inclusionAI/Ring-lite-rl-data huggingface dataset:
I-2 Consider a chessboard $n \times n$ where $n>1$ is a positive integer. We select the centers of $2 n-2$ squares. How many selections are there such that no two selected centers lie on a line parallel to one of the diagonals of the chessboard?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n}}
Determine the asymptotic bounds for the sum \( \sum\limits_{k=1}^n k^k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^n)}
Let $\{a_n\}$ be a sequence of real numbers. Is it possible to find a subsequence $\{a_{n_k}\}$ such that the limit of the ratio of consecutive terms, $\lim \dfrac{a_{n_{k+1}}}{a_{n_k}}$, exists and is a member of the set $\{-1, 0, 1, \infty\}$? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Consider a matrix \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that:
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\). Determine the maximum number of matrices that satisfy these conditions such that no two of them are similar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673}
Given the numbers 0, 1, 2, 3, 4, 5, 6, determine the total number of 3-digit numbers that can be formed from these digits without repetition and divided by 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Consider a well-defined set $S$ and a subset $A$ of $S$ defined as follows: \[ A = \{x \mid (x \in S) \wedge (x \not\in S)\}. \] Determine whether $A$ represents the empty set $\varnothing$ due to the contradictory predicate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\varnothing}
Let $S$ be a subset of $\{1,2,3,\cdots,1989\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest number of elements $S$ can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{905}
Evaluate the integral of \( \log(z) \) over the unit circle in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Minh enters the numbers $1$ through $81$ into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$11$}
Let $k$ be a field, and consider the set $S = A_1 \cup \{(0,0)\}$, where $A_1 = \{(x,y) \in k^2: x \neq 0\}$. Determine whether $S$ is open or closed in the affine plane $\mathbb{A}^2$. Provide your answer as either "open", "closed", or "neither".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{neither}
The circle touches the sides $AB$ and $BC$ of triangle $ABC$ at points $D$ and $E$ respectively. Find the height of triangle $ABC$ dropped from point $A$, given that $AB = 5$, $AC = 2$, and points $A$, $D$, $E$, and $C$ lie on the same circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{6}}{5}}
Let $N = \lceil\delta^{-\gamma}\rceil$, with $0 < \gamma < 1$. Determine whether \( \lim_{\delta \to 0} f(t,\delta,\lceil \delta^{-\gamma}\rceil)=0 \) for all \( t > 0 \), where \( f(t,\delta, N)= e^{-t/\delta} \sum_{k=0}^{N-1}\frac{(t/\delta)^k}{k!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Tommy goes for a run around his neighborhood and decides to calculate how many wheels he saw. All the trucks in his neighborhood have 4 wheels, all the cars have 4 wheels, bicycles have 2 wheels, and buses have 6 wheels. However, some vehicles were partially hidden, so Tommy could only see 75% of their wheels. If he saw 12 trucks, 13 cars, 8 bicycles, and 3 buses, and assuming all hidden wheels were distributed evenly among the vehicle types, how many wheels did Tommy see in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Consider a $(-1)$-curve $E \subseteq S$ inside a non-singular complex projective surface, where $E \cong \mathbb{P}^1_\mathbb{C}$ and $E^2 = -1$. Let $\sigma \in \text{Aut}(\mathbb{C})$ be a field automorphism, and define the fibered product $E^\sigma := E \times_{\text{Spec } \mathbb{C}} \text{Spec } \mathbb{C}$. Is it true that ${(E^\sigma)}^2 = -1$? In other words, is the auto-intersection preserved by the action of $\text{Aut}(\mathbb{C})$ on $E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let \(X \in \mathbb{R}^{m}\) and \(W \in \mathbb{R}^{p}\) be random vectors. Determine whether the statement \(E[X|W] = 0\) implies \(E[XW^{T}]=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E[XW^T] = 0}
Let $ABCD$ be a convex quadrilateral with $AB=CD=10$, $BC=14$, and $AD=2\sqrt{65}$.  Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of $\triangle APB$ and $\triangle CPD$ equals the sum of the areas of $\triangle BPC$ and $\triangle APD$.  Find the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
112
}
Find the number of solutions, in degrees, to the equation $\sin^{10}x + \cos^{10}x = \frac{29}{16}\cos^4 2x,$ where $0^\circ \le x^\circ \le 2007^\circ.$


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The number of solutions is 45.}
Determine if the Mellin transform of the function \( f(t) = e^{-t} \int_0^t e^w g(w)\,dw \) converges in the strip \( a<\Re(z)<b \), given that the Mellin transform of \( g(t) \) converges in the same strip.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < \Re(z) < b}
Evaluate the limit: $$\lim_{n\to\infty}\int_0^n \frac{e^{-t} \sin{\frac{t}{n}} dt}{(1+\frac{t}{n}^2)\arctan{\frac{t}{n}}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$ then $x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} = $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51.005

}
How many nine-digit numbers composed of eight 3's and one 0 leave a remainder of 1 when divided by 4? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let \(\{ x \}\) denote the nearest integer to \(x\), so that \(x - \frac{1}{2} \leq \{ x \} < x + \frac{1}{2}\). Define the sequence \(u_1, u_2, u_3, \ldots\) by \(u_1 = 1\) and \(u_{n+1} = u_n + \{ u_n \sqrt{2} \}\). For example, \(u_2 = 2\), \(u_3 = 5\), \(u_4 = 12\). Find the units digit of \(u_{1985}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given the differential equation $ty''(t)+2y'(t)+ty=0$ for $t>0$, with initial conditions $y(0+)=1$ and $y'(0+)=0$, find the value of $Y(1)$, where $Y(s)$ is the Laplace transform of $y(t)$. Assume that $Y(s) = -\tan^{-1}(s) + C$ and determine the constant $C$ to find $Y(1)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Mom gave Vasya money for 30 pencils. It turned out that the pencil factory is running a promotion in the store: in exchange for a receipt for the purchase of a set of 20 pencils, $25\%$ of the cost of the set is refunded, and in exchange for a receipt for the purchase of a set of 5 pencils, $10\%$ is refunded. What is the maximum number of pencils that Vasya can buy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If $D$ is a dense subset (not a subspace) of a Hilbert space $H$, is the orthogonal complement of $D$ equal to $\{ 0 \}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Determine whether the integral \( \int_0^\infty \frac{\sin ax}{x} \, dx = \frac{\pi}{2} \) holds true when \( a = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the limit \( \mathcal{T} = \lim\limits_{n\to\infty} nt_n \), where \( t_n = \frac{1}{2n+1} - \frac{1}{2n+2} + \frac{1}{2n+3} - \frac{1}{2n+4} + \ldots + \frac{1}{4n-1} - \frac{1}{4n} \). Relate this limit to the Riemann sums for the function \((1+x)^{-2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Given an arithmetic sequence \(\{a_n\}\), where the sum of the first \(m\) terms is \(S_m = \frac{m}{n}\) and the sum of the first \(n\) terms is \(S_n = \frac{n}{m}\) with \(n \neq m\), determine whether \(S_{m+n}\) is greater than 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_{m+n} > 4}
Jirka had three identical rectangles. First, he placed them next to each other as shown in the picture and obtained a rectangle with a perimeter of $20 \mathrm{~cm}$. He then arranged them differently and obtained a rectangle with a different perimeter. What was the perimeter of this rectangle? (In each arrangement, no two rectangles overlap and there are no gaps between them.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).

Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of \( g(3) \). Find \( m \times t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6019}{3}}
There are three triangles: an acute triangle, a right triangle, and an obtuse triangle. Sasha took one triangle, and Borya took the remaining two. It turned out that Borya can place one of his triangles against the other (without overlapping) to form a triangle identical to Sasha's. Which triangle did Sasha take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Right}}
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1,507,509$}
Let $a,b \in \mathbb{R}_+$ be real positive numbers with $\frac{1}{2}<a<1$ and let $I=[0,b]$ be a closed real interval. Define the sequence of functions $f_n(x) : I \to \mathbb{R}$ by \[ f_n(x) = { \dfrac{n}{(2n-1)^{a+x}} - \dfrac{n}{(2n)^{a+x}} } \] and the sequence of functions $g_n(x) : I \to \mathbb{R}$ by \[ g_n(x) = \dfrac{n \cdot \ln(2n)}{(2n)^{a+x}} - \dfrac{n \cdot \ln(2n-1)}{(2n-1)^{a+x}}. \] Let $h : I \to \mathbb{R}$ be the function \[ h(x)= \sum_{n=1}^\infty \left| \dfrac{f_n(x)-f_n(0)}{x} - g_n(0) \right|^2. \] Determine whether \[ \lim_{x \to 0^+} h(x)=0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let  $a_0, a_1,\dots, a_{19} \in \mathbb{R}$  and   $$ P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}. $$   If  $P(x)=P(-x)$  for all  $x \in \mathbb{R}$ , and   $$ P(k)=k^2, $$   for  $k=0, 1, 2, \dots, 9$  then find   $$ \lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}. $$  
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-(9!)^2 + 1}
5. Determine the number of all coverings of a chessboard $3 \times 10$ by (nonoverlapping) pieces $2 \times 1$ which can be placed both horizontally and vertically.

(Stanislava Sojáková)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{571}
Find a root of the polynomial \( f(x) = x^4 + 10x^2 + x + 3 \) modulo \( 3^4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Which element is the major component in solar cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Si
}
There are 62 boxes in total, with small boxes being 5 units per ton and large boxes being 3 units per ton. A truck needs to transport these boxes. If the large boxes are loaded first, after loading all the large boxes, exactly 15 small boxes can still be loaded. If the small boxes are loaded first, after loading all the small boxes, exactly 15 large boxes can still be loaded. How many large boxes are there among these boxes in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
How long does a train 375 m long running at the speed of 78.5 km/hr take to cross a bridge 450 m in length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.83}
Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

\[ 
\sqrt[4]{a(b+2c)} + \sqrt[4]{b(c+2d)} + \sqrt[4]{c(d+2a)} + \sqrt[4]{d(a+2b)} 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt[4]{3}}
In the Cartesian coordinate system, let A, B, and C be three distinct points on the curve $y= \frac {1}{x-1}$. Let D, E, and F respectively be the midpoints of BC, CA, and AB. Then, the circle passing through points D, E, and F always passes through a fixed point \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 0)}
17. [10] How many ways are there to color every integer either red or blue such that $n$ and $n+7$ are the same color for all integers $n$, and there does not exist an integer $k$ such that $k, k+1$, and $2 k$ are all the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let ellipse $C$:$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ have foci $F_{1}(-c,0)$ and $F_{2}(c,0)$. Point $P$ is the intersection point of $C$ and the circle $x^{2}+y^{2}=c^{2}$. The bisector of $\angle PF_{1}F_{2}$ intersects $PF_{2}$ at $Q$. If $|PQ|=\frac{1}{2}|QF_{2}|$, then find the eccentricity of ellipse $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}-1}
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 15x\]for all integers $x.$ Find the smallest possible value of $a.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
At 8:10 AM, Feifei walked from home to school. After 3 minutes, the dog started running to catch up with her and caught up with her 200 meters away from home. The dog immediately ran back home after catching up with her, and then turned back to chase Feifei again, catching up with her 400 meters away from home. The dog once again ran back home after catching up with her, and then started chasing Feifei again, finally catching up with her at the school. Feifei reached the school at __ minutes past 8 o'clock. 

(Note: Fill in the blank with the appropriate time)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28 \text{ minutes}}
Find the sum of all integers $n$ with $2 \le n \le 999$ that have the following property: if $x$ and $y$ are randomly selected without replacement from the set $\{ 1, 2, \dots, n \}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Find the operator norm of the linear operator $A$ defined by \begin{equation} Ax=(x_1+x_2, x_2+x_3, \ldots) \end{equation} for all $x \in \ell_2$. It has been shown that this operator is bounded with \begin{equation} \sum_{i=0}^\infty|x_i+x_{i+1}|^2 \leq 6\sum_{i=0}^\infty|x_i|^2. \end{equation} Determine the norm of $A$ by considering the supremum of $Ax$ for $|x|=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the three internal angles \( A, B, C \) of triangle \( \triangle ABC \) satisfy the equations \( \cos A = \sin B = 2 \tan \frac{C}{2} \), find the value of \( \sin A + \cos A + 2 \tan A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the remainder when \( \frac{3^{2015} + 1}{4} \) is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{227}
After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7721}
Tracy has been collecting novels from her friends to donate to the Children For The Future charity organization. In the first week, she collects a certain number of books on her own, and then two of her friends volunteer to help her in the following weeks. The book collection process for the following weeks is as follows:

- In the second week, Tracy collects twice as many books as she did in the first week, while her first friend collects three times as many books as Tracy did in the first week. 
- In the third week, Tracy collects three times as many books as she did in the first week, while her first friend collects four times as many books as Tracy did in the first week, and her second friend starts collecting and gathers half as many books as Tracy did in the first week.
- In the fourth week, Tracy collects four times as many books as she did in the first week, her first friend collects five times as many books as Tracy did in the first week, and her second friend collects as many books as Tracy did in the first week.
- In the fifth week, Tracy collects five times as many books as she did in the first week, her first friend collects six times as many books as Tracy did in the first week, and her second friend collects twice as many books as Tracy did in the first week.
- In the sixth week, Tracy collects six times as many books as she did in the first week, her first friend collects seven times as many books as Tracy did in the first week, and her second friend collects three times as many books as Tracy did in the first week.

If they have collected a total of 1,025 books now, how many books did Tracy collect in the first week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC = DC = 3$, $OD = 2$, and $AB = 6$, find the radius of the circle. If the radius is \(\sqrt{x}\), what is the value of \(x\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Determine the minimum number of pairs \((x, R(x))\) required to fully characterize a rotation \(R\) in \(\mathbb{R}^n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Determine the degree of the splitting field of the polynomial \((x^2+x+1)(x^3+x+1)\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let point $P$ be a moving point on the curve $y=2x^2$. The tangent line to the curve $y=2x^2$ at point $P$ is denoted as $l$. The line passing through point $P$ and perpendicular to line $l$ intersects the curve $y=2x^2$ at another point $Q$. Find the minimum value of $|PQ|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3}}{4}}
Find the number of ways to order the integers $1, 2, 3, 4, 5, 6, 7$ from left to right so that each integer has all its divisors, besides itself, appearing to the left of it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
V is the pyramidal region defined by the inequalities \( x, y, z \geq 0 \) and \( x + y + z \leq 1 \). Evaluate the integral:

\[ \int_V x y^9 z^8 (1 - x - y - z)^4 \, dx \, dy \, dz. \
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9! 8! 4!}{25!}}
Find the angle between the non-intersecting diagonals of two adjacent lateral faces of a right quadrilateral prism, given that the plane in which they lie forms an angle $\alpha$ with the base plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \arctan \cos \alpha}
若存在实数 a 及正整数 n 使得 f(x)=\cos2x-a\sin x 在 (0,n\pi) 内恰有 2022 个零点，则满足条件的正整数 n 的值有__________个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \notin \mathbf{Q}$, $f(x)=0$; and when $x=\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\left(\frac{7}{8}, \frac{8}{9}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{16}{17}$}
Compute the limit \( \lim_{n \to \infty} n\left(\frac{{\pi}^2}{6} - \sum_{k=1}^{n} \frac{1}{k^2} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum number of equally spaced subintervals, N, required such that the difference between the Upper and Lower Riemann sums for the integral \( \int_{1}^{17} \frac{1}{x^3} \, dx \) is less than \( \frac{1}{100} \). The Upper and Lower Riemann sums are given by:

\[ L_{n} = \frac{16}{N} \sum_{i=1}^{N}\frac{1}{(2i+1)^3} \]
\[ U_{n} = \frac{16}{N} \sum_{i=1}^{N}\frac{1}{(2i-1)^3} \]

The difference between them is:

\[ U_{n} - L_{n} = \frac{16}{N} \left[1-\frac{1}{(2N)^3} - \frac{1}{(2N +1)^3}\right] \]

Find the smallest N such that:

\[ \frac{16}{N} \left[1-\frac{1}{(2N)^3} - \frac{1}{(2N +1)^3}\right] < \frac{1}{100} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1600}
The circumferences of Lebanese cedar cones are modeled as Gamma-distributed random variables with a mean of 5.40 cm and a variance of 3.24 cm². A botanist is interested in collecting cones with a circumference greater than 6 cm. Calculate the expected number of cones she needs to inspect to find her third cone with a circumference greater than 6 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.01}
The length of a red bus is four times the length of an orange car. The length of the orange car is also 3.5 times shorter than the length of a yellow bus. A green truck is parked between the red bus and the yellow bus, and the green truck is as long as two orange cars. There is a row of trees that covers 25% of the view between the red bus and the yellow bus. If the red bus is 48 feet long, and the yellow bus and green truck are parked end to end next to the red bus, what length of the yellow bus may the red bus driver see, considering the trees that partially obstruct the view?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.5}
如果非负整数 $m$ 及其各位数码之和均为 6 的倍数，则称 $m$ 为 "六合数". 求小于 2012 的非负整数中六合数的个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Determine the minimum value of \( n \) such that there exist real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) satisfying \( x_{1}+2 x_{2}+\cdots+ n x_{n}=2009 \) and \( x_{1}+x_{2}+\cdots+x_{n}=0 \), where each \( x_{i}= \pm 7 \) for \( i=1,2, \ldots, n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}