inclusionAI/Ring-lite-rl-data huggingface dataset:
For each positive integer \( n \geq 1 \), we define the recursive relation given by
\[ a_{n+1} = \frac{1}{1 + a_{n}}. \]

Suppose that \( a_{1} = a_{2012} \). Find the sum of the squares of all possible values of \( a_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For each positive digit $D$ and positive integer $k$, we use the symbol $D_{(k)}$ to represent the positive integer having exactly $k$ digits, each of which is equal to $D$. For example, $2_{(1)}=2$ and $3_{(4)}=3333$. There are $N$ quadruples $(P, Q, R, k)$ with $P, Q$ and $R$ positive digits, $k$ a positive integer with $k \leq 2018$, and $P_{(2k)}-Q_{(k)}=\left(R_{(k)}\right)^{2}$. What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
James can do a farmer's walk with 300 pounds per hand for 20 meters. He can lift a certain percentage more if he only goes 10 meters. In addition, lifting straps give him another 20%. If he increases his 20-meter distance without straps by 50 pounds and everything else increases in proportion, he can move 546 pounds with lifting straps for 10 meters. What is the percentage increase in weight he can lift if he only goes 10 meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3}
In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
A bank serves a million clients, and Ostap Bender knows the list of these clients. Each client has a unique six-digit PIN code. On each move, Ostap Bender can choose a client he has not yet chosen and peek at the digits of the code in any $N$ positions (he can choose different positions for different clients). Ostap wants to find out the PIN code of millionaire Koreiko. What is the smallest number, $N$, that guarantees he can do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
Define the infinite products:  
\[ A = \prod_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \]  
If \(\frac{A}{B} = \frac{m}{n}\) where \(m,n\) are relatively prime positive integers, determine \(100m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
A pad of paper comes with 60 sheets. Evelyn uses a pad of paper for writing notes at work every week. She usually works from Tuesday to Thursday, but takes Monday and Friday off. However, during her workdays, she alternates her paper usage. On the first working day of the week, she uses 2 sheets, on the second day, she uses 4 sheets, and on the third day, she uses 6 sheets. 

Considering that Evelyn works for 48 weeks a year, but she takes 8 additional days off for holidays or sick days, how many sheets of paper does she use on average per day she works?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the volume of the solid generated by rotating the region bounded by the curves $$y=e^x,\>\>\>\>\>y=\cos(\pi x^2/2),\>\>\>\>\>x=0,\>\>\>\>\>x=1$$ about the $y$-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(\pi-1)}
40. (5 points)
Form a ten-digit number without repeating digits using numbers $0 \sim 9$, where the number can be divided by 11. There are $\qquad$ such numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285120}
Find the number of positive integers $n$ such that 
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) > 0.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A regular tetrahedron with 2 unit edges has two of its vertices on the circumference of the base and the top of a right circular cylinder. What is the volume of the cylinder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}\pi}
How many integers from 1 to 2001 have a digit sum that is divisible by 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{399}
7th Australian 1986 Problem A2 What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let a 1 = 3, a 2 = 3 3 , and a n+1 = 3 a n . Similarly, b 1 = 100, b 2 = 100 100 etc. What is the smallest n for which b n > a 100 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
There are many $n$-fold coverings (For instance $\Bbb R/n\Bbb Z$ as $S^1$ cover) but I can't find one with trivial automorphism group. What is the smallest $n$ for which there is a covering with trivial automorphism group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{9}}
Find the correlation function of the stationary random function $X(t)$, given its spectral density $s_{x}(\omega)=s_{0}$ in the interval $-\omega_{0} \leq \omega \leq \omega_{0}$, and outside this interval $s_{x}(\omega)=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{2s_0 \sin(\omega_0 \tau)}{\tau} }
Given the sets \( A = \{(x, y) \mid ax + y = 1, x, y \in \mathbb{Z}\} \), \( B = \{(x, y) \mid x + ay = 1, x, y \in \mathbb{Z}\} \), and \( C = \{(x, y) \mid x^2 + y^2 = 1\} \), find the value of \( a \) when \( (A \cup B) \cap C \) is a set with four elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a triangular prism $M-ABCD$ with a square base and $MA = MD$, $MA \perp AB$, if the area of $\triangle AMD$ is 1, find the radius of the largest sphere that can be placed within the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} - 1}
Let $x,$ $y,$ $z,$ $v,$ $w$ be positive real numbers such that $x^2 + y^2 + z^2 + v^2 + w^2 = 2016.$  Let $M$ be the maximum value of
\[xz + 2yz + 3zv + 7zw,\]and let $x_M,$ $y_M$, $z_M,$ $v_M,$ $w_M$ be the values of $x,$ $y,$ $z,$ $v,$ $w,$ respectively, that produce the maximum value of $M.$  Find $M + x_M + y_M + z_M + v_M + w_M.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52 + 3036 \sqrt{7}}
a person travels from p to q a speed of 80 km / hr and returns by increasing his speed by 10 % . what is his average speed for both the trips ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83.81}
A ball inside a rectangular container of width 7 and height 12 is launched from the lower-left vertex of the container. It first strikes the right side of the container after traveling a distance of $\sqrt{53}$ (and strikes no other sides between its launch and its impact with the right side). How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Mrs. Hilt measured the distance from her desk to the water fountain. It was 30 feet. If she goes to the water fountain 4 times today , Mrs. Hilt walked _____ feet on her trips to the fountain .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
In an obtuse triangle \(ABC\), a point \(L\) is chosen on the side \(AB\), which is equal to 14, such that it is equidistant from the lines \(AC\) and \(BC\). On the segment \(AL\), there is a point \(K\) that is equidistant from the vertices \(A\) and \(B\). Find the sine of angle \(ACB\) if \(KL = 1\) and \(\angle CAB = 45^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 - \sqrt{2}}{6}}
在复平面上,复数 $z_{1}, 1+\mathrm{i}, 1+a \mathrm{i}(a \in \mathbf{R})$ 对应的点分别是 $A, B, C$, 点 $A$ 在线段 $B C$ 上运动, 复数 $z_{2}$ 满足 $\left|z_{2}\right|=1$. 若复数 $z_{1}+z_{2}$ 对应的点组成的图形的面积是 $\pi+4$ ，请给出 $a$ 可能的取值的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5n+1}
Let a, b, c, and d represent the current ages of four individuals. In five years, the total age of a and b will be 10 years more than the total age of b and c at that time. Currently, the total age of c and d is 12 years less than the total age of a and d. 

Let R be the difference in years between the individuals a and c. Also, let L(a) and L(c) represent the linear functions of the ages of a and c after five years, respectively:

L(a) = a + 5
L(c) = c + 5

Given this information, find the value of R - the difference in years between individuals a and c in the current year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
一枚不均匀的硬币，若随机抛掷它两次均得到正面的概率为 \frac{1}{2} ，则随机抛掷它两次得到正面、反面各一次的概率的近似整数值是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There exists a polynomial $P$ of degree 5 with the following property: if $z$ is a complex number such that $z^{5}+2004 z=1$, then $P(z^{2})=0$. Calculate the quotient $P(1) / P(-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2010012 / 2010013}
A circle is tangent to the extensions of two sides \( AB \) and \( AD \) of square \( ABCD \) with side length \( 6 + 2 \sqrt{5} \) cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 36^{\circ} \), and it is known that \( \sin 18^{\circ} = \frac{\sqrt{5} - 1}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(2 \sqrt{2} + \sqrt{5} - 1)}
In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = 2 \) and \( BC = 5 \), then \( BX \) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8041}
Given the function $f\left(x\right)=2a^{x}-ex^{2}+18$, where $a \gt 0$ and $a\neq 1$. If $f\left(x\right)$ has two critical points $x_{1}$ and $x_{2}$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{1}{e}, 1) \cup (1, e)}
The light glows for some seconds. It glowed 292.29411764705884 times between 1:57:58 and 3:20:47 am. How many seconds does the light glow for?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Isabel wants to save 40 files onto disks, each with a capacity of 1.44 MB. 5 of the files take up 0.95 MB each, 15 files take up 0.65 MB each, and the remaining 20 files each take up 0.45 MB. Calculate the smallest number of disks needed to store all 40 files.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Six positive numbers, not exceeding 3, satisfy the equations \(a + b + c + d = 6\) and \(e + f = 2\). What is the minimum value of the expression

$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
In the given acute triangle ABC, the incenter I and circumcenter O intersect the line BC at point X. Let M be the midpoint of the arc BC (not containing point A) of the circumcircle of triangle ABC. If the points A, O, M, and X are concyclic, calculate the measure of angle BAC in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In an equilateral triangle \( ABC \) with side length 1, points \( D \) and \( E \) are taken on sides \( AB \) and \( AC \) respectively such that when the triangle is folded along line segment \( DE \), vertex \( A \) falls exactly on side \( BC \). The minimum length of \( AD \) is \( \qquad \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3} - 3}
100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1979}
The rabbit and the turtle are traveling from point $A$ to the forest amusement park. The rabbit starts at 9:00 AM and jumps 40 meters in one minute, playing for 2 minutes after every 3 minutes of jumping. The turtle starts at 6:40 AM and crawls 10 meters per minute without any breaks. If the turtle arrives at the amusement park 15 seconds earlier than the rabbit, how many meters is it from point $A$ to the forest amusement park?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2370 \text{ meters}}
The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A 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 square root of 4. Express your answer in the form ±y, where y is a positive number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2}
Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.19}
Given a square \(ABCD\) with side length 1, determine the maximum value of \(PA \cdot PB \cdot PC \cdot PD\) where point \(P\) lies inside or on the boundary of the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{16}}
Rowena is able to mow 2/5 of a lawn in 18 minutes. If she began the job at 10:00 a.m., and mowed at this same constant rate, what time did she finish mowing the entire lawn? Express your answer in 12-hour time format (e.g., 10:45 a.m.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:45 a.m.}
A young man bought a gift priced at $21 but costing $18 from Mr. Wang's store using a $100 bill. Mr. Wang gave the young man $79 in change but later had to return the $100 bill to a neighbor because it was counterfeit. Calculate Mr. Wang's total financial loss in this transaction. Express your answer in dollars.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
A number is chosen uniformly at random from the set of all positive integers with at least two digits, none of which are repeated. Find the probability that the number is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41}{81}}
Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. Gwendoline
repeatedly rolls the two dice and records the product of the two values until one of the values she records
appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Jack received 10 emails and 12 letters in the morning. He then received 3 emails and some letters in the afternoon. Jack received 7 more emails in the morning than in the afternoon. How many letters did Jack receive in the afternoon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Four students are to complete a 100-meter relay race, with the condition that the distance run by each person cannot exceed three times the distance run by any other student. If the distance run by a certain student is x meters, find the range of values for x. Express your answer as an inequality in the format a ≤ x ≤ b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 ≤ x ≤ 50}
Alex is an entrepreneur who measures success by the financial growth of their business and the assets they accumulate. Last year, Alex's business made a profit of $120,000. This year, Alex set a goal to increase the profit by 25%. Additionally, Alex plans to invest $50,000 in a new project, expecting a 10% return on this investment by the end of the year. What is the total expected financial gain for Alex this year from both the increased profit and the investment return?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35000}
已知抛物线 $x^{2}=4 y$ 上的两个动点 $A\left(x_{1}, y_{1}\right) 、 B\left(x_{2}, y_{2}\right)\left(y_{1}+y_{2}=2, y_{1} \neq y_{2}\right)$ 。若线段 $A B$ 的垂直平分线与 $y$ 轴交于点 $C$, 则 $\triangle A B C$ 面积的最大值为某个复杂分数形式，标准答案格式为$\frac{a\sqrt{b} }{c}$，则a+b+c的值为多少。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Emma has saved $1,200 in her bank account, which earns a 3% compound annual interest. On January 1st, she withdrew 8% of her balance to buy a new pair of shoes. The day after, she deposited 25% of the amount she withdrew. On February 5th, Emma received her paycheck and deposited 150% of the amount she spent on shoes. On March 16th, one week after her birthday, she withdrew 5% of her current balance to contribute to a friend's birthday gift. Her bank compounds interest monthly. 

What is the balance in her bank account on April 1st, considering her transactions and interest earned on her account?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1,217.15}
Consider the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. On the right branch of the hyperbola $C$, there exists a point $P$ such that the inscribed circle of $\triangle PF\_1F\_2$ has a radius of $a$. The center of the circle is denoted as $M$, the centroid of $\triangle PF\_1F\_2$ is $G$, and $MG \parallel F\_1F\_2$. Determine the eccentricity of the hyperbola $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the sum of all positive integers $n$ with the following properties:

- $n$ is not divisible by any primes larger than $10$.
- For some positive integer $k$, the positive divisors of $n$ are ordered as:
  \[1 = d_1 < d_2 < d_3 < \cdots < d_{2k} = n.\]
- The divisors of $n$ satisfy the condition:
  \[d_1 + d_2 + \cdots + d_k = 3k.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{416}
The natural number  $a_n$  is obtained by writing together and ordered, in decimal notation , all natural numbers between  $1$  and  $n$ . So we have for example that  $a_1 = 1$ , $a_2 = 12$ ,  $a_3 = 123$ ,  $. . .$  ,  $a_{11} = 1234567891011$ ,   $...$  . Find all values of  $n$  for which  $a_n$  is not divisible by  $3$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n \equiv 1 \pmod{3} }
A certain fraction of the people in the room are under the age of 21, and exactly 5/12 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, and there are 36 people in the room under the age of 21, what is the fraction of people under the age of 21?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/7}
You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Suppose that $F(8) = \frac{43}{256}$, and $F(10)$ can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1195}
The distance from the top of the can on the floor to the top of the bottle on the table is $150 \textrm{ cm}$. The distance from the top of the bottle on the floor to the top of the can on the table is $110 \textrm{ cm}$. Find the height of the table. Express your answer in centimeters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Given \( z \in \mathbf{C} \). If the equation with respect to \( x \):
$$
4 x^{2}-8 z x+4 i+3=0
$$
has real roots, then the minimum value of \( |z| \) is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A person walks at 4 km/h and misses the bus by 10 minutes. If they walk at a different speed, they reach 5 minutes before the arrival of the bus. The distance they walk to reach the bus stand is 5 km. What is their walking speed when they reach 5 minutes before the bus arrives?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
For what the smallest \(\$ n \$\) is it true that for any set \(\mathrm{A}\) of \(2007\) sets, there exists a set \(\mathrm{B}\) of \( n \) sets such that every set in \(\mathrm{A}\) is the intersection of two different sets in \(\mathrm{B}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}
If $\det \mathbf{A} = -1,$ then find $\det (\mathbf{7A}).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-49}
用 $0,1,2,3,4,5$ 这六个数字组成没有重复数字的四位偶数, 并将这些偶数从小到大排列起来, 则第 99 个数是 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{所求的第 99 个偶数为 4052}
A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the area of the total surface of the cone to the area of its lateral surface, if the side of the base of the pyramid is 4, and the angle between the height of the pyramid and the plane of the lateral face is $30^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}