inclusionAI/Ring-lite-rl-data huggingface dataset:
In the following equation, the same Chinese characters represent the same digit, different Chinese characters represent different digits, and no Chinese character represents 7. "迎", "春", and "杯" are not equal to 1. What is the sum of the three digits represented by "迎", "春", and "杯"?
学数学 $\times 7 \times$ 迎 $\times$ 春 $\times$ 杯 $=$ 加油加油 $\times$ 吧.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine if the congruence $n \equiv 1 \pmod{\lambda^5}$, where $\lambda = 1 - \zeta_5$ and $\zeta_5$ is a 5th root of unity, can be simplified to a congruence modulo an integer in $\mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Two trains are moving on parallel tracks in the same direction: Train A is 400 meters long and has a speed of 63 km/hr, while Train B is 300 meters long and has a speed of 54 km/hr. At the same time a man is walking in the direction of the moving trains with a speed of 3 km/hr. How long will it take for the faster Train A to completely overtake Train B and the man walking, considering that Train A starts from behind the man and has to completely cross both the man and Train B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Given a positive integer N, count the number of pairwise coprime integer triples (a, b, c) such that 1 ≤ a, b, c ≤ N. For example, when N = 3, what is the count of valid triples?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Consider a Poisson process $N_t$ with $\\langle N \\rangle_t = N_t$, and a continuous process of finite variation $X_t$ with $\\langle X \\rangle_t = 0$. Determine if the quadratic covariation $\\langle X, N \\rangle_t = 0$ is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Last summer \( {30}\% \) of the birds living on Town Lake were geese, \( {25}\% \) were swans, \( {10}\% \) were herons, and \( {35}\% \) were ducks.
 What percent of the birds that were not swans were geese?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40
}
方程$( x ^ { 2 } - 6 x + c _ { 1 } ) ( x ^ { 2 } - 6 x + c _ { 2 } ) ( x ^ { 2 } - 6 x + c _ { 3 } ) = 0$的自然数解集为$\{ x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$,$x _ { 5 } , x _ { 6 } \}$，那么$c _ { 1 } - c _ { 2 }$的最大值为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the quadratic equation $2x^2-2x+3m-1=0$ with two real roots $x_1$ and $x_2$, and the inequality $x_1 \cdot x_2 > x_1 + x_2 - 4$, determine the range of the real number $m$ that satisfies both conditions. Express your answer as an inequality in terms of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{3} < m \leq \frac{1}{2}}
In a regular tetrahedron \( ABCD \) with edge length \( a \), points \( A_1 \), \( B_1 \), \( C_1 \), and \( D_1 \) are chosen in the planes \( BCD \), \( CDA \), \( DAB \), and \( ABC \), respectively, such that the line \( A_1B_1 \) is perpendicular to the plane \( BCD \), the line \( B_1C_1 \) is perpendicular to the plane \( CDA \), the line \( C_1D_1 \) is perpendicular to the plane \( DAB \), and the line \( D_1A_1 \) is perpendicular to the plane \( ABC \). Find the volume of the tetrahedron \( A_1B_1C_1D_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^3 \sqrt{2}}{162}}
A train 180 meters long running at the rate of 54 kmph takes 55.99552035837134 seconds to cross a bridge of certain length. How long is the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{659.93}
给定正整数 $n=3$. 求最大的实数 $\lambda$, 使得不等式 $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ 对任何满足 $a_{1}<a_{2}<\cdots<a_{n}$ 的正整数 $a_{1}, a_{2}, \cdots, a_{n}$ 均成立.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Miyuki texted a six-digit integer to Greer. Two of the digits of the six-digit integer were 3s. Unfortunately, the two 3s that Miyuki texted did not appear and Greer instead received the four-digit integer 2022. What is the number of possible six-digit integers that Miyuki could have texted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Calculate the measure \( \mu(A) \), where \( \mu = L^1 \otimes \sum_{k = 1}^{\infty}k\delta_k \) and \( A = \{(x, y) \in \mathbb{R}^2: 3 - |x| > y > 2|x|\} \). Use Fubini's theorem to evaluate the integral \( \mu(A) = \int_{A} d\mu \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
\( S \) is the sum of the first 10 terms of an increasing arithmetic progression \( a_{1}, a_{2}, a_{3}, \ldots \) consisting of integers. It is known that \( a_{6} a_{12} > S + 1 \) and \( a_{7} a_{11} < S + 17 \). Indicate all possible values of \( a_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_1 \in \{-6, -5, -4, -2, -1, 0\}}
Determine whether the statement "$$\forall\varepsilon\gt0,\exists N\in\mathbb{Z^+},\text{s.t.}\forall n\gt N,\left| a_n-a_N\right|\lt\varepsilon$$" is equivalent to the convergence of the sequence $\{a_n\}$. Answer 'yes' if they are equivalent, otherwise answer 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
The right focus of the hyperbola $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ is $F$, and its left and right vertices are $A_{1}$ and $A_{2}$, respectively. A perpendicular line to $A_{1}A_{2}$ passing through $F$ intersects the hyperbola at points $B$ and $C$. If $A_{1}B \perp A_{2}C$, then the slope of the asymptotes of this hyperbola is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Suppose \( \{a_1,a_2,\ldots,a_7\} \) is a set of pair-wisely different positive integers. If \( a_1, 2a_2, \ldots, 7a_7 \) can form an arithmetic series (in this order), find the smallest positive value of \(|a_7-a_1|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
For what values of $ \alpha $, does this integral converge? $$ \int_0^\infty x^\alpha \left( 1- e^{-\frac{1}{\sqrt{x}}} \right) \, \mathrm d x $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\alpha \in (-1, -\frac{1}{2})\)}
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that $$\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}$$whenever $\tan 2023x$ is defined. What is $a_{2023}?$




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1
}
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Consider two totally finite measure spaces $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$. Let $A\subseteq \Omega_1\times \Omega_2$ satisfy the given measurability conditions. If \(\int_{\Omega_1} \mu_2(A_x) \, d\mu_1 = \int_{\Omega_2} \mu_1(A^y) \, d\mu_2 = 0\), determine whether we can conclude that \(\mu_1\times\mu_2(A) = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The 61st question: Find the maximum value of the positive integer $n$, such that for any simple graph of order $n$ with vertices $v_{1}, v_{2}, \ldots, v_{n}$, there always exist $n$ subsets $A_{1}, A_{2}, \ldots, A_{n}$ of the set $\{1,2, \ldots, 2020\}$, satisfying: $A_{i} \cap A_{j} \neq \varnothing$ if and only if $v_{i}$ is adjacent to $v_{j}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
A regular tetrahedron has two spheres that are inscribed in and circumscribed around it. Additionally, in each face of the regular tetrahedron and between it and the circumscribed sphere, there are four spheres touching the center of the face. Given that point P is inside the circumscribed sphere, calculate the probability that P lies inside one of the five small spheres. Express your answer as a decimal value between 0 and 1, rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.19}
Bin $A$ has one white ball and four black balls.  Bin $B$ has three balls labeled $\$1$ and one ball labeled $\$7$.  Bin $W$ has five balls labeled $\$8$ and one ball labeled $\$500$.  A game is played as follows: a ball is randomly selected from bin $A$.  If it is black, then a ball is randomly selected from bin $B$; otherwise, if the original ball is white, then a ball is randomly selected from bin $W$.  You win the amount printed on the second ball selected.  What is your expected win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$20}
Suppose $f_n$ is a sequence of bounded measurable functions on a probability space, and $f_n$ does not converge to $0$ almost everywhere. Determine whether there exists a set of positive measure $E$ such that $\limsup\limits_{n\to\infty}|f_n(x)|>2\epsilon$ for all $x\in E$. Answer "yes" or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
4.1. Two identical cylindrical vessels are connected at the bottom by a small cross-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg \(/ \mathrm{m}^{3}\), and the density of oil is 700 kg \(/ \mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Provide the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \cdot 6^{3/2}}{3 \cdot 5^{5/4}}}
Consider an analytic function $f$ defined on the open unit disk $\mathbb{D}$ with the power series representation $$f(z) = \sum_{n = 0}^\infty a_n z^n.$$ Define the sequence of functions $f_n(z)$ as $$f_n(z) = a_0 + a_1z + \ldots + a_nz^n.$$ If this sequence converges uniformly to $f$ on the closed unit disk, determine whether the radius of convergence of the power series is strictly greater than 1 or can be exactly 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The diagram shows an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Find a positive real number $R$ such that for the complex function $p(z) = z^{10} + 100z + 9$, the inequality $|p(z)| > 1$ holds whenever $|z| > R$. Provide a rationale for your choice of $R$ without using complex number manipulations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Regular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find the minimum possible value of $(AM + MN + NA)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Find the limit of the following expression as $n$ approaches infinity: $$\lim_{n \to +\infty} \frac{1}{\sqrt{(n^2)}} + \frac{1}{\sqrt{(n^2+n)}} + \frac{1}{\sqrt{(n^2+2n)}}......+ \frac{1}{\sqrt{(n^2+2n^2)}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(sqrt(3)-1)}
A circle of radius 2 has center at (2,0). A circle of radius 1 has center at (5,0). A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2 \sqrt{2}$}
Evaluate the following sum, where $a, b, c$ are natural numbers and $a \leq c$:
$$\sum_{i=0}^{b}\frac{a\binom{b}{i}\binom{c}{a}}{(i+a)\binom{b+c}{i+a}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The volume of a cube \( ABCD-A_1B_1C_1D_1 \) is 512. \( M \) and \( N \) are the midpoints of edges \( CB \) and \( BB_1 \), respectively. A plane passing through points \( A \), \( M \), and \( N \) cuts a triangular prism \( B-AMN \). Find the surface area of the triangular prism \( B-AMN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Determine the linear minimum mean squared error (MMSE) estimator for \( y \) given \( x \) for a 2D random point \((x,y)\) uniformly distributed on a shaded region. Express the estimator in the form \( \hat{y} = wx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\hat{y} = \frac{1}{2}x}
Let $K = \mathbb{C}(x_1, \ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$, where $G \cong \mathbb{Z}$. The field of $G$-invariants, $K^G$, is an extension of $\mathbb{C}$ with transcendence degree $d$. For a generic $n$-tuple $x \in \mathbb{C}^n$, consider the Zariski closure of the $G$-orbit $\{ g(x) \}_{g \in G} \subset \mathbb{C}^n$. Can you conclude that the dimension of this Zariski closure is generically $n-d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - d}
In how many ways can each square of a $2 \times 9$ board be colored red, blue, or green so that no two squares that share an edge are the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39366}
If $A$ and $B$ are subsets of $\mathbb{N}$ with $\delta(A) = \delta(B) = 0$, does the natural density $\delta(A \cup B)$ exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the group to which $Gal(\mathbb{Q}(i+\sqrt{2}):\mathbb{Q})$ is isomorphic. The order of this group is 4. Is it isomorphic to the Klein four-group or the cyclic group of order 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V_4}
设$S={1,2,⋯,2005}$，若$S$中任意$n$个两两互素的数组成的集合中都至少有一个素数，试求$n$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find all values of $n$ such that there exists a rectangle with integer side lengths, perimeter $n$, and area $2n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$50,36,32$
}
Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
7点多少分的时候，分针落后时针100度．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given three unit vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ in a plane that are not collinear and satisfy $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = \mathbf{0}$. If $0 \leq t \leq 1$, what is the range of values of $|-2 \boldsymbol{a} + t \boldsymbol{b} + (1-t) \boldsymbol{c}|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{5}{2}, \sqrt{7}\right]}
Given $D \in \mathbb{R}^{N\times M}$, $\psi\in \mathbb{R}^M$, and $s^0 \in \mathbb{R}^N$, if for all $h\in\mathbb{R}^N$ the relationship $h^T s^0 = h^T D\psi$ holds, does it imply that $s^0 = D\psi$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s^0 = D\psi}
Calculate the value of $\displaystyle\frac{245^2-225^2}{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{470}
A collection of $8$ cubes consists of one cube with edge - length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: 
Any cube may be the bottom cube in the tower. 
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ 
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by $1000$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{458}
In a ring $R$, if there exists an element $z \in R$ such that $za = az = z$ for all $a \in R$, does it necessarily follow that $z = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In isosceles $\triangle A B C, A B=A C$ and $P$ is a point on side $B C$. If $\angle B A P=2 \angle C A P, B P=\sqrt{3}$, and $C P=1$, compute $A P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given that $\log_{10}2 \approx 0.30103$, ﬁnd the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$20\text{ dollars}$}
If the inequality $\cos \alpha_{1} \cos \alpha_{2} \cdots \cos \alpha_{n} + \sin \alpha_{1} \sin \alpha_{2} \cdots \sin \alpha_{n} \leqslant M$ always holds, then what is the minimum value of $M$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
An eight-sided polygon ABCDEFGH is divided into a rectangle and a square. The side lengths are integers, the area of the square is greater than the area of the rectangle, and the product of the two areas is 98. Calculate the possible perimeter of the polygon ABCDEFGH. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
10.4. Let's call a snake in a convex $n$-gon an open, non-self-intersecting broken line with $n-1$ segments, the set of vertices of which coincides with the set of all vertices of the $n$-gon. Find the number of different snakes in an $n$-gon. (Snakes are equal if they coincide as geometric loci of points. For example, the number of snakes in a triangle is 3)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n\cdot2^{n-3}}
Given three distinct non-zero numbers, Petya and Vasya create quadratic equations by using these numbers as coefficients in various orders. If a quadratic equation has at least one root, Petya gets a candy wrapper. If it has no roots, Vasya gets a candy wrapper. The first three candy wrappers went to Petya, and the next two to Vasya. Determine who will get the last, sixth candy wrapper.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
Find the greatest positive real number  $M$  such that for all positive real sequence  $(a_n)$  and for all real number  $m < M$ , it is possible to find some index  $n \ge 1$  that satisfies the inequality  $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ M = 4 }
The cricket team of 11 members is 25 years old and the wicket keeper is older by a certain number of years. If the ages of these 2 are excluded, the average age of the remaining players is 1 year less than the average age of the whole team. The average age of the team is 22 years. How much older is the wicket keeper?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A lock code is made up of four digits that satisfy the following rules: - At least one digit is a 4, but neither the second digit nor the fourth digit is a 4. - Exactly one digit is a 2, but the first digit is not 2. - Exactly one digit is a 7. - The code includes a 1, or the code includes a 6, or the code includes two 4s. How many codes are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Find the minimum value of the expression \(\left\lfloor \frac{8(a+b)}{c} \right\rfloor + \left\lfloor \frac{8(a+c)}{b} \right\rfloor + \left\lfloor \frac{8(b+c)}{a} \right\rfloor\), where \(a\), \(b\), and \(c\) are arbitrary natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
King Qi and Tian Ji are having a horse race. Tian Ji's top horse is better than King Qi's middle horse but worse than King Qi's top horse. Tian Ji's middle horse is better than King Qi's lower horse but worse than King Qi's middle horse. Tian Ji's lower horse is worse than King Qi's lower horse. Now, if a horse is randomly selected from both sides for a race, what is the probability of Tian Ji winning? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
A dishonest dealer professes to sell his goods at cost price, but uses varying weights for different items. For potatoes, he uses a weight of 900 grams instead of a kilogram. For onions, he uses a weight of 850 grams instead of a kilogram, and for carrots, he uses a weight of 950 grams for each kilogram. In one day, he sells 10 kilograms of potatoes, 15 kilograms of onions, and 25 kilograms of carrots at their respective cost prices. Calculate his overall gain percentage for that day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.89\%}
For a positive integer n, 5^n is a factor of a certain factorial, but 5^(n+1) is not a factor of the same factorial. The value of n is 4. What is the factorial?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24!}
Given a bounded sequence $\{x_n\}$ with limit superior $x^*$ and limit inferior $x_*$, determine whether for every $c \in [x_*,x^*]$, there exists a subsequence of $\{x_n\}$ that converges to $c$. Answer "yes" or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Square \(ABCD\) has sides of length \(4\), and \(M\) is the midpoint of \(\overline{CD}\). A circle with radius \(2\) and center \(M\) intersects a circle with radius \(4\) and center \(A\) at points \(P\) and \(D\). Find the distance from \(P\) to \(\overline{AD}\). 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{21}