inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:

1. each chosen number is not divisible by 6, by 7 and by 8;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers 6, 7 or 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
The cross-section of a regular hexagonal pyramid SABCDEF is formed by a plane passing through vertex $C$ of the base $ABCDEF$ and parallel to the median $BM$ of the lateral face $SAB$ and the apothem $SN$ of the lateral face $SAF$. The side of the base of the pyramid is 2, and the distance from vertex $S$ to the cutting plane is 1. Find the cosine of the angle between the plane of the cross-section and the plane of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Evaluate the integral \(\int_\Gamma \frac{2z+j}{z^3(z^2+1)}\,\mathrm{d}z\) where \(\Gamma\) is the positively oriented circle \(|z-1-i| = 2\). Use residue theory to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Determine the conditions on \(\lambda\) such that the operator \(T: C([0,1]) \to C([0,1])\) defined by \((Tx)(t) = x(0) + \lambda \int_0^t x(\tau) \, d\tau\) is a contraction in the closed ball of radius \(\frac{1}{4}\) in \(C([0, 1])\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|\lambda| < 1}
A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m\geq 2, n\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{370}
Let $\|\cdot\|$ be a norm over $\mathbb{R}^2$ with $\|e_1\|=\|e_2\|=1$, where $e_1, e_2$ are the standard basis of $\mathbb{R}^2$. If $\|e_1+e_2\|=2$, find the range of possible values for $\|e_1-e_2\|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2]}
Let  $G$  be the centroid of triangle  $ABC$  with  $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from  $G$  to the three sides of the triangle.

Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.

*2019 CCA Math Bonanza Individual Round #11*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2348}{195}}
Determine the number of distinct ways to paint two identical regular tetrahedrons using the colors red, blue, green, and yellow, with each face of a tetrahedron painted a different color. Consider two colorings to be the same if one can be obtained from the other by rotation or reflection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
11. (China Mathematical Olympiad) Let $S=\{1,2,3, \cdots, 98\}$, find the smallest natural number $n$, such that in any $n$-element subset of $S$, one can always select 10 numbers, and no matter how these 10 numbers are divided into two groups, there is always a number in one group that is coprime with the other four numbers, and a number in the other group that is not coprime with the other four numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Consider the integer   $$ N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}. $$   Find the sum of the digits of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{342}
The Fibonacci sequence is defined as
$$
a_0 = 0, \; a_1 = a_2 = 1, \; a_{n+1} = a_n + a_{n-1} \; (n \geq 1).
$$

Find the greatest common divisor of the 1960th term and the 1988th term.
(Note: This is from the 29th International Mathematical Olympiad, 1988)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{317811}
Determine the first 1963 digits after the decimal point in the decimal expansion of \((\sqrt{26} + 5)^{1963}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{000\ldots000}
Grandma has 8 grandchildren and 4 different types of popsicles: 6 Vanilla, 6 Strawberry, 5 Banana, and 3 Chocolate. Each grandchild requests one popsicle of a particular flavor. What is the total number of different sets of requests that Grandma can fulfill?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57750}
A line \( l \) passes through the point \( P(3,0) \) with slope \( k \) and intersects the right branch of the hyperbola \( C: x^{2}-\frac{y^{2}}{3}=1 \) at points \( A \) and \( B \). The point \( F \) is the right focus of the hyperbola \( C \). Given that \( |AF| + |BF| = 16 \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 3}
Consider a $4 \times 4$ grid of squares, each of which are originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minutes it will take Piet to change the entire grid to blue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Sara makes a staircase out of toothpicks as shown:
[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } } [/asy]
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \leq f(b)$ whenever $a$ and $b$ are positive integers with $a \leq b$. (c) $f(2a)=f(a)+1$ for all positive integers $a$. How many possible values can the 2014-tuple $(f(1), f(2), \ldots, f(2014))$ take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1007}
Given an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, point $P$ is a point on the ellipse, and line segment $F_{1}P$ intersects the $y$-axis at point $Q$. If $|PQ|=2|QF_{1}|$ and triangle $\triangle PF_{1}F_{2}$ is an isosceles triangle, then the eccentricity of the ellipse is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3} - 1}{2}}
Calculate the rate of mass flow through the sphere defined by \( \rho = 1 \) in the outward direction, given the fluid density \( f(x, y, z) = x^{2} + 2y^{2} + z^{2} \) and velocity \( v(x, y, z) = x^{2}i + xy^{2}j + zk \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8\pi}{5}}
Find the limit of the Binomial probability distribution function evaluated at \(x = k\) as \(n\) approaches infinity, given that \(p = \sqrt{\lambda/n}\). Express your answer in terms of \(\lambda\) and \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the double integral: $$\int_{-\pi/2}^{\pi/2} \int_{2\cos\theta}^{2}2r\sqrt{4-r^2} \,dr\,d\theta$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{64}{9}}
The areas of the triangles formed by the diagonals of a trapezoid and its bases are $S_{1}$ and $S_{2}$.

Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{S_1} + \sqrt{S_2})^2}
Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular
$n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along
the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum
possible value of $d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{180}{mn}}
Determine the number of positive integers \( n < 10^4 \) such that \( 2^n - n^2 \) is divisible by 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2857}
Consider two independent, identically distributed (IID) standard normal random variables $X$ and $Y$. Given that $E(X|X+Y>0) = \frac{1}{\sqrt{\pi}}$, calculate $E(XY|X+Y>0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given two sets of points \(A = \left\{(x, y) \mid (x-3)^{2}+(y-4)^{2} \leqslant \left(\frac{5}{2}\right)^{2}\right\}\) and \(B = \left\{(x, y) \mid (x-4)^{2}+(y-5)^{2} > \left(\frac{5}{2}\right)^{2}\right\}\), the number of lattice points (i.e., points with integer coordinates) in the set \(A \cap B\) is ...
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the game of *Winners Make Zeros*, a pair of positive integers \((m,n)\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:

- If \(m \geq n\), the player chooses a positive integer \(c\) such that \(m-cn \geq 0\), and replaces \((m,n)\) with \((m-cn,n)\).
- If \(m < n\), the player chooses a positive integer \(c\) such that \(n-cm \geq 0\), and replaces \((m,n)\) with \((m,n-cm)\).

When \(m\) or \(n\) becomes \(0\), the game ends, and the last player to have moved is declared the winner. If \(m\) and \(n\) are originally \(2007777\) and \(2007\), find the largest choice the first player can make for \(c\) (on his first move) such that the first player has a winning strategy after that first move.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
Given that $\phi=\frac{1+\sqrt{5}}{2}$. Let $$n=\frac{1}{1}+\frac{1}{1+\phi}+\frac{1}{1+\phi+\phi^2}+\frac{1}{1+\phi+\phi^2+\phi^3}+\dots$$ Find the value of $\lfloor2n\rfloor+\lceil2n\rceil$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If 45% of z is 120% of y, y is 75% of the difference between x and the half of w, and z is 80% of 150% of w, what percent of x is the product of w and y minus one-third of z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.55\%}
Let $X, Y, Z$ be independent and identically distributed random variables, each following a standard normal distribution, $\mathcal{N}(0,1)$. Given that $X > Z$, find the conditional probability $\mathbb{P}(Y > Z \mid X > Z)$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!" 

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Max and Minnie each add up sets of three-digit positive integers. Each of them adds three different three-digit integers whose nine digits are all different. Max creates the largest possible sum. Minnie creates the smallest possible sum. What is the difference between Max's sum and Minnie's sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1845}
Given the parametric equations \( x = \cos^{-1} (8t^4 - 8t^2 + 1) \) and \( y = \sin^{-1} (3t - 4t^3) \), find \( \frac{dy}{dx} \) for \( 0 < t < \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
What is the minimum distance between  $(2019, 470)$  and  $(21a - 19b, 19b + 21a)$  for  $a, b \in Z$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{101}}
Determine the largest open ball around zero where the matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is a local homeomorphism or diffeomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Evaluate the infinite series: $$\sum_{k=1}^{\infty} \sum_{j=1, j \neq k}^{\infty} \sum_{i=1, i \neq j, i \neq k}^{\infty} \frac{1}{4^i4^j4^k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{945}}
Determine the cardinality of the set of all bijective functions from the set of natural numbers to itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Evaluate the limit: \[ \lim_{x \to 0} \frac{\tan^{12}x-x^{12}}{x^{14}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the minimum value of the function $f(x, y, z) = x^2 + y^2 + z^2$ subject to the constraints $x^2 + 2y^2 = 1$ and $xy + xz = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Given a random variable $X$ with probability density function (pdf) \( f_X(x) = \begin{cases} \frac{3}{x^4} & \text{if } x > 1 \\ 0 & \text{if } x \leq 1 \end{cases} \), find a function of a uniform random variable \( U \sim \text{Unif}(0,1) \) that has the same distribution as \( X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 - U)^{-1/3}}
Find the number of nondegenerate triangles whose vertices lie in the set of points $(s, t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, where $s$ and $t$ are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2148}
Determine the value of \( d \) for which a generalized Cantor set, constructed by retaining intervals of size \( d < 1 \) at both ends and removing an interval of size \( 1-2d \) at each stage, has Lebesgue measure zero. The process continues such that at the \( n^{th} \) stage, \( 2^{n-1} \) intervals of size \( 1-2d^n \) are removed. For which value of \( d \) does the following equation hold? 
\[ \sum_{n=1}^{\infty} 2^{n-1} (1-2d^n) = 1. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Find the vector \( \mathbf{s} = (s_1, s_2, s_3, s_4)^T \), where each \( s_i \in \{-1, 1\} \) and the number of 1's and -1's in \( \mathbf{s} \) is equal, that minimizes the maximum of the following correlation values:

\[ c_1 = \mathbf{s} \cdot (1, -1, 1, -1)^T \]
\[ c_2 = \mathbf{s} \cdot (-1, 1, -1, s_1) \]
\[ c_3 = \mathbf{s} \cdot (1, -1, s_1, s_2) \]
\[ c_4 = \mathbf{s} \cdot (-1, s_1, s_2, s_3) \]

Determine the vector \( \mathbf{s} \) that achieves this minimization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, -1, -1, 1)}
Find the rank of the image of the homomorphism $H_2(Q) \to H_2(\tilde{X})$ in the blow-up of $\mathbb{CP}^{3}$ at a planar nodal cubic curve, where $Q$ is the smooth quadric surface replacing the node in the blow-up.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Provide an example of a continuous function defined on the entire complex plane, \(\mathbb{C}\), that is not holomorphic at any point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = \overline{z}}
How many natural numbers \( x \) satisfy \( 3 \cdot 10^{13} < x < 10^{14} \) such that each digit from 1 to 9 appears exactly once in \( x \) and no zeros are adjacent to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35562240}
5. A sphere with radius $\frac{4}{9}$ lies inside a regular quadrilateral pyramid $SABCD$ with a base side of 8 and height 3. This sphere touches the base plane $ABCD$ of the pyramid and the lateral faces $SBC$ and $SCD$. The plane $\gamma$ touches the sphere, passes through point $B$, the midpoint $K$ of edge $CD$, and intersects edge $SC$ at point $M$. Find the volume of the pyramid $MBCK$. (20 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{192}{37}}
Minnie and Penny are cycling on the same road. Minnie's cycling speeds on flat road, downhill, and uphill are 20 km/h, 30 km/h, and 5 km/h respectively. Penny's cycling speeds on flat road, downhill, and uphill are 30 km/h, 40 km/h, and 10 km/h respectively. Minnie travels from Town A to Town B, cycling uphill for 10 km, then from Town B to Town C, cycling downhill for 15 km, and finally from Town C back to Town A on a flat road for 20 km. Penny travels the same entire distance but in a different route, starting from Town A, through Town C, then Town B, and back to Town A. Calculate the time difference in minutes between Minnie and Penny to complete the 45 km journey. Express your answer as a single numerical value in minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87.5}
Bob chooses a  $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If  $N$  is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of  $N$ . For example, if Bob’s string is  $0000$  and the stream of bits begins  $101000001 \dots$ , then  $N = 7$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a matrix \( A \in \mathbb{R}^{5 \times 5} \) and a parameter \( \alpha \in \mathbb{R} \), consider the linear system:
\[ Ax = \begin{pmatrix} 1 \\ -1 \\ \alpha^2 + \alpha \\ 1 \\ -1 \end{pmatrix} \]
The general solution to this system is:
\[ x = \begin{pmatrix} t+u+1 \\ s+t-u \\ s-t+1 \\ -s-t+u \\ -s-u-1 \end{pmatrix} \]
Additionally, the following vectors are in the column space of \( A \):
\[ \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ \alpha^2-3 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\ \alpha^2-2 \end{pmatrix} \]
Find the value of \( \alpha \) such that \( A^2 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Points \( M \) and \( K \) are taken on the sides \( CB \) and \( CD \) of the square \( ABCD \) such that the perimeter of triangle \( CMK \) is equal to twice the side length of the square. Find the measure of angle \( MAK \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Determine the number of inflection points of the function \((x-2)^6(x-3)^9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: \[ \lim_{n \to \infty} \int_0^1 \sqrt{\frac{1}{x} + n^2 x^{2n}} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Express the given data "$20$ nanoseconds" in scientific notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \times 10^{-8}}
Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other.
Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2020 \times 505$}
It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. Finally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, it follows that $E$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{AM - SH}{M - H}$

}
Determine the number of possible orientations for a smooth manifold with $n$ maximal connected components, considering both orientable and non-orientable cases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
$\begin{aligned}
 & \text{For certain pairs }(m,n)\text{ of positive integers with }m\geq n\text{ there are exactly }50\text{ distinct positive integers }k\mathrm{~such~that} \\
 & |\log m-\log k|<\log n.\text{ Find the sum of all possible values of the product }mn.
\end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{177}
In the symmetric group $S_{13}$, determine the number of permutations $\tau$ such that $\tau\alpha\tau^{-1}=\gamma$, where $\alpha = (1,2,4,8)(3,5,7,9,11,13)(6,12)$ and $\gamma = (1,2,3,4)(5,6,7,8,9,11)(12,13)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
What is the maximum possible dimension of the annihilator $W°$ of the subspace $W= \{A^{m}; m \geq 0\}$ of the vector space $V= M_{n}(R)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - 1}
Suppose $X$ and $Y$ are independent random variables with distribution Exp(2). Define $U=e^{2X}$ and $V=X-Y$. Calculate $E(U\mid\,V=2)$. \(\boxed{\text{Your answer here}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2e^4}
Compute the derivative of the distribution \( f_a(x) = \frac{H(x+a) - H(x-a)}{2a} \), where \( H(x) \) is the Heaviside step function, with respect to \( a \) in the sense of distributions:

\[ \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} \frac{ \left <f_{a+h} ,\phi \right > - \left <f_{a} ,\phi \right > }{h} \]

Then, evaluate the limit as \( a \to 0 \):

\[ \lim_{a\to 0} \frac{\partial f_a}{\partial a} \]

for a given test function \( \phi(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Farmer John has a (flexible) fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he call enclose an maximum possible area of $A_0$, and when he uses one of the walls or both walls, he gets a maximum of area of $A_1$ and $A_2$ respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4000
}
4. On each of the lines $x=0$ and $x=2$, there are 62 points with ordinates $1, 2, 3, \ldots, 62$. In how many ways can three points be chosen from the marked 124 points so that they form the vertices of a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["7908"]}
The leading coefficient of the quadratic trinomial \( f(x) \) is 1. All three coefficients, in some order, form a geometric progression with a common ratio \( q \). Find all possible values of \( q \) given that it is a rational number and the difference between the roots of \( f(x) \) is \( q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}