inclusionAI/Ring-lite-rl-data huggingface dataset:
Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1028}
For finite sets  $A,M$  such that  $A \subseteq M \subset \mathbb{Z}^+$ , we define   $$ f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}. $$   Given a positive integer  $k$ , we call  $M$  *k-colorable* if it is possible to color the subsets of  $M$  with  $k$  colors so that for any  $A \subseteq M$ , if  $f_M(A)\neq A$  then  $f_M(A)$  and  $A$  have different colors.
Determine the least positive integer  $k$  such that every finite set  $M \subset\mathbb{Z}^+$  is k-colorable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 2 }
In a regular truncated quadrilateral pyramid, a section is made through the diagonals of the bases, and another section passes through a side of the lower base and the opposite side of the upper base. The angle between the intersecting planes is $\alpha$. Find the ratio of the areas of the sections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cos \alpha}
The perimeter of a right-angled triangle is 160 cm. If two of the sides are 40 cm and 50 cm, and the triangle is isosceles, find the length of the third side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find the largest positive integer \(d\) for which there exists a degree $d$ polynomial $P$ with real coefficients such that there are at most $d$ different values among $P(0), P(1), \ldots, P\left(d^{2}-\right.$ $d)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The lateral edges of a triangular pyramid are pairwise perpendicular, with one of them equal to \(a\) and also equal to the sum of the other two. Find the radius of the sphere that touches the base of the pyramid and its lateral faces' extensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2}}
In an infinite sequence of tosses of a fair coin, what is the expected number of tosses required for the pattern HTTH to appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Calculate the present value of a continuously varying, continuously paid annuity with payments $(8k + tk)$ made to an account, where $0 \leq t \leq 10$. The interest is credited at a force of interest $\delta_t = \frac{1}{8+t}$. After 10 years, the account's value is $20,000$. Find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1000}{9}}
Given the equation \( px^2 + y^2 + qz^2 + 2yz + zx + 3xy = 0 \) represents a pair of perpendicular planes, find the value of \( p - q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the greatest integer \( c \) such that the function \( f(x) = \frac{x-1}{c-x^2+1} \) does not take any values in the interval \([-1, -\frac{1}{3}]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
The perimeter of triangle $ABC$ is 1. Circle $\omega$ is tangent to side $BC$, the extension of side $AB$ at point $P$, and the extension of side $AC$ at point $Q$. The line passing through the midpoints of $AB$ and $AC$ intersects the circumcircle of triangle $APQ$ at points $X$ and $Y$. Find the length of segment $XY$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Find the number of pairs  $(n,C)$  of positive integers such that  $C\leq 100$  and  $n^2+n+C$  is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Let  $N$  be an odd number,  $N\geq 3$ .  $N$  tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called *suprising* if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher. 

It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(N-1)(3N-1)}{8}}
Given an equilateral triangular prism \( S-ABC \) with height \( SO = 3 \) and a base edge length of 6. A perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with the foot of the perpendicular at \( O' \). On \( AO' \), a point \( P \) is chosen such that \( AP:PO' = 8 \). Find the area of the cross-section passing through point \( P \) and parallel to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$20$}
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$.  How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{279}
In a certain pond, a biologist conducts an experiment over three days to estimate the fish population. On the first day, 80 fish were caught, tagged, and returned to the pond. On the second day, 100 fish were caught, of which 20 were found to have been tagged. On the third day, 120 fish were caught, and 36 of them were found to have been tagged. If the percent of tagged fish caught on each day approximates the percent of tagged fish in the pond, what is the approximate number of fish in the pond?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{320}
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? Express your answer in the form \(\frac{k}{m}\), where k and m are integers with no common factors other than 1. Find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Brendan works online and is paid every week. Once he gets the money, he recharges his debit card with half of his pay. In June, he earned $5000 and bought himself a used car. His total amount of remaining money at the end of the month is $1000. What is the worth of the used car he bought?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$4000}
9. The base of the pyramid $\mathrm{TABCD}$ is an isosceles trapezoid $\mathrm{ABCD}$, the midline of which is equal to $5 \sqrt{3}$. The ratio of the areas of the parts of the trapezoid $\mathrm{ABCD}$, into which it is divided by the midline, is $7: 13$. All lateral faces of the pyramid $\mathrm{TABCD}$ are inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the pyramid TAKND, where points $\mathrm{K}$ and $\mathrm{N}$ are the midpoints of edges $\mathrm{TB}$ and $\mathrm{TC}$, respectively, and AD is the larger base of the trapezoid ABCD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \(ABCD\) be an isosceles trapezoid with \(AB = 1\), \(BC = DA = 5\), and \(CD = 7\). Let \(P\) be the intersection of diagonals \(AC\) and \(BD\), and let \(Q\) be the foot of the altitude from \(D\) to \(BC\). Let \(PQ\) intersect \(AB\) at \(R\). Compute \(\sin \angle RPD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{5}}
Determine whether there are any solutions to the functional equation \( f(f(x)-x) = f(f(x)) \) other than the constant solution \( f(x) = k \), where \( k \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = k}
Randomly fill the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 into a $3 \times 3$ grid, with each small square filled with one number and all numbers being distinct. What is the probability that the sum of the numbers in each row and each column is odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{14}}
3. For how many positive integers $n$ less than 200 is $n^{n}$ a cube and $(n+1)^{n+1}$ a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Calculate the flux of the vector field \( F = g(y,z)i + 8j - (2z+3)k \) through the surface of a curved pipe \( S \), where \( S \) is bounded by two circles: \( C_1 \) with radius 3 in the plane \( y=0 \) and \( C_2 \) with radius 1 in the plane \( z=4 \). The volume of \( S \) is 12. Assume \( g \) has continuous partial derivatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83\pi - 24}
How many essentially different proportions can be made from the four letters $x, y, z, t$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Call a positive integer  $n\geq 2$  *junk* if there exist two distinct  $n$  digit binary strings  $a_1a_2\cdots a_n$  and  $b_1b_2\cdots b_n$  such that


- $a_1+a_2=b_1+b_2,$
- $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$  for all  $2\leq i\leq n-1,$  and
- $a_{n-1}+a_n=b_{n-1}+b_n$ .

Find the number of junk positive integers less than or equal to  $2016$ .

[i]Proposed by Nathan Ramesh
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Mary's income is 60% more than Tim's income, and Tim's income is 20% less than Juan's income. Mary saves 15% of her income, while Tim and Juan save 12% and 18% of their incomes, respectively. Mary's expenses are 25% of her income, while Tim's expenses are 30% of his income, and Juan spends 20% of his income on rent. Calculate the percentage difference between Mary's and Juan's total savings and expenses combined.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.74\%}
A quadrilateral and a pentagon are given on the plane (both may be non-convex), such that no vertex of one lies on the side of the other. What is the maximum possible number of points of intersection of their sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2019\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
If 5% more is gained by selling an article for Rs. 350 than by selling it for Rs. 345, what is the cost of the article?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $ x_{0} \equal{} x_{1995}$, such that
\[ x_{i \minus{} 1} \plus{} \frac {2}{x_{i \minus{} 1}} \equal{} 2x_{i} \plus{} \frac {1}{x_{i}},
\]
for all $ i \equal{} 1,\cdots ,1995$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{997} }
Shapovalov A.V.

Through the origin, lines (including the coordinate axes) are drawn that divide the coordinate plane into angles of $1^{\circ}$.

Find the sum of the abscissas of the points of intersection of these lines with the line $y=100-x$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8950}
Seven children sit in a circle. The teacher distributes pieces of candy to the children such that:

- Every child gets at least one piece of candy.
- No two children receive the same number of pieces of candy.
- The numbers of candy pieces given to any two adjacent children have a common factor other than 1.
- There is no prime dividing every child's number of candy pieces.

What is the smallest number of pieces of candy that the teacher must have ready for the children?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Given a triangular pyramid $P-ABC$, where the volume ${V}_{P-ABC}=\frac{4\sqrt{3}}{3}$, $\angle APC=\frac{\pi }{4}$, $\angle BPC=\frac{\pi }{3}$, $PA\perp AC$, $PB\perp BC$, and the plane $PAC$ is perpendicular to the plane $PBC$, find the volume of the circumscribed sphere of the triangular pyramid $P-ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{32\pi }{3}}
2018年9月10日，全国教育大会在北京召开，习近平总书记在会上提出“培养德智体美劳全面发展的社会主义建设者和接班人”.某学校贯彻大会精神，为学生开设了一门模具加工课，经过一段时间的学习，拟举行一次模具加工大赛，学生小明、小红打算报名参加大赛.

$参考数据 \left(\text{其中}t_i=\frac{1}{x_i}\right)：$

| $\sum \limits^{7}_{{i=1}}t_iy_i$ | $\overline{t}$ | $\sum \limits^{7}_{{i=1}}t_i^2-7\overline{t}^2$ |
| ------ | ------ | ------ |
| 1845   | 0.37   | 0.55   |

$参考公式：对于一组数据(u1,v1)，(u2,v2)，\ldots ，(un,vn)，其线性回归直线v=\alpha +\beta u的斜率和截距的最小二乘估计公式分别为 \hat{\beta } = \frac{\sum \limits^{n}_{{i=1}}u_iv_i-n\overline{u} \cdot \overline{v}}{\sum \limits^{n}_{{i=1}}u^2_i-n\overline{u}^2} ， \hat{\alpha } = \overline{v} - \hat{\beta } \cdot  \overline{u}.$
$小明和小红拟先举行一次模拟赛，每局比赛各加工一个模具，先加工完成模具的人获胜.两人约定先胜4局者赢得比赛.若小明每局获胜的概率为\frac{3}{5}，已知在前3局中小明胜2局，小红胜1局.若每局不存在平局，请你估计小明最终赢得比赛的概率.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{513}{625}$}
Find the angle within the range of -360° to 0° that has the same terminal side as the angle 1250°.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-190^\circ}
If one takes 6 different letters from "CN Dream" and arranges them in a row, how many different permutations are there that include the letter combination "ea" (in the same order)? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
How many digits in the product of the two eight-digit numbers 11111111 and 99999999 are odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
12 is 4% of a, and a certain percentage of b is 4. c equals b / a. What is the value of the percentage of b?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Philatelist Andrey decided to distribute all his stamps equally into 3 envelopes, but there was one extra stamp left. When he distributed them equally into 5 envelopes, there were 3 stamps left; finally, when he distributed them equally into 7 envelopes, 5 stamps were left over. How many stamps does Andrey have in total, given that he recently bought an additional album that can hold 150 stamps since his old album was no longer sufficient?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208}
During the summer, Mike mowed lawns for 20 days. He earned 5 dollars for each lawn mowed, and on average, he mowed 2 lawns every weekday and 3 lawns on each weekend day. Mike spent 24 dollars on new mower blades and an additional 15 dollars on gasoline. How many 5 dollar games could he buy with the money he had left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let $x_1, x_2, x_3$ be the solutions to the equation $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of
\[
\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12033}
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$363$}
A train passes a platform in 34 seconds. The same train passes a man standing on the platform in 20 seconds. The speed of the train is some km/hr, and the length of the platform is 210.0168 meters. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.00432}
设 $a=1+10^{-4}$ 。在 $2023 \times 2023$ 的方格表的每个小方格中填入区间 $[1, a]$ 中的一个实数。设第 $i$ 行的总和为 $x_{i}$ ，第 $i$ 列的总和为 $y_{i}$ ， $1 \leq i \leq 2023$ 。 用$a$表示$\frac{y_{1} y_{2} \cdots y_{2023}}{x_{1} x_{2} \cdots x_{2023}}$ 的最大值为$\frac{(m a+n)^{p}}{p^{p} a^{m}}$，求$m+n+p$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4046}
Xonathan Jue goes to the casino with exactly $1000. Each week, he has a $\frac{1}{3}$ chance of breaking even and a $\frac{2}{3}$ chance of losing $500. Evaluate the expected amount of weeks before he loses everything.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the minimum positive integer value of the expression (x + y^2)(x^2 - y)/(xy), where x and y are distinct positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given a circle with equation $x^2 + y^2 - 4x - 4y - 10 = 0$, there are at least three distinct points on the circle whose distances to the line $l : ax + by = 0$ are $2\sqrt{2}$. Determine the range of possible values for the angle of inclination of line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{\pi}{12}, \frac{5\pi}{12}]}
Determine the number of homomorphisms from a group $G$ of order $2^n(2m+1)$ to an abelian group of order $2^s$, where $0 \leq s \leq n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^s$}
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{11}{30}$}
6. Find the largest possible number of integers that can be selected from the set $\{1,2,3, \ldots, 100\}$ such that no two of them differ by 2 or 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Inscribe a rectangle of base  $b$  and height  $h$  in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of  $h$  do the rectangle and triangle have the same area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ h = \frac{2}{5} }
I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
From the natural numbers between 1 and 99, 50 different numbers were chosen such that no two of them sum to 100 and no two of them sum to 99. Which numbers were chosen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50, 51, 52, \ldots, 99}
There are $2022$ equally spaced points on a circular track $\gamma$ with a circumference of $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.

Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labelings of the $2022$ points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2042222}
Call a set of integers <i>spacy</i> if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\ldots,12\},$ including the empty set, are spacy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$129$}
Determine the maximum length of a repeating loop that can be obtained by repeatedly applying the function \( f(x) \) to any natural number \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 5, 6, 6, 7, 7\}$, with no digit used more times than it appears in the set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Find the total derivative of \( u(x,z) = \ln\left(\frac{x^\alpha + z^\alpha}{z^\alpha}\right) \) with respect to \( x \), given that \( z = h(x) = (ax^4 + b)^{1/3} \) and \( u'(x,z) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the angle between the tangents to the curve \( y = \frac{x^{2} \sqrt{3}}{24} \) that pass through the point \( M(4, -2 \sqrt{3}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Sir Bedevir will only participate in a jousting tournament if he knows that he has at least a $1 / 2$ probability of winning. In any given match, the probability of the opponents winning is proportional to their combat skills. Bedevir's combat skill is $1$, while the combat skill of the $n$-th opponent is $\frac{1}{2^{n+1}-1}$. How many knights must have registered for the tournament if, after careful calculation, Sir Bedevir decided to participate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Counting from the right end, what is the $2500$th digit of $10000!$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the expansion of \((1+x^3+x^5)^{20}\), how many distinct terms are present? Express the powers of \(x\) as \(P = 3b + 5c\) with \(a + b + c = 20\). Determine the number of distinct terms in the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Find the expected value of $g(X) = (X+1)f(X+2) - Xf(X+1)$ for a Poisson random variable $X$ with mean $\lambda$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \(\displaystyle\oint_C\left(\frac{1}{\sin(z)-1}\right)dz\) where \(C\) is a circle with radius 5, oriented clockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}