inclusionAI/Ring-lite-rl-data huggingface dataset:
Petya marked 5 points on one face of a cube, then turned the cube and marked 6 points on the adjacent face, then turned again and marked 7 points on the next face, and so on. He marked points on each face. What is the maximum number of points that can be on two opposite faces combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let  $\mathcal{A}$  be the set of finite sequences of positive integers  $a_1,a_2,\dots,a_k$  such that  $|a_n-a_{n-1}|=a_{n-2}$  for all  $3\leqslant n\leqslant k$ . If  $a_1=a_2=1$ , and  $k=18$ , determine the number of elements of  $\mathcal{A}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1597}
In the convex quadrilateral $AEBC$, it is given that $\angle BEA = \angle CAE = 90^{\circ}$, $AB = 15$, $BC = 14$, and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to the line $\overline{AB}$. If the ray $CD$ intersects $\overline{AE}$ at $F$, compute the product $AE \cdot AF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Mr. Thompson is a firm believer in traditional gasoline cars and refuses to embrace electric vehicles. He drives his car 20 miles to work every day, five days a week. His car consumes 1 gallon of gasoline for every 25 miles driven, and the current price of gasoline is $4 per gallon. 

Calculate how much Mr. Thompson spends on gasoline for his work commute in one week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Circle $\Omega$ has radius 5. Points $A$ and $B$ lie on $\Omega$ such that chord $A B$ has length 6. A unit circle $\omega$ is tangent to chord $A B$ at point $T$. Given that $\omega$ is also internally tangent to $\Omega$, find $A T \cdot B T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the identity \( 1+2+\cdots+n=\frac{1}{2} n(n+1) \). If we set \( P_{1}(x)=\frac{1}{2} x(x+1) \), then it is the unique polynomial such that for all positive integer \( n \), \( P_{1}(n)=1+2+\cdots+n \). In general, for each positive integer \( k \), there is a unique polynomial \( P_{k}(x) \) such that
\[ P_{k}(n)=1^{k}+2^{k}+3^{k}+\cdots+n^{k} \quad \text { for each } n=1,2, \ldots . \]

Find the value of \( P_{2010}\left(-\frac{1}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the sum of the values of $x$ for which $\binom{x}{0} - \binom{x}{1} + \binom{x}{2} - \cdots + \binom{x}{2008} = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017036}
The side lengths of a right triangle are integers, with no common divisor greater than 1. The radius of the circle tangent to the hypotenuse is 420 units. Determine the side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{399, 40, 401}
Let $f: \mathbb R \to \mathbb R$ be a non-constant continuous function such that $(e^x-1)f(2x)= \left(e^{2x}-1\right)f(x)$. If $f'(0) =1 $, then $\lim_{x\to 0}\left(\dfrac {f(x)}{x}\right)^{\frac 1x}= ? $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{1/2}}
Sets $A$, $B$, and $C$ satisfy the following conditions:  
- $|A| = 92$  
- $|B| = 35$  
- $|C| = 63$  
- $|A \cap B| = 16$  
- $|A \cap C| = 51$  
- $|B \cap C| = 19$  

Compute the number of possible values of $|A \cap B \cap C|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given real numbers $a$ and $b$ satisfying ${3}^{a-2}+a=\frac{1}{2}$ and $\frac{1}{3}{b}^{3}+\log_{3}b=-\frac{1}{2}$, find the value of $\frac{{3}^{a}}{{b}^{3}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
There are 28 students in the class. On March 8, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given if it is known that there are 4 times more roses than daffodils, but 10 times fewer than tulips?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains. Calculate the number of tetrads that can be produced by this cell during meiosis. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given $p$: for all $x \in \mathbb{R}$, the inequality $x^2 - mx + \frac{3}{2} > 0$ always holds true, and $q$: the foci of the ellipse $\frac{x^2}{m - 1} + \frac{y^2}{3 - m} = 1$ lie on the x-axis. If the proposition $p \land q$ is true, determine the range of values for the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 < m < \sqrt{6}}
Suppose we choose two numbers $x, y \in [0,1]$ uniformly at random. If the probability that the circle with center $(x, y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1] \times [0,1]$ is written as $\frac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $f(x)$ be a cubic polynomial with roots $x_{1}, x_{2}$, and $x_{3}$. Assume that $f(2 x)$ is divisible by $f^{\prime}(x)$ and compute the sum of the absolute values of the ratios $x_{1}: x_{2}: x_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Zyaka decided to buy a krumblik. In the store, they also sold kryambliks. Zyka bought a kryamblik and received coupons worth $50\%$ of the cost of the kryamblik. He used these coupons to pay $20\%$ of the cost of the krumblik. By paying the remaining amount, he also bought the krumblik. By what percentage did Zyka's expenses for buying both the kryamblik and the krumblik exceed his initially planned expenses for buying only the krumblik?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x^4} + \frac{1}{y^4} + \frac{1}{z^4} = 1.\]Find the minimum value of
\[\frac{x^4 y^4 + x^4 z^4 + y^4 z^4}{x^3 y^2 z^3}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{2}}
A boy needs to cover a distance of 10000 meters in 28 minutes. Find his average speed in centimeters per second.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{595.24}
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative while the sum of any 11 consecutive terms is positive. What is the maximum number of terms in such a sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $n$ be a positive integer. 
Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left\lfloor \frac{n}{2} \right\rfloor + 1 }
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy] import cse5; pathpen=black; pointpen=black; size(6cm); pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689); filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]
Hint
$[ABC] = \frac{1}{2}ab \text{sin} C$ is your friend for a quick solve. If you know about homotheties, go ahead, but you'll still need to do quite a bit of computation. If you're completely lost and you have a lot of time left in your mocking of this AIME, go ahead and use analytic geometry.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=1$, $b=\sqrt{7}$, and $c=\sqrt{3}$, then $B=$ _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute \[ \prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right) \] in closed form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{7}}
A positive integer is called *fancy* if it can be expressed in the form   $$ 2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}}, $$   where  $a_1,a_2, \cdots, a_{100}$  are non-negative integers that are not necessarily distinct. Find the smallest positive integer  $n$  such that no multiple of  $n$  is a *fancy* number.

*Senior Problems Committee of the Australian Mathematical Olympiad Committee*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{101} - 1}
Renaldo drove 15 kilometers. Ernesto drove 7 kilometers more than one-third of Renaldo's distance. Marcos drove a negative 5 kilometers. What is the absolute difference between the total distances driven by Renaldo and Ernesto combined, and the distance driven by Marcos?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
If 2³ - 7² = 1 ; 3³ - 6² = 9 ; 5³ - 9² = 16 then 4³ - 8² = ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the function $f(x)=e^x+2x-a$ (where $a \in \mathbb{R}$, and $e$ is the base of natural logarithms). If there exists a point $(x_0, y_0)$ on the curve $y=\sin x$ such that $f(f(y_0))=y_0$, find the range of value for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1+e^{-1}, e+1]}
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$10$}
We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{859}
Twelve colored cubes are arranged in a row. There are 3 blue cubes, 2 yellow cubes, 3 red cubes, and 4 green cubes, but not in that order. A yellow cube is at one end, and a red cube is at the other end. The red cubes are all together within the row, and the green cubes are also all together within the row. The tenth cube from the left is blue. What is the color of the sixth cube from the left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{green}}
For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$, the resulting number is a multiple of $30$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Evaluate: $\displaystyle \int_{0}^{\infty}\frac{\sin\left(x^{2}\right) + \cos\left(x^{2}\right) - 1} {x^{2}}\,\mathrm{d}x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{272}
Mr. and Mrs. A are opposite to each other. The distance between them is unknown. The speed of Mr. A and Mrs. A are 30 kmph and 10 kmph, respectively. One bee is running between Mr. A's nose to Mrs. A's nose and returning back to Mr. A's nose. The speed of the bee is 60 kmph. The bee traveled 180 km. What is the distance between Mr. A and Mrs. A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given that the function f(x) = A sin(ωx + φ) is monotonic on $[0, \frac{\pi}{3}]$ and f(0) = f($\frac{5\pi}{6}$) = -f($\frac{\pi}{3}$), find the positive value of ω.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For what values of $c$ does the sequence $x_n=(2c-3)(c^{n-2}+c^{n-3}+\cdots + c+1)+2$ converge as $n\to\infty$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,1)∪{3/2}}
Let  $ABCDEF$  be a regular hexagon, and let  $G$ ,  $H$ ,  $I$ ,  $J$ ,  $K$ , and  $L$  be the midpoints of sides  $AB$ ,  $BC$ ,  $CD$ ,  $DE$ ,  $EF$ , and  $FA$ , respectively. The intersection of lines  $\overline{AH}$ ,  $\overline{BI}$ ,  $\overline{CJ}$ ,  $\overline{DK}$ ,  $\overline{EL}$ , and  $\overline{FG}$  bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of  $ABCDEF$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/7}
If (10 ^ a certain exponent * 3.456789) ^ 10 is written as a single term, there would be 20 digits to the right of the decimal place. What is the certain exponent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle \(ABC\), the medians drawn to the sides \(AC\) and \(BC\) intersect at a right angle. Given that \(AC = b\) and \(BC = a\), find \(AB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\dfrac{a^2 + b^2}{5}}}
Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. Calculate the time Leo thinks it is. Express your answer in the format HH:MM, including AM/PM designation if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12:30}
Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide the sum of the possible characteristics.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the following limit:
$$	\lim_{x \to 0}x \tan (xa+ \arctan \frac{b}{x})$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b}{1 - ab}}
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \sqrt{2}}
Five students made the following statements about how many people studied math yesterday:

- Zhang: "Nobody studied math."
- Li: "One person studied math."
- Wang: "Two people studied math."
- Zhao: "Three people studied math."
- Liu: "Four people studied math."

The teacher knows that some of them studied and some didn’t. Those who studied told the truth, and those who didn’t lied. Determine the exact number of students who studied math yesterday. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If the sum of the lengths of the six edges of a trirectangular tetrahedron $PABC$ (i.e., $\angle APB=\angle BPC=\angle CPA=90^o$) is $S$, determine its maximum volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{S^3(\sqrt{2}-1)^3}{162}}
Ali chooses one of the stones from a group of 2005 stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. The game continues until two stones remain, and Ali confesses the marked stone. What is the minimum number of moves Betül needs to guarantee finding the marked stone? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The average age of 12 men is increased by years when two of them whose ages are 21 years and 23 years are replaced by two new men. What is the average age of the two new men?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Find the integral of $$\int \frac{1}{1+\tan x}dx$$ without using partial fractions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{1}{2}x+\frac{1}{2}\ln|\sin x+\cos x|+C$$}
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \) which satisfy the equation \( a x + b y + c z = 1 \), where \( a, b, c \) are positive integers with \( a < b < c \), find \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1987}
一个不均匀的骰子，掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列。独立地先后郑该骰子两次，所得的点数分别记为 $a, b$ 。若事件 " $a+b=7$ " 发生的概率为 $\frac{1}{7}$ ，则事件 " $a=b$ "发生的概率为 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given the following balanced chemical equation:

2 HClO2 + 3 H2O2 → 4 H2O + 2 ClO2

Determine the weight of 6 moles of HClO2 if 4 moles of H2O2 are reacted together in this reaction under equilibrium conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{410.76}
Let $P$ be a moving point on the circle $(x+3)^{2}+(y-1)^{2}=2$, and $Q$ be a moving point on the line $y=x$. Determine the minimum value of $|PQ|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Plane A and Plane B are 500 kilometres apart and flying towards each other . If plane A is travelling at a constant speed of 240 k/m and plane B is travelling at a constant speed of 360 k/m when will the planes meet ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Malcolm has 240 followers on Instagram and 500 followers on Facebook. The number of followers he has on Twitter is half the number of followers he has on Instagram and Facebook combined. Meanwhile, the number of followers he has on TikTok is 3 times the number of followers he has on Twitter, and he has 510 more followers on Youtube than he has on TikTok. Additionally, he has 120 followers on Pinterest which is the double of his Snapchat followers. Over the next month, he gains 15% more followers on Instagram and 20% more followers on Facebook. In that same period, he gains 30 more followers on Twitter, 45 on TikTok, and loses 10 followers on Snapchat. How many followers does Malcolm have on all his social media accounts after this one month period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4221}
设集合 $A=\left\{n \left\lvert\, \frac{n}{3} \in \mathbf{N}_{+}\right.\right\}$，$B=\left\{y \mid y=x+4+\sqrt{5-x^{2}}\right\}$. 求集合 $A \cap B$ 中元素的个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There is a calculator with digits from 0 to 9 and two operation symbols. Initially, the display shows the number 0. You can press any buttons. The calculator performs actions in the sequence of button presses. If an operation symbol is pressed multiple times in a row, the calculator only remembers the last press. A scattered scientist pressed a lot of buttons in a random sequence. Estimate the probability that the result of the resulting chain of operations is an odd number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
14. If $a, b, c \in \mathbf{R}^{+}$, and satisfy $\frac{k u b c}{a+b+c} \geqslant(a+b)^{2}+(a+b+4 c)^{2}$. Find the minimum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Consider a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $x, y$, the equation $f(x f(y)) = y f(x)$ holds. Find the smallest possible value of $f(2007)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:



- All connections operate in both directions
- If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.


Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Call a positive integer "mild" if its base-3 representation never contains the digit 2. How many values of \( n \) (where \( 1 \leq n \leq 1000 \)) have the property that both \( n \) and \( n^{2} \) are mild?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For any complex number $w=a+bi$, $|w|$ is defined to be the real number $\sqrt{a^2+b^2}$. If $w=\cos40^\circ+i\sin40^\circ$, then $|w+2w^2+3w^3+...+9w^9|^{-1}$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2}{9}\sin20^\circ$}
设 $a, b, c$ 为正数， $a<b$ 。若 $a, b$ 为一元二次方程 $a x^{2}-b x+c=0$ 的两个根，且 $a, b, c$ 是一个三角形的三边长，则 $a+b-c$ 的取值范围是$\left(\frac{m}{n}, \sqrt{p}-q\right)$，求$m+n+p+q$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}