=\frac{b-a}{n}$ and $x_k = a+kh$. Assume that the derivatives are bounded. What is the error for this method?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Pui Ching Invitational Mathematics Competition, consider the polynomials $f(x,y)$ and $g(x,y)$ in two variables. Given that $g(0,0)=6$ and the equation \((x^2+2xy+3y^2)f(x,y)=(4x^2+5xy+6y^2)g(x,y)\) holds for all real numbers $x$ and $y$ satisfying \(x^2y+6y=xy^3+5x^2+2x\), find the value of $f(0,0)$. \(\boxed{?}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
An alphabet consists of  $n$  letters. What is the maximal length of a word if we know that any two consecutive letters  $a,b$  of the word are different and that the word cannot be reduced to a word of the kind  $abab$  with  $a\neq b$  by removing letters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximal time interval $J$ for which the Existence and Uniqueness Theorem guarantees the existence and uniqueness of a solution $y : J \to \mathbb{R}$ for the initial value problem (IVP): \[ y' = y^2 + \frac{1}{t}, \quad y(1) = 1. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
An $n$-string is a string of digits formed by writing the numbers $1,2, \ldots, n$ in some order (in base ten). For example, one possible 10-string is $$35728910461$$ What is the smallest $n>1$ such that there exists a palindromic $n$-string?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a joint Gaussian random vector \(X\) with distribution:
\[
X \sim \mathcal{N}\left(\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 & 1 & 2 \\ 1 & 4 & 4 \\ 2 & 4 & 16 \end{bmatrix}\right)
\]
Calculate the expected value \(E[X_1 \mid X_2 + X_3 = 0]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each of the three circles with radius \( r \) is tangent to the other two. Find the area of the triangle formed by the common external tangents to these circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a monotonic function f(x) defined on (0,+\infty), for all x\in(0,+\infty), there is f[f(x)-\ln x]=e+1. Find the interval where the zero point of the function g(x)=f(x)-f′(x)-e is located. Express your answer as an interval (a,b).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of points on the quadric $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ in the projective space $\mathbb{P}_3$ over the finite field $\mathbb{F}_9$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the sides opposite to the internal angles A, B, and C of triangle ABC are a, b, and c respectively, if -c cosB is the arithmetic mean of $\sqrt {2}$a cosB and $\sqrt {2}$b cosA, find the maximum value of sin2A•tan²C.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $x, y$, and $z$ are real numbers such that $2 x^{2}+y^{2}+z^{2}=2 x-4 y+2 x z-5$, find the maximum possible value of $x-y+z$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle passes through the intersection points of any two of the lines \( y = x \), \( y = 2x \), and \( y = 15 - 0.5x \). What is the length of the chord of the circle that lies on the line \( x + y = 16 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} \) divide a circle of radius 1 into six equal parts. A ray \( l_{1} \) is drawn from \( A_{1} \) in the direction of \( A_{2} \), a ray \( l_{2} \) is drawn from \( A_{2} \) in the direction of \( A_{3} \), and so on, with a ray \( l_{6} \) is drawn from \( A_{6} \) in the direction of \( A_{1} \). From a point \( B_{1} \) on the ray \( l_{1} \), a perpendicular is dropped onto the ray \( l_{6} \), from the foot of this perpendicular another perpendicular is dropped onto \( l_{5} \), and so on. The foot of the sixth perpendicular coincides with \( B_{1} \). Find the segment \( B_{1} A_{1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
10. (20 points) Let
$$
f(x)=x^{5}-3 x^{3}+2 x^{2}+3 x+6 \text {, }
$$

and let $A_{n}=\prod_{k=1}^{n} \frac{(4 k-3) f(4 k-3)}{(4 k-1) f(4 k-1)}$. Find the value of $A_{25}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a, b, c \in (0,1]\) and \(\lambda\) be a real number such that \(\frac{\sqrt{3}}{\sqrt{a+b+c}} \geq 1+\lambda(1-a)(1-b)(1-c)\) is always satisfied. Find the maximum value of \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the average of the first 12 even numbers, including both positive and negative even numbers in the sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $F$ ,  $D$ , and  $E$  be points on the sides  $[AB]$ ,  $[BC]$ , and  $[CA]$  of  $\triangle ABC$ , respectively, such that  $\triangle DEF$  is an isosceles right triangle with  hypotenuse  $[EF]$ . The altitude of  $\triangle ABC$  passing through  $A$  is  $10$  cm. If  $|BC|=30$  cm, and  $EF \parallel BC$ , calculate the perimeter of  $\triangle DEF$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral $$\int_0^1 \int_0^1 \left\{\frac{y+x}{y-x}\right\}\,\mathrm{d}y\,\mathrm{d}x$$ where { } denotes the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Anya wants to buy ice cream that costs 19 rubles. She has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin in her pocket. Anya randomly picks three coins from her pocket without looking. Find the probability that the selected coins will be enough to pay for the ice cream.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume f(x) is an even function defined on the real numbers and is a decreasing function on the interval (0, ∞). Given that x1 < 0 and x1 + x2 > 0, compare f(-x1) and f(-x2). Express your answer as a comparison (e.g., >, <, =) between f(-x1) and f(-x2).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular quadrilateral pyramid has all its edges of equal length. There are two spheres that touch every edge of the pyramid. What is the ratio of their volumes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f$ defined as follows: \[ f(0) = 0, \] \[ f(1) = 1, \] and for every integer $n \geq 1$, \[ f(3n-1) = f(n) - 1, \] \[ f(3n) = f(n), \] \[ f(3n+1) = f(n) + 1. \] Find the value of $f(2011)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a free group $G$ generated by two elements $a$ and $b$, consider a random walk starting at the identity element $1$. At each step, multiply the current word by one of the generators $a$, $b$, $a^{-1}$, or $b^{-1}$, each with probability $\frac{1}{4}$. What is the probability $\rho$ that the reduced word $1$ is ever reached during this process?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of infinitesimal of the function \( h(x) = \frac{\ln^5(1+x\ln^{1/8}(x))}{x^x-1} \) with respect to \( x \) as \( x \to 0^+ \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle with dimensions \(24 \times 60\) is divided into unit squares by lines parallel to its sides. Into how many parts will this rectangle be divided if its diagonal is also drawn?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a field $K$ and a positive integer $n$, if $a^n = a$ for all $a \in K$, determine the possible values of the characteristic of $K$ when $n$ is odd.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A function \( f: \mathbb{N^+} \rightarrow \mathbb{N^+} \), defined on the set of positive integers \( \mathbb{N^+} \), satisfies the following properties:
\[f(n)=\begin{cases} f(n/2) & \text{if } n \text{ is even}\\ f(n+5) & \text{if } n \text{ is odd} \end{cases}\]
Let \( R=\{ i \mid \exists{j} : f(j)=i \} \) be the set of distinct values that \( f \) takes. What is the maximum possible size of \( R \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $f\left(x\right)=(e^{x-a}-1)\ln \left(x+2a-1\right)$, if $f\left(x\right)\geqslant 0$ always holds for $x\in \left(1-2a,+\infty \right)$, then the real number $a=$____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest integer solution to the equation \( y = x^2 + \frac{(1-x)^2}{(1+x)^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $13$ cities in a certain kingdom. Between some pairs of the cities a two-way direct bus, train or plane connections are established. What is the least possible number of connections to be established so that choosing any two means of transportation one can go from any city to any other without using the third kind of vehicle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the total number of zeros in the expression \((10^{10} + 10^{11} + 10^{12} + \ldots + 10^{2009})^{2010}\).
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 \(2x + y = 2035\) 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:
Let  $a,b,c>0$  such that  $a \geq bc^2$  ,  $b \geq ca^2$  and  $c \geq ab^2$  . Find the maximum value that the expression :  $$ E=abc(a-bc^2)(b-ca^2)(c-ab^2) $$   can acheive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
During summer break, some kids from Lawrence county go to camp and the other 668278 kids stay home. There are 150780 more kids who spent their summer break at the camp compared to those who stayed home. How many kids went to camp?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The medians of one triangle are equal to the sides of another triangle. Find the ratio of the areas of these triangles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is an icosikaipentagon. How many diagonals can you draw from one vertex?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_S (x + 2y - z) \, dV \) over the tetrahedron \( S \) in \( \mathbb{R}^3 \) with vertices \((0,0,0), (0,1,2), (1,2,3), \text{ and } (-1,1,1)\). Consider using a linear diffeomorphism to transform \( S \) into the unit cube.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A coin is flipped 110 times, with equal probabilities for heads and tails on each flip. Let \( p \) be the probability that heads appear at least 61 times, and \( q \) be the probability that heads appear less than 49 times. Calculate \( p - q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any real sequence $\left\{x_{n}\right\}$, define the sequence $\left\{y_{n}\right\}$ as follows:
$$
y_{1} = x_{1}, \quad y_{n+1} = x_{n+1} - \left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} \quad (n \geqslant 1).
$$

Find the smallest positive number $\lambda$ such that for any real sequence $\left\{x_{n}\right\}$ and any positive integer $m$, the following inequality holds:
$$
\frac{1}{m} \sum_{i=1}^{m} x_{i}^{2} \leqslant \sum_{i=1}^{m} \lambda^{m-i} y_{i}^{2}.
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\mathbb{R} } \frac{\sin^4(\frac{t}{2}) }{t^2} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to use diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\alpha$ and $\beta$ are the roots of the equation $x^2 - 3x - 2 = 0,$ find the value of $5 \alpha^4 + 12 \beta^3.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many positive three-digit integers $abc$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $bc$ corresponds to the day in that month, or $ab$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.
Please reason step by step, and put your final answer within \boxed{}.


Context:
数列 $\left\{a_{n}\right\}$ 为等差数列, 且满足 $3 a_{5}=8 a_{12}>0$, 数列 $\left\{b_{n}\right\}$ 满足 $b_{n}=a_{n} \cdot a_{n+1} \cdot a_{n+2}\left(n \in \mathbf{N}^{*}\right),\left\{b_{n}\right\}$ 的前 $n$ 项和记为 $S_{n}$. 问: $n$ 为何值时, $S_{n}$ 取得最大值？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F$ be a field and let $a, b, c$ be indeterminates. Consider the field extension $L = F(ac, bc, ab) \subseteq K = F(a, b, c)$. Determine the degree of the extension $[K:L]$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Numbers 1, 2, 3, ..., 10 are written in a circle in some order. Petya calculated the sums of all triplets of neighboring numbers and wrote the smallest of these sums on the board. What is the largest number that could have been written on the board?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Brian goes fishing twice as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, they caught 13600 fish altogether. How many times did Chris go fishing?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$[\quad$ Evaluation + example $\quad]$

The hostess baked a pie for her guests. She may have either 10 or 11 guests. Into what smallest number of slices should she cut the pie in advance so that it can be evenly divided among either 10 or 11 guests?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all pairs \((n, p)\) of natural numbers \(n\) and prime numbers \(p\) that satisfy the equation \(p(p-1) = 2(n^3 + 1)\). Calculate the sum \(\sum (n_i + p_i)\) for all such pairs \((n_i, p_i)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer that has 144 positive divisors, and among these divisors, there are ten consecutive ones.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设$\frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 1 + \cdots } + \frac { 1 } { 1 } } } = \frac { m } { n } $,

其中$m$和$n$是互质的自然数，而等式左边含有1988条分数线，试计算$m ^ { 2 } + m n - n ^ { 2 }$的值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathcal{T}$ be the set of all linear maps $A$ from $\mathbb{R}^2$ such that $A^4 - 5A^2 + 4I = 0$. Determine the maximum number of elements in $\mathcal{T}$ such that no two maps are similar.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sequence formed from the numbers $1, 2, 3, 4, 5, 6, 7$, where all $7!$ permutations are equally likely. What is the probability that there will be at least five consecutive positions in which the numbers are in increasing order?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{\epsilon \to 0} \int_\mathbb {R} \frac{1}{x^2-(1-\ic\epsilon)^2} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given independent exponential random variables $X_1, X_2, X_3, X_4$ with rates 1, 2, 3, and 4, respectively, calculate the probability $P(X_1<X_2<X_3<X_4|X_2<X_4)$. Use the formula for conditional probability to find your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A particle moves along a straight line with a position function $x(t)$. At any instant, the product of its position and acceleration equals the square of its velocity. Given that the particle starts at position $x(0) = 1$ with initial velocity $v(0) = 0.1$, find the position function $x(t)$. Provide your answer as a function of $t$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two teams, Team A and Team B, each consist of 10 people. Team A selects and removes n people from its team, where 1 ≤ n ≤ 9. Team B then selects m people from its team to replace with m of the n people removed from Team A, where 0 ≤ m ≤ n. Determine the number of distinct possible combinations of teams after these changes, and find the remainder when this number is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of intersections between the polynomials \( g(x) = a_1x + a_2x^2 + \ldots + a_kx^k \) and \( f(x) = b_jx^j \), where \( a_1, a_2, \ldots, a_k > 0 \), \( j \in \{1, 2, \ldots, k-1\} \), \( b_j > 0 \), and \( x \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U$ be an open set of the Riemann sphere, and let $z_i$ be $n$ distinct points in $U$. Consider $E$, the vector space of meromorphic functions on $U$ with poles of order no more than 2, and $F$, the subspace of $E$ consisting of functions that are holomorphic in a neighborhood of each $z_i$. Determine if the quotient space $E/F$ has finite dimension, and if so, find its dimension.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the variance of the expression \(12(X_1 + X_2 + X_3)\), where \(X_1\) is a random variable uniformly distributed on \((0,1)\), \(X_2\) is uniformly distributed on \((0,X_1)\), and \(X_3\) is uniformly distributed on \((0,X_2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the eigenvalues of the operator $D^2$ for which the eigenfunctions are 2π-periodic.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four siblings ordered an extra large pizza. Alex ate \( \frac{1}{5} \) ,Beth \( \frac{1}{3} \) ,and Cyril \( \frac{1}{4} \) of the pizza. Dan got the leftovers. What is the
 sequence of the siblings in decreasing order of the part of pizza they consumed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a qualification group with  $15$  volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding  $N$ . If there are at least  $7$  teams qualified, find the possible least value of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the region in the first octant (where $x \ge 0$, $y \ge 0$, $z \ge 0$) bounded by the cylinder $x^2 + y^2 = 4$ and the planes $z = 2$ and $y + z = 4$. Evaluate the integral \( \int_0^2 \int_0^{4-z} \int_0^{\sqrt{4-y^2}} dx \, dy \, dz \) to find the volume.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( z \) be a complex number satisfying \( \left( {z + \frac{1}{z}}\right) \left( {z + \frac{1}{z} + 1}\right)  = 1 \) . Evaluate
 \( \left( {3{z}^{100} + \frac{2}{{z}^{100}} + 1}\right) \left( {{z}^{100} + \frac{2}{{z}^{100}} + 3}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Count the number of permutations in $S_5$ that are a product of exactly two disjoint cycles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cyclic quadrilateral \(ABCD\), it is known that \(\angle ADB = 48^{\circ}\) and \(\angle BDC = 56^{\circ}\). Inside the triangle \(ABC\), a point \(X\) is marked such that \(\angle BCX = 24^{\circ}\), and the ray \(AX\) is the angle bisector of \(\angle BAC\). Find the angle \(CBX\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit:

$$\lim \limits_{x \to 0} \frac{(\sin{x})^{x-1}}{\frac{1}{x}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle  CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$.
Note: $[ABCD]$ denotes the area of the polygon $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Cars A and B start from points A and B simultaneously and move towards each other. Car A travels at a higher speed than Car B. After Car A has traveled 60 kilometers, they meet at point C. Car A continues forward while Car B turns around and follows in the same direction as Car A. Calculate the maximum distance between the two cars when Car A reaches point B. Express your answer in kilometers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose the state of Georgia uses a license plate format "LLDLLL", and the state of Nebraska uses a format "LLDDDDD". Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Nebraska issue than Georgia?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC$. The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1$. On the line passing through $C$ and parallel to $MD$, there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150^\circ$. Find the $\angle MTD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The set \( A \) consists of integers, with the smallest element being 1 and the largest being 100. Except for 1, every element is equal to the sum of two (not necessarily distinct) elements from the set \( A \). Find the minimum number of elements in set \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the region of points $(x,y,z)$ such that $$(x^2+y^2+z^2+8)^2\leq36(x^2+y^2).$$ The original answer is in the format $m\pi^{n}$, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man can row his boat with the stream at 22 km/h and against the stream at some speed. The man's rate is 6 km/h. What is his speed against the stream?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that in triangle $\triangle ABC$, $a=2$, $\angle A=\frac{π}{6}$, $b=2\sqrt{3}$, find the measure of angle $\angle C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two individuals, A and B, participate in an exam that consists of a pool of 10 questions. A can answer 6 of these questions correctly, while B can answer 8 correctly. For each test, 3 questions are randomly selected from the pool. The passing criterion is to answer at least 2 questions correctly. Calculate the probability that at least one of A or B passes the test. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of a circle is given by $A = \pi r^2$. If the area is $A = \pi(3m^2 + 30m + 75)$, what is the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在三棱锥 $D-A B C$ 中, $\angle B A D=30^{\circ}$, $\angle A C B=\angle A B D=90^{\circ}, C A=C B$. 若点 $C$ 在平面 $A B D$ 上的射影恰在 $A D$ 上, 则二面角 $C$ $A B-D$ 的平面角的正弦值大小为 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
The functions \( f(x) \) and \( g(x) \) are defined for all \( x > 0 \). The function \( f(x) \) is equal to the greater of the numbers \( x \) and \( \frac{1}{x} \), and \( g(x) \) is equal to the smaller of the numbers \( x \) and \( \frac{1}{x} \). Solve the equation \( f(5x) \cdot g(8x) \cdot g(25x)=1 \). In the answer, specify the solution if it is unique, or the sum of the solutions if there are several. Round the answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of lines that pass through the point $(0,1)$ and intersect the parabola $y^2=4x$ at only one point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any integer $k \geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\{x_n\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p(x_n)$ for $n \geq 0.$ Find the smallest positive integer $t$ such that $x_t = 2090.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the system of equations $|x_i-x_j|=\alpha$ with $x_i \in \mathbb{R}^n$, what is the minimum number of equations that can be satisfied for a general $n$, assuming that the $x_i$ values cannot be all equal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the acute triangle ABC, the sides opposite to the angles A, B, and C are a, b, and c respectively, with a > b > c. If the side PQ of the inscribed square PQRS falls on side a of triangle ABC, and the other two vertices R and S lie on sides b and c respectively, let the side length of square PQRS be x_a. Similarly, define x_b and x_c. Determine the correct relationship among x_a, x_b, and x_c. Express your answer using inequality symbols.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Maria ordered a certain number of televisions at $R$ \$ 1994.00 each. She noticed that in the total amount to be paid, there are no digits 0, 7, 8, or 9. What was the smallest number of televisions she ordered?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are  $n\geq 3$  pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number  $M$  such that after  $M$  times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils  $ A$  and  $ B$ , and if  $ A$  initially stands on the left side of  $ B$  then  $ A$  will finally stand on the right side of  $ B$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_0^1 \log \Gamma(x) \cos (2\pi n x)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a factory, the average monthly salary of laborers and supervisors is Rs. 1250 per month. There are 6 supervisors with an average monthly salary of Rs. 2450. There are 42 laborers. What is the average monthly salary of the laborers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the convex quadrilateral $ABCD$, it is known that $\angle CBD = 58^\circ$, $\angle ABD = 44^\circ$, and $\angle ADC = 78^\circ$. Find the angle $\angle CAD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\,\frac {n\left( 3\,{{\rm e}^{3}} \left( \frac {n+1}{n} \right) ^{-3\,n}n-1-3\,n \right)}{n+1} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The perimeter of an isosceles triangle is 24. A median to one of the equal sides divides the perimeter into two parts in the ratio 5:3. Find the length of the base of this triangle. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\mathbb{R}} (F(x+b)-F(x+a)) \, dx \) for any real numbers \(a\) and \(b\), where \(F\) is a distribution function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the following limit: $$\lim\limits_{x\to0}\frac{\left(1+x\cdot e^x\right)^{\frac{1}{x^2}}}{\left(1+x\cdot5^x\right)^{\frac{1}{x^2}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any real numbers sequence  $\{x_n\}$  ,suppose that  $\{y_n\}$  is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$   ${(n \ge 1})$   .
Find the smallest positive number   $\lambda$  such that  for any real numbers sequence  $\{x_n\}$  and all positive integers  $m$  , have  $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ordered triples \((a, b, c)\) of positive integers are there such that none of \(a, b, c\) exceeds 2010 and each of \(a, b, c\) divides \(a + b + c\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define  $\phi_n(x)$  to be the number of integers  $y$  less than or equal to  $n$  such that  $\gcd(x,y)=1$ . Also, define  $m=\text{lcm}(2016,6102)$ . Compute  $$ \frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand? (We can wear multiple rings in one finger)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{23} = \frac{a}{23!}\), find the remainder when \(a\) is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct arrangements of the letters in the word "MATHEMATICS" such that no two vowels are adjacent and the letter 'T' appears before the letter 'S'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the derivative \( f'(e^{i 2\pi/3}) \) for the function \( f(z) = z^{1/4} \), where the branch is defined in \( \mathbb{C} - (\infty,0] \) and satisfies \( f(1) = -i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set of points \(S\) where each point \((x, y)\) has integer coordinates with \(1 \leq x, y \leq 26\). Determine the number of ways to select a convex subset of \(S\) that contains exactly 98 points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Nathan is buying decorations for his wedding reception. The reception hall will have 30 tables. Each table needs a linen tablecloth ($25 to rent), 6 place settings ($12 each to rent), and a centerpiece. Each centerpiece will have 15 roses ($6 each, but there is a 10% discount for bulk purchase), 20 lilies ($5 each, but there is a 5% discount for bulk purchase), 5 daisies ($3 each), and 3 sunflowers ($4 each). Additionally, Nathan needs to purchase lighting for the hall at a cost of $450, and tablecloth rentals come with a 15% service charge. How much will the decorations cost after all the discounts and additional charges are applied?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Gary is baking cakes, one at a time. However, Gary’s not been having much success, and each failed cake will cause him to slowly lose his patience, until eventually he gives up. Initially, a failed cake has a probability of $0$ of making him give up. Each cake has a $1/2$ of turning out well, with each cake independent of every other cake. If two consecutive cakes turn out well, the probability resets to $0$ immediately after the second cake. On the other hand, if the cake fails, assuming that he doesn’t give up at this cake, his probability of breaking on the next failed cake goes from p to $p + 0.5$. If the expected number of successful cakes Gary will bake until he gives up is$ p/q$, for relatively prime $p, q$, find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the order of $257$ in $\Bbb Z/660$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a six-digit decimal number $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$, each digit $a_{i}(1 \leqslant i \leqslant 6)$ is an odd number, and the digit 1 is not allowed to appear consecutively (for example, 135131 and 577797 satisfy the conditions, while 311533 does not satisfy the conditions). Find the total number of such six-digit numbers. $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three people, A, B, and C, are standing in a row from left to right. If two people are randomly selected for a position swap, what is the probability that A will be to the left of B after two such swaps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the quotient group \( \frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle(1,1,1),(1,3,2)\rangle} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_{D}\left(54 x^{2} y^{2}+150 x^{4} y^{4}\right) d x d y \) where the region \(D\) is bounded by the lines \(x=1, y=x^{3}\), and \(y=-\sqrt{x}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1{:}00 \text{ PM}$ and were able to pack $4$, $3$, and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45 \text{ PM}$. At what time did Daria join the group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-6 n+7}{3 n^{2}+20 n-1}\right)^{-n+1}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Beth, Carolyn, and George take turns reading a 20-page story in the order Beth, Carolyn, then George. One evening, Beth is absent, and Carolyn and George read the story in turns starting with Carolyn. How many pages are read by the same person in both the original and modified reading sequences? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product of uncommon real roots of the two polynomials \( {x}^{4} + 2{x}^{3} - 8{x}^{2} - {6x} + {15} \) and \( {x}^{3} + 4{x}^{2} - x - {10} \) is ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the community center has 8 cans of soup and 2 loaves of bread, with each can of soup feeding 4 adults or 7 children and each loaf of bread feeding 3 adults or 4 children, and the center needs to feed 24 children, calculate the number of adults that can be fed with the remaining resources.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$as a rational number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) be an even function such that \( f(1)=1 \) and \( f(2017) \neq 1 \). For any \( x, y \in \mathbf{Z} \), the following inequality holds:
$$
2 f(x+y)-f(x)-f(y) \leqslant|f(x)-f(y)|.
$$

Find all possible values of \( f(2018) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If ${a}^{2}=3{b}^{2}+3{c}^{2}-2\sqrt{3}bc\mathrm{sin}A$, then angle $C=$_______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( A B C \), the angle \( \angle B \) is equal to \( \frac{\pi}{3} \). A circle with radius 3 cm is drawn through points \( A \) and \( B \), touching the line \( A C \) at point \( A \). Another circle with radius 4 cm is drawn through points \( B \) and \( C \), touching the line \( A C \) at point \( C \). Find the length of side \( A C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$P(x)$ is a polynomial of degree at most $6$ such that $P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$, and $P(7)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P(8)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(\theta \in \mathbb{R}\) for which the set \(S = \{x \in \mathbb{R}^n : |x-a| \leq \theta|x-b|\}\) is convex.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$ \int\limits_0^{\pi/2} \frac{\cos^3x + \sin^3x}{\sqrt{\sin 2x}}dx $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
设由 1,2,\cdots,9 的所有排列 X=(x_1,x_2,\cdots,x_9) 构成的集合为 A. 对 X\in A，记 f(X)=x_1+2x_2+\cdots+9x_9，M=\{f(X)\ |\ X\in A\}. 求 |M|.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine \[ \left\lfloor \prod_{n=2}^{2022} \frac{2n+2}{2n+1} \right\rfloor, \] given that the answer is relatively prime to 2022.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field $\mathbb{v} = \operatorname{curl}\mathbb{u}$, where $\mathbb{u} = (xy, xz^2, x^2y)$, downwards through the paraboloid $z = 1 - x^2 - y^2$ for $x \geq 0$ and $z \geq 0$. What is the value of this flux?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \(a \in [0, \infty)\) for which there exists a continuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that \(f(f(x)) = (x-a)^2\) for all \(x \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the expression \[ \left[\cos ^{2}\left(\dfrac{x }{2}\right) + \,\sqrt{\,{1 - \alpha}\,}\,\sin^{2}\left(\dfrac{x }{2}\right)\right] ^{2} + \dfrac{\alpha }{4}\,\sin^{2}\left(x\right) \] where \( 0 \le \alpha \le 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $K = \mathbb{Z}_2[x]/\langle x^4 + x^2 + x \rangle$. Find an element $g \in K$ such that $g^2 = x^3 + x + 1$. Provide your answer as a polynomial in $K$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of nodes in a hexagonal grid required to ensure that among any selection of nodes, there will always be two nodes such that the midpoint of the segment joining them is also a node of the hexagonal grid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rachel has two identical basil plants and an aloe plant. She also has two identical white lamps and two identical red lamps she can put each plant under (she can put more than one plant under a lamp, but each plant is under exactly one lamp). How many ways are there for Rachel to put her plants under her lamps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the $p$-adic valuation of $|a-b|_p$, where $a$ is a primitive $p$-th root of unity in $\overline{\mathbb{Q}_p}$ and $b$ is a root of $X^{p-1}+p$ in $\overline{\mathbb{Q}_p}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
2. 12 friends agree to go out for a meal once a week, each time evenly divided into 3 tables. Each table seats 4 people, until any two people have eaten at the same table at least once. How many weeks are needed at minimum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{1}\sum_{n=1}^{\infty}\frac{9}{\pi n^{2}}\cos^{3}\left(\frac{2 \pi n x}{3}\right) d x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral: \[ \int_0^1\int_0^1 \frac {x^2-y^2}{(x^2+y^2)^2} \, dy \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_{[0,1]^2} (1+x^2+y^2)^{-3/2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\frac{1}{4}$ of the candies. Then, he shared $\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike had. Later, Mike’s sister took between one to four candies, leaving Mike with five candies in the end. Calculate the number of candies Mike started with initially.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_{|z|=2} \sqrt{z^2-1} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.

Determine the maximum possible number of positive integers among the 2018 integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Benny goes to the market for buying some apples to be distributed between her 9 kids equally. She takes a certain amount of money with her. The cost of each apple is 4 dollars. She buys enough apples to share 5 apples equally between her 18 kids. How much money did Benny take with her to the market?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 n=5，L_n(x)=\df{1}{n!}\cdot\df{\dd^n}{\dd x^n}(x^2-1)^n. 记 S=\displaystyle\int_{-1}^1|L_n(x)|^2\ \dd x，求 [S].
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a \textit{subsequence} of a string \( \mathcal{S} \) of letters to be a positive-length string using any number of the letters in \( \mathcal{S} \) in order. For example, a subsequence of \( HARRISON \) is \( ARRON \). Compute the number of subsequences in \( HARRISON \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set
$A=\{n \in \mathbb{N}: P(n)$ is non-zero, square free and $S(n)$ is a proper divisor of $P(n)\}$.
Find the maximum possible number of digits of the numbers in $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the plane Cartesian coordinate system, consider the points \((i, j)\) where \(i, j = 1, 2, \cdots, 5\). Calculate the number of triangles with vertices at these points that have a positive area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya has thought of six natural numbers: \( a, b, c, d, e, f \). For one ruble, it is possible to specify any two of them and find out their product. Petya knows that any two of the thought numbers are coprime (i.e., they have no common divisors greater than 1). For what minimum amount can he find out all the thought numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Zhukov G.

A teacher is planning to give the children a problem of the following type. He will inform them that he has thought of a polynomial $P(x)$ of degree 2017 with integer coefficients, the leading coefficient of which is 1. Then he will tell them $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$ and separately inform them of the value of the expression $P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right)$. Based on this data, the children are supposed to find the polynomial that the teacher could have thought of. For what smallest $k$ can the teacher formulate such a problem so that the polynomial found by the children will definitely match the one thought of?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = \frac{2^2 + 3 \cdot 2 + 1}{3! \cdot 4!} + \frac{3^2 + 3 \cdot 3 + 1}{4! \cdot 5!} + \frac{4^2 + 3 \cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2 + 3 \cdot 10 + 1}{11! \cdot 12!} \). Find the remainder when \( 11! \cdot 12! \cdot A \) is divided by 11.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c$ be real nonzero numbers such that $a + b + c = 12$ and \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = 1. \] Compute the largest possible value of $abc - \left(a + 2b - 3c\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a lower bound for the integral $\int_{0}^{1}\frac{1}{x^p+1}dx$ for $p<-1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
While reading about the history of different towns, Jared learned that Euston had 20 more carriages than Norfolk. Norwich had 100 carriages, and Flying Scotsman had 20 more carriages than Norwich. Additionally, Jared found out about two more towns - Victoria, which had 15 carriages fewer than Euston, and Waterloo, which had twice the number of carriages as Norwich. If Euston had 130 carriages, how many carriages are there in all these towns combined?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer \( n \) (\( n \geqslant 6 \)), let set \( A \) consist of sums of any 5 consecutive positive integers not greater than \( n \), and let set \( B \) consist of sums of any 6 consecutive positive integers not greater than \( n \). If the number of elements in the intersection of sets \( A \) and \( B \) is 2016, find the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
When $p = \sum\limits_{k=1}^{6} k \text{ ln }{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 (n+1)^{\alpha+1}-n^{\alpha+1}<n^{\alpha}(\alpha+1)<n^{\alpha+1}-(n-1)^{\alpha+1}，-1<\alpha<0. 设 x=\ds{k=4}{10^6}\df{1}{\sqrt[3]{k}}，则 x 的整数部分为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider all prisms whose base is a convex 2015-gon.

What is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices?
Please reason step by step, and put your final answer within \boxed{}.


Context:
[ Dirichlet's Principle (finite number of points, lines, etc.).]  [ Central angle. Arc length and circumference length ]

On a circle of radius 1, a point $O$ is marked and a notch is made to the right with a compass of radius $l$. From the resulting point $O_{1}$, another notch is made in the same direction with the same radius, and this is done 1968 times. After this, the circle is cut at all 1968 notches, resulting in 1968 arcs. How many different arc lengths can result from this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\rightarrow \infty} n \int_{-n}^{n} \cos^2(\pi x) \left(\frac{1}{(x-3)(x-4)} + \frac{1}{(x+3)(x+4)}\right) dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying
$$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $f(x) = x^5 + 2x^3 + 3x^2 + x + 1$, calculate the value of $v_3$ when evaluating the value of $f(3)$ using Horner's method.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the dimension of the family of zero-dimensional subschemes of degree 25 in \( \mathbb{A}^5_K \) such that \( V(\mathfrak{m}^3) \subset \Gamma \subset V(\mathfrak{m}^4) \), where \( \mathfrak{m} \) is the maximal ideal of the origin in \( \mathbb{A}_K^5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether there exists a natural number $k$ such that for any selection of 180 vertices from a regular 360-gon centered at point $O$, there are two vertices $A$ and $B$ among the selected vertices such that the angle $AOB$ measures $k$ degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum length of a closed, non-self-intersecting polyline that follows the grid lines of a $6 \times 10$ checkered field?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya and Vasya play a game. First, Petya thinks of a polynomial \( P(x) \) with integer coefficients. Then, several moves are made. During a move, Vasya pays Petya one ruble and names any integer \( a \) of his choice, which he has not named before, and Petya responds by telling how many integer solutions the equation \( P(x) = a \) has. Vasya wins as soon as Petya states the same number of solutions twice (not necessarily consecutively). What is the minimum number of rubles Vasya needs to guarantee a win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n \) such that the inequality \[ \sum_{i=n}^{100} \binom{100}{i} \left(\frac{80}{100}\right)^i \left(\frac{20}{100}\right)^{100-i} \geq 0.9 \] holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the expression \( \frac{1}{n^2} \sum_{i = 0}^{n} \ln\binom{n}{i} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$. Find $100m+n$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:

$$\lim_{x\to 0} \frac {1-(\cos 2x)^3(\cos 5x)^5(\cos 7x)^7(\sec 4x)^9(\sec 6x) ^{11}}{x^2}$$

Using a Taylor series expansion or L'Hôpital's Rule, find the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2a+3b+5c$ 的最大值和最小值之差。
Please reason step by step, and put your final answer within \boxed{}.


Context:
The age difference between Asaf and Alexander's age is half the total number of pencils Asaf has. The sum of their ages is 140. Alexander has 60 more pencils than Asaf, and they have together 220 pencils. How old is Asaf?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of flips needed to estimate the probability of heads $p$ of a potentially biased coin with an error margin of $\frac{1}{1000}$ and a confidence level of at least $0.99$. What is the minimum number of coin flips required?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A particle is located at some point on a line. During the first second, it splits in half and the halves move away from each other in opposite directions by a distance of 1 from the original position. During the next second, the resulting particles again split in half and the halves move away from each other in opposite directions by a distance of 1 from their previous positions. When any two particles collide, they annihilate each other. For example, after two seconds, only two particles remain. How many particles will remain after 129 seconds?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知四边形 $A B C D$ 为圆柱轴截面，母线 $A B$ 长为 1, 底面直径 $B C$ 长为 $\sqrt{2}$, $E$ 为底面圆周上的动点， $P$ 、 $Q$ 分别为点 $B$ 在 $A E 、 A C$ 上的射影。则当三棱锥 $A-P B Q$ 的体积最大时， $C E$ 的原始答案为 \(\dfrac{\sqrt{m}}{n}\) 的形式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of all point $(x,y)$ in the coordinate plane such that $0 \le x \le \frac{\pi}{2}$ and $0 \le y \le \frac{\pi}{2}$. What is the area of the subset of $S$ for which \[\sin^2x-\sin x \sin y + \sin^2y \le \frac34?\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train of a certain length is running with a speed of 65 km/hr. In 5.4995600351971845 seconds, it passes a man who is running at 7 km/hr in the direction opposite to that in which the train is going. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length \(1\). The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. The length of one of the two congruent sides of one of the isosceles triangles is given in the form \(\frac{\sqrt{k}}{m}\). Please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle is cut at one corner to form a pentagon with side lengths of 13, 19, 20, 25, and 31. Calculate the area of the resulting pentagon. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X Y Z$ be a triangle with $\angle X Y Z=40^{\circ}$ and $\angle Y Z X=60^{\circ}$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $X Y Z$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $Y Z$, and let ray $\overrightarrow{X I}$ intersect side $Y Z$ at $B$. Determine the measure of $\angle A I B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x| \leq 1$. Compute the number of distinct valid paths the frog can take.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the region given by the inequality
\[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 4.\]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:
Given that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$, if the graph of $y=g(x)$ is symmetric about the line $x=2$ and $g(2) = 4$, find $\sum _{k=1}^{22}f(k)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the module $M=\mathbb{Z}^3/N$ as a direct sum of cyclic modules, where $N=\{(x,y,z)\in \mathbb{Z}^3\;|\; 2x+3y-5z=0\} \subset \mathbb{Z}^3$. Provide the decomposition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the following summation in matrix form: \[ D_{βν} = \sum^{n}_{µ=1}\sum^{n}_{a=1}A_{µν}B_{αµ}C_{αβ} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the alternating group $A_5$. What is the probability that the subgroup generated by two randomly chosen elements $a$ and $b$ in $A_5$, denoted by $\langle a, b \rangle$, is a cyclic group?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( M = \{\mu : \mathcal{B}(\mathbb{R}) \to \mathbb{R} : \mu \text{ is a measure}\} \). What is \(|M|\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the power function $f(x)=({m}^{2}-2m-2){x}^{{m}^{2}-2}$ is increasing on $(0,+\infty)$, the value of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a darts game, Don throws 5 darts in total. The first dart hits a bullseye and is worth triple points due to a special bonus in play, totaling 150 points. The second dart completely misses the target so receives no points. His third dart is worth half the points of the bullseye, without any special bonuses. The fourth dart hits a double ring, which is twice the points of a bullseye without the bonus. The fifth dart, not only hits another bullseye but also triggers a rule called 'Golden Bullseye', which adds an additional point for every earned point so far. What is the final total score from these 5 throws?
Please reason step by step, and put your final answer within \boxed{}.


Context:
39 students participated in a math competition. The exam consisted of 6 problems, and each problem was worth 1 point for a correct solution and 0 points for an incorrect solution. For any 3 students, there is at most 1 problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the 39 students. Find the smallest possible value of $B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $a$ for which the inequality \( \int_0^\infty e^{2a W_s^2} \, ds < \infty \) holds almost surely, where \( W_s \) is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immediately return and meet again at point C. Given that A walks 60 meters per minute and B walks 80 meters per minute, how far apart are points A and B?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x \] for all real numbers \( x \) and \( y \).

Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\sin x$ and $y=\sin (2x+ \frac {\pi}{3})$, determine the horizontal shift required to transform the graph of the function $y=\sin x$ into the graph of the function $y=\sin (2x+ \frac {\pi}{3})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $n$ is an odd number, find the remainder of $7^n+C_n^17^{n-1}+C_n^27^{n-2}+\ldots+C_n^{n-1}7$ when divided by 9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose \(X\) and \(Y\) have a joint density \(f_{X, Y}(x, y) = c\) for all \(x, y\) satisfying:
\(0<y<x<1\) or \(0<y<1-x\) and \(0<x<1\). Find the value of \(c\) such that the total probability is 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For an ordered  $10$ -tuple of nonnegative integers  $a_1,a_2,\ldots, a_{10}$ , we denote
    \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When  $i=1$ , we take  $a_1+a_2+\cdots+a_{i-1}$  to be  $0$ . Let  $N$  be the average of  $f(a_1,a_2,\ldots,a_{10})$  over all  $10$ -tuples of nonnegative integers  $a_1,a_2,\ldots, a_{10}$  satisfying
    \[a_1+a_2+\cdots+a_{10}=20.\]
    Compute the number of positive integer divisors of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1 = a_2 = \frac{1}{5}$, and for $n \ge 3$,
\[a_n = \frac{a_{n-1} + a_{n-2}}{1 + a_{n-1}a_{n-2}}.\]
Find the smallest integer $n$ such that $a_n > 1 - 5^{-2022}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the optimal constant \( C_{nr} \) such that for all matrices \( A, B \) in \( \mathbf{M}_n(\mathbb{C}) \), the inequality \( r([A,B]) \le C_{nr} r(A)r(B) \) holds, where \( r(A) \) is the numerical radius of \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Start with an angle of \(60^\circ\) and bisect it, then bisect the lower \(30^\circ\) angle, then the upper \(15^\circ\) angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original \(60^\circ\) angle into two angles. Find the measure (degrees) of the smaller angle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a hyperbola with one focus at $(0,-5)$ and the nearest directrix $d_1: x-3y-4=0$, and an eccentricity of $\sqrt{10}$, find the equation of the second directrix of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer solutions to the equation \(x^2 + 4xy = 10^{100}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of vertices for a polyhedron defined by the constraints given by the identity matrix and a vector with all entries equal to 0.5.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five months ago, Mike earned 10 times more money than Fred. At that time, Fred's salary was $1000, and he received a $500 bonus. Fred invested the entire amount (salary + bonus) in a fund that gave a 20% return over five months. Mike also got a 10% bonus on his salary and invested the entire amount (salary + bonus) in a fund that gave a 25% return over the same period. Now, Mike's salary has increased by 40%, and he is required to pay a 15% tax on his current salary. Calculate Mike's net salary (after tax) now.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sum function of the series $$\sum g_n$$ where $g_n(x)=\frac{(n+1)x}{(1+(n+1)^2x^2)}-\frac{nx}{[1+n^2x^2]}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit using the Central Limit Theorem:
\[ \lim_{n \to \infty} \frac{8^n}{27^n}\sum_{k=0}^n\binom{3n}{k}\frac{1}{2^k} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. What is the group key $K$ that can be shared by four individuals with keys $k_{1}=11, k_{2}=12, k_{3}=17, k_{4}=19$ using the modulus $p=1009$ and base $a=3$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the functions $f(x) = [x]^2 - [x+6]$ and $g(x) = 3kx^2 + 2x + 4(1-3k)$, where $[x]$ denotes the floor function. Let $A = \{x \mid f(x) = 0\}$. Determine the interval $[a, b]$ for $k$ such that every element of set $A$ satisfies the inequality $g(x) \ge 0$. Calculate the value of $6b - 3a$. \( \text{Find } 6b - 3a. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
We say that a positive integer $m$ covers the number $1998$ if the digits $1, 9, 9, 8$ appear in this order as digits of $m$. For instance, $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.

Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n \ge 5$), with all digits different from $0$. What is the remainder of $k(n)$ when divided by $8$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose you play the following game: You roll a fair die at most 60 times and get an amount of dollars equal to the last number rolled. You can decide to stop the game at any time. What is the (approximate) value of this game for N=60?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$[a]$ denotes the greatest integer less than or equal to $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{\mathrm{k}}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, find the largest positive integer $k$ not exceeding 48.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The negation of the proposition "$\forall x \geqslant 2, x^2 \geqslant 4$" is _______ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\exists x_0 \geqslant 2, x_0^2 < 4}


Context:
All the milk in container A, which was filled to its brim, was poured into three containers: B, C, and D. The quantity of milk in container B was 45% less than the capacity of container A, the quantity in container C was 12.5% more than the capacity of container A, and the quantity in container D was 20% less than the capacity of container A. If 150 liters was transferred from C to B and 100 liters from D to C, then all three containers, B, C, and D would have equal quantities of milk. What was the initial quantity of milk in container A?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). Calculate the reflex angle at \( G \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}$, where $a$ and $b$ are different four-digit positive integers (natural numbers) and $c$ is a five-digit positive integer (natural number). What is the number $c$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数 $f(x)=x \ln x-a x^{2}, a \in \mathbf{R}$，设函数 $F(x)=|f(x)|(x \in[1, \mathrm{e}])$ 有极小值,  $a$ 的取值范围是$\left(a, \frac{b}{\mathrm{e}}\right) \cup\left(\frac{b}{\mathrm{e}}, \frac{b}{c}\right)$，求$a+b+c$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
14. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns,
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\cdots \cdots$ & $a_{1 n}$ \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\cdots \cdots$ & $a_{2 n}$ \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\cdots \cdots$ & $a_{3 n}$ \\
$a_{41}$ & $a_{42}$ & $a_{43}$ & $a_{44}$ & $\cdots \cdots$ & $a_{4 n}$ \\
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots \cdots$ & $\cdots$ \\
$a_{n 1}$ & $a_{n 2}$ & $a_{n 3}$ & $a_{n 4}$ & $\cdots \cdots$ & $a_{n n}$
\end{tabular}

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+\cdots+a_{n n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ ABCD$  be an isosceles trapezoid with  $ \overline{AD}\parallel{}\overline{BC}$  whose angle at the longer base  $ \overline{AD}$  is  $ \dfrac{\pi}{3}$ . The diagonals have length  $ 10\sqrt {21}$ , and point  $ E$  is at distances  $ 10\sqrt {7}$  and  $ 30\sqrt {7}$  from vertices  $ A$  and  $ D$ , respectively. Let  $ F$  be the foot of the altitude from  $ C$  to  $ \overline{AD}$ . The distance  $ EF$  can be expressed in the form  $ m\sqrt {n}$ , where  $ m$  and  $ n$  are positive integers and  $ n$  is not divisible by the square of any prime. Find  $ m \plus{} n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered pairs of real numbers $(a,b)$ such that $(a+bi)^{2002} = a-bi$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
It is found through experiments under the same conditions that the germination rate of mung bean seeds at the Taihu Agricultural College is $97.1\%$. Please estimate the number of seeds that cannot germinate in about $1000$ pounds of mung bean seeds in the Taihu area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms. But if Anya instead gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went to pick mushrooms?
Please reason step by step, and put your final answer within \boxed{}.


Context:
New York recorded 5000 new coronavirus cases on a particular week. In the second week, some new coronaviruses cases were recorded by the state. In the third week, 2000 more cases were recorded in the state. The total number of recorded new coronaviruses in the state after the three weeks is 9500. What is the ratio of the number of new cases recorded in the second week to the number of new cases recorded in the first week?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of rectangle PRTV is divided into four rectangles, PQXW, QRSX, XSTU, and WXUV. Given that the area of PQXW is 9, the area of QRSX is 10, and the area of XSTU is 15, find the area of rectangle WXUV.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task B-1.4.

In each cell of a $4 \times 4$ table, a number is written. For each cell, the sum of the numbers in its adjacent cells is equal to the same natural number $x$ (two cells are adjacent if they share a side). Determine the number $x$ such that the sum of all numbers in the table is 282.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer  $n$ , define  $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer  $k$  for which  $7?9?=5?k?$ .

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


Context:
Divide the natural numbers from 1 to 8 into two groups such that the difference between the sum of the numbers in each group is equal to 16. How many different ways are there to do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a function $g(x)$ defined on the interval $I$, denote $g(I) = \{y | y = g(x), x \in I\}$. Given that the function $y = f(x)$, which has a domain of $[0, 3]$, has an inverse function $y = f^{-1}(x)$, and $f^{-1}([0, 1)) = [1, 2)$, $f^{-1}((2, 4]) = [0, 1)$. If the equation $f(x) - x = 0$ has a solution $x$, then $x =$ __.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a Cartesian coordinate system \( xOy \), there are 2018 circles \( \odot A_{1}, \odot A_{2}, \cdots, \odot A_{2018} \), where the center of \( \odot A_{k} \) (for \( k=1,2, \cdots, 2018 \)) is \( A_{k}\left(a_{k}, \frac{1}{4} a_{k}^{2}\right) \) and the radius is \( \frac{1}{4} a_{k}^{2} \). Given that \( a_{1}>a_{2}>\cdots>a_{2018}=\frac{1}{2018} \), and \( \odot A_{k+1} \) is externally tangent to \( \odot A_{k} \) (for \( k=1,2, \cdots, 2017 \)), find the value of \( a_{1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( x, y, z \) be three distinct natural numbers arranged in ascending order, and the sum of their reciprocals is an integer. Find \( x, y, z \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the irregular pyramid \(ABCD\), the sum of the plane angles at the vertex \(A\) is \(180^\circ\). Find the surface area of the pyramid if the area of the face \(BCD\) is \(s\) and \(AB = CD\), \(AD = BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that $\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.$ What is $k?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC$. Suppose that $BC = 27$, $CD = 25$, and $AP = 10$. If $MP = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carly is a pet groomer, and today her task was to trim the nails on dogs' paws. She trimmed 164 nails in total but was faced with some irregularities among the dogs. Three of the dogs had only three legs, two dogs had only three nails on one of their paws, and one dog had an extra nail on one paw, making it five nails on that particular paw. Given these conditions, how many dogs did Carly work on today?
Please reason step by step, and put your final answer within \boxed{}.


Context:
[ Angles between lines and planes ]  $\left[\begin{array}{l}\text { Equation of a plane } \\ {[\text { d }}\end{array}\right]$

Given points $A(1 ; 0 ; 1), B(-2 ; 2 ; 1), C(2 ; 0 ; 3)$ and $D(0 ; 4 ;-2)$. Find the angle between the line $A B$ and the plane $B C D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
I live in a five-story building with four entrances, with four apartments on each landing. What is the minimum number of questions you can ask to determine which apartment I live in, if I answer all your questions truthfully, but only with "yes" or "no"?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all integer triples \((x, y, z)\) such that \(x^3 + y^3 + z^3 - 3xyz = 2003\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
5 实数$a _ { 1 } ,$ $a _ { 2 } , \cdots , a _ { n } ( n > 3 )$满足：$a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \geq n ,$且$a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \cdots +$ $a _ { n } ^ { 2 } \geq n ^ { 2 }$.求$\max \{ a _ { 1 } ,$ $a _ { 2 } ,$,⋯,$a _ { n } \}$的最小值．（第28届美国数学奥林匹克试题）
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a convex quadrilateral such that $m (\widehat{ADB}) = 15^{\circ}$ and $m (\widehat{BCD}) = 90^{\circ}$. The diagonals of the quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC| = 4$, $|EA| = 8$, and $|EP| = 2$. If the answer is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the expression $\cos (-\frac{25}{3}\pi ) + \sin (\frac{25}{6}\pi ) - \tan (-\frac{25}{4}\pi )$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute $\operatorname{tg} 46^{\circ}$, using the value of the function $y=\operatorname{tg}x$ at $x=45^{\circ}$ and replacing its increment with a differential.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a complex number $z$ satisfies $|z - 2 + i| = 1$, find the minimum value of $|z + 1 - 2i|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the graph of the function $f(x,y) = xy$ over the unit circle $D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( x \) and \( y \) are two coprime positive integers, find the number of pairs \( (x, y) \) that satisfy the equation:

\[ x + \frac{2016}{x} = 32y + \frac{63}{y} \]

Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the ring $k[x,y]/(y-x^2,y^2-x^3)$. Find a polynomial representative of $\frac{1}{1-y}$ in this ring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $g'(x) \geq -5$ for $-2 \leq x \leq 3$ and $g(3) = 7$, what is the minimum possible value of $g(-2)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Johnny is out walking his two dogs at night, and his son joins him for the walk. Along the way, they are joined by their wheelchair-using friend and their service dog. How many legs' worth of organisms and wheels are traveling together for this walk?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(x, y,\) and \(z\) are real numbers satisfying \(x + y + z = 10\) and \(x^2 + y^2 + z^2 = 50\), find the maximum possible value of \((x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In \(\triangle ABC\), \(AB = BC\), and \(\overline{BD}\) is an altitude. Point \(E\) is on the extension of \(\overline{AC}\) such that \(BE = 10\). The values of \(\tan \angle CBE\), \(\tan \angle DBE\), and \(\tan \angle ABE\) form a geometric progression, and the values of \(\cot \angle DBE,\) \(\cot \angle CBE,\) \(\cot \angle DBC\) form an arithmetic progression. The area of \(\triangle ABC\) is in the form \(\frac{m}{n}\). Please calculate the value of \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lynne is tiling her long and narrow rectangular front hall. The hall is 2 tiles wide and 13 tiles long. She is going to use exactly 11 black tiles and exactly 15 white tiles. Determine the number of distinct ways of tiling the hall so that no two black tiles are adjacent (that is, share an edge).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Which natural numbers cannot be expressed as the sum of multiple consecutive natural numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
We inscribe spheres with a radius of \(\frac{1}{2}\) around the vertices of a cube with edge length 1. There are two spheres that touch each of these eight spheres. Calculate the difference in volume between these two spheres.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
When the polynomial \( x^{1951}-1 \) is divided by \( x^{4}+x^{3}+2x^{2}+x+1 \), a quotient and a remainder are obtained. Find the coefficient of \( x^{14} \) in the quotient.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carter has a 14-hour road trip. He wants to stop every certain number of hours to stretch his legs. He also wants to make 2 additional stops for food and 3 additional stops for gas. If each pit stop takes 20 minutes, his road trip will become 18 hours. How often does he want to stop to stretch his legs?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the least common multiple of three consecutive natural numbers is 660, find these three numbers. Express your answer as three specific numbers in ascending order.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle with radius \( R \) and center at point \( O \) is divided by points \( A, B, C, D, E, F \) into six equal parts. Determine the area of the figure \( COE \), bounded by the arc \( OC \) centered at point \( B \), the arc \( OE \) centered at point \( F \), and the arc \( CE \) centered at point \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Draw a line on a plane so that it is equidistant from three given points that are not collinear. How many such lines can be drawn?
Please reason step by step, and put your final answer within \boxed{}.