inclusionAI/Ring-lite-rl-data huggingface dataset:
You have four charged batteries and four uncharged batteries, but you do not know which are which. A radio requires two charged batteries to operate. What is the minimum number of attempts needed to ensure the radio works? An attempt consists of inserting two batteries into the radio and checking if it operates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A point is randomly thrown onto the interval $[6, 10]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3k-10\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
The inclination angle of the line $x - y + 3 = 0$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
A variable plane intersects the coordinate planes to form a tetrahedron with a constant volume of \(96k^3\). Determine the locus of the center of gravity of this tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{xyz = 9k^3}
From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}$}
Construction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, with each project taking exactly one more day than the previous one to complete. The first project takes 3 days, the second takes 4 days, and so on.

Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Tara is saving up to buy a new clarinet. She already has $10 saved. The clarinet costs $90. She plans to sell her old books to make the money. She sells each book for a certain amount. However, when she is halfway towards her goal, she loses all her savings and has to start over. She sells a total of 25 books by the time she reaches her goal. How much does she sell each book for?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$4.80}
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
By placing addition signs between some of the digits of the number 123456789, we can obtain various sums. What is the minimum number of addition signs necessary to obtain the number 54 in this manner? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many Young diagrams are there with at most $p$ rows and $q$ columns, without restrictions on the weight of the diagrams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\binom{p+q}{p}$}
Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: "Multiply my number by two and subtract two from the result," "Multiply my number by three and then add four," and lastly, "Add seven to my number!" The iPhone knows that initially, Vova's number was 1. How many four-digit numbers could the iPhone XXX theoretically achieve by obediently following Vova's commands?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9000}
What is the smallest number that must be added to 956734 to make it exactly divisible by a combination of prime numbers, such as the product of 751 and 503?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176525}
The points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( S A B, S A C, S B C, \) and \( A B C \) of the tetrahedron \( S A B C \). It is known that \( A B = S C = 5 \), \( A C = S B = 7 \), \( B C = S A = 8 \). Find the volume of the tetrahedron \( K L M N \). If necessary, round your answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.66}
Determine the coordinates of the point \( P(m, n) \) on the ellipse \( \frac{x^2}{8} + \frac{y^2}{18} = 1 \) such that the tangent line at \( P \) intersects the coordinate axes to form a triangle \( OAB \) with the origin \( O \) and the least possible area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
Two equally strong tennis players play against each other until one player wins three consecutive games. Each game is independent, and each player has a probability of \( \frac{1}{2} \) of winning any given game. What is the expected number of games they will play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine if there exists a positive integer $k$ such that the inequality \( \sum_{i=1}^{k} \frac{y_i^2}{x_i} \leq \frac{(\sum_{i=1}^{k}y_i)^2}{\sum_{i=1}^{k}x_i} \) holds, where all variables $x_i$ and $y_i$ are positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The price of Margit néni's favorite chocolate was increased by 30%, and at the same time her pension increased by 15%. By what percentage does Margit néni's chocolate consumption decrease if she can spend only 15% more on chocolate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.54\%}
Use Green's Theorem to evaluate the line integral \( \int_C P\,dx + Q\,dy \), where \( P = xy \) and \( Q = x^2 \). The curve \( C \) is the loop in the first quadrant described by the polar equation \( r = \sin 2\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{105}}
Find the integers \( n \) such that there exist \( x \), \( y \), and \( k \) with \( x \) and \( y \) being coprime and 
\[ 
3^n = x^k + y^k 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$.
Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7
}
Let $x_1, x_2, \dots, x_n$ be a positive integer sequence such that each term is less than or equal to $2001$ and for every $i\geq 3$, $x_i = |x_{i-1}-x_{i-2}|$. What is the largest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3002
}
The frame of a painting has the form of a $105^{\\prime\\prime}$ by $105^{\\prime\\prime}$ square with a $95^{\\prime\\prime}$ by $95^{\\prime\\prime}$ square removed from its center.  The frame is built out of congruent isosceles trapezoids with angles measuring $45^\\circ$ and $135^\\circ$.  Each trapezoid has one base on the frame\u0027s outer edge and one base on the frame\u0027s inner edge.  Each outer edge of the frame contains an odd number of trapezoid bases that alternate long, short, long, short, etc.  What is the maximum possible number of trapezoids in the frame?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
76
}
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$27$}
Evaluate the integral \( \int_0^{\infty} \frac{\cos^2 x \sin 2x}{x e^{x \sqrt{2}}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Find the smallest positive integer $n$ such that an $n \times n$ grid can be colored with red, yellow, and blue, with each color appearing the same number of times, and satisfying the condition that if a row contains a yellow cell, it must also contain a blue cell and cannot contain a red cell.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the limit as $n$ approaches infinity for the following expression, given a sequence of independent and identically distributed (i.i.d.) non-negative random variables $X_i$ with mean $\mu = 4$ and variance $\sigma^2 = 16$:
$$ \lim_{n \to \infty} E\left[\cos\left(\sqrt{S_n} - 2\sqrt{n}\right)\right],$$
where $S_n = \sum_{i=1}^n X_i$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1/2}}
Evaluate the integral \( \int_0^{30} \frac{1}{48!} t^{48} e^{-t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - e^{-30} \sum_{k=0}^{48} \frac{30^k}{k!}}
Find an explicit bijection between the set of equivalence classes of the equivalence relation defined on \( \mathbb{R}[X] \) by \( P_1 \sim P_2 \) if \( P_1 - P_2 \) is divisible by \( X \), and the set of real numbers \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[P] \mapsto P(0)}
Find the spectrum of the operator $T = S + S^*$, where $S$ is the unilateral shift operator on $\mathcal{l}^2$ and $S^*$ is its adjoint, the backward shift operator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 2]}
Find the total number of different integers the function

\[ f(x) = \left\lfloor x \right\rfloor + \left\lfloor 2 \cdot x \right\rfloor + \left\lfloor \frac{5 \cdot x}{3} \right\rfloor + \left\lfloor 3 \cdot x \right\rfloor + \left\lfloor 4 \cdot x \right\rfloor \]

takes for $0 \leq x \leq 100.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{734}
In a game played on a 4x4 checkerboard, two players, Cat and Mouse, each place a checker on a cell of the board, with Cat going first. The players then take turns moving their checkers to an adjacent square, either vertically or horizontally, with Cat moving first. If, after any move, the two checkers occupy the same square, Cat wins. If each checker has made 2013 moves without this occurring, Mouse wins. Determine which player has a winning strategy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Mouse}
Simplify the following expression:

$$
\sqrt{2-\sqrt{3}} \cdot \sqrt{2-\sqrt{2-\sqrt{3}}} \cdot \sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3}}}} \cdot \sqrt{2+\sqrt{2-\sqrt{2-\sqrt{3}}}}
$$

Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Using the digits 1, 2, 3, 4, and 5 without repetition, how many 5-digit wave numbers can be formed such that the tens and thousands digits are each larger than the digits adjacent to them? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Colour a $20000 \times 20000$ square grid using 2000 different colours, with one colour per square. Two squares are considered neighbours if they share a vertex. Define a path as a sequence of squares where each pair of successive squares are neighbours.

Mark $k$ squares on the grid. For each unmarked square $x$, there is exactly one marked square $y$ of the same colour such that $x$ and $y$ are connected by a path consisting of squares of the same colour.

Furthermore, for any two marked squares of the same colour, any path connecting them must pass through squares of all the different colours.

Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
A square board with dimensions 30 cm by 30 cm is divided along the grid lines into 225 smaller squares of equal area. Determine the maximum possible total length of the cuts made on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
A baseball team played 45 games in a season, out of which 15 were away games. They won 6 home games and 7 away games, with 3 of their wins at home being in extra innings. What is the ratio of the number of away game losses to the number of home game losses that did not go into extra innings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
Given that the function $f(x) = ax + \sin x$ has extreme points, find the range of the real number $a$. Express your answer as an interval or inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Rewrite the expression $y = \frac{(s-r)(q-p)}{(q-r)(s-p)}$ in terms of $x = \frac{(r-p)(s-q)}{(r-q)(s-p)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - x}
The probability that a fire will occur is 0.001. If there is a fire, the amount of damage, $X$, follows a Pareto distribution with $P(X>x) = \left(\frac{2(10)^6}{2(10)^6+x} \right)^2$. An insurance policy covers the excess loss over a deductible of $100,000$. The one-time insurance premium is 110% of the expected payment. Calculate the insurance premium.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2095.24}
Given the equation \(ax + by + 2c = 0\) with conditions \(c \neq 0\) and \(ab - c^2 \geq 0\), find the maximum value of \(xy\). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $a, b, c$ be the roots of the equation $x^3 - 9x^2 + 11x - 1 = 0$, and define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Compute $s^4 - 18s^2 - 8s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-37}
Given that
\[\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!}\] 
for some positive integer $m$, find $m \pmod{71}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
将等差数列 $1,5,9,13, \cdots, 2017$ 排成一个大数 $15913 \cdots 2017$ ，求该数值被 9 除的余数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the sequence ${a_n}=C_{200}^n⋅{({\sqrt[3]{6}})^{200-n}}⋅{({\frac{1}{{\sqrt{2}}}})^n}$, where $n$ ranges from $1$ to $95$, find the number of integer terms in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The hands of a clock are straight 22 times in a certain period. What is the period in which the hands of a clock are straight 22 times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Bobby buys two packets of candy. He eats two candies every day from Monday through Friday and takes a certain number of candies during the remaining days of the week. It takes him 3 such weeks to finish the 2 packets, and there are 18 candies in a packet. How many candies does Bobby take during the remaining days of the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
## Task Condition

Find the derivative.

$y=5 x-\ln \left(1+\sqrt{1-e^{10 x}}\right)-e^{-5 x} \cdot \arcsin \left(e^{5 x}\right)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5e^{-5x}\cdot\arcsin(e^{5x})}
s is a set of 81 consecutive multiples of 7. If the sum of the two smallest numbers in s is 145, then find the sum of the greatest number in s and the median of set s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{978}
Given the sets \( A=\{(x, y) \mid y=-|x|-2\} \) and \( B=\{(x, y) \mid (x-a)^2+y^2=a^2\} \) such that \( A \cap B = \varnothing \), determine the range of the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\sqrt{2} - 2 < a < 2\sqrt{2} + 2}
Given that the solution set of the inequality $x^2+ax+4<0$ is empty, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-4, 4]}
Given real numbers \( x \) and \( y \) such that 

\[ \frac{x^2 + y^2}{x^2 - y^2} + \frac{x^2 - y^2}{x^2 + y^2} = k, \]

find \( \frac{x^8 + y^8}{x^8 - y^8} + \frac{x^8 - y^8}{x^8 + y^8} \) in terms of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^4 + 24k^2 + 16}{4k^3 + 16k}}
Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and notices that both coins are heads. What is the probability that the coin Carl removed was also heads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/5}
On the side \( AD \) of a cyclic quadrilateral \( ABCD \), there is the center of the circle that touches the other three sides of the quadrilateral. Find \( AD \) if \( AB = 2 \) and \( CD = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
[Example 2.2.1] Simplify: $M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{5}+1}
Given that \( x \) and \( y \) are integers and satisfy the equation
$$
\left(\frac{1}{x} + \frac{1}{y}\right) \left(\frac{1}{x^2} + \frac{1}{y^2}\right) = -\frac{2}{3} \left(\frac{1}{x^4} - \frac{1}{y^4}\right),
$$
how many possible values of \( x + y \) are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
The angles $\alpha$ that satisfy the condition $|\sin\alpha| = -\sin\alpha$ are.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2k\pi-\pi, 2k\pi], k \in \mathbb{Z}}
9-6. Petya wants to place 99 coins in the cells of a $2 \times 100$ board so that no two coins are in cells that share a side, and no more than one coin is in any cell. How many ways are there to place the coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{396}
From a list of integers from $1$ to $2022$, inclusive, delete all numbers in which at least one of its digits is a prime. How many numbers remain without erasing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{431}
A heavy concrete platform anchored to the seabed in the North Sea supported an oil rig that stood 40 m above the calm water surface. During a severe storm, the rig toppled over. The catastrophe was captured from a nearby platform, and it was observed that the top of the rig disappeared into the depths 84 m from the point where the rig originally stood. 

What is the depth at this location? (Neglect the height of the waves.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68.2 \text{ meters}}
Let  $ABC$  be triangle such that  $|AB| = 5$ ,  $|BC| = 9$  and  $|AC| = 8$ . The angle bisector of  $\widehat{BCA}$  meets  $BA$  at  $X$  and the angle bisector of  $\widehat{CAB}$  meets  $BC$  at  $Y$ . Let  $Z$  be the intersection of lines  $XY$  and  $AC$ . What is  $|AZ|$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set \{0,1,\dots,2010\}. Call such a polynomial "good" if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20$, $P(m_{15})-15$, $P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{460}
Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Over a period of six months, the number of animal sightings and family visits to a national park fluctuates due to various factors such as seasonal changes, animal behavior, and park events. The pattern of family visits and animal sightings during this time is as follows:

- In January, 100 families visit the park and see animals 26 times.
- In February, family visits increase by 50% and they see animals three times as many as in January.
- In March, animal sightings decline due to the change in season, and families see animals only half as many times as in February. Additionally, the number of families visiting the park decreases by 20% compared to February.
- In April, the park hosts a special event, resulting in a 70% increase in family visits compared to March. During this month, families see animals 40% more often compared to March.
- In May, the number of families remains the same as in April, but animal sightings decrease by 25% due to the warmer climate.
- In June, with the onset of summer vacations, there is a 30% increase in family visits, but animal sightings remain the same as in May due to the hot weather.

Assuming that each family visits the park only once per month, calculate the total number of animal sightings by families during these six months.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
A  quartic funtion  $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$  touches the line  $ y \equal{} px \plus{} q$  at  $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$  Find the area of the region bounded by these graphs in terms of  $ a,\ \alpha ,\ \beta$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a (\beta - \alpha)^{5}}{30}}
设 X=\{1,2,\cdots,10\}，f:X\rightarrow X 的映射满足① f\circ f=I_X，其中 f\circ f 是复合映射，I_X 为 X 上的恒等映射；② |f(i)-i|\leq 2，对任意 i\in X.求映射 f 的数目.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{401}
Paolo has 14 coconuts, while Dante has a certain number of times more coconuts than Paolo. If Dante sold 10 of his coconuts and had 32 coconuts left, what is the ratio of Dante's coconuts to Paolo's coconuts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16:7}
Let  $f_1(x)=x^2-1$ , and for each positive integer  $n \geq 2$  define  $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial  $f_{2004}$  have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
A regular hexagonal paper with a side length of 1 is joined with a regular triangular paper with a side length of 1. The pieces must be aligned edge-to-edge without overlapping. Find the number of edges in the resulting new shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}