inclusionAI/Ring-lite-rl-data huggingface dataset:
For all positive integers $n$, let
\[ f(n) = \sum_{k=1}^n \varphi(k) \left\lfloor \frac{n}{k} \right\rfloor^2. \]
Compute $f(2019) - f(2018)$. Here, $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11431}
In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the term before that to the previous one:

$$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8, \ldots$$

As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12246}
A worker's daily wage is increased by 60%, and after the increase, they also receive a productivity bonus of $10 per day. The new total wage, including the bonus, is $45 per day. If the worker worked 8 hours per day, what was the worker's hourly wage before the increase and bonus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$2.73}
Evaluate the constant $C$ in terms of $p$ without using summation notation, given the following formula: $$C=\sum_{n_{L}=0}^{\infty}\left[\sum_{n_{G}=n_{L}}^{\infty}\left(\left(-1\right)^{n_{G}+n_{L}+1}\cdot\frac{p^{n_{G}}}{n_{L}!}\right)\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{e^p}{1+p}}
In a business meeting, each participant shakes hands with every other participant exactly once, except for Mr. L. Mr. L arrives late and only shakes hands with those still present. With a total of 100 handshakes recorded, how many people had already left the meeting before Mr. L's arrival?

(The question is from the CMI2011 Undergraduate Entrance Examination.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Points \( A, B, C, D \) lie on the circumference of a circle, and \( BC = DC = 4 \), \( AE = 6 \). The lengths of segments \( BE \) and \( DE \) are both integers. Find the length of \( BD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Kaylee studies in a school where the bell rings once a class starts and again when a class gets over. There is a break of 15 minutes after every class. On most days, Kaylee's classes in order are: Maths, History, Geography, Science, and Music. 

However, on Monday, the schedule was altered due to a special school assembly in the morning that lasted 45 minutes, which delayed the start of the classes and consequently the ringing of the bell. The assembly added another instance of the bell ring at the start and end of it. Furthermore, after the History class, there was a surprise pop quiz which also had a start and an end bell ring. When the surprise quiz was happening, the regular class bells did not ring. 

If right now she is in her Geography class, how many times would the bell have rung by now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A man buys 350 pens at the marked price of 270 pens from a wholesaler. If he sells these pens giving a discount of 4.3%, what is the profit percent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24.06\%}
Rachel had to complete some pages of math homework, 7 pages of reading homework and 3 more pages of biology homework. She had to complete 11 pages of math and biology homework. How many pages of math homework did she have to complete?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For how many positive integers $n$ does $1+2+...+n$ evenly divide from $6n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5$}
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, find the value of $y$. Express your answer in degrees, rounded to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67.5}
On a lengthy, one-way, single-lane highway, cars travel at uniform speeds and maintain a safety distance determined by their speed: the separation distance from the back of one car to the front of another is one car length for each 10 kilometers per hour of speed or fraction thereof. Cars are exceptionally long, each 5 meters in this case. Assume vehicles can travel at any integer speed, and calculate $N$, the maximum total number of cars that can pass a sensor in one hour. Determine the result of $N$ divided by 100 when rounded down to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In the logging yard, there were logs 6 meters and 8 meters long of the same thickness. Which logs are more advantageous to take in order to obtain more pieces with fewer cuts to saw them into 1-meter pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{It is more advantageous to use 6-meter logs.}
Vincent has 72 inches of rope that he wants to use for a project, but he needs to cut it into smaller pieces first. He cuts it into 12 equal length pieces, but then he realizes it's too short, so he ties three pieces together. The knots then make each piece 1 inch shorter. How long are his pieces of rope after all this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
There are  $15$  (not necessarily distinct) integers chosen uniformly at random from the range from  $0$  to  $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is  $\frac mn$  for relatively prime positive integers  $m,n$ . Find  $100m+n$ *Proposed by Yannick Yao*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
展览馆中有 2015 幅画在展出. 每位观众选择一对画，并将它们的名字写在同一张纸上. 一个艺术家从这张纸中选择一些画对，并评判每对中哪幅画更好. 之后助手按下述规则继续评判：在每一步中，如果对某个正整数 n，在画对 (A_1,A_2) 中 A_1 比 A_2 好，在画对 (A_2,A_3) 中 A_2 比 A_3 好，\cdots\cdots，在画对 (A_{n-1},A_n) 中 A_{n-1} 比 A_n 好，则助手可以在画对 (A_1,A_n) 中评判 A_1 比 A_n 好. 求最小的正整数 k，使得无论纸上写了哪些画对，艺术家都可以对其中的 k 对评判，使得助手能在有限步内对余下所有的画对评判.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
设函数$f _ { 0 } ( x ) = \vert x \vert , f _ { 1 } ( x ) = \vert f _ { 0 } ( x ) - \vert \vert , f _ { 2 } ( x ) = \vert f _ { 1 } ( x ) - 2 \vert$ 则函数$y = f _ { 2 } ( x )$的图像与x轴所围成图形中的封闭部分的面积是.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
对正整数 $n$ 及实数 $x(0 \leqslant x<n)$, 定义
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1}
$$
其中 $[x]$ 表示不超过实数 $x$ 的最大整数， $\{x\}=x-[x]$ 。若整数 $m, n \geqslant 2$ 满足
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$
求 $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Six circles with radii equal to 1 are placed on a plane such that the distance between the centers of any two of them is greater than \( d \). What is the smallest \( d \) for which there exists a line that does not intersect any of the circles, and on each side of which lie three circles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a ticket sale event, there is a tiered discount system. For the first 5 tickets purchased, 1 ticket is given for free. For every additional ticket purchased after the first 5, 2 extra tickets are given for free. If a customer purchases 10 tickets under this sale, what percentage of the original price of all the tickets received is saved?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.38\%}
What is the limit of the following expression as \( n \) approaches infinity? $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{e}\)}
The product of three prime numbers is 5 times greater than their sum. Find all such triples of numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 5, 7}
If $cos\frac{π}{5}$ is a real root of the equation $ax^{3}-bx-1=0$ (where $a$ and $b$ are positive integers), then $a+b=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The set \( S \) contains 6 distinct positive integers. The average of the two smallest numbers is 5 and the average of the two largest is 22. What is the greatest possible average of all the numbers in the set \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.5}
Peter Pan and Crocodile are each getting hired for a job. Peter wants to get paid 6.4 dollars daily, but Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, 80 cents on day 4, and so on. After how many whole days will Crocodile's total earnings exceed that of Peter's?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$, 以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K$, 已知 $B C=25, B D=20, B E=7$, 求 $A K$ 的长。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{241}
On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
Evaluate the sum $\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$ for a positive real number $x$, without using the Gamma function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{x}\)}
Find the fourth derivative of the inverse function $f^{-1}(y)$ evaluated at $y=1$, where $f(x) = \sin{x} + x^2 + 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral: $$\int_0^\infty \frac{1}{x^{2n+5}} \left ( \cos x - \sum_{k=0}^n \frac{(-1)^k x^{2k+2}}{(2k + 2)!} \right ) \mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a group of people participating in some curriculum, 25 of them practice yoga, 15 study cooking, 8 study weaving, 2 of them study cooking only, some of them study both the cooking and yoga, 3 of them participate all curriculums. If 3 people study both cooking and weaving, how many people study both cooking and yoga?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $r=H_{1}$ be the answer to this problem. Given that $r$ is a nonzero real number, what is the value of $r^{4}+4 r^{3}+6 r^{2}+4 r ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given the binomial expansion $(3x-2)^{6}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}$, find the value of $a_{0}+a_{1}+2a_{2}+3a_{3}+4a_{4}+5a_{5}+6a_{6}=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82}
**
Evaluate 
\[
\frac{3}{\log_8{3000^4}} + \frac{4}{\log_9{3000^4}},
\]
giving your answer as a fraction in lowest terms.

**
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Aaron has some brothers. Bennett has 6 brothers. What is the relationship between the number of Bennett's brothers and twice the number of Aaron's brothers if Aaron has 4 brothers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The set \( S \) consists of 9 distinct positive integers. The average of the two smallest integers in \( S \) is 5. The average of the two largest integers in \( S \) is 22. Calculate the greatest possible average of all of the integers of \( S \). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers, whereas $69$ and $157$ are not. 

Let $A$ and $B$ be $4$-digit palindrome numbers, and $C$ be a $3$-digit palindrome number. Given that $A - B = C$, what is the value of $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
14. $A, B, C$ three people went to Teacher $D$'s house to play. Teacher $D$ gave each of them a hat and wrote a four-digit number on each hat. It is known that these three four-digit numbers are all perfect squares (for example, $4=2^{2}, 100=10^{2}, 4.100$ are all squares of some number, such numbers are called perfect squares), and the tens digit of these three four-digit numbers is 0, and the units digit is not 0. Each child can only see the numbers on others' hats. These three children are very smart and honest, and the following conversation took place:
$A$ said: “The units digit of the numbers on $B$ and $C$'s hats are the same.”
$B$ and $C$ said at the same time: “After hearing what $A$ said, I know my number.”
$A$ said: “After hearing what $B$ and $C$ said, I also know my number, and the units digit of my number is an even number.”
Find: The sum of the numbers on $A, B, C$'s hats.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14612}
In a rectangle  $ABCD$ , two segments  $EG$  and  $FH$  divide it into four smaller rectangles.  $BH$  intersects  $EG$  at  $X$ ,  $CX$  intersects  $HF$  and  $Y$ ,  $DY$  intersects  $EG$  at  $Z$ . Given that  $AH=4$ ,  $HD=6$ ,  $AE=4$ , and  $EB=5$ , find the area of quadrilateral  $HXYZ$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The number of $x$-intercepts on the graph of $y=\sin(1/x)$ in the interval $(0.0001,0.001)$ is closest to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2865}
Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A \), \( B \), and \( C \) such that \( A B = B C \). From points \( A \), \( B \), and \( C \), perpendiculars \( A D \), \( B E \), and \( C F \) are drawn to line \( m \) with feet at \( D \), \( E \), and \( F \) respectively. Given \( A D = \sqrt{15} \), \( B E = \frac{7}{2} \), and \( C F = \sqrt{10} \), find the distance between \( l \) and \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
A batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \( v \) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at least \( \left(\frac{v}{20}\right)^{2} \) kilometers, how many hours will it take to transport all the supplies to the disaster area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the country Máxico, there are two islands: the island "Mayor" and the island "Menor". The island "Mayor" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offers direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city on the island "Mayor" to any other city on the island "Mayor".

If the percentage of flights that connect two cities in the same state is the same for the flights of each airline, what is the least number of cities that can be on the island "Mayor"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
The function f(n) defined on the set of natural numbers N is given by:

f(n) = {
n - 3  (n >= 1000)
f[f(n + 7)]  (n < 1000)
}

Calculate the value of f(90). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
3. What is the smallest radius of a sphere from which a regular quadrilateral pyramid with a base edge of 14 and an apothem of 12 can be cut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\sqrt{2}}
In the diagram, rectangle \( PRTV \) is divided into four rectangles. The area of rectangle \( PQXW \) is 9. The area of rectangle \( QRSX \) is 10. The area of rectangle \( XSTU \) is 15. Calculate the area of rectangle \( WXUV \). Express your answer as a fraction or whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{2}}
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. What is the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
In $\triangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\triangle ABC$ so that each $n$-gon shares a side with a distinct side of $\triangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $a=\sqrt{3}$, $b=\log_{3}2$, $c=\cos{2}$. Arrange $a$, $b$, and $c$ in increasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c < b < a}
Concentric circles  $\Omega_1$  and  $\Omega_2$  with radii  $1$  and  $100$ , respectively, are drawn with center  $O$ . Points  $A$  and  $B$  are chosen independently at random on the circumferences of  $\Omega_1$  and  $\Omega_2$ , respectively. Denote by  $\ell$  the tangent line to  $\Omega_1$  passing through  $A$ , and denote by  $P$  the reflection of  $B$  across  $\ell$ . Compute the expected value of  $OP^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10004}
In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that the two roots of the equation x^2 + (k^2 - 4)x + k - 1 = 0 are opposites of each other, find the value of k. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Let  $n > 1$  be an integer and  $X = \{1, 2, \cdots , n^2 \}$ . If there exist  $x, y$  such that  $x^2\mid y$  in all subsets of  $X$  with  $k$  elements, find the least possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n + 1}
已知$z _ { 1 } , z _ { 2 } , z _ { 3 } , \cdots , z _ { 6 }$是2+i的六次方根，而它们的辐角主值分别为$\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } , \cdots , \alpha _ { 6 }$ 求$\tan \alpha _ { 1 } \tan \alpha _ { 2 } + \tan \alpha _ { 2 } \tan \alpha _ { 3 } + \cdots + \tan \alpha _ { 6 } \tan \alpha _ { 1 }$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
A roulette wheel can land on any number from 0 to 2007 with equal probability. The roulette wheel is spun repeatedly. Let \( P_{k} \) denote the probability that at some point the sum of the numbers that have come up in all the spins made equals \( k \). Which number is greater: \( P_{2007} \) or \( P_{2008} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P_{2007}}
Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself).

*Proposed by Yang Liu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9901}
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{147}{1024}}
一城市设计成分为若干方块的矩形形状，n条街道互相平行，另外m条街道与它们相交成直角.警察在城市的街道上（不是十字路口）值勤.每个警察报告从他身旁开过去的汽车的号码、运行方向及通过时间.为了能根据警察提供的数据同时再现沿着闭合线路（即不重复通过同一地段的线路）行驶的任何汽车的路线，至少需要多少警察值勤？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ $(m-1)(n-1)$ }
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1990}
For which $n$ can an $n \times n$ grid be divided into one $2 \times 2$ square and some number of strips consisting of five cells, in such a way that the square touches the side of the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 2 \pmod{5}}
Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, requiring that any two adjacent digits have different parities, and 1 and 2 are adjacent. The number of such six-digit numbers is (answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In how many ways can 10 fillér and 50 fillér coins be placed side by side (with all centers on a straight line) to cover a $1 \mathrm{~m}$ long segment (not more), using at least 50 coins, and considering the order of the two types of coins? (Coins of the same value are not distinguished. The diameter of the 10 fillér is $19 \mathrm{~mm}$, and the diameter of the 50 fillér is $22 \mathrm{~mm}$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270725}
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the numbers on the three faces adjacent to that vertex. If the sum of the numbers on the vertices is $1512$, and the sum of the numbers on one pair of opposite faces is $8$, what is the sum of the numbers on all the faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}