inclusionAI/Ring-lite-rl-data huggingface dataset:
In the game "Clap 7", natural numbers starting from 1 that contain the digit "7" are called "visible 7s", and multiples of 7 are called "invisible 7s". Among the natural numbers from 1 to 100, what is the total number of "visible 7s" and "invisible 7s"? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given that $i$ is the imaginary unit, and the complex number $z = \frac{a+i}{1-i}$, where $a$ is a real number, and $|z| = 1$, calculate the value of $a$. Express your answer as a real number, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
求最小的质数 $p$, 满足 $(p, N)=1$,其中, $N$ 为所有满足以下条件的 $\left(a_{0}, a_{1}, \cdots\right.$, $a_{2012}$ ）的个数：
(1) $\left(a_{0}, a_{1}, \cdots, a_{2012}\right)$ 是 $0,1, \cdots, 2012$的一个排列;
(2) 对 2013 的任意一个正约数 $m$ 及所有的 $n(0 \leqslant n<n+m \leqslant 2012)$ 有
$m \mid\left(a_{n+m}-a_{n}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Find the value of $\displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by 14% while buying and by 20% while selling. What is his percentage profit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5\%}
In a right tetrahedron \(ABCD\), \(DA\), \(DB\), and \(DC\) are mutually perpendicular. Let \(S\) and \(R\) represent its surface area and the radius of the circumscribed sphere, respectively. What is the maximum value of \(\frac{S}{R^2}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2}{3}(3+\\sqrt{3})"]}
6.1. Find the sum of all integers \( x \in [-3 ; 13] \) that satisfy the inequality

$$
\left(1-\operatorname{ctg}^{2} \frac{\pi x}{12}\right)\left(1-3 \operatorname{ctg}^{2} \frac{\pi x}{12}\right)\left(1-\operatorname{tg} \frac{\pi x}{6} \cdot \operatorname{ctg} \frac{\pi x}{4}\right) \leqslant 16
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Given that $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 + 2\sqrt{3}}
I have  $8$  unit cubes of different colors, which I want to glue together into a  $2\times 2\times 2$  cube.  How many distinct  $2\times 2\times 2$  cubes can I make?  Rotations of the same cube are not considered distinct, but reflections are.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
The power function $$y=(m^{2}-m-1)x^{m^{2}-2m-3}$$ is monotonically decreasing on the interval \[0, +∞), then the value of the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Define \( a_3 = \frac{2 + 3}{1 + 6} \), and for \( n \geq 4 \), \( a_n = \frac{a_{n-1} + n}{1 + n a_{n-1}} \). Find \( a_{1995} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1991009}{1991011}}
Given points C and D on segment AB, where M and N are the midpoints of AC and DB respectively, and given that AB = 10 and CD = 2, calculate the length of MN. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{7^{\sqrt{n+1}-\sqrt{n}}\cdot(\frac{n+1}{2})!\cdot(\frac{n+1}{2})!}{(n+1)\cdot(\frac{n}{2})!\cdot(\frac{n}{2})!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179}
Given a complex polynomial \(P_n(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0\), where \(\alpha_i\) are its complex roots, calculate the product \((\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^n P_n(-1)}
Determine the cardinality of the set of $\mathbb{Q}$-morphisms from the algebraic closure of $\mathbb{Q}$ to $\mathbb{C}$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
After the "I Love Beijing" knowledge competition, A, B, and C made predictions about their scores. A said: "My score is higher than B's." B said: "C's score is higher than mine and A's." C said: "My score is higher than B's." After the scores were announced, the scores of the three people are all different, and only one person predicted correctly. In this case, the order of the three people from highest to lowest score is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A, B, C}
At the same time, a car departs from point \( A \) to point \( B \) at a speed of \( 90 \) km/h, and a train departs from point \( B \) to point \( C \) at a speed of \( 60 \) km/h. After 2 hours from the start, they are at the shortest distance from each other. Find the distance between the points if all three points are equidistant from each other and connected by straight roads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210 \text{ km}}
Evaluate: $\displaystyle\int_{0}^{1}\frac{\ln(x+1)}{x^2+1}\,\mathrm{d}x$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{8} \ln(2)$}
What is the output of the given program when the input is 6? Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
A container can hold a certain number of orange slices. If a company had three hundred twenty-nine orange slices to put into containers, they would need 3 more slices to fill up the last container. How many orange slices can one container hold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let $\Omega,\Omega^*$ be bounded domains in $\mathbb{R}^n$, and $B_r(y_0)$ be a small ball of radius $r$ centered at $y_0\in\Omega^*$. In $\Omega^*$, define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on $B_r(y_0)$. We want to calculate the function obtained by applying the Legendre transform on $\psi$, namely $$u_0(x)=\sup\{x\cdot y-\psi(y):y\in B_r(y_0)\}$$ What is the explicit expression of $u_0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \cdot y_0 + r \sqrt{1+|x|^2}}
Find $\int \frac{\sin ^{3} x}{\cos ^{5} x} \, dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4 \cos^4 x} - \frac{1}{2 \cos^2 x} + C}
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$.  What is $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
Logan makes $65,000 a year. He spends a certain amount on rent every year, $5000 on groceries every year, and $8000 on gas every year. If he wants to have at least $42000 left and needs to make $10,000 more each year, how much does he spend on rent every year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$20,000}
In tetrahedron \(ABCD\), the dihedral angle between face \(ABC\) and face \(BCD\) is \(60^\circ\). The orthogonal projection of vertex \(A\) onto face \(BCD\) is \(H\), which is the orthocenter of \(\triangle BCD\). Point \(G\) is the centroid of \(\triangle ABC\). Given that \(AH = 4\) and \(AB = AC\), find the length of \(GH\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{21}}{9}}
Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}$. Find the value of $CE^2+DE^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Evaluate the sum of all positive irreducible fractions less than $1$ with the denominator $1991$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{900}
Between them, Mark, Sarah, and John have 36 traffic tickets. Mark has twice as many parking tickets as Sarah, and they each have an equal number of speeding tickets. John has one third the number of parking tickets as Mark, but twice as many speeding tickets as Sarah. If Sarah has 6 speeding tickets, how many parking tickets does Mark have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a cylinder with a base radius of $r$ and a height of $h$, its volume is 2, and its surface area is 12. Find the value of $\frac {1}{r} + \frac {1}{h}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In triangle $PQR,$ $PQ = 25$ and $QR = 10.$ Find the largest possible value of $\tan B.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\sqrt{21}}}
For a unitary matrix $U$, determine the condition on the $i$-th column vector of $U$ that ensures the $i$-th entry of the product $U\mathbf{x}$ is non-zero, where $\mathbf{x}$ is the $i$-th unit vector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U_{ii} \neq 0}
Determine the radius of convergence of the power series \( \displaystyle \sum_{n=0}^{\infty} \left(\int_{|z|=1} \frac{\cos{\left(\frac{1}{z-10}\right)}}{z^{n+1}} \,dz\right) z^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the number of real-valued solutions to the equation \( f(f(x)) = 25 \) for \( 0 \le x \le 2023 \), where \( f(x) = x\{x\} \) and \( \{x\} \) is the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1997001}
Calculate the volume of the subset \( \left\{ \begin{pmatrix} 2 & 33 & 444\\ 0 & 3 & 4 \\ 0 & 0 & 4 \end{pmatrix}\begin{pmatrix}x \\y \\ z \end{pmatrix}:x^2+y^2+z^2 \leq 1\right\} \) of \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi}
Two circles, each with a radius of 6 cm, have their centers on the surface of a sphere with a radius of 10 cm. These centers are also the points of tangency between the circles and the sphere. Determine if the circles can intersect, and if so, find the number of possible intersection points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the Ramsey number \( R(T, K_{1,n}) \) where \( T \) is a tree with \( m \) vertices and \( m-1 \) is a divisor of \( n-1 \), with \( m, n \ge 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 1}
Given the integers from 1 to 25, Ajibola wants to remove the smallest possible number of integers so that the remaining integers can be split into two groups with equal products. What is the sum of the numbers which Ajibola removes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Evaluate the improper integral: $$\int_{0}^{\infty}\text{sech}^2(x+\tan(x))dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( n \) be a given positive integer. Find the smallest positive integer \( u_n \) such that, for every positive integer \( d \), the number of numbers divisible by \( d \) in any \( u_n \) consecutive positive odd numbers is no less than the number of numbers divisible by \( d \) in the set of odd numbers \( 1, 3, 5, \ldots, 2n-1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
3. Let $\Sigma(n)$ denote the sum of the digits of the number $n$. Find the smallest three-digit $n$ such that $\Sigma(n)=\Sigma(2 n)=\Sigma(3 n)=\ldots=\Sigma\left(n^{2}\right)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he "only" solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively, with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1100}
Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be the sequence defined by
$$
a_{0}=1, a_{n+1}=\frac{1}{n+1} \sum_{k=0}^{n} \frac{a_{k}}{n-k+2}
$$
Find the limit
$$
\lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{a_{k}}{2^{k}}
$$
if it exists. The original answer is in the form $f\left(\frac{a}{b}\right)=\frac{e}{c}$, please give the value of a+b+c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
There are  $800$  marbles in a bag. Each marble is colored with one of  $100$  colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.

Suppose Anna has not stopped after drawing  $699$  marbles. Compute the probability that she stops immediately after drawing the  $700$ th marble.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{99}{101}}
The negation of the proposition "$\forall x > 2$, $x^{2} > 4$" is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exists x_{0} > 2, x_{0}^{2} \leqslant 4}
The given expression $\boldsymbol{F}(n)$, where $n$ denotes a variable positive integer, should be assigned a plus sign when $n$ is even and a minus sign when $n$ is odd. How can this be expressed concisely in an algebraic form without providing any verbal explanation of the formula?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^n F(n)}
Divide the sequence $\{2n+1\}$ cyclically into one-term, two-term, three-term, four-term groups as follows: $(3), (5,7), (9,11,13), (15,17,19,21), (23), (25,27), (29,31,33), (35,37,39,41), (43), \cdots$. What is the sum of the numbers in the 100th group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1992}
The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$  Find $a + b + c + d.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Calculate the value of \( \sum_{n=2}^{\infty} \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - e}
Let $P(x)=x^{3}+a x^{2}+b x+2015$ be a polynomial all of whose roots are integers. Given that $P(x) \geq 0$ for all $x \geq 0$, find the sum of all possible values of $P(-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9496}
A country is shaped like a square and is divided into 25 identical square counties. In each county rules either a knight-count, who always tells the truth, or a liar-count, who always lies. One day, each count said: "Among my neighbors, there are an equal number of knights and liars." What is the maximum number of knights that there could be? (Counts are considered neighbors if their counties share a common side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2  \cdot a_3 \cdot  ...  \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{61}{16}}
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times does $111$ appear in this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The line connecting the center of the circumscribed circle and the intersection point of the altitudes of a scalene triangle is parallel to the bisector of one of its angles. What is the measure of this angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects $AC$ at $X$. Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
An aquarium holds clownfish, blowfish, and angelfish in a ratio of 1:1:2, respectively. 26 of the blowfish stay in their own tank, and the remaining blowfish swim into a display tank. An equal number of clownfish join the blowfish in the display tank, but then a third of these clownfish swim back into their own tank. The number of angelfish in the display tank is 1.5 times the number of clownfish that remain in the display tank, and the angelfish that stay in their own tank is half the initial number of blowfish. If the aquarium holds a combined total of 180 fish, how many clownfish are now in the display tank?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Niles is collecting his book club's annual fees. Each of the six members pays $150/year towards snacks, plus $30 each for six hardcover books and $12 each for six paperback books. There is a 10% discount on snacks for long-standing members and three of them qualify for this discount. Additionally, members need to pay a 5% sales tax on all books. Moreover, the book club chooses three special event activities during the year, costing a total of $60 for each member. Lastly, in case the chosen books are of the mystery genre, the hardcover books cost $35 each, while if the books are pre-2010 publications, the paperback books cost only $10 each. During this year, four hardcover books are mysteries and two paperback books are pre-2010 publications. Calculate the total amount of money Niles collects from all the members.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$2903.40}
(9) (20 points) The angles of triangle $ABC$ satisfy: $\frac{A}{B}=\frac{B}{C}=\frac{1}{3}$; Find the value of $T=\cos A+\cos B+\cos C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1+\sqrt{13}}{4}}
How can we find the closed form of the following series? $$\sum_{k=0}^n\frac{1}{k+1}{\binom n k}^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(2n+1)!}{{(n+1)!}^2}}
The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x) \cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Golu started from his house towards north. After covering a distance of 8 km, he turned towards left (west) and covered a distance of 6 km. Then, he took another left (south) and walked for 10 km. Following that, he turned right (west) and walked for 3 km. Finally, he took a 45-degree turn towards southwest and covered an additional 5 km. What is the shortest distance now from his house?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.7}
How many distinct sets are there such that each set contains only non-negative powers of  $2$  or  $3$  and sum of its elements is  $2014$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \leq k \leq N$, with each possible $k$ having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let $p$ be the probability that the number of rocks left in the pile after each round is a multiple of 5. If $p$ is of the form $5^{a} \cdot 31^{b} \cdot \frac{c}{d}$, where $a, b$ are integers and $c, d$ are positive integers relatively prime to $5 \cdot 31$, find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-501}
Let $\pi_1$, $\pi_2$, $\pi_3$ be three planes in a projective space such that $\pi_1 \cap \pi_2 \cap \pi_3$ = $\emptyset$, $\dim\,(\pi_1 \cap \pi_2) = 1$, $\dim\,(\pi_1 \cap \pi_3) = \dim (\pi_2 \cap \pi_3) =0.$ Find the dimension of $\pi_1 \lor \pi_2\lor \pi_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the hyperbola $C$: $mx^{2}+ny^{2}=1(mn < 0)$, one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Determine the eccentricity of $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {5}{4}}
Given positive integers \(a_{1}, a_{2}, \ldots, a_{2023}\) such that \(a_{k}=\sum_{i=1}^{2023}\left|a_{k}-a_{i}\right|\) for all \(1 \leq k \leq 2023\), find the minimum possible value of \(a_{1}+a_{2}+\cdots+a_{2023}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046264}
The greatest common divisor of 217 and 155 is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
The Mad Hatter's clock gains 15 minutes per hour, and the March Hare's clock loses 10 minutes per hour. Once, they set their clocks by the Dormouse's clock (which stopped and always shows 12:00) and agreed to meet at 5 o'clock in the evening for a traditional five-o'clock tea. How long will the Mad Hatter wait for the March Hare if each arrives exactly at 17:00 by their own clocks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \text{ hours}}