inclusionAI/Ring-lite-rl-data huggingface dataset:
What is the greatest number of natural numbers, not exceeding 2016, that can be selected such that the product of any two selected numbers is a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
for any integer k > 1 , the term “ length of an integer ” refers to the number of positive prime factors , not necessarily distinct , whose product is equal to k . for example , if k = 24 , the length of k is equal to 4 , since 24 = 2 × 2 × 2 × 3 . if x, y, and z are positive integers such that x > 1 , y > 1, z > 1, and x is a product of two distinct prime numbers, y is a product of three distinct prime numbers, and x + 3y + 5z < 5000 , what is the maximum possible sum of the length of x, the length of y, and the length of z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the constant \( k \) such that for any sequence of positive numbers \( a_i \), the inequality \[ \sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 + \ldots + a_n} \leq k \sum_{n=1}^{\infty} \frac{1}{a_n} \] holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设 a 是正整数，数列 \{b_n\} 定义为 b_n=\left(1+\df{1}{n}\right)^{2024n+a}. 若 \{b_n\} 单调递减，则 a 的最小值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Given that the probability of drawing a first-class product is 0.65, the probability of drawing a second-class product is 0.2, and the probability of drawing a third-class product is 0.1, calculate the probability of drawing a non-first-class product. Express your answer as a decimal value between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.35}
Calculate the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 3 = 0,\] given that all roots are real and nonnegative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
$ 500 is divided amongst a, b, c, d, and e so that a may get 5 / 7 as much as b, c, d, and e together, b may get 11 / 16 as much as a, c, and d together, c may get 3 / 8 as much as a, b, and e together, and d may get 7 / 12 as much as a, b, and c together. Find the share of a.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$208.33}
Let $V$ be an infinite-dimensional vector space over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. If the dimension of $V$ is $\alpha$, a cardinal number, determine the dimension of $V^*$, the algebraic dual of $V$. Express your answer in terms of $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^\alpha}
A BWM tree is defined recursively:

- An empty tree is a BWM tree of height 0 and size 0.
- A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest.
- The height of a nonempty BWM tree is one more than the height of its tallest subtree, and the size of a nonempty BWM tree is one more than the sum of the sizes of the subtrees.

What is the minimum size of a height-10 BWM tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{154}
Find the scalar value of \( a \) such that the vector set \( \{u_1+au_2, u_2+au_3, u_3+au_4, u_4+au_1\} \) is linearly independent, given that the vector set \( \{u_1, u_2, u_3, u_4\} \) is linearly independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq \pm 1}
$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{900}{109}$}
Let \( X \sim \text{Poisson}(\lambda) \), where \( \lambda > 0 \) is unknown. If \( \delta(X) \) is the unbiased estimator of \( g(\lambda)=e^{-\lambda}(3\lambda^2+2\lambda+1) \), find the value of \( \sum_{k=0}^\infty \delta(k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
There are 10 ping pong players participating in a round-robin tournament. The results show that there are no draws, and among any group of 5 players, there is one player who defeats the other 4 and one player who loses to the other 4. How many players have exactly two wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of solutions, counting multiplicity, in the domain \(\{z \in \mathbb{C} : 1 < |z| < 2\}\) for the equation:
\[z^9 + z^5 - 8z^3 + 2z + 1 = 0\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a triangle ΔABC with a fixed side length, determine the number of inscribed equilateral triangles that can be drawn with vertices on different sides of ΔABC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $k$ be a positive integer. A sequence $a_0, a_1, \ldots, a_n$, where $n > 0$, of positive integers satisfies the following conditions:

1. $a_0 = a_n = 1$;
2. $2 \leq a_i \leq k$ for each $i = 1, 2, \ldots, n-1$;
3. For each $j = 2, 3, \ldots, k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0, a_1, \ldots, a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
4. For any $i = 1, 2, \ldots, n-1$, $\gcd(a_i, a_{i-1}) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.

Suppose there is another sequence $b_0, b_1, \ldots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} > \frac{b_i}{a_i}$ for all $i = 0, 1, \ldots, n-1$. Find the minimum value of $b_n - b_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In an isosceles triangle \(ABC\), the angle at the base \(AC\) is \(\alpha\). The circle inscribed in this triangle touches the sides of the triangle at points \(A_1\), \(B_1\), and \(C_1\). Find the ratio of the area of triangle \(A_1B_1C_1\) to the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos \alpha (1 - \cos \alpha)}
9.4. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{3-\sqrt{2}} \sin x+1) \cdot(3+2 \sqrt{7-\sqrt{2}} \cos y-\cos 2 y)$. If the answer is not an integer, round it to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
将时钟盘面上标有数字 $1,2, \cdots, 12$ 的十二个点分别染红、黄、蓝、绿四种颜色，每色三个点，以这些点为顶点作 $n$ 个凸四边形，使其满足：  
$(1)$每个四边形的四个顶点四色都有；  
$(2)$对于其中任何三个四边形，都存在某一色，染有该色的三个顶点所标数字互不相同。  
求 $n$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In an Abelian category $\mathscr{A}$, determine the category in which the Hom functor $\mathrm{Hom}_\mathscr{A}(A,-)$ takes values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{Ab}}
Shapovalov A.B.

Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them gets nine handfuls, after which the other takes all the remaining coins (the division may also end if the coins are divided before anyone gets nine handfuls). Hapok can grab any number of coins in a handful. What is the maximum number of coins he can guarantee himself regardless of Glazok's actions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Determine the cardinality of the set of all surjective functions from \( \mathbb{N} \) to \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the integral of \( f(x) = \frac{1}{1 + \tan(\sqrt{2} x)} \) over the interval from 0 to \(\pi/2\). In other words, find \(\int_0^{\pi/2} f(x) \, dx\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$.

Note: The circumcircle of a triangle is the circle that passes through its three vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$R = \dfrac{24}{7}$
}
When Jeffrey walks, for every three steps forward, he takes two steps backwards. However, after he covers the first 30 steps, he changes his pattern and starts taking only one step backwards for every four steps forward. If the distance between the house and the mailbox is 66 steps, what is the total number of steps Jeffrey takes when he goes from the house to the mailbox, considering the additional constraint and the change in his stepping pattern after the first 30 steps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x < 2010$ such that $x^2 - n$ is divisible by $2010$. Determine the remainder when $\sum_{n = 0}^{2009} n \cdot \theta(n)$ is divided by $2010$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{335}
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$.
Find $ \lim\limits_{n\to\infty}(a_0+a_1+\ldots+a_n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{

\[ \frac{1}{3} \]}
The function  $g\left(x\right)$  is defined as  $\sqrt{\dfrac{x}{2}}$  for all positive  $x$ .  $ $   $$ g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right) $$    $ $ can be expressed as  $\cos(b)$  using degrees, where  $0^\circ < b < 90^\circ$  and  $b = p/q$  for some relatively prime positive integers  $p, q$ . Find  $p+q$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( n = p_1^{k_1} + p_2^{k_2} + \ldots + p_m^{k_m} \), where \( p_1, p_2, \ldots, p_m \) are distinct prime numbers and \( k_1, k_2, \ldots, k_m \) are positive integers. Determine the number of ways \( n \) can be expressed as a product of two positive integers that have no common factors, considering that the order of the factors matters (i.e., \( a \cdot b \) and \( b \cdot a \) are regarded as different).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^m}
A general population follows an exponential distribution with parameter \( \lambda \). A sample of six elements was drawn: \( 3.2, 2.5, 0.1, 5.1, 4.7, 4.1 \). Using the most-powerful test, verify the null hypothesis that \( \lambda = \frac{1}{2} \) against the alternative hypothesis that \( \lambda = \frac{1}{3} \). Assume a significance level of \( \alpha = 0.05 \). Determine the critical value \( c_2 \) for the test statistic \( \sum_{i=1}^{n} x_{i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.026}
In the three phrases "尽心尽力", "力可拔山", and "山穷水尽", each Chinese character represents a digit between 1 and 8. Identical characters represent the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each phrase is 19, and "尽" > "山" > "力", then the maximum value for "水" is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the expectation of the falling factorial of a geometrically distributed random variable $X$ with success probability $p$, denoted by $E[(X)_3]$. The probability generating function of $X$ is $G(z) = \sum_{k=1}^{\infty} kp(1-p)^{k-1}z^k$, where $X$ represents the number of trials needed for the first success.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6(1-p)^2}{p^3}}
A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \_\_\_\_\_\_ (answer in numbers).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Compute\[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\](Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101\cdot 2^{49}-50}
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
If $e^{i \theta} = \frac{3 + i \sqrt{8}}{4},$ then find $\sin 6 \theta.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{855 \sqrt{2}}{1024}}
Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Point $P$ is the unique point in the plane of $\triangle ABC$ such that $\triangle PBM \sim \triangle PNC$. What is $AP$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Find the following limit.
 \( \mathop{\lim }\limits_{{n \rightarrow  \infty }}\frac{1}{n}\log \)  \( \left( {\mathop{\sum }\limits_{{k = 2}}^{{2}^{n}}{k}^{1/{n}^{2}}}\right) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(\ln 2\).

}
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $|d - \sqrt{n}|$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
## Problem Statement

Find the indefinite integral:

$$
\int \frac{\sqrt[3]{1+\sqrt[5]{x}}}{x \sqrt[15]{x^{4}}} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{15(1+\sqrt[5]{x})^{4/3}}{4x^{4/15}}+C}
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the angles $A$, $B$, and $C$ respectively. If $\dfrac {a\sin A + b\sin B - c\sin C}{a\sin B} = 2\sin C$, find the size of $\angle C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
For any positive integer n, let d(n) denote the number of positive divisors of n; and let φ(n) denote the number of elements from the set {1, 2, · · · , n} that are coprime to n. (For example, d(12) = 6 and φ(12) = 4.) Find the smallest positive integer n such that d(φ(n)) = 2017.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^2017}
Ben "One Hunna Dolla" Franklin is flying a kite KITE such that $I E$ is the perpendicular bisector of $K T$. Let $I E$ meet $K T$ at $R$. The midpoints of $K I, I T, T E, E K$ are $A, N, M, D$, respectively. Given that $[M A K E]=18, I T=10,[R A I N]=4$, find $[D I M E]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Use Stokes' Theorem to compute the line integral \( \int_C (y \, \mathrm{d}x + z \, \mathrm{d}y + x \, \mathrm{d}z) \), where \( C \) is the curve of intersection of the cylinder \( x^2 + y^2 = 1 \) and the surface \( z = x \cdot y \), oriented counterclockwise around the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
In a small pond there are eleven lily pads in a row labeled $0$ through $10.$ A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability \(\frac{N}{10}\) and to pad \(N+1\) with probability \(1-\frac{N}{10}\). Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Calculate the value of \(k+m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
Evaluate the line integral \( \int_C y^2 x^2 \, dy + 2xy^3 \, dx \) using Green's Theorem, where \( C \) is the negatively oriented triangle with vertices at \((0,0), (3,1), (3,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{12}{5}}
In the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots\). Find \(a_{1999}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3331}
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\frac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{446}
Determine the number of elements of order 10 in the group \( \mathbb{Z}_{50\mathbb{Z}} \times \mathbb{Z}_{20\mathbb{Z}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
What is the value of the definite integral $\int_{0}^{1} \left(1 + x + x^2 + x^3 + \ldots + x^{n-1}\right) \left(1 + 3x + 5x^2 + \ldots + (2n-1)x^{n-1}\right) dx$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Determine the number of distinct ways to color the faces of a regular icosahedron using two colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17824}
Find the last two digits of the sum of the eighth powers of the integers from 1 to 100, inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. Find the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$. The original answer is in the form of $k\sqrt{m}$, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
There is a unique positive integer \( x \) such that \( {x}^{x} \) has 703 positive factors. What is \( x \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \( n \) be a positive integer. Determine the largest integer \( f(n) \) such that there exists a polynomial \( p \in \mathbb{Z}[x, y] \) satisfying \((x+2y)^{2n} - x^{2n} = 2^{f(n)} p\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \nu_2(n)}
Given a sequence of independent random variables \(\{X_i, i \geq 1\}\), where \(\mathbb{P}(X_i = i^2 - 1) = i^{-2}\) and \(\mathbb{P}(X_i = -1) = 1 - i^{-2}\), determine the constant \(c\) to which \(\frac{1}{n} \sum_{i=1}^n X_i\) converges almost surely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
João wrote down all the powers of 2, 3, and 5 greater than 1 and less than 2017 on a sheet of paper. He then calculated all possible products of two distinct numbers from that sheet and recorded them on another sheet of paper. How many distinct integers did João write on the second sheet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Given the set {1, 4, 7, 10, 13, 16, 19}, how many different integers can be expressed as the sum of three distinct members of this set? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given a sequence $\{a_{n}\}$, where ${a}_{n}=n•{a}^{n}$, $n$ is a positive integer, and the constant $a \gt 0$, $a\neq 1$. If $\{a_{n}\}$ is a strictly decreasing sequence, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac{1}{2})}
In triangle $XYZ$ with right angle at $Z$, $\angle XYZ < 45^\circ$ and $XY = 6$. A point $Q$ on $\overline{XY}$ is chosen such that $\angle YQZ = 3\angle XQZ$ and $QZ = 2$. Determine the ratio $\frac{XQ}{YQ}$ in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7 + 3\sqrt{5}}{2}}
Given a sequence $\{a\_n\}$ that satisfies: for any $n∈N^{*}$, $a_{n+1}=ka_{n}+2k-2$, where $k$ is a constant not equal to $0$ or $1$. If $a_{i}∈\{-272,-32,-2,8,88,888\}$, where $i=2$, $3$, $4$, $5$, then the sum of all possible values of $a_{1}$ is $\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2402}{3}}
The convex pentagon ABCDE has all sides of equal length, and angles A and B are both 90 degrees. Calculate the degree measure of angle E. Express your answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
In a random tournament with $n$ nodes, where each pair of distinct nodes $a$ and $b$ has an edge connecting $a$ to $b$ or $b$ to $a$ with equal probability of 0.5, determine the expected length of the longest transitive chain. If a simple closed-form expression is not available, provide a tight upper bound for this expected length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \log_2 n}
Martha receives a daily allowance of $15. She decides to save varying portions of this amount every day for three weeks. In the first week, she saves half of this amount every day, except for one day when she saves only a third. During the second week, she saves 60% of her daily allowance for the first 5 days, and only 25% on the last 2 days. In the third week, Martha decides to save 75% of her allowance every day, but she has to pay for a $10 expense on the first day and a $20 expense on the fifth day of that week. How much has she saved at the end of these three weeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$151.25}
A point $A = (a,b)$ lies on the graph $y = x^2 + 1$. A point $B = (c,d)$ lies within the area defined by $(x+2)^2 + (y+2)^2 = r^2$. Define a matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $M$ always has an inverse (i.e., the determinant of $M$ is not zero). What is the maximum possible value of $50r^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Determine the actual growth rate of \( \left|\frac{\zeta'}{\zeta}\left(-\frac{1}{2}+it\right)\right| \) as \( t \to \infty \), where \( \zeta \) is the Riemann zeta-function and \( t > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log t}
Find the sum of the digits of the smallest positive integer $k$ such that the sum of the coefficients $a_0 + a_1 + \ldots + a_{k-1}$ in the expansion of the polynomial \((1 + x)(1 + 2x)(1 + 3x) \cdots (1 + kx) = a_0 + a_1x + a_2x^2 + \cdots + a_kx^k\) is divisible by 2005.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}