inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the approximate interval where the zero of the function $f(x) = \ln x - \frac{3}{x}$ is located. Express your answer as an interval in the format (a, b).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
There are 3 kids waiting for the swings and twice as many kids waiting for the slide. Each kid waits 2 minutes for the swings and a certain amount of seconds for the slide. The shorter wait is 270 seconds shorter. How many seconds do the kids wait for the slide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630}
Let  $E$  and  $F$  be the respective midpoints of  $BC,CD$  of a convex quadrilateral  $ABCD$ . Segments  $AE,AF,EF$  cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of  $\Delta BAD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of $40 \mathrm{~km} / \mathrm{h}$ and the minivan travels at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass between the time at which the minivan arrives in Betatown and the time at which the car arrives in Betatown?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For how many two-digit natural numbers \( n \) are exactly two of the following three statements true: (A) \( n \) is odd; (B) \( n \) is not divisible by 3; (C) \( n \) is divisible by 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
The line $y = a$ and the graph of the function $y = x^3 - 3x$ have three distinct intersection points. Then, the range of $a$ is \_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Identify a monomial that is of the same type as 2ab, meaning it contains exactly the same variables. Coefficients do not matter in this comparison. Provide your answer as a monomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab}
Of the natural numbers greater than 1000 that are composed of the digits $0, 1, 2$ (where each digit can be used any number of times or not at all), in ascending order, what is the position of 2010?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Some mat-weavers can weave 4 mats in 4 days. At the same rate, 14 mat-weavers would weave 49 mats in 14 days. How many mat-weavers were initially weaving?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $f(x)=\int_{0}^{x}\frac{\sin(t)}{t^{3/2}}dt$ find $\lim_{k \to \infty} \int_{0}^{\infty}k^{3/2}f(x)e^{-kx} dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two-hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. At what time will the privateer overtake the merchantman? Provide your answer in the format of H:MM PM, where H and MM are two-digit numbers (possibly with leading zeros). Please give the value of H + MM.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
The minimum positive period of the function $y=\sin x \cdot |\cos x|$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
In a round goblet whose section is the graph of the function $y = x^4$, a cherry — a ball of radius r — is dropped. What is the largest r for which the ball can touch the lowest point of the bottom? (In plain math words: what is the maximal radius of the disc lying in the domain $y ≥ x^4$ and containing the origin?) The original answer is in the format $\frac{m}{n}\sqrt[3]{2}$, please give the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Points \( M \) and \( N \) are marked on the sides \( AB \) and \( BC \) of triangle \( ABC \) respectively such that \(\angle CMA = \angle ANC \). The segments \( MC \) and \( AN \) intersect at point \( O \), and \( ON = OM \). Find \( BC \), if \( AM = 3 \, \text{cm} \) and \( BM = 4 \, \text{cm} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \, \text{cm}}
$\frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\frac{1}{64}$}
3. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $120^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:\sqrt{3}}
A hospital's internal medicine ward has 15 nurses, who work in pairs, rotating shifts every 8 hours. After two specific nurses work the same shift together, calculate the maximum number of days required for them to work the same shift again.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
For each three-digit number, calculate the sum of its digits. The total sum of all these sums is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12600}
Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(T_k - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{451}
Find the minimum value of the expression $$A=\frac{x}{y}+2\sqrt{1+\frac{y}{z}}+3\sqrt[3]{1+\frac{z}{x}}$$ given that $x, y, z > 0$ and $x \geq \max\{y, z\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2\sqrt{2} + 3\sqrt[3]{2}}
A school volleyball team played several matches. After winning the next match, the share of victories increased by $1 / 6$. In order to increase the share of victories by another $1 / 6$, the team had to win two more consecutive matches. What is the minimum number of wins the team needs to achieve to increase the share of victories by another $1 / 6$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6} here the answer}
Let $\omega$ and $\Gamma$ be circles such that $\omega$ is internally tangent to $\Gamma$ at a point $P$. Let $AB$ be a chord of $\Gamma$ tangent to $\omega$ at a point $Q$. Let $R\neq P$ be the second intersection of line $PQ$ with $\Gamma$. If the radius of $\Gamma$ is $17$, the radius of $\omega$ is $7$, and $\frac{AQ}{BQ}=3$, find the circumradius of triangle $AQR$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{170}}
有 20 张卡片上分别写有数字1,2,…,20, 将它们放入一个盒子内.有4个人从中不放回地各抽取一张卡片,抽到两个较小数字的两人在同一组,抽到两个较大数字的两人在同一组.现其中有两人抽到5、14,则此两人在同一组的概率等于___(用最简分数作答)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{58}
A palindrome is a word that reads the same backwards as forwards, such as "eye", "race car", and "qwertyytrewq". 

How many letters are in the smallest palindrome containing the letters b, o, g, t, r, and o, not necessarily in that order and not necessarily adjacent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( m \) be a positive real number such that for any positive real numbers \( x \) and \( y \), the three line segments with lengths \( x + y \), \( \sqrt{x^2 + y^2 + xy} \), and \( m \sqrt{xy} \) can form the three sides of a triangle. Determine the range of values for \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2-\sqrt{3}, 2+\sqrt{3})}
Find the closed form of the sum: $$g(n)=\sum_{k=0}^{n-1}\left(\left(\sum_{i=k+1}^{n}(-1)^i\binom{n}{i}\right)\cdot\left(\sum_{j=0}^{k}(-1)^j\binom{n}{j}\right)\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\binom{2(n-1)}{n-1}}
Let \( h_a = \beta_b = m_c \), where \( h_a = AP \), \( \beta_b = BL \), and \( m_c = CF \) are, respectively, the altitude, angle bisector, and median drawn from three different vertices of a triangle \( ABC \). What value can the ratio of the length of the longest side of triangle \( ABC \) to the length of the shortest side take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $V$ be a 10-dimensional real vector space and $U_1, U_2$ two linear subspaces such that $U_1 \subseteq U_2$, $\dim U_1 = 3$, and $\dim U_2 = 6$. Let $\varepsilon$ be the set of all linear maps $T: V \rightarrow V$ which have $T(U_1) \subseteq U_1$ and $T(U_2) \subseteq U_2$. Calculate the dimension of $\varepsilon$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Let $A B C$ be a triangle with $A B=4, B C=8$, and $C A=5$. Let $M$ be the midpoint of $B C$, and let $D$ be the point on the circumcircle of $A B C$ so that segment $A D$ intersects the interior of $A B C$, and $\angle B A D=\angle C A M$. Let $A D$ intersect side $B C$ at $X$. Compute the ratio $A X / A D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{9}{41}$}
How many integers  $n$  with  $10 \le n \le 500$  have the property that the hundreds digit of  $17n$  and  $17n+17$  are different?

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
How many noncongruent triangles are there with one side of length 20, one side of length 17, and one \(60^\circ\) angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Distribute 7 doctors, including Doctors A and B, into three medical teams with one team having 3 members and the other two teams having 2 members each. In how many ways can this be done so that Doctors A and B are not in the same team?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
The integer sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Determine the value of the expression \(\left[\frac{a_{2}}{a_{1}}\right]\left\{\left[\frac{a_{3}}{a_{2}}\right\}\left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left\{\frac{a_{20225}}{a_{2024}}\right\}\left[\frac{a_{20224}}{a_{2}}\right]\right\}\right.\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number $\overline{523abc}$ divisible by $7, 8$ and $9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{523152}
61. In quadrilateral $A B C D$ with angles $\angle A=60^{\circ}, \angle B=90^{\circ}, \angle C=120^{\circ}$, the point of intersection of the diagonals $M$ is marked. It turns out that $B M=1$ and $M D=2$. Find the area of $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a $2$-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$. The original answer is in the format $n\pi$, please give the value of n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
On the lateral side \( CD \) of the trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. A perpendicular \( AH \) is dropped from vertex \( A \) to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
已知直线$l:x+my-2-m=0(m\in R)$恒过定点$\textit{A}$，点$\textit{B}$，$\textit{C}$为圆$O:{{x}^{2}}+{{y2}}=25$上的两动点，满足$\angle BAC=90{}^\circ $，则弦$\textit{BC}$长度的最大值为___________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4\sqrt{5}$}
Mr. Caiden wants to do repairs to his house and requires 300 feet of metal roofing to do this. The metal roofing costs $8 per foot. The supplier offers a tiered discount system: a 10% discount is applied if purchasing 200-299 feet, and a 15% discount is applied if purchasing 300 feet or more. The shipping fee varies as well: the first 200 feet are delivered for free, but there is a $150 charge for each additional batch of 100 feet. Additionally, there is a 5% sales tax on the order if the total cost is less than or equal to $2000, and a 7% sales tax if the total cost is more than $2000.

Given that the supplier is providing the first 250 feet of metal roofing for free, how much money will Mr. Caiden be required to pay for the remaining metal roofing, taking into account the tiered discounts, varying shipping fees, and applicable sales tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$357}
Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?

(M. V. Karlukova)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1234.4321}
Two circles of the same radius 9 intersect at points \(A\) and \(B\). A point \(C\) is chosen on the first circle, and a point \(D\) is chosen on the second circle. It turns out that point \(B\) lies on the segment \(CD\), and \(\angle CAD = 90^\circ\). On the perpendicular to \(CD\) passing through point \(B\), a point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on opposite sides of the line \(CD\)). Find the length of the segment \(CF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1763}
Let  $f(x)=x^{2021}+15x^{2020}+8x+9$  have roots  $a_i$  where  $i=1,2,\cdots , 2021$ . Let  $p(x)$  be a polynomial of the sam degree such that  $p \left(a_i + \frac{1}{a_i}+1 \right)=0$  for every  $1\leq i \leq 2021$ . If  $\frac{3p(0)}{4p(1)}=\frac{m}{n}$  where  $m,n \in \mathbb{Z}$ ,  $n>0$  and  $\gcd(m,n)=1$ . Then find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
I made a mistake in dividing a number by 4, and instead added by 4 to get 40. Find the correctly calculated value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
We have a collection of identical spheres that we stack to form a tetrahedron where all edges have $n$ spheres. Calculate, as a function of $n$, the total number of points of tangency (contacts) between the spheres in the pile.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^3 - n}
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A basketball team won 50 percent of its first 40 games in a particular season, 60 percent of the next 30 games, and 85 percent of its remaining games. If the team won a total of 62 percent of its games that season, what was the total number of games that the team played?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
Find the percentage of people with a grade of "excellent" among the selected individuals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
在正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=$ $B C=C A=A A_{1}=2$, 棱 $A_{1} C_{1}$ 上的点 $D$ 满足 $B C_{1} / /$平面 $A B_{1} \dot{D}$ 。以 $D$ 为球心、 $\frac{\sqrt{7}}{2}$ 为半径的球面与侧面 $A A_{1} B_{1} B$ 的交线长的$\frac{3}{\pi}$倍是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A man buys an article for $100, but he got a 10% discount on the initial purchase. After buying, he pays a 5% sales tax on the discounted price. Later, he sells the article for $130 including a 15% sales tax. Find the actual gain percent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.62\%}
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\%}
In triangle $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
11. (12 points) Five students, Jia, Yi, Bing, Ding, and Wu, ranked in the top 5 (no ties) in a math competition and stood in a row for a photo. They each said the following:

Jia said: The two students next to me are both ranked behind me;
Yi said: The two students next to me are both ranked adjacent to me;
Bing said: All the students to my right (at least 1) are ranked ahead of me;
Ding said: All the students to my left (at least 1) are ranked behind me;
Wu said: I am standing in the second position from the right.

Given that they are all honest children, Jia, Yi, Bing, Ding, and Wu ranked $A, B, C, D, E$ respectively. What is the five-digit number $\overline{\mathrm{ABCDE}}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23514}
For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0$, $f(2) = 1$, and $f(4) = f(6) = 2$. Let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
Given the ellipse $C$: $\dfrac{x^{2}}{4} + \dfrac{y^{2}}{3} = 1$ with its right focus $F$, and $P$ is a moving point on the ellipse $C$, and a fixed point $A(2,4)$, then the minimum value of $|PA| - |PF|$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( A = \{x \mid -2 \leq x \leq a\} \), \( B = \{y \mid y = 2x + 3, x \in A\} \), and \( C = \{z \mid z = x^2, x \in A\} \). If \( C \subseteq B \), find the range of the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2} \leq a \leq 3\)}
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire length of the pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe, if Gavrila's step is 80 cm? Round your answer to the nearest whole meter. The speed of the tractor is constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
The graph of the function y = sinx - $\sqrt{3}$cosx can be obtained by shifting the graph of the function y = cosx + $\sqrt{3}$sinx at least \_\_\_\_\_\_ units to the left.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{2}}
Johann originally had 120 oranges. He decided to eat 25% of them. After eating, his friend Emily borrowed 1/3 of the remaining oranges but returned 10 later. Meanwhile, two other friends, Carson and Beth, stole 1/4 and 1/5 of the remaining oranges respectively. However, feeling guilty, Carson returned half of the stolen oranges and Beth returned 8. How many oranges does Johann have now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
函数 $f(x) = \left|\log_{\frac{1}{2}} x\right|$ 的单调增区间为何?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
All three-digit numbers from 100 to 999 are written consecutively without spaces. Kostya underlined $k$ consecutive digits in this sequence, and Andrey underlined another $k$ consecutive digits in this sequence. The $k$-digit numbers underlined by the boys are equal. What is the largest possible value of $k$ for which this could happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that $x$, $y$ are positive integers such that $x(x+1)$ divides $y(y+1)$, but neither $x$ nor $x+1$ divides either $y$ or $y+1$. Find the smallest possible value of $x^2 + y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{596}
Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate the $\int \int_SF* dS$ for the given vector field F and the oriented surface S. for closed surfaces, use the positive (outward) orientation. F(x,y,z) = xi +yj +5k. S is the boundary of the region enclosed by the cylinder $x^2 + z^2$ = 1 and the planes y = 0 and x + y = 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4π}