inclusionAI/Ring-lite-rl-data huggingface dataset:
Stephen had a 10:00 a.m. appointment 60 km from his home. He averaged 80 km/h for the trip and arrived 20 minutes late for the appointment. At what time did he leave his home? Express your answer in 12-hour clock format (e.g., 9:00 a.m.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:35 a.m.}
Find the number of local extrema of the function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let  $S$  be a finite set of points in the plane, such that for each  $2$  points  $A$  and  $B$  in  $S$ , the segment  $AB$  is a side of a regular polygon all of whose vertices are contained in  $S$ . Find all possible values for the number of elements of  $S$ .

Proposed by *Viktor Simjanoski, Macedonia*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 3 }
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\[7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\[N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Let point O be located inside a regular tetrahedron ABCD, and let G be the centroid of the tetrahedron. The line OG intersects each face of the tetrahedron at points A1, B1, C1, and D1. Find the value of the sum of the ratios A1O to A1G, B1O to B1G, C1O to C1G, and D1O to D1G.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The mean of $\frac12 , \frac34$ , and $\frac56$ differs from the mean of $\frac78$ and $\frac{9}{10}$ by $\frac{m}{n}$ , 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{859}
Given a geometric sequence {a_n} where a_2 * a_6 = 36, find the value of a_4. Express your answer in the form ±n, where n is the absolute value of a_4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±6}
In quadrilateral \(ABCD\), the diagonals intersect at point \(O\). It is known that \(S_{ABO} = S_{CDO} = \frac{3}{2}\), \(BC = 3\sqrt{2}\), and \(\cos \angle ADC = \frac{3}{\sqrt{10}}\). Find the minimum area of such a quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the power function $f(x)=(m-2)x^{m^2-2m}$, find $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a positive integer $n (n \geq 2)$, select $m$ different numbers from $1, 2, \ldots, 3n$. Among these numbers, there must exist four distinct numbers $a, b, c, d$ such that $a = b + c + d$. Find the minimum value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n + 2}
The manufacturing cost of a shoe is Rs. 200, and the transportation cost is Rs. 500 for 100 shoes. The selling price is Rs. 246. What is the percentage gain on the selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}
A man walking at a constant rate of 5 miles per hour is passed by a woman traveling in the same direction along the same path at a certain rate. The woman stops to wait for the man 2 minutes after passing him, while the man continues to walk at his constant rate. The woman must wait 4 minutes until the man catches up. What is the woman's traveling rate in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
How many equivalence relations on the set $S=\{a,b,c,d,e\}$ satisfy the conditions $a \sim b, b \sim c$, and $a \nsim e$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
[ Tangent lines and tangent circles ] Described quadrilaterals $\quad]$

Two circles of radii 4 and 3 touch each other externally. Common external tangents $P Q$ and $R S$ are drawn to these circles such that points $P$ and $S$ lie on the circle of the larger radius, and points $Q$ and $R$ lie on the circle of the smaller radius. Find the radius of the circle that touches segments $R S$, $S P$, and $P Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{24}{7}}
Expand and find the sum of the coefficients of the expression $-(2x - 5)(4x + 3(2x - 5))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-15}
Given School A and School B each have 3 teachers signing up for volunteer teaching, with School A having 2 males and 1 female, and School B having 1 male and 2 females, calculate the probability that the selected 2 teachers have the same gender.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{9}}
In the arithmetic sequence $\{a_n\}$, if $\frac{a_{1010}}{a_{1009}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, then the maximum positive integer value of $n$ for which $S_n > 0$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
The lines \(x = \frac{1}{4}y + a\) and \(y = \frac{1}{4}x + b\) intersect at the point \((1,2)\). The original answer is in \(\frac{k}{m}\) format, please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
There are several rows in the stands of a hockey arena, each row with 168 seats. For the final match, 2016 students from several sports schools were invited as spectators, with no more than 45 from each school. Students from any school must be seated in the same row. What is the minimum number of rows required in the arena to ensure this can always be achieved?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given $m \in \mathbb{R}$, if $f(x) = (m + 1)x^{\frac{2}{3}} + mx + 1$ is an even function, find the interval where $f(x)$ is monotonically increasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, +\infty)}
7. (10 points) The Spring Festival is approaching. Starting from January 17, 2011 (Monday), workers in the factory gradually return home for the New Year to reunite with their families. If the number of workers leaving the factory each day is the same, by January 31, there are still 121 workers left in the factory. During this 15-day period, the total workdays (one person working one day counts as 1 workday, and workers do not need to be counted after they leave the factory) recorded in the factory is 2011 workdays, with weekends off and no one absent. How many workers have returned home by January 31? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
A farmer bought 164 rabbits. If the farmer wanted to fill a certain number of cages with the same number of rabbits and needed to buy 6 more rabbits, how many cages did the farmer want to fill?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Using the inequality $1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+1}}{2k} \approx \ln 2$ and $1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+2}}{2k+1} \leq \ln 2$, determine the minimum number of terms needed to be summed in the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ to approximate $\ln 2$ within an error of $0.001$ without computing the exact value of $\ln 2$. The initial attempt involved comparing the differences for $k=1$, resulting in a difference of $-\frac{1}{6}$. How can this be used to find the required number of terms ($k$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Hamilton Avenue has eight houses. On one side of the street are the houses numbered $1, 3, 5, 7$ and directly opposite are houses $2, 4, 6, 8$ respectively. An eccentric postman starts deliveries at house $1$ and delivers letters to each of the houses, finally returning to house $1$ for a cup of tea. Throughout the entire journey he must observe the following rules:  
1. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout.  
2. Each house except house $1$ is visited exactly once (house $1$ is visited twice).  
3. The postman at no time is allowed to cross the road to the house directly opposite.  

How many different delivery sequences are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let \( \triangle ABC \) be a triangle with \( AB = 3 \), \( AC = 8 \), \( BC = 7 \). \( M \) and \( N \) are the midpoints of \( \overline{AB} \) and \( \overline{AC} \), respectively. Point \( T \) is selected on side \( BC \) such that \( AT = TC \). The circumcircles of triangles \( BAT \) and \( MAN \) intersect at \( D \). Compute \( DC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7\sqrt{3}}{3}}
A train 120 m in length crosses a telegraph post in 6 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and the height from $BC$ is $\frac{a}{2}$. The maximum value of $\frac{c}{b}+\frac{b}{c}$ is \_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{2}}
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x)+f\left(1-\dfrac{1}{x}\right)=\arctan x$$ for all real $x\neq0.$ (As usual, y=arctan $x$ means $-\pi/2<$ $y<\pi/2$ and tan$y=x.)$ Find $$\int_0^1f(x)\:dx.$$ The original answer is in the format \(\frac{m \pi}{n}\). Please give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
3rd Centromerican 2001 Problem A2 C and D are points on the circle diameter AB such that ∠AQB = 2 ∠COD. The tangents at C and D meet at P. The circle has radius 1. Find the distance of P from its center.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\sqrt{3}}}
Two kinds of vodka are mixed in the ratio 1 : 2 and 2 : 1 and they are sold fetching the profit 40 % and 20 % respectively. If the vodkas are mixed in equal ratio and the individual profit percent on them are increased by 4 / 3 and 5 / 3 times respectively, what is the profit percentage of the mixture?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.33\%}
已知抛物线 $\Gamma: y^{2}=8 x$ 的焦点为 $F$ ，过 $F$ 作一条直线 $l$ 与抛物线 $\Gamma$ 交于 $A 、 B$ 两点，分别过 $A 、 B$ 作抛物线 $\Gamma$ 的切线，与 $y$ 轴交于 $P 、 Q$ 两点. 求四边形 $A P Q B$ 面积的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
There are 10 positive even integers less than a certain number that contain digits 5 or 9. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given a geometric sequence {a_n} with real terms, and the sum of the first n terms is S_n. If S_{10} = 10 and S_{30} = 70, then find the value of S_{40}. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
From an equal quantity of squares with sides 1, 2, and 3, create a square of the smallest possible size.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
1. [4] Submit an integer $x$ as your answer to this problem. The number of points you receive will be $\max (0,8-|8 x-100|)$. (Non-integer answers will be given 0 points.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let the function $f(x)$ satisfy $f(x) = f\left(\frac{1}{{x+1}})$. The domain is $D=\left[0,+\infty \right)$, and the range is $A$. If the set $\{y|y=f(x), x\in \left[0,a\right]\}$ can take on all values in $A$, then the range of parameter $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{\sqrt{5}-1}{2}, +\infty)}
Let set $T = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Find the number of all nonempty subsets $A$ of $T$ such that $3 \mid S(A)$ and $5 \nmid S(A)$, where $S(A)$ is the sum of all the elements in $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45486}
In a line formation of 5 students, including 2 male students and 3 female students, if male student A does not stand at either end and only two of the three female students stand next to each other, then the number of different possible arrangements is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
11. Stephanie enjoys swimming. She goes for a swim on a particular date if, and only if, the day, month (where January is replaced by '01' through to December by '12') and year are all of the same parity (that is they are all odd, or all are even). On how many days will she go for a swim in the two-year period between January 1st of one year and December 31st of the following year inclusive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
Let $1,\alpha_1,\alpha_2,...,\alpha_{10}$ be the roots of the polynomial $x^{11}-1$. It is a fact that there exists a unique polynomial of the form $f(x) = x^{10}+c_9x^9+ \dots + c_1x$ such that each $c_i$ is an integer, $f(0) = f(1) = 0$, and for any $1 \leq i \leq 10$ we have $(f(\alpha_i))^2 = -11$. Find $\left|c_1+2c_2c_9+3c_3c_8+4c_4c_7+5c_5c_6\right|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
9.3. Find the value of the expression $x-\sqrt{2022 x}+2023$,

if $x-\sqrt{\frac{2022}{x}}=2023$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
When a plane intersects a cube, what type of regular polygon cannot be formed as a cross-section? Express your answer as the name of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Pentagon}}
On the side \( AB \) of the convex quadrilateral \( ABCD \), point \( M \) is chosen such that \( \angle AMD = \angle ADB \) and \( \angle ACM = \angle ABC \). The quantity three times the square of the ratio of the distance from point \( A \) to the line \( CD \) to the distance from point \( C \) to the line \( AD \) is equal to 2, and \( CD = 20 \). Find the radius of the inscribed circle of triangle \( ACD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt{10} - 2 \sqrt{15}}
Define the sequence of positive integers $\{a_n\}$ as follows:
- Let $a_1=1$, $a_2=3$.
- For each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$).

For example:
- $a_2=3_{10}=11_2$, so $a_3=11_3+2_3=6_{10}$.

Express $a_{2013}$ in base $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23097}
Let \(ABCD\) be a square with side length 5, and let \(E\) be the midpoint of side \(AB\). Let \(P\) and \(Q\) be the feet of perpendiculars from \(B\) and \(D\) to \(CE\), respectively, and let \(R\) be the foot of the perpendicular from \(A\) to \(DQ\). The segments \(CE, BP, DQ\), and \(AR\) partition \(ABCD\) into five regions. What is the median of the areas of these five regions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ |a| \geq 2 }
In quadrilateral $ABCD$, $\overrightarrow{AB}=(1,1)$, $\overrightarrow{DC}=(1,1)$, $\frac{\overrightarrow{BA}}{|\overrightarrow{BA}|}+\frac{\overrightarrow{BC}}{|\overrightarrow{BC}|}=\frac{\sqrt{3}\overrightarrow{BD}}{|\overrightarrow{BD}|}$, calculate the area of the quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let \( f(x) \) be an odd function defined on \( \mathbf{R} \) such that \( f(x) = x^2 \) for \( x \geq 0 \). If the inequality \( f(x+a) \geq 2 f(x) \) holds for any \( x \in [a, a+2] \), determine the range of the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\sqrt{2}, +\infty)}
When $x \in (0, +\infty)$, the inequality $c^{2}x^{2}-(cx+1)\ln x+cx\geqslant 0$ always holds, then the range of the real number $c$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac {1}{e},+\infty\right)}
On a line, there are blue and red points, with at least 5 red points. It is known that on any segment with endpoints at red points that includes at least one red point inside, there are at least 3 blue points. Additionally, on any segment with endpoints at blue points that includes exactly 2 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on a segment with endpoints at red points, not containing other red points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The jury, when preparing versions of the district math olympiad problems for grades $7, 8, 9, 10, 11$, aims to ensure that each version for each grade contains exactly 7 problems, of which exactly 4 do not appear in any other version. What is the maximum number of problems that can be included in the olympiad?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
十边形内任意三条对角线都不会在其内部交于同一点，问这个十边形所有对角线可以把这个十边形划分为 $\qquad$个区域.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{246}
The equation
$$
1 - 3 \sin \theta + 5 \cos 3\theta = 0 \quad (0 < \theta < 2\pi)
$$
has how many distinct solutions? Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A number is supposed to be multiplied by 5 and divided by 2, then squared. However, instead of multiplying the irrational number pi (π) by 5 and dividing it by 2, the number is divided by 5 and multiplied by 2 before being squared. What is the percentage of error obtained in the final result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97.44\%}
Given the function $y = \cos\left(x+ \frac {\pi}{5}\right)$, where $x\in\mathbb{R}$, determine the horizontal shift required to obtain this function's graph from the graph of $y=\cos x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{5}}
If $3x=2x+6$, then $3x \_\_\_\_\_\_=6.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $x,y,z>0$ and $x\geq max\{y,z\}$, find the minimum value of: $$A=\frac{x}{y}+2\sqrt{1+\frac{y}{z}}+3\sqrt[3]{1+\frac{z}{x}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+2\sqrt{2}+3\sqrt[3]{2}}
A necklace has a total of 99 beads. Among them, the first bead is white, the 2nd and 3rd beads are red, the 4th bead is white, the 5th, 6th, 7th, and 8th beads are red, the 9th bead is white, and so on. Determine the total number of red beads on this necklace.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
30 pikes were released into a pond, and they gradually eat each other. A pike is considered to be full if it has eaten at least three pikes (whether full or hungry). What is the maximum number of pikes that can be full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let  $P_1$ ,  $P_2$ ,  $P_3$ ,  $P_4$ ,  $P_5$ , and  $P_6$  be six parabolas in the plane, each congruent to the parabola  $y = x^2/16$ .  The vertices of the six parabolas are evenly spaced around a circle.  The parabolas open outward with their axes being extensions of six of the circle's radii.  Parabola  $P_1$  is tangent to  $P_2$ , which is tangent to  $P_3$ , which is tangent to  $P_4$ , which is tangent to  $P_5$ , which is tangent to  $P_6$ , which is tangent to  $P_1$ .  What is the diameter of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
11.069. The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the section made through this lateral edge and the height of the base is $Q$. Determine the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q\sqrt{Q/3}}
In the sequence \(\{a_n\}\), \(a_1 = -1\), \(a_2 = 1\), and \(a_3 = -2\), for all \(n \in \mathbb{N_+}\) it holds that \(a_n a_{n+1} a_{n+2} a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3}\) and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\). Determine the value of the sum of the first 4321 terms of the sequence \(S_{4321}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4321}
$1 \cdot 411,2,3,4,5,6$ 的排列 $\left(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}\right)$ 具有如下性质: 对于 $i=1,2,3,4,5,\left(n_{1}, n_{2}, \cdots, n_{i}\right)$ 都不是 $(1,2, \cdots, i)$ 的排列. 求这种排列的个数.

The permutation $\left(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}\right)$ of $1, 2, 3, 4, 5, 6$ has the following property: for $i=1,2,3,4,5, \left(n_{1}, n_{2}, \cdots, n_{i}\right)$ is not a permutation of $(1,2, \cdots, i)$. Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{461}
Let  $P(x)$  be a real quadratic trinomial, so that for all  $x\in \mathbb{R}$  the inequality  $P(x^3+x)\geq P(x^2+1)$  holds. Find the sum of the roots of  $P(x)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given two parallel lines $3x - 2y - 1 = 0$ and $6x + ay + c = 0$, the distance between them is $\frac{2\sqrt{13}}{13}$. Find the value of $\frac{c + 2}{a}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Three cards have numbers on their front and back sides as follows: 1 and 2, 3 and 4, 5 and 6. If these three cards are placed side by side, how many different three-digit numbers can be formed, given that 6 cannot be used as 9? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}