inclusionAI/Ring-lite-rl-data huggingface dataset:
One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:
(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching either end.
(2) A bus departs from both Station $A$ and Station $B$ every 6 minutes. Determine the minimum number of buses required for this bus route.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1481480}
Find the value of the integral $$\begin{align}I=\int_{0}^{1}\frac{\ln(x)\,dx}{1-x^2}\end{align}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\(\frac{\pi^2}{8}\)}
6. Given the set
$$
A=\left\{n \in \mathbf{Z} \left\lvert\,\left(1+\frac{1}{n}\right)^{n+1}=\left(1+\frac{1}{2020}\right)^{2000}\right.\right\} \text {. }
$$

Then the number of elements in set $A$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
5th Putnam 1942 Problem B4 A particle moves in a circle through the origin under the influence of a force a/r k towards the origin (where r is its distance from the origin). Find k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If a taxpayer has an annual income before tax of $42,400 from three different sources - salary ($28,000), freelance work ($10,000), and investments ($4,400) - with a standard deduction of $12,000, a personal exemption of $4,000, and additional deductible expenses (such as medical and child care) of $6,500, and they are filing as a single individual, what is the differential savings in their tax liability if the income tax rates change in the following manner for each tax bracket:

Current Tax Rates:
- 10% on income up to $9,850
- 15% on income between $9,851 and $23,250
- 25% on income between $23,251 and 37,000
- 42% on income over $37,001

New Tax Rates:
- 10% on income up to $13,500
- 15% on income between $13,501 and $25,000
- 25% on income between $25,001 and $39,000
- 32% on income over $39,001

Calculate the differential savings for the tax payer considering these additional factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$182.50}
The variables x and y have a linear correlation. When x takes the values of 16, 14, 12, and 8, the observed values of y are 11, 9, 8, and 5 respectively. Using the reference formulas for linear regression, predict the approximate value of x when y = 10. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Insert a digit in the middle of a two-digit number to form a three-digit number. For some two-digit numbers, the resulting three-digit number can be $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
If "+" or "-" is added in front of the numbers $\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$, making their algebraic sum equal to $1$, then there are a total of _______ ways to do this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Clever Cat sleeps under the oak tree from noon till midnight and is awake telling stories from midnight till noon. A poster on the tree above him says "Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time". Calculate the number of hours a day the poster tells the truth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
If both \( 7n+1 \) and \( 8n+1 \) can be represented as the sum of three distinct positive integers in a geometric progression, what is the smallest value of the positive integer \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find all odd integers \( n > 1 \) for which there is at least one permutation \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) such that the sums \( a_1 - a_2 + a_3 - \ldots + a_n \), \( a_2 - a_3 + a_4 - \ldots - a_n + a_1 \), \( a_3 - a_4 + \ldots + a_n - a_1 + a_2 \), \( \ldots \), \( a_n - a_1 + a_2 - \ldots + a_{n-1} \) are all positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 1 \pmod{4}}
A circular disk is divided by 2n equally spaced radii (n > 0) and one chord. Calculate the maximum number of non-overlapping regions into which the disk can be divided. Express your answer as a formula in terms of n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n + 1}
The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9/14}
Let the set \( S \) contain 2012 elements, where the ratio of any two elements is not an integer. An element \( x \) in \( S \) is called a "good element" if there exist distinct elements \( y \) and \( z \) in \( S \) such that \( x^2 \) divides \( y \cdot z \). Find the maximum possible number of good elements in \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
In equilateral triangle $ABC$ with side length 2, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_{1}$ and $B_{2}$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $CB$ at $C_{1}$ and $C_{2}$, respectively. Find the perimeter of the triangle formed by lines $A_{1}A_{2}, B_{1}B_{2}, C_{1}C_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66-36\sqrt{3}}
Task 4 - 250914 Three circles with the given radius $r$ may lie in a plane such that each touches the other two. A common tangent is drawn to each pair of these three circles, which does not have any point in common with the third circle. With these three tangents, an equilateral triangle is constructed.

Calculate the area of this triangle in dependence on $r$!

Hint: For numerical values that occur in the area specification, the use of approximate values is permitted (but not required); if so, however, with an indication - and justification - of how many decimal places the approximate value is accurate to.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^2\cdot(4\sqrt{3}+6)}
What is the maximum length of a closed self-avoiding polygon that can travel along the grid lines of an $8 \times 8$ square grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2^{2^{n-1}}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ are positive integers. Find $b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
What is the last two digits of the decimal representation of  $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Alice is once again very bored in class. On a whim, she chooses three primes $p, q, r$ independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of $p x^{2}+q x+r$. What is the probability that at least one of her roots is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{200}}
Let $A B C D$ be a rectangle with $A B=8$ and $A D=20$. Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to $A B$ and $A D$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112-25 \pi}
Gus has to make a list of  $250$  positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if  $15$  is a number from the list so the list contains  $15$  numbers other than  $15$ . Determine the maximum number of distinct numbers the Gus list can contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
An archipelago consisting of an infinite number of islands is spread out along the southern shore of an endless sea in a chain. The islands are connected by an infinite chain of bridges, and each island is also connected to the shore by a bridge. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that, after a strong earthquake, it will be possible to travel from the first island to the shore using the remaining bridges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Find the number of positive integers less than or equal to  $2017$  that have at least one pair of adjacent digits that are both even. For example, count the numbers  $24$ ,  $1862$ , and  $2012$ , but not  $4$ ,  $58$ , or  $1276$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{738}
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{415}
Edward starts at his house located at \((0,0)\) and needs to reach the point \((6,4)\), which is the coordinate for his school. However, there is a park shaped as a square with coordinates \((2,1), (2,3), (4,1), (4,3)\). Edward cannot walk inside the park, but there is a road along the perimeter of the square. Determine how many different shortest routes exist from Edward's house to his school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determine the length of the portion of $\overline{XY}$ contained in the cube with edge length $4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3}}
Four points, $A, B, C$, and $D$, are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{35}{12 \pi^2}}
Given that $M$ is a point on the parabola $x^2 = 4y$, and $F$ is its focus. Point $A$ is on the circle $C: (x+1)^2 + (y-5)^2 = 1$. Find the minimum value of $|MA| + |MF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a sample set with a capacity of 20 that is divided into groups, the group intervals and their respective frequencies are as follows: (10, 20), 2; (20, 30), 3; (30, 40), 4; (40, 50), 5; (50, 60), 4; (60, 70), 2. Calculate the frequency of the sample in the interval $(-∞, 50)$. Express your answer as a decimal value, rounded to 2 decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.70}
The coordinates of the point $P(2,3)$ with respect to the $x$-axis are ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,-3)}
The solution set of the inequality $(x-2)\sqrt{x+3} \geq 0$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-3\}\cup[2,+\infty)}
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. Find the number of positive integers $k$ less than $2014$ for which $f(k)=8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$.  Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The problem of transporting goods from a riverside city A to another location B. The shortest distance from B to the river bank is BC = b kilometers, and the distance of AC along the river is a kilometers. If the water transportation cost is \(\frac{1}{n}\) times the road transportation cost (where \(n\) is a number greater than 1), to minimize the total transportation cost from A to B, what is \(DC = \qquad\) kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b}{\sqrt{n^2 - 1}}}
Given two lines $l_{1}$: $(a+2)x+(a+3)y-5=0$ and $l_{2}$: $6x+(2a-1)y-5=0$ are parallel, then $a=$        .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{2}}
## Task 11/78

A mathematician writes to another: "This year I will be 100 years old, next year 200."

Obviously, the age references are not in the decimal system. How old is the mathematician?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49_{(10)}}
Tanika is selling boxes of crackers for her scout troop's fund-raiser. On Thursday, she sold 60 boxes at a 10% discount. On Friday, she sold 50% more boxes than on Thursday at a 15% discount. On Saturday, she sold 80% more boxes than on Friday at a 20% discount. On Sunday, she sold 30% less boxes than on Saturday at a 25% discount. How many boxes did she sell, in total, over the four days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{425}
Evaluate the flux of the vector field $\mathbf{F} = 5x\hat{i} + y\hat{j} + z\hat{k}$ through the surface $S_1$ of the paraboloid $x = y^2 + z^2$ oriented in the negative $x$ direction, bounded by the plane $x = 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24π}
Find the number of strictly increasing sequences of nonnegative integers with the following properties:
- The first term is 0 and the last term is 12. The sequence has at least two terms.
- Among any two consecutive terms, exactly one of them is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Write down a set of Pythagorean triples (three positive integers that can form the three sides of a right triangle): $\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 5}
Let $(b_1,b_2,b_3,\ldots,b_{10})$ be a permutation of $(1,2,3,\ldots,10)$ for which
$b_1>b_2>b_3>b_4 \mathrm{\ and \ } b_4<b_5<b_6<b_7<b_8<b_9<b_{10}.$
Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Which of the following expressions describes the number of cans of soda that can be purchased for $D$ dollars, where $1$ dollar is worth $4$ quarters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{4DS}{Q}$}
Suppose  $a_0,a_1,\ldots, a_{2018}$  are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers  $x$ .  Compute the remainder when  $a_0^2 + a_1^2 + \cdots + a_{2018}^2$  is divided by  $2017$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
$2 \cdot 107$ Let the natural number $n \geqslant 5, n$ different natural numbers $a_{1}, a_{2}, \cdots, a_{n}$ have the following property: for any two different non-empty subsets $A$ and $B$ of the set
$$
S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}
$$

the sum of all numbers in $A$ and the sum of all numbers in $B$ will not be equal. Under the above conditions, find
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}
$$

the maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2-\frac{1}{2^{n-1}}}
Find the value of the surface integral $$\int_S yds$$ where $S$ is the part of the plane $$z=1+y$$ that lies inside the cone $$z=\sqrt{2(x^2+y^2)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2π}
Find the volume of the pyramid \( PABCD \), which has a quadrilateral \( ABCD \) with sides 5, 5, 10, and 10 as the base, its shorter diagonal being \( 4\sqrt{5} \), and all the lateral faces inclined at an angle of \( 45^\circ \) to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{500}{9}}
A particular novel contains $40,\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Suppose that the probabilities of person A and person B hitting the target in each shot are $\frac{3}{4}$ and $\frac{4}{5}$, respectively, and that each shot is independent. If they take shots in the order of A, B, A, B, until one of them hits the target and then stop shooting, what is the probability that A has taken two shots when they stop shooting?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{400}}
There are  $20$  geese numbered  $1-20$  standing in a line. The even numbered geese are standing at the front in the order  $2,4,\dots,20,$  where  $2$  is at the front of the line. Then the odd numbered geese are standing behind them in the order,  $1,3,5,\dots ,19,$  where  $19$  is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered  $1,2,\dots,20$  (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given an infinite sequence $\{a_n\}$, where $a_1=1$, $a_2=2$, and for any $n\in N^*$, $a_{n+2}=a_n$. The sequence $\{b_n\}$ satisfies $b_{n+1}-b_n=a_n$ ($n\in N^*$). If any term in the sequence $\left\{ \frac{b_{2n}}{a_n} \right\}$ repeats infinitely many times within the sequence, then the value of $b_1$ that meets the requirement is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If three angles \(x, y, z\) form an arithmetic sequence with a common difference of \(\frac{\pi}{2}\), then \(\tan x \tan y + \tan y \tan z + \tan z \tan x = \) ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Two equal circles intersect at point \( C \). Through point \( C \), two lines are drawn, intersecting the circles at points \( A, B \) and \( M, N \) respectively. The line \( AB \) is parallel to the line of centers, and the line \( MN \) forms an angle \( \alpha \) with the line of centers. Given that \( AB = a \), find \( NM \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \cos \alpha}
The opening of a pit is a circle with a diameter of $100 \mathrm{~m}$, and its cross-sections passing through the center of the circle are equilateral triangles. Let $A$ and $B$ be two diametrically opposite points on the opening, $C$ the deepest point of the pit, and $H$ the midpoint of $BC$. What is the length of the shortest path connecting $A$ with $H$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\sqrt{5} \approx 111.8 \text{ m}}
17 Find the number of all positive integers $a$ that satisfy the following condition: there exist non-negative integers $x_{0}, x_{1}, x_{2}, \cdots$, $x_{2001}$, such that $a^{x_{0}}=a^{x_{1}}+a^{x_{2}}+\cdots+a^{x_{2001}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A divisor of a natural number is called proper if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is a prime number, and the other is a perfect square, and their sum is also a perfect square. How many interesting numbers are there that do not exceed 1000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
5. Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2}=36$ lines. Assume that all 36 lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}
A circle is circumscribed around $ABCD$ as follows: [asy]
pair pA, pB, pC, pD, pO;
pO = (0, 0);
pA = pO + dir(-40);
pB = pO + dir(40);
pC = pO + dir(130);
pD = pO + dir(190);
draw(pA--pB--pC--pA);
draw(pA--pD--pC--pA);
label("$A$", pA, SE);
label("$B$", pB, NE);
label("$C$", pC, NW);
label("$D$", pD, SW);
draw(circle(pO, 1));
label("$30^\circ$", pA + dir(150) * .45);
label("$40^\circ$", pC + dir(-20) * .35);
[/asy] How many degrees are in $\angle CAB + \angle ACD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
15 points $P_i$ are placed within the unit sphere. Find the maximum possible value of \(\sum_{1\leq i < j \leq 15} |P_i P_j|^2\) given that \(x_i^2+y_i^2+z_i^2 \leq 1\) for each point \(P_i = (x_i, y_i, z_i)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Let \(ABCD\) be a square of side length 2. Let points \(X, Y\), and \(Z\) be constructed inside \(ABCD\) such that \(ABX, BCY\), and \(CDZ\) are equilateral triangles. Let point \(W\) be outside \(ABCD\) such that triangle \(DAW\) is equilateral. Let the area of quadrilateral \(WXYZ\) be \(a+\sqrt{b}\), where \(a\) and \(b\) are integers. Find \(a+b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}