inclusionAI/Ring-lite-rl-data huggingface dataset:
Two distinct geometric progressions have first terms both equal to 1. The sum of their common ratios is 3, and the sum of their sixth terms is 573. Find the sum of the fifth terms of these progressions. If the answer is ambiguous, provide the sum of all possible values of the required quantity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^8 - 4a^6 + 6a^4 - 4a^2 + 1}
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665}
Let $U$ be a bounded open set in $\mathbb{R}^N$, and let $f_n: U \rightarrow \mathbb{R}$ be a sequence of positive Lebesgue-measurable functions such that $\int_U f_n \to 0$. Suppose $b: U \rightarrow \mathbb{R}$ is a continuous function with $0 < b < 1$ in $U$. Determine whether $\int_U b f_n \to 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A uniform ladder of mass $m$ and length $\mathcal{L}$ is resting on a wall. A man of mass $m$ climbs up the ladder and is in perfect equilibrium with the ladder when he is $\frac{2}{3}\mathcal{L}$ the way up the ladder. The ladder makes an angle of $ \theta = 30^\circ $ with the horizontal floor. If the coefficient of static friction between the ladder and the wall is the same as that between the ladder and the floor, which is $\mu$, what is $\mu$, expressed to the nearest thousandth? 

[i](Proposed by Ahaan Rungta)[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \mu \approx 0.678 }
The polynomial $x^{19} - x - 2$ has $19$ distinct roots. Let these roots be $a_1, a_2, ..., a_{19}$. Find $a^{37}_1 + a^{37}_2+...+a^{37}_{19}$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
An athlete with a mass of 78.75 kg is testing a net used by firefighters for rescuing people. The net sagged by 100 cm when the athlete jumped from a height of 15 meters. Assuming the net behaves like a spring, calculate how much it will sag when a person with a mass of 45 kg jumps from a height of 29 meters.

Given:

$m_1 = 78.75$ kg

$x_1 = 1$ m

$h_1 = 15$ m

$m_2 = 45$ kg

$h_2 = 29$ m
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.04 \, \text{m}}
Determine a non-trivial lower bound for the 'de Polignac constant' \( K_{Pol} = \sum_{i>0}2^{-k_{i}} \), where \( k_{i} \) is the \( i \)-th positive integer such that \( 2k_{i} \) is a Polignac number. A Polignac number is a number that is the difference of two consecutive primes in infinitely many ways. Is there a known non-trivial lower bound for \( K_{Pol} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{-123}}
Find all values of $n$ such that the sum of the digits of $2^n$ equals the sum of the digits of $5^n$. It is given that $n=3$ is a solution and that $3$ divides $n$ when considering the problem modulo $9$. Provide your answer as a set of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0, 3\}}
Determine whether the equation \( \lim\limits_{r \to \infty} [ (x - r/\sqrt{2})^2 + (y + r/\sqrt{2})^2 = r^2 ] \) converges to the line equation \( y = x \) as \( r \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = x}
Let $ABCD$ be a square of side length $2$. A semicircle is drawn with diameter $\overline{AC}$ that passes through point $B$. Find the area of the region inside the semicircle but outside the square.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi - 2}
# Problem 5. (3 points)

$A B C D$ is a cyclic quadrilateral. The extension of side $A B$ beyond point $B$ and the extension of side $C D$ beyond point $C$ intersect at point $P$. The extension of side $A D$ beyond point $D$ and the extension of side $B C$ beyond point $C$ intersect at point $Q$. It turns out that angles $B P C$ and $C Q D$ are equal. It is also known that $C Q=20, D Q=12, B P=3$. Find the square of the length of diagonal $A C$. If there are multiple possible answers, list them in ascending order separated by commas or semicolons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1040}
In an isosceles trapezoid, the angle at the base is $50^{\circ}$, and the angle between the diagonals, facing the lateral side, is $40^{\circ}$. Is the center of the circumscribed circle inside or outside the trapezoid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Outside}
A $3491$ by $3491$ square has its length decreased by $60$ and its width increased by $60$.  By how much does its area change?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3600}
Given two lines $l_1$ and $l_2$ with known vector equations, a point $P(x, y, z)$ in space, and a point $Q(x_1, y_1, z_1)$ lying on $l_1$, if it is specified that $PQ = PR$, where $R$ is a point on $l_2$ but its coordinates are unknown, is it possible to determine the coordinates of point $R$? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
在计算□82/8时，□里填多少，商是三位数．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine whether there is only one function \( f: \mathbb{Z}^+ \to \mathbb{Z}^+ \) that satisfies the equation \( f(n) + 2f(f(n)) = 3n + 5 \) for all positive integers \( n \), or if there are multiple functions that satisfy this relation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(n) = n + 1}
Fermat determines that since his final exam counts as two tests, he only needs to score a 28 on it for his test average to be 70 . If he gets a perfect 100 on the final exam, his average will be 88 . What is the lowest score Fermat can receive on his final and still hav an average of 80 ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
68
}
For all points $P$ in the coordinate plane, let $P'$ denote the reflection of $P$ across the line $y=x$. For example, if $P=(3,1)$, then $P'=(1,3)$. Define a function $f$ such that for all points $P$, $f(P)$ denotes the area of the triangle with vertices $(0,0)$, $P$, and $P'$. Determine the number of lattice points $Q$ in the first quadrant such that $f(Q)=8!$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84
}
In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that $1$ kilometer equals $m$ miles, and $1$ gallon equals $\ell$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles per gallon?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \frac{100\ell m}{x} $}
Determine the cardinality of the set \( C = \{g : \mathbb{Q} \to \mathbb{Q} \mid \forall x \in \mathbb{Q}, g(\lfloor x \rfloor) = g(x)\} \), where \(\lfloor x \rfloor\) is the floor function of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
In triangle $ABC$, point $K$ is chosen on the bisector $AL$ such that $\angle BKL = \angle KBL = 30^{\circ}$. Let $M$ be the intersection of lines $AB$ and $CK$, and $N$ be the intersection of lines $AC$ and $BK$. Find the measure of angle $\angle AMN$. *Proposed by D. Shiryaev, S. Berlov*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Determine if the density of the set of numbers expressible as \(a^2 + b^2 + c^2 - abc\) for positive integers \(a, b, c\) is exactly 50%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
Given matrices X of size 3x3 and Y of size 3x1, reformulate the machine equation into a linear system of equations where the unknown parameters are the entries of X and Y. Determine the minimum number of input/output pairs required to solve this linear system uniquely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the remainder when the number \(2^{2} + 222^{222} + 22222^{22222} + 2222222^{2222222} + 222222222^{222222222}\) is divided by 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let  $p$  be a prime number and let  $\mathbb{F}_p$  be the finite field with  $p$  elements. Consider an automorphism  $\tau$  of the polynomial ring  $\mathbb{F}_p[x]$  given by
\[\tau(f)(x)=f(x+1).\]
Let  $R$  denote the subring of  $\mathbb{F}_p[x]$  consisting of those polynomials  $f$  with  $\tau(f)=f$ . Find a polynomial  $g \in \mathbb{F}_p[x]$  such that  $\mathbb{F}_p[x]$  is a free module over  $R$  with basis  $g,\tau(g),\dots,\tau^{p-1}(g)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ g = x^{p-1} }
For some positive integers $n$, the numbers $2^n$ and $5^n$ have the same leading digit in their decimal representation. Find all such leading digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are 1955 points. What is the maximum number of triples that can be chosen such that each pair of triples has exactly one point in common?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{977}
Find the Hankel transform of the function \( \frac{1}{1+x_1^2+x_2^2} \) in two variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K_0(k)}
It is given that the roots of the polynomial  $P(z) = z^{2019} - 1$  can be written in the form  $z_k = x_k + iy_k$  for  $1\leq k\leq 2019$ .  Let  $Q$  denote the monic polynomial with roots equal to  $2x_k + iy_k$  for  $1\leq k\leq 2019$ .  Compute  $Q(-2)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-1 - 3^{2019}}{2^{2018}}}
What is the number of pairs of nonnegative integers $(x,y)$ which satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1998}$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that \(\lim_{n \to \infty} \sum^n_{k =0} \frac{nC_k}{n^k(k+3)} =ae+b\), find the value of \(a-b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For each integer $n \geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4 a_5 \cdots a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{962}
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least \( x \) times as large as the shorter piece is

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( \frac{2}{x + 1} \)}
Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter of radius 3 cm at the origin of the coordinate plane, and her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
The  $48$  faces of  $8$  unit cubes are painted white. What is the smallest number of these faces that can be repainted black so that it becomes impossible to arrange the  $8$  unit cubes into a two by two by two cube, each of whose  $6$  faces is totally white?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $n$ be a positive integer greater than $4$ such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. Find the sum of the digits of $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
30% 

}
A train passes a platform in 40 seconds, while it passes a man standing 5 meters from the edge of the platform in 20 seconds. If the initial speed of the train is 54 km/hr and it accelerates at a constant rate of 0.5 m/s^2, find the length of the platform.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{605}
Let \(X_1, X_2, \ldots, X_n\) be a sequence of i.i.d. random variables with the common characteristic function \(f(t) = e^{-t^2 - \sqrt{|t|}}\). Determine the value of \(\alpha\) such that \(\eta_n = n^{\frac{1}{\alpha}}\) and find the limiting distribution of \(\frac{X_1 + X_2 + \cdots + X_n}{\eta_n}\) as \(n \to \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $G$ be a group of order $p^n$, where $p$ is a prime number, and let $H$ be a subgroup of $G$ with order $p^{n-2}$ that is not normal in $G$. Determine whether the index $[G:N_G(H)]$ is equal to $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
Given the equations x/a + y/b + z/c = 1 and a/x + b/y + c/z = 0, find the value of the algebraic expression x^2/a^2 + y^2/b^2 + z^2/c^2. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given an affine variety $X$ with a known dimension, determine the maximum value of $m$ such that there exists a dominant morphism from $X$ to $\mathbb{A}_{k}^{m}$. Provide your answer as the maximum value of $m$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim X}
The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least 2 larger than an adjacent number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{3}}
Which function grows faster as $n$ becomes larger: $f(n)=100^{100^{100^n}}$ or $g(n)=2^{2^{2^{2^n}}}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(n)}
In the spectrum of a commutative ring \( A \) with unity, is the closure of the set \( \{p\} \), where \( p \) is a prime ideal in Spec \( A \), equal to \( V(p) \), the set of all prime ideals containing \( p \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V(p)}
Find the number of ordered 8-tuples $(a_7,a_6,\ldots,a_0)$ of nonnegative integers with $0 \le a_i \le 4$ for all $i$, such that the polynomial $a_7n^7 + a_6n^6 + a_5n^5 + a_4n^4 + a_3n^3 + a_2n^2 + a_1n + a_0$ is divisible by 5 for all integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Determine if there is a closed form for the integral \( I = \int_{-\infty}^{\infty} e^{-e^x-e^{-x}+x}~ dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 K_1(2)}
Evaluate the following limit using the correct approach:
$$\lim_{x\to 0} \left(\frac{\sec^2{x}}{x^2} - \cot^2{x}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
4. Let the polynomial $R(x)$ have a degree less than 4, and there exists a polynomial $P(x)$ such that
$$
\begin{array}{l}
7 \sin ^{31} t+8 \sin ^{13} t-5 \sin ^{5} t \cdot \cos ^{4} t-10 \sin ^{7} t+5 \sin ^{5} t-2 \\
\equiv P(\sin t)\left[\sin ^{4} t-(1+\sin t)\left(\cos ^{2} t-2\right)\right]+R(\sin t),
\end{array}
$$

where $t \in \mathbf{R}$. Try to find such an $R(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13x^{3}+5x^{2}+12x+3}
For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider an infinite set of paint colors, each assigned to a different positive integer. If the set of real numbers is colored with these paints, must there exist three distinct real numbers \(x, y, z\), painted with the same color, such that \(x \cdot y = z\)? Answer "yes" or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Determine if the Galois group of the polynomial \(x^6+x^5+2x^4-2x^3-x^2+1=0\) is solvable and find its order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
In the expansion of $\left( \sqrt[3]{x}+\frac{1}{\sqrt{x}} \right)^{20}$, the total number of rational terms in $x$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the expression \( n^{\aleph_0} \) for \( n \in \mathbb{N} \) and determine if it can be expressed as \( \aleph_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Find the gradient of the expression \( \| z - Zx \|^2_2 \) with respect to \( x \), where \( z \in \mathbb{R}^n \), \( Z \in \mathbb{R}^{n\times n} \), and \( x = [x_1,x_2,\ldots,x_n]^T \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2Z^\top (Zx - z)}
Find the smallest positive integer $n$ such that for all $x$ in the interval $\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]$, the inequality \(\left|\cos(x) - p_{n}(x)\right| < 10^{-k}\) holds, where $p_{n}(x)$ is the Taylor polynomial of degree $n$ of $\cos(x)$ about $a=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Islandland consists of ten islands, some of which are connected by two-way air routes. If you select any 9 islands, it is possible to fly around them one by one and return to the starting island. Find the minimum number of air routes that can exist in this country.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In the multiplication question, the sum of the digits in the four boxes is:

[Multiply $879 \times 492$ using long multiplication. Find the sum of the four numbers in the thousands place column.]



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12

}
Let $z_1, z_2, \ldots, z_{20}$ be the twenty (complex) roots of the equation:

$z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \cdots + 441 = 0$

Calculate the value of\[\cot{\left(\sum_{k = 1}^{20} \cot^{ - 1}{z_k}\right)}.\]


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{241}{220}$}
例5 对任意$k \in N _ { + }$，令f(k)表示k的各位数字的和的平方，且对于n≥2，令$f ^ { ( n ) } ( k ) = f ( f ^ { ( n - 1 ) } ( k ) )$，求$f ^ { ( 2 0 0 5 ) } ( 2 ^ { 2 0 0 2 } )$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Determine whether the set \( E' = \{x \in [0,1] : |x - \frac{p}{q}| < q^{-2} \text{ for finitely many } p,q \in \mathbb{N} \} \) has Lebesgue measure 0 or 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For a positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$. Determine all positive integers $n$ for which $d(n)$ is the second largest divisor of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$8,9,12$
}
Find all positive integers $n$ satisfying the following.
$$2^n-1 \text{ doesn't have a prime factor larger than } 7$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,2,3,4,6
}
Evaluate the limit as \( n \to \infty \):
\[ \lim_{n \to \infty} \sqrt[n]{n^{(4n)}+(4n)^n}\left[\left(2+\frac{1}{n^2}\right)^{18}-\left(4+\frac{4}{n^2}\right)^9\right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{589824}
Example 5 Let $n \geqslant 3$ be a positive integer, and $a_{1}, a_{2}, \cdots, a_{n}$ be any $n$ distinct real numbers with a positive sum. If a permutation $b_{1}, b_{2}, \cdots, b_{n}$ of these numbers satisfies: for any $k=1,2, \cdots, n$, $b_{1}+b_{2}+\cdots+b_{k}>0$, then this permutation is called good. Find the minimum number of good permutations. (2002


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)!}
Determine the number of distinct ways to number the faces of an icosahedron with numbers from 1 to 20, considering that the faces are indistinguishable due to the icosahedron's symmetries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20!}{120}}
If 2 / 5 of a number decreased by 3 times 1 / 4 of the number increased by 7 is 14, then the number is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-20}
In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged so that for any two villages, there is exactly one way to travel from one to the other without traveling along any road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. To maintain secrecy, he will not visit any village twice, and will not visit two consecutive villages that are directly connected by a road. How many villages can he be guaranteed to fly over? He can start from any village.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Determine whether the partial derivative of \( f(x, y)=\frac{x^3y^2}{x^6 +y^2} \) with respect to \( y \) exists at the point \( (x, y) = (0, 0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that $E[X^n] = n^2$, determine the value of $E[e^{2X}]$. Express your answer in terms of known constants or functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 6e^2}
Let $\{a_k\}^{2011}_{k=1}$ be the sequence of real numbers defined by$$a_1=0.201, \quad a_2=(0.2011)^{a_1},\quad a_3=(0.20101)^{a_2},\quad a_4=(0.201011)^{a_3},$$and more generally\[ a_k = \begin{cases}(0.\underbrace{20101\cdots0101}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is odd,} \\ (0.\underbrace{20101\cdots01011}_{k+2 \ \text{digits}})^{a_{k-1}}, &\text {if } k \text { is even.}\end{cases} \]

Rearranging the numbers in the sequence $\{a_k\}^{2011}_{k=1}$ in decreasing order produces a new sequence $\{b_k\}^{2011}_{k=1}$. What is the sum of all the integers $k$, $1\le k \le 2011$, such that $a_k = b_k$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1341}
For each integer \( 1 \leq n \leq 1000 \), let \( f(n) \) be the number of integers \( 1 \leq \mathcal{N}_0 \leq 5000 \) such that Anne has a winning strategy in the game described. How many values of \( n \) satisfy \( f(n) \geq 2520 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}