inclusionAI/Ring-lite-rl-data huggingface dataset:
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{}.
Answer: \boxed{\dfrac{4}{5}}
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{}.
Answer: \boxed{70
}
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{}.
Answer: \boxed{5}
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{}.
Answer: \boxed{-14\pi}
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{}.
Answer: \boxed{47}
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{}.
Answer: \boxed{600}
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{}.
Answer: \boxed{-3}
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{}.
Answer: \boxed{2 - \sqrt{2}}
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{}.
Answer: \boxed{3432}
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{}.
Answer: \boxed{2}
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{}.
Answer: \boxed{$181$}
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{}.
Answer: \boxed{10800}
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{}.
Answer: \boxed{0}
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{}.
Answer: \boxed{16}
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{}.
Answer: \boxed{\text{isosceles right triangle}}
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{}.
Answer: \boxed{5}
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{}.
Answer: \boxed{\$5}
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{}.
Answer: \boxed{153
}
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{}.
Answer: \boxed{5}
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{}.
Answer: \boxed{
$\frac{1}{6}$}
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{}.
Answer: \boxed{Sphere}
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{}.
Answer: \boxed{10}
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{}.
Answer: \boxed{16}
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{}.
Answer: \boxed{\dfrac{3025}{2}}
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{}.
Answer: \boxed{277.1}
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{}.
Answer: \boxed{O(h^2)}
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{}.
Answer: \boxed{19}
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{}.
Answer: \boxed{ 2n - 1 }
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{}.
Answer: \boxed{(0, 2)}
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{}.
Answer: \boxed{19}
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{}.
Answer: \boxed{-\dfrac{3}{14}}
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{}.
Answer: \boxed{2 r^2 (2\sqrt{3} + 3)}
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{}.
Answer: \boxed{(1,2)}
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{}.
Answer: \boxed{100}
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{}.
Answer: \boxed{3 - 2\sqrt{2}}
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{}.
Answer: \boxed{4}
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{}.
Answer: \boxed{8\sqrt{2}}
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{}.
Answer: \boxed{2}
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{}.
Answer: \boxed{\frac{1}{1000001}}
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{}.
Answer: \boxed{64/27}
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{}.
Answer: \boxed{-1}
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{}.
Answer: \boxed{12\sqrt{2} + 12}
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{}.
Answer: \boxed{0}
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{}.
Answer: \boxed{0.4}
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{}.
Answer: \boxed{f(-x1) > f(-x2)}
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{}.
Answer: \boxed{1:27}
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{}.
Answer: \boxed{3}
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{}.
Answer: \boxed{\dfrac{1}{3}}
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{}.
Answer: \boxed{4}
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{}.
Answer: \boxed{1512}
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{}.
Answer: \boxed{2}
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{}.
Answer: \boxed{2}
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{}.
Answer: \boxed{\frac{2}{3}}
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{}.
Answer: \boxed{13}
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{}.
Answer: \boxed{36}
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{}.
Answer: \boxed{20100}
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{}.
Answer: \boxed{1018}
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{}.
Answer: \boxed{0}
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{}.
Answer: \boxed{409529}
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{}.
Answer: \boxed{4:3}
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{}.
Answer: \boxed{22}
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{}.
Answer: \boxed{\dfrac{1}{6}}
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{}.
Answer: \boxed{0.04}
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{}.
Answer: \boxed{2}
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{}.
Answer: \boxed{\dfrac{\pi}{4}}
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{}.
Answer: \boxed{12}
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{}.
Answer: \boxed{672.5 + 31.5\sqrt{17}}
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{}.
Answer: \boxed{273}
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{}.
Answer: \boxed{$16$}
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{}.
Answer: \boxed{2}
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{}.
Answer: \boxed{15}
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{}.
Answer: \boxed{10}