inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the conditions under which the function \( \frac{f(z)}{g(z)} \), where \( f(z)=\sum_{k=m}^\infty a_kz^k \) and \( g(z)=\sum_{k=p}^\infty b_kz^k \), is holomorphic in a neighborhood of 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq p}
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Find the number of $4 \times 4$ matrices $A$ such that $A^2$ equals a given matrix and $a_{11}, a_{33} > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P(x)\equiv 0}
Determine the radius of convergence of the series represented by the limit \( L_1 \) given below, for a positive integer \( k \):
\[ L_1=\lim_{n\rightarrow\infty} \left|\sqrt[n]{\frac{(n!)^k}{(kn)!}}\right| \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^k}
Petya cut three $2 \times 2$ squares from an $8 \times 8$ checkered square as shown in the figure. Olya wants to cut as many $1 \times 3$ rectangles as possible from the remaining part of the square (following the grid lines). What is the maximum number of such rectangles she can cut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Restore the digits. On the board, the product of three consecutive even numbers was written. During the break, Vasya erased some of the digits. As a result, the board shows $87*****8$. Help Petya find the missing digits in the product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87526608}
Let \( f \) be a function that takes in a triple of integers and outputs a real number. Suppose that \( f \) satisfies the equations
\[ 
f(a, b, c) = \frac{f(a+1, b, c) + f(a-1, b, c)}{2} 
\]
\[ 
f(a, b, c) = \frac{f(a, b+1, c) + f(a, b-1, c)}{2} 
\]
\[ 
f(a, b, c) = \frac{f(a, b, c+1) + f(a, b, c-1)}{2} 
\]
for all integers \( a, b, c \). What is the minimum number of triples at which we need to evaluate \( f \) in order to know its value everywhere?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the limit of the sum of the areas of triangles formed by a line segment of length 1, with a ratio of 0.9, as the number of triangles approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{38}}
Determine the value of $b$ ($b \neq a$) such that the causal linear shift-invariant system described by the difference equation \( y(n) - ay(n-1) = x(n) - bx(n-1) \) is an allpass system, meaning the magnitude of its frequency response is constant, independent of frequency.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{a}}
Calculate the variance of the expression \((W_1W_2 - Z_1Z_2)^2\), where \(W_1, W_2, Z_1,\) and \(Z_2\) are independent random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
An infinite sequence of positive integers $a_1, a_2, \ldots$ is defined such that for $k \geq 2$, the $k^\text{th}$ element is equal to $k$ plus the product of the first $k-1$ elements of the sequence. Given that $a_1 = 1$, determine the smallest prime number that does not divide $a_{2010}$. \( \text{Express your answer as a single prime number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $p$ be a prime number and consider a group $G$ with order $p^4$ such that the center $Z(G)$ has order $p^2$. Determine the number of conjugacy classes of $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^3 + p^2 - p}
Calculate the double integral \( \int_{\mathbb{R}}\left(\int_{\mathbb{R}} f(x,y) \, dx\right)dy \) for the function \( f(x,y) \) defined as follows:

\[ f(x,y) = \begin{cases} 
      1 & x\geq 0, \ x\leq y\leq 1+x \\
      -1 & x\geq 0, \ 1+x\leq y\leq 2+x \\
      0 & \text{otherwise}
   \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the expected value of the following limit:
\[ \lim_{N \to \infty} \left( \prod_{i = 1}^{N} X_{i} \right)^{\dfrac{1}{N}} \]
where each random variable $X_i$ for $i = 1, \ldots, N$ is an independently and identically distributed random integer from 1 to 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.147}
Determine the conditions on the coefficients \(a\), \(b\), and \(c\) such that the integral \(\int \frac{ax^2+bx+c}{x^3(x-1)^2} \, dx\) represents a rational function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + 2b + 3c = 0}
Evaluate the integral \( \int_C (x - y)(dx + dy) \) using Green's Theorem, where \( C \) is the semicircular part of \( x^2 + y^2 = 4 \) above the line \( y = x \), from \((-\sqrt{2}, -\sqrt{2})\) to \((\sqrt{2}, \sqrt{2})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
Suppose you have  $27$  identical unit cubes colored such that  $3$  faces adjacent to a vertex are red and the other  $3$  are colored blue. Suppose further that you assemble these  $27$  cubes randomly into a larger cube with  $3$  cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as  $\frac{1}{2^n}$  for some positive integer  $n$ . Find  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
In how many ways can \((n+1)\) people be seated at two indistinguishable circular tables, ensuring that both tables have at least one person? Express your answer in terms of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n! \sum_{k=1}^{n} \frac{1}{k}}
Determine the number of distinct ways to insert $k$ vertices along the edges of a polygon with $n$ sides, considering that the order of insertion matters and vertices can be placed anywhere on the edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^k}
Given a sequence of $n$ independent and identically distributed random variables $X_1, X_2, \ldots, X_n$, each following an exponential distribution with rate parameter $\lambda = 1$, let $M_n = \max(X_1, X_2, \ldots, X_n)$. The cumulative distribution function of $M_n$ is $F_{M_n}(x) = (1 - e^{-x})^n$ for $x \geq 0$. Calculate the expected value of $e^{M_n}$, i.e., $\mathbb{E}(e^{M_n})$. \( \text{Express your answer in terms of } n. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In 1740, Knowall discovered a comet and calculated that it would be visible to humans in 1823, 1906, and so on, with a recurrence every 83 years. From now until the year 3000 AD, the number of times humans can see this comet is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Compute $$\sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{137}{24}-8 \ln 2}
Determine the range of values of \(\alpha\) for which the function \(f(x) = |x|^{\alpha}\) belongs to the Sobolev space \(H^1(B(0,1))\) in \(\mathbb{R}^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \geq 0}
The tangent of the angle between the median and the altitude drawn to the side of an isosceles triangle is equal to $\frac{1}{2}$. Find the sine of the vertex angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Find the limit of the sequence \( b_n = \sqrt[n]{\sum_{k=1}^{n}{\left(1 + \frac{1}{k}\right)^k}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the set $X=\{a,b,c,d\}$ and the topology $\tau=\{\emptyset, \{a,b\}, \{a,b,c\}, \{c,d\}, X\}$, find the connected component $C(a)=\bigcup \{A \mid a \in A , A \subset X, \text{$A$ connected}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{a, b\}}
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college, one-third of the freshman class drops out of school, 40 students in the sophomore class drop out of school, and one-tenth of the junior class drops out of school. Given that the college only admits new freshman students and wants to begin each school year with 3400 students enrolled, how many students does it need to admit into the freshman class each year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1185}
Given is a chessboard 8x8. We have to place  $n$  black queens and  $n$  white queens, so that no two queens attack. Find the maximal possible  $n$ . 
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a non-constant sequence {a_i} satisfying a_{i+1}^2 - a_i a_{i+1} + a_i^2 = 0, with boundary conditions a_1 = a_{n+1} = 1, find the value of the sum ∑_{i=0}^{n-1} a_i. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Six different digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. Calculate the sum of the six digits used.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying
$\forall x,y\in\mathbb{R}: \begin{cases} f(x+y)=f(x)f(y), \\ f(x)=xg(x)+1 & \end{cases}\quad\mathrm{and}\quad\lim_{x\to0}g(x)=1.$
Find the derivative of $f$ in an arbitrary point $x.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f'(x)=e^x}
A and B play a game where each has a 50% chance of winning any given game. The game continues until one player wins n consecutive games. What is the expected total number of games played until this occurs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
Let \( n \) be a natural number. A volleyball team consisting of \( n \) women and \( n \) men lines up for a game. Each team member occupies one of the positions \( 1, 2, \ldots, 2n \), whereby exactly positions \( 1 \) and \( n+1 \) are outside the court. During the game, all team members rotate, switching from position \( i \) to position \( i+1 \) (or from \( 2n \) to \( 1 \). How many possibilities are there for the starting line-up so that there are always at least \( n-1 \) women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n!)^2 \cdot 2^n}
Evaluate the limit \( \lim_{x\to0}{\frac{x^2+2\ln(\cos x)}{x^4}} \) without using L'Hopital's rule or Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex, the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{245}
Given that the sequence $\{a_n\}$ is an increasing sequence, and for any $n \in \mathbb{N}^*$, $a_n = 2n^2 + kn$ always holds, then the range of values for the real number $k$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-6, +\infty)}
Compute the residue of the function \( f(z) = e^{z + \frac{1}{z}} \) at \( z = 0 \) using the definition of a residue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_1(2)}
Evaluate the double integral of the function \( f(x,y) = x^2 - y^2 \) over the region \( D \) bounded by the curves \( y = x \), \( y = x + 1 \), \( xy = 1 \), and \( xy = 2 \) in the first quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Consider the quadratic function f(x) = ax^2 + bx + c, where a, b, and c are integers and x is a real number. Let p be a positive even integer with a positive units digit, such that 10 ≤ p ≤ 50. If the units digit of p ^ 3 minus the units digit of p ^ 2 is equal to 0, f(p^2) is equal to f(p^3), and f(p) = 0, what is the units digit of p + 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the minimal polynomial of the linear map $T_P: M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ defined by $T_P(X) = P^{-1}XP$, where $P = \begin{pmatrix} i & 2 \\ -1 & -i \end{pmatrix}$ and $P^2 + 3I = 0$. Provide your answer as a polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - 1}
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts.  Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group.  Each of them has the same number of internet friends.  In how many different ways can this happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{170}
Zsspasii A.A.

The number of the current Olympiad (70) is formed by the last digits of the year of its holding, written in reverse order.

How many more times will such a situation occur in this millennium?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Convert the equation $2x(x-1)=3x$ into the general form of a quadratic equation, $ax^2 + bx + c = 0$, and provide the coefficients $a$, $b$, and $c$ in that order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, -5, 0)}
Evaluate the limit: $$ \lim_{n \to \infty} \frac{\left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2}\right)^n }{ 2 \left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 + 1}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2 (n^2 +1)}\right)^n } $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Evaluate the expression \( \sum\limits_{n=1}^{1009} \gcd(2n, 2020) - \sum\limits_{n=0}^{1008} \gcd(2n+1, 2019) \) without using calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6481}
Consider the set of complex numbers z satisfying $|1+z+z^{2}|=4$. The largest possible value for the imaginary part of z can be written as $\frac{\sqrt{m}}{n}$, where m and n are positive integers and m is not divisible by the square of any prime. What is m+n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$21$}
For a Brownian motion $W_t\sim\mathcal{N}(0,t)$, determine the values of $c$ and $d$ such that $cW_2+W_4$ and $W_5-dW_7$ are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 1}
Calculate the integral \( \int_T db \wedge dt \) in \( \mathbb{R}^5 \) with coordinates \((p_1,q_1,p_2,q_2,t)\), where \( db = dp_1 \wedge dp_2 \) and \( T = \{ t \in [0,1], p_1^2 + q_1^2 = p_2^2 + q_2^2 \leq t^2 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers  $n < 100$  going through and "flipping" every  $n$ th locker. How many lockers are open after this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.

[i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2n-1)!!}
Determine the Hausdorff dimension of the set $A$ of non-recurrent walks on $\mathbb{Z}$, where walks are identified with infinite binary sequences in the Cantor space $2^\mathbb{N}$. A non-recurrent walk is defined as a sequence $x$ for which $\frac{\sum_{k=1}^n x(k)}{n} = \frac{1}{2}$ for only finitely many $n$. What is the Hausdorff dimension of this set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that 
$\frac{x^2}{a^2} + \frac{y^2}{a^2-16} = \frac{(x-20)^2}{b^2-1} + \frac{(y-11)^2}{b^2} = 1.$ 
 Find the least possible value of $a+b.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A stock investment increased by $15\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.04\%}
Arrange three identical black balls and three identical white balls from left to right into a row. A "valid arrangement" is defined as an arrangement where, starting from any position (including this position) to the right, up to the last ball, the number of black balls is greater than or equal to the number of white balls. Calculate the probability of having a "valid arrangement". Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Determine the order of convergence of the modified Newton-Raphson method given by the iteration formula: \[ x_{n+1} = x_n - m \frac{f(x_n)}{f'(x_n)} \] where \( m \) is the multiplicity of the root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the dimension of the vector space \( \mathcal{L}_F(K) \) over \( K \), where \( K/F \) is a finite-dimensional field extension with \([K:F]=n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the limit of the function \( \cos\left(e^{x^{1/3}+\sin x}\right) \) as \( x \to \pm \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the set of discontinuities \( D_f \) for the function \( f: \mathbb{R} \to \mathbb{R} \) defined by:
\[
f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \frac{1}{n} & x=r_n \in \mathbb{Q} \end{cases}
\] where \( \mathbb{Q} = \{r_1, r_2, \ldots\} \) is the set of rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}}
$\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. Find the angle $\angle EDF$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
${66^\circ} $}
Initially, a natural number  $n$  is written on the blackboard. Then, at each minute, *Neymar* chooses a divisor  $d>1$  of  $n$ , erases  $n$ , and writes  $n+d$ . If the initial number on the board is  $2022$ , what is the largest composite number that *Neymar* will never be able to write on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2033 }
Company P had 15 percent more employees in December than it had in January. If Company P had 426.09 employees in January, how many employees did it have in December?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
Determine the optimal reserve price \( r \) that maximizes the seller's expected revenue in a second-price sealed-bid auction with two buyers, where each buyer's valuation is independently drawn from \( \{0, 1, 2\} \) with equal probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
给定点 $A(-2,2)$, 已知 $B$ 是椭圆 $\frac{x^2}{25}+\frac{y^2}{16}=1$ 的动点, $F$ 是左焦点, 当 $|A B|+\frac{5}{3}|B F|$ 取最小值时, 则点 $B$ 的坐标 $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left(-\frac{5}{2} \sqrt{3}, 2\right)$}
Given the matrix \(K_{3\times3} = \begin{bmatrix} 1&1&1 \\ 3&2&1 \\ 1&2&1 \end{bmatrix}\) with \(|K| = 2\), find the determinant of the matrix expression \(|2K^3 - 2K^4|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-384}
Let $Q$ be a point on or inside the boundary of a square $ABCD$. Determine the minimum value of the sum $\angle QAB + \angle QBC + \angle QCD + \angle QDA$. Provide your answer in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
In a circle, $AB$ is a diameter, and $CD$ is a chord parallel to $AB$. Point $E$ is the intersection of lines $AC$ and $BD$, and $\angle AED = \beta$, but now $\beta$ is such that $\angle AED + \angle BED = 180^\circ - 2\beta$. The ratio of the area of triangle $\triangle CDE$ to that of triangle $\triangle ABE$ is required.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos^2 \beta}
Mary chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340}
A restricted path of length $n$ is a path of length $n$ such that for all $i$ between 1 and $n-2$ inclusive, if the $i$th step is upward, the $i+1$st step must be rightward. Find the number of restricted paths that start at $(0,0)$ and end at $(7,3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
A triangular piece of paper with an area of 1 is folded along a line parallel to one of its sides and pressed flat. What is the minimum possible area of the resulting figure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}