inclusionAI/Ring-lite-rl-data huggingface dataset:
Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
将与 105 互素的所有正整数从小到大排成数列, 试求出这个数列的第 1000 项.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2186}
Add 78.621 to 34.0568 and round to the nearest thousandth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112.678}
Find the limit of the sequence $\{b_n\}$ defined recursively as $b_n = \sqrt{b_{n-1}b_{n-2}}$ for $n≥3$, where $b_1$ and $b_2$ are positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b_1^{\frac{1}{3}}b_2^{\frac{2}{3}}}
For each real number \(a\) with \(0 \leq a \leq 1\), let numbers \(x\) and \(y\) be chosen independently at random from the intervals \([0, a]\) and \([0, 1]\), respectively, and let \(P(a)\) be the probability that
\(\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\).
Find the maximum value of \(P(a)\). The original answer is in the form \(k - \sqrt{m}\), please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
已知集合 $A=\{1,5\}, B=\left\{x^{2}-3 x+1, x^{2}-4 x+5\right\}$ ，若集合 $A \cup B$ 的元素个数是 3 ，请给出可能的实数 $x$ 中的最大值与最小值之差。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(ABCD\) be a cyclic quadrilateral, and suppose that \(BC = CD = 2\). Let \(I\) be the incenter of triangle \(ABD\). If \(AI = 2\) as well, find the minimum value of the length of diagonal \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{3}}
Machine A takes some hours to complete a certain job and starts that job at 7 am. After two hours of working alone, Machine A is joined by Machine B and together they complete the job at 12 pm. It would have taken Machine B 18.07228915662651 hours to complete the job if it had worked alone for the entire job. How long does it take for Machine A to complete the job alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the integral $$\iint_D \frac{\partial}{\partial y}\frac{y}{(x^2+y^2)^2} \, dx \, dy$$ where the domain $D$ is defined as $D=\left\{ (x,y) \in \mathbb{R}^2 \mid 1\leq x^2+y^2 \leq 4 \right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3\pi}{4}}
The sequence $\{a_n\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\cos \frac {2n\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Calculate $S_{24}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{304}
If the circle $x^2+y^2=r^2$ ($r>0$) intersects with the circle $(x+3)^2+(y-4)^2=36$, then the range of $r$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,11)}
Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57.3}
Milo's parents tell him that he can win cash rewards for good grades. He will get $5 times the average grade he gets. He gets three 2s, four 3s, a 4, and a certain number of 5s. He gets $15 cash. How many 5s did he get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The solution to the inequality

$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$

is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.

Recall that $[x]$ denotes the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78.00}
Natasha climbs up a hill with varying elevation levels and changing weather conditions. The hill has three distinct elevation levels: the first section has an inclination of 15 degrees, the second section has an inclination of 30 degrees, and the third section has an inclination of 45 degrees. Additionally, during her journey, Natasha experiences four weather conditions: clear weather, light rain, heavy rain, and strong winds. 

The weather conditions affect her average speed as follows:
- Clear weather: no change in speed
- Light rain: reduces her speed by 10%
- Heavy rain: reduces her speed by 35%
- Strong winds: reduces her speed by 25%

It takes Natasha 4 hours to reach the top, during which she experiences clear weather in the first section, light rain in the second section, and strong winds in the third section. On her descent, it takes her 2 hours, with heavy rain affecting her speed throughout the entire descent. If her average speed along the whole journey is 3.5 kilometers per hour, what was her average speed (in kilometers per hour) while climbing to the top under these varying conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.625}
There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2040}
94. The base of the pyramid $ABCD$ is an equilateral triangle $ABC$ with a side length of 12. The edge $BD$ is perpendicular to the plane of the base and equals 10 $\sqrt{3}$. All vertices of this pyramid lie on the lateral surface of a right circular cylinder, whose axis intersects the edge $BD$ and the plane $ABC$. Determine the radius of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{6}}{2}}
已知三个月球探测器$\alpha $，$\beta $，$\gamma $共发回三张月球照片$A$，$B$，$C$，每个探测器仅发回一张照片．甲说：照片$A$是$\alpha $发回的；乙说：$\beta $发回的照片不是$A$就是$B$；丙说：照片$C$不是$\gamma $发回的．若甲、乙、丙三人中有且仅有一人说法正确，则照片$B$是探测器_____发回的．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\alpha $}
Triangle $ABC$ has $\overline{AB} = 5$, $\overline{BC} = 4$, $\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$, and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
Determine the value of $f_{2023}(2^{2023})$ given that $f_n(k)=f_1(f_{n-1}(k))$ and that $f_x(2^x) = 1$ whenever $x$ is a multiple of 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A rectangular table of size \( x \) cm by 80 cm is covered with identical sheets of paper of size 5 cm by 8 cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Given that points \( B \) and \( C \) are in the fourth and first quadrants respectively, and both lie on the parabola \( y^2 = 2px \) where \( p > 0 \). Let \( O \) be the origin, and \(\angle OBC = 30^\circ\) and \(\angle BOC = 60^\circ\). If \( k \) is the slope of line \( OC \), find the value of \( k^3 + 2k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
The monotonic decreasing interval of the function $f(x) = x^2 - 2|x|$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1] \cup [0, 1]}
Given two quadratic functions $y=x^{2}-2x+2$ and $y=-x^{2}+ax+b$ $(a > 0,b > 0)$, if their tangent lines at one of their intersection points are perpendicular to each other, find the maximum value of $ab$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{16}}
Kira has  $3$  blocks with the letter  $A$ ,  $3$  blocks with the letter  $B$ , and  $3$  blocks with the letter  $C$ . She puts these  $9$  blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence  $ABCAABCBC$  the blocks with the letter A have distances  $1, 3$ , and  $4$  between one another, the blocks with the letter  $B$  have distances  $2, 4$ , and  $6$  between one another, and the blocks with the letter  $C$  have distances  $2, 4$ , and  $6$  between one another. Altogether, we got distances of  $1, 2, 3, 4$ , and  $6$ ; these are  $5$  distinct distances. What is the maximum number of distinct distances that can occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Ancient Greek mathematicians from the Pythagorean school studied various polygonal numbers, such as triangular numbers 1, 3, 6, 10, ..., with the $n$-th triangular number being $\frac{n(n+1)}{2} = \frac{1}{2}n^{2} + \frac{1}{2}n$. Let the $n$-th $k$-sided polygon number be denoted as $N(n, k)$ ($k \geq 3$). Below are the expressions for the $n$-th number of some $k$-sided polygon numbers:  
Triangular numbers $N(n, 3) = \frac{1}{2}n^{2} + \frac{1}{2}n$  
Square numbers $N(n, 4) = n^{2}$  
Pentagonal numbers $N(n, 5) = \frac{3}{2}n^{2} - \frac{1}{2}n$  
Hexagonal numbers $N(n, 6) = 2n^{2} - n$  
...  
From this, we can deduce the expression for $N(n, k)$ and calculate $N(8, 12) = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Given that point $P$ is a moving point on circle $C_{1}$: $\left(x-1\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-1\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{26}-3}
In the Cartesian coordinate plane $(xOy)$, point $A(-2,0)$ is given. Point $B$ is on the circle $C$: $(x-2)^{2}+y^{2}=4$. Point $M$ is the midpoint of $AB$. If there exists a point $P$ on the line $l$: $y=kx-\sqrt{5}k$ such that $\angle OPM = 30^{\circ}$, find the range of values for the real number $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 2]}
The diagonal \( BD \) of quadrilateral \( ABCD \) is the diameter of the circle circumscribed around this quadrilateral. Find the diagonal \( AC \) if \( BD = 2 \), \( AB = 1 \), and \( \angle ABD : \angle DBC = 4 : 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2} + \sqrt{6}}{2}}
On the chessboard, $ 32$ black pawns and $ 32$ white pawns are arranged. In every move, a pawn can capture another pawn of the opposite color, moving diagonally to an adjacent square where the captured one stands. White pawns move only in upper-left or upper-right directions, while black ones can move in down-left or in down-right directions only; the captured pawn is removed from the board. A pawn cannot move without capturing an opposite pawn. Find the least possible number of pawns which can stay on the chessboard.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Evaluate the second-order partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text {if }(x,y) \neq (0,0) \\ 0 & \text{otherwise}\end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A 275 meter long train crosses a man standing on the platform in 7 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{141.43}
Equilateral \(\triangle ABC\) has side length \(1\), and squares \(ABDE\), \(BCHI\), \(CAFG\) lie outside the triangle. Find the area of hexagon \(DEFGHI\). The answer is in the form \(k+\sqrt{m}\), please give the value of \(k+m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a sequence of length 15 consisting of zeros and ones, find the number of sequences that have either all zeros consecutive or all ones consecutive, but not both.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{238}
Machine A can finish a job in some hours, machine B can finish the job in 2 hours, and machine C can finish the job in 6 hours. It takes 1.090909090909091 hours for A, B, and C together to finish the job. How many hours does it take for machine A to finish the job alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A company occupies three floors in a tower: the 13th, 14th, and 25th floors. A cleaning crew is hired to clean these spaces. They work for four hours on the 13th and 14th floors. Then, the crew splits in half: one half cleans the 25th floor while the other half continues cleaning the 13th and 14th floors. After four more hours, the cleaning of the lower two floors is complete, but the 25th floor requires an additional cleaner who will finish it in 8 more hours the next day. Assuming that the cleaning time is directly proportional to the area being cleaned, how many cleaners were in the crew initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
在三角形 $CAT$ 中，$\angle ACT =\angle ATC$ 且 $\angle CAT = 36^\circ$。如果 $\overline{TR}$ 平分 $\angle ATC$，求 $\angle CRT$ 的度数。如果答案是 $x^\circ$，求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Simplify $\tan \frac{\pi}{24} + \tan \frac{7 \pi}{24}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{6} - 2 \sqrt{3}}
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfies the equation \( f\left(x^{2}\right) f^{\prime \prime}(x) = f^{\prime}(x) f^{\prime}\left(x^{2}\right) \) for all real \( x \). Given that \( f(1) = 1 \) and \( f^{\prime \prime \prime}(1) = 8 \), find the value of \( f^{\prime}(1) + f^{\prime \prime}(1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
定义在实数上的函数 $f(x)=\frac{\sin \pi x}{\sqrt{1+x+x^{2}}}$ 的最小值是以 \frac{m \sqrt{n}}{k} 的形式表示，请给出 m + n + k 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Ray B is the perpendicular bisector of AC. Triangle ABC has a perimeter of $41.82$. What is the measure of AB?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.94}
Given an arithmetic sequence {a_n} with common difference d, if the variance of a_1, a_2, a_3, a_4, a_5, a_6, a_7 is 1, find the value of d. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\frac{1}{2}}
Given a random variable $\xi$ that follows the normal distribution $N(0, \sigma^2)$, if $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.954}
Triangles are drawn between the rays of a 30-degree angle as shown in the figure. Points \(A_{i}\) lie on one ray, and points \(B_{i}\) lie on the other ray. The angles \(\angle O A_{i} B_{i}\) are 120 degrees, and the angles \(\angle O B_{i} A_{i+1}\) are right angles. Determine the ratio of the areas of triangles \(A_{1} B_{1} A_{2}\) and \(A_{48} B_{48} A_{49}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2^{94}}
For which \( n \) can we find a cyclic shift \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) (i.e., \( i, i+1, i+2, \ldots, n, 1, 2, \ldots, i-1 \) for some \( i \)) and a permutation \( b_1, b_2, \ldots, b_n \) of \( 1, 2, 3, \ldots, n \) such that \( 1 + a_1 + b_1 = 2 + a_2 + b_2 = \ldots = n + a_n + b_n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is odd}}
In a class of 40 students, the first group of 15 students has an average score of 70% on an exam. The scores of the second group of 10 students have a weighted average of 85%, where each of their score counts twice as much as the scores of the first group. The third group of 10 students has an average of 92%, where each of their score counts thrice as much as the scores of the first group. The last group of 5 students have an average score of 60%, with each of their score counting half as much as the scores of the first group. What is the overall weighted average score in percentage for the entire class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83.85\%}
Evaluate the integral $\int_0^{\frac{\pi}{2}} \sqrt{\tan\theta} + \sqrt{\cot\theta} d\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt2 \pi}
Find the maximum value of the expression $$\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$$ given that $a$, $b$, and $c$ are positive real numbers such that $3a + 4b + 5c = 12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The sequence $(a_n)$ is defined by the following initial conditions and recurrence relations:

- Initial conditions: 
  \[ a_1 = 1, \quad a_2 = 0 \]

- Recurrence relations for $k \in \mathbb{N}$:
  \[ a_{2k+1} = a_k + a_{k+1}, \quad a_{2k+2} = 2a_{k+1} \]

Find the value of $a_m$ for $m = 2^{19} + 91$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
On the New Year's table, there are 4 glasses arranged in a row: the first and third contain orange juice, and the second and fourth are empty. In anticipation of guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings, after which the configuration becomes the reverse: the first and third are empty, and the second and fourth are full.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the tetrahedron \(ABCD\), \(\angle ACB = \angle CAD = 90^{\circ}\) and \(CA = CB = AD / 2\) and \(CD \perp AB\). What is the angle between the faces \(ACB\) and \(ACD\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
A 250 m long train running at a certain speed crosses another train running in opposite direction at the speed of 80 km/hr in 9 sec. The length of the other train is 250.04 m. What is the speed of the first train in km/hr?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120.016}
Given that point $O$ is a point in the plane of $\triangle ABC$, $4\overrightarrow{OA}•\overrightarrow{AC}={\overrightarrow{OC}^2}-1$, $4\overrightarrow{OC}•({\overrightarrow{OB}+\overrightarrow{OC}})=1-{\overrightarrow{OB}^2}$, then the maximum value of the angle between the vectors $4\overrightarrow{OA}+\overrightarrow{OB}$ and $2\overrightarrow{OA}-\overrightarrow{OC}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
For how many positive integers $x$ is $(x-2)(x-4)(x-6) \cdots(x-2016)(x-2018) \leq 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1514}
Peter wants to create a unique die where each face displays a different positive integer. For any two adjacent faces, the numbers must differ by at least two. Determine the minimum possible sum of the six numbers on the die.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
To shift the graph of the function $y=\sin \left(2x- \frac{\pi}{4}\right)$ to the graph of the function $y = \sin(2x)$, determine the horizontal shift required.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{8}}
Zhenya took a $3 \times 3$ board and placed a column of blue and red cubes on each cell. Then he drew a diagram of the resulting arrangement: he wrote down the number of cubes of both colors in each column (the order of the cubes is unknown).

What is the maximum number of blue cubes Zhenya can see if he looks at the construction from the front? (For example, if a column of 8 cubes stands behind a column of 5, he will see all 5 cubes of the front column and only 3 upper cubes of the rear column.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{683}
The time on a digital clock reads 3:33. What is the shortest length of time, in minutes, until all of the digits are again equal to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Let \( S \) be a subset of \(\{1,2,3, \ldots, 199,200\}\). We say that \( S \) is pretty if, for every pair of elements \( a \) and \( b \) in \( S \), the number \( a - b \) is not a prime number. What is the maximum number of elements in a pretty subset of \(\{1,2,3, \ldots, 199,200\}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
In a dice game, you start by rolling a single fair six-sided die. If you roll an even number, you then roll two dice. If the sum of the two dice is even, you roll three dice, and this process continues, increasing the number of dice by one each time, until you roll an odd sum. The first odd sum you roll is your score. Calculate the average score per turn in this game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{4}}
A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is 35 7/17. What is the value of the erased number? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the value of the 2-form $dxdy+3dxdz$ on the oriented triangle with vertices $(0,0,0)$, $(1,2,3)$, and $(1,4,0)$ in that order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7/2}
Xiaohong walks up a mountain, resting for 10 minutes every 30 minutes. Walking down the mountain, she rests for 5 minutes every 30 minutes. The speed of walking down is $1 \frac{1}{2}$ times the speed of walking up. If Xiaohong took 3 hours and 50 minutes to walk up, how much time did it take to walk down?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \text{ hours } 15 \text{ minutes}}