inclusionAI/Ring-lite-rl-data huggingface dataset:
I have two arithmetic sequences. The first term of the first sequence is $0$. The second term of the first sequence is the first term of the first sequence plus the first term of the second sequence. Similarly, the third term of the first sequence is the second term of the first sequence plus the second term of the second sequence. If the fifth term of the second sequence is $3$, what is the fifth term of the first sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that \( n \) is a natural number less than 50, find all \( n \) such that \( 3n + 5 \) and \( 5n + 4 \) have a common divisor greater than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7, 20, 33, 46}
On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
What is the minimum number of circles required to completely cover a circle with twice the radius?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $A B, A C, A D, B C, B D, C D$. If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2001}
求所有次数不超过 $2$ 的整系数多项式 $f$, 使得存在整系数多项式 $g$, 满足
$$  
x^{3}-1 \mid f(x) g(x)-1  
$$
当x=1时请提供所有满足条件的 $f(x)$ 的绝对值和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( n \) be a positive integer. A sequence of \( n \) positive integers (not necessarily distinct) is called full if it satisfies the following condition: For each positive integer \( k \geq 2 \), if the number \( k \) appears in the sequence, then so does the number \( k-1 \), and moreover, the first occurrence of \( k-1 \) comes before the last occurrence of \( k \). For each \( n \), how many full sequences are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
A train 130 m long passes a man, running at a certain speed in the direction opposite to that of the train, in 6 seconds. The speed of the train is 71.99376049916006 kmph. What is the speed of the man in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.006}
Two athletes start running simultaneously—one from point $A$ to point $B$ and the other from point $B$ to point $A$. They run at different but constant speeds and meet at a distance of 300 meters from point $A$. After reaching points $A$ and $B$, they immediately turn around and meet again at a distance of 400 meters from point $B$. Find the length of the segment $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500 \text{ meters}}
Determine the smallest prime number that does not divide any five-digit number whose digits are in strictly increasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Points \( A, B, C, \) and \( D \) lie on a straight line in that order. For a point \( E \) outside the line,

\[ \angle AEB = \angle BEC = \angle CED = 45^\circ. \]

Let \( F \) be the midpoint of segment \( AC \), and \( G \) be the midpoint of segment \( BD \). What is the measure of angle \( FEG \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
The trapezoid $ABCD$ is divided by the line segment $CE$ into a triangle and a parallelogram, as shown in the diagram. Point $F$ is the midpoint of the segment $CE$, the line $DF$ passes through the midpoint of the segment $BE$, and the area of the triangle $CDE$ is $3 \text{ cm}^{2}$.

Determine the area of the trapezoid $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \text{ cm}^2}
The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). Find how many $4$-digit numbers include $4$ consecutive digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Given a point \( P \) in the plane, let \( S_P \) be the set of points whose distance from \( P \) is irrational. What is the smallest number of such sets whose union is the entire plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
从0,1,2,3,4,5,6,7,8,9这10个数中取3个数，使其和为不小于10的偶数，则不同的取法有种. （1998年全国高中联赛试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
How many permutations of 1, 2, 3, ... , n have each number larger than all the preceding numbers or smaller than all the preceding numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
A pentagonal prism is used as the base of a new pyramid. One of the seven faces of this pentagonal prism will be chosen as the base of the pyramid. Calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting structure after this pyramid is added.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming each planet is unique?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
Determine the values of k for which the function $f(x)=x^k|x|$ is four times differentiable at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k≥4}
Given a regular $n$-sided prism, where the area of the base is $S$. Two planes intersect all the lateral edges of the prism in such a way that the volume of the part of the prism between the planes is $V$. Find the sum of the lengths of the segments of the lateral edges of the prism that are enclosed between the planes, given that the planes do not have any common points inside the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{nV}{S}}
Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{13}}
先把一个半径为5，弧长为$6\pi $的扇形卷成一个体积为最大的空心圆锥，再把一个实心的铁球融化为铁水倒入此圆锥内(假设圆锥的侧面不渗漏，且不计损耗)，正好把此空心的圆锥浇铸成了一个体积最大的实心圆锥，则此球的半径为________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt[3]{9}$}
What is the difference between 70% of 120 and 6/9 of 150 divided by 80% of 250, when compared to 18% of 180 multiplied by 5/7 of 210?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4776.5}
4.	Find the biggest positive integer $n$, lesser thar $2012$, that has the following property:
If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1944}
There are three commissions of bureaucrats. It is known for each pair of bureaucrats from different commissions that among the members of the remaining commission, there are exactly 10 bureaucrats who are acquainted with both and exactly 10 bureaucrats who are not acquainted with either. Find the total number of bureaucrats in the commissions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
In the diagram, \( AE \) and \( BD \) are straight lines that intersect at \( C \). If \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Given a triangle  $ABC$  with angles  $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let  $H$  be its orthocentre, and  $O$  be its circumcenter. Let  $F$  be the midpoint of side  $AB$ , and  $Q$  be the foot of the perpendicular from  $B$  onto  $AC$ . Denote by  $X$  the intersection point of the lines  $FH$  and  $QO$ . Suppose the ratio of the length of  $FX$  and the circumradius of the triangle is given by  $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of  $1000a + 100b + 10c + d$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1132}
For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set \(A=\{n \in \mathbb{N}: P(n)\) is non-zero, square-free, and \(S(n)\) is a proper divisor of \(P(n)\}\). Find the maximum possible number of digits of the numbers in \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
The pentagon PQRST is divided into four triangles with equal perimeters. The triangle PQR is equilateral. PTU, SUT, and RSU are congruent isosceles triangles. Find the ratio of the perimeter of the pentagon PQRST to the perimeter of the triangle PQR. Express your answer as a simplified ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
Find the smallest positive integer $a$ such that $x^4+a^2$ is not prime for any integer $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There is a certain number of boxes in a warehouse that can be divided evenly into 5 equal shipments by boat or some equal shipments by truck. The smallest number of boxes that could be in the warehouse is 120. How many equal shipments can be made by truck?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the following limit: $$\lim_{n \to \infty} \frac{1}{(\ln n)^2} \sum_{i=1}^n \ln \left(\frac{i}{\sqrt[i]{e}}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given that $\sqrt {3}\sin x+\cos x= \frac {2}{3}$, find the value of $\tan (x+ \frac {7\pi}{6})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{4}}
Call a date \( mm/dd/yy \) \textit{multiplicative} if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, \( 01/21/21 \), \( 03/07/21 \), and \( 07/03/21 \) are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Last year, Nicki dedicated herself to a strict exercise regime comprising of running, swimming, and hiking. She started with 10 miles of running per week for the first three months, increased it to 20 miles per week for the next three, and then 30 miles per week for the last six months. For swimming, she maintained 5 miles per week for the first six months but didn't pursue it during the second half of the year. Additionally, every fourth week was a rest week, where she would only engage in hiking, covering a distance of 15 miles each time. Considering these variations and rest weeks, what is the total distance she covered in all her exercises during the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1095}
From home to work and back. If a person walks to work and takes transport back, it takes him a total of one and a half hours. If he takes transport both ways, the entire journey takes him 30 minutes. How much time will it take for the person to walk both to work and back?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5 \text{ hours}}
This year, February $29$ fell on a Sunday. In which year will February $29$ next fall on a Sunday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2032}
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In a football tournament, 15 teams participated, each of which played exactly once with all the others. A win awarded 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it turned out that some 6 teams each scored at least $N$ points. What is the largest possible integer value for $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
On the sides \( AB \) and \( CD \) of an inscribed quadrilateral \( ABCD \), which is not a trapezoid, points \( P \) and \( R \) are marked, respectively, such that \( AP:PB = CR:RD \). On the segment \( PR \), a point \( Q \) is chosen such that \( PQ:QR = AB:CD \). Find the ratio of the areas of triangles \( AQD \) and \( BQC \) if it is known that \( AD = x \) and \( BC = y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x}{y}}
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$83.33$}
已知椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ 与圆 $x^{2}+y^{2}=b^{2}$, 过椭圆上一点 $M$ 作圆的两切线, 切点分别为 $P, Q$, 直线 $P Q$ 与 $x, y$轴分别交于点 $E, F$, 当$a=2,b=2$时，求 $S_{\triangle E O F}$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
8(7/18)-(2(11/13)+1(7/18))
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\left(\frac{2}{13}\right)}
Let  $ABC$  be an equilateral triangle with side length  $1$ . Points  $A_1$  and  $A_2$  are chosen on side  $BC$ , points  $B_1$  and  $B_2$  are chosen on side  $CA$ , and points  $C_1$  and  $C_2$  are chosen on side  $AB$  such that  $BA_1<BA_2$ ,  $CB_1<CB_2$ , and  $AC_1<AC_2$ . 
Suppose that the three line segments  $B_1C_2$ ,  $C_1A_2$ ,  $A_1B_2$  are concurrent, and the perimeters of triangles  $AB_2C_1$ ,  $BC_2A_1$ , and  $CA_2B_1$  are all equal. Find all possible values of this common perimeter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
two trains of length 250 m and 120 m are 50 m apart . they start moving towards each other on parallel tracks , at speeds 64 kmph and 42 kmph . after how much time will the trains meet ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.26}
一个正方体的8个顶点可以组成 个非等边三角形.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order.  Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1005}{1006}}
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:36}
How many distinct sets of 8 positive odd integers sum to 20 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$. Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$. Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$, 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{73}
Find the smallest positive interger number  $n$  such that there exists  $n$  real numbers  $a_1,a_2,\ldots,a_n$  satisfied three conditions as follow:
a.  $a_1+a_2+\cdots+a_n>0$ ;
b.  $a_1^3+a_2^3+\cdots+a_n^3<0$ ;
c.  $a_1^5+a_2^5+\cdots+a_n^5>0$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Captain Rusczyk tracked down a pirate who had stolen $2345_{6}$ dollars worth of goods from his ship. After winning an epic duel, the Captain demands that the pirate return $41324_{5}$ dollars. How much has the pirate gone in debt due to his two encounters with Rusczyk? Express your answer in base 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2145}
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the \textit{floor} function, which returns the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1973}
Let $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For how many integers $n \neq 1$ does $(n-1)^3$ divide $n^{2018(n-1)} - 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that the solution set for the quadratic inequality $ax^2+bx+c<0$ is $(1, 2)$, find the solution set for the quadratic inequality $cx^2+bx+a<0$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{2}, 1\right)}
A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of m. If m is replaced by m+10, the new sequence has a mean of 24 and a median of m+10. If m is replaced by m-8, the new sequence has a median of m-4. Find the value of m. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The line of sight of a circular cone is 1, and the angle it forms with the base is $\theta$. The edge length of the inscribed cube in this cone is $\qquad$
$\qquad$ (4 vertices of the cube are on the base of the cone, and the other 4 vertices are on the side surface of the cone).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sin\theta}{2+\sqrt{2}\tan\theta}}
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <tt>HH</tt>, and etc. For example, in the sequence <tt>TTTHHTHTTTHHTTH</tt> of 15 coin tosses we observe that there are two <tt>HH</tt>, three <tt>HT</tt>, four <tt>TH</tt>, and five <tt>TT</tt> subsequences. How many different sequences of 15 coin tosses will contain exactly two <tt>HH</tt>, three <tt>HT</tt>, four <tt>TH</tt>, and five <tt>TT</tt> subsequences?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$560$}
Positive integers  $a_0<a_1<\dots<a_n$ , are to be chosen so that  $a_j-a_i$  is not a prime for any  $i,j$  with  $0 \le i <j \le n$ . For each  $n \ge 1$ , determine the smallest possible value of  $a_n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4n + 1}
Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible by 3 when she stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3-\sqrt{3}}{3}}
11.183. A regular triangular prism is described around a sphere, and a sphere is described around it. Find the ratio of the surfaces of these spheres.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:1}
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. (A set of integers is "pairwise relatively prime" if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any "conspiratorial" subset of the integers 1 through 16?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A pentagon  $ABCDE$  is circumscribed about a circle. The angles at the vertices  $A{}$ ,  $C{}$  and  $E{}$  of the pentagon are equal to  $100^\circ$ . Find the measure of the angle  $\angle ACE$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40^\circ}
For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}+1$, how many multiplications in the field $F$ are required when multiplying $f(x)$ and $g(x)$ using the Karatsuba method?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^r}
A circle intersects the  $y$ -axis at two points  $(0, a)$  and  $(0, b)$  and is tangent to the line  $x+100y = 100$  at  $(100, 0)$ . Compute the sum of all possible values of  $ab - a - b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}