inclusionAI/Ring-lite-rl-data huggingface dataset:
Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], calculate the range of the function g(x) on the interval [-2020,2020]. Express your answer as an interval [min, max].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-4043,4041]}
Given the sequence $a_1, a_2, \ldots, a_n, \ldots$, insert 3 numbers between each pair of consecutive terms to form a new sequence. Determine the term number of the 69th term of the new sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
There are 4 prime numbers between 1 and a certain number that are factors of 210. What is the ending number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers \( n \) satisfy \( f(n) = f(2005) \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digit of $M$, then we call this number $M$ a "heart's desire number"; if this two-digit number can also be divided by the sum of the thousands digit and units digit of $M$, then we call this number $M$ not only a "heart's desire" but also a "desire fulfilled". ["Heart's desire, desire fulfilled" comes from "Analects of Confucius. On Governance", meaning that what is desired in the heart becomes wishes, and all wishes can be fulfilled.] For example, $M=3456$, since $45 \gt 3+6$, and $45\div \left(3+6\right)=5$, $3456$ is not only a "heart's desire" but also a "desire fulfilled". Now there is a four-digit natural number $M=1000a+100b+10c+d$, where $1\leqslant a\leqslant 9$, $1\leqslant b\leqslant 9$, $1\leqslant c\leqslant 9$, $1\leqslant d\leqslant 9$, $a$, $b$, $c$, $d$ are all integers, and $c \gt d$. If $M$ is not only a "heart's desire" but also a "desire fulfilled", where $\frac{{10b+c}}{{a+d}}=11$, let $F\left(M\right)=10\left(a+b\right)+3c$. If $F\left(M\right)$ can be divided by $7$, then the maximum value of the natural number $M$ that meets the conditions is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5883}
Find the inverse Laplace transform of the following function:
$$\sum_{k=0}^\infty \frac{(-1)^k (s+1)^{2k}}{ (2k)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos(t+1)}
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$) with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a line passing through point $F_{2}$ intersects the right branch of the hyperbola at two distinct points $M$ and $N$. If $\triangle MNF_{1}$ is an equilateral triangle, then the eccentricity of this hyperbola is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {3}}
A trapezoid field is uniformly planted with wheat. The trapezoid has one pair of parallel sides measuring 80 m and 160 m, respectively, with the longer side on the bottom. The other two non-parallel sides each measure 120 m. The angle between a slanted side and the longer base is $45^\circ$. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
In △ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. If (2b-c)cosA=acosC, calculate the value of angle A in degrees. Express your answer in the format \boxed{[answer]°}, where [answer] is the numerical value of the angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60°}
Three noncollinear points and a line $\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\ell$ (or $\ell$ itself). There are exactly $n$ lines perpendicular to $\ell$ with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Define the binary operation $a \Delta b = ab + a - 1$. Compute the following expression:
\[
10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10))))))))
\]
where $10$ is written $10$ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10999999999}
5. Given a positive integer $n$ greater than 2004, fill the numbers $1,2,3, \cdots, n^{2}$ into the squares of an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns) such that each square contains exactly one number. If a square contains a number that is greater than the numbers in at least 2004 squares in its row and at least 2004 squares in its column, then this square is called a “good square”. Find the maximum number of “good squares” on the chessboard.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n-2004)}
Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that Timur counted as the 105th, Alexander counted as the 8th?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
There has been a successive increase of some percentage in the price of gas from the previous month, and then a 20% increase. A driver should reduce gas consumption by 35.89743589743589% so that the expenditure does not change. What was the percentage increase in the price of gas in the first month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
Given a set of consecutive positive integers $I=\{1, 2, 3, \ldots, 238\}$, if $T$ is a subset of $I$ and satisfies the condition: if $x \in T$, then $7x \notin T$, find the maximum number of elements in the set $T$. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208}
What is the sum of the first 1234 terms of the sequence where the number of 2s between consecutive 1s increases by 1 each time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2419}
A pond has a rare diatom present that is randomly distributed in its surface water with a rate of \( \lambda = 3 \) cells/ml. If three 1.5 ml samples of surface water are independently taken, what is the probability that exactly two of these three samples will contain four or more diatoms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.444}
Find the asymptotic bounds in the form of \(\Theta(\cdot)\) for the recurrence relation: \[T(n) = T\left(\frac{n}{\log n}\right) + \log \log n\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(\log n)}
1. A column of scouts has a length of $1 \mathrm{~km}$ and is moving uniformly. The courier, who is at the front of the column, runs to the end of the column, delivers the message, and then returns to the front of the column. During this time, the column travels a distance of $1 \mathrm{~km}$. How far did the courier travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}+1}
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, what is the largest integer value of $x$ such that the area of this new triangle is less than 2004?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Determine the Galois group of the polynomial $T^3-2$ over the finite field $\mathbb{F}_5$. Provide your answer as a group description or notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
On the Richter scale, which measures the total amount of energy released during an earthquake, a reading of x - 1 indicates one-tenth the released energy as is indicated by a reading of x. The frequency at which an earthquake occurs is represented by f(x) for a given Richter scale value x. Earthquake A occurred at location M with a Richter scale reading of 5, while earthquake B occurred at location N with a reading of 3. If the frequency of earthquake events at location M is 2 times the frequency of those at N, find the ratio of the energy release of earthquake A to earthquake B in terms of f(x).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200:1}
Stacy and Heather are some miles apart and walk towards each other along the same route. Stacy walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. Heather starts her journey 24 minutes after Stacy. How far apart were Stacy and Heather initially if Heather has walked 5.7272727272727275 miles when the two meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the angle \(\angle BCD\) is \(120^\circ\). A circle with radius 1 is inscribed in this angle, passing through points \(A\), \(B\), and \(D\). Find the area of triangle \(ABD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{4}}
If the chord cut by the line $ax-by+2=0$ on the circle $x^{2}+y^{2}+2x-4y+1=0$ is $4$, find the minimum value of $\dfrac{2}{a}+\dfrac{3}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4+2\sqrt{3}}
Given that $i$ is the imaginary unit, and the complex number $z = (x^2 - 1) + (x + 1)i$ is purely imaginary. Find the value(s) of the real number $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Eastern rattlesnakes have 6 segments in their tails, while Western rattlesnakes have 8 segments. Additionally, Southern rattlesnakes have 7 segments, and Northern rattlesnakes have 9 segments in their tails. Calculate the average percentage difference in tail sizes of the Eastern, Southern, and Northern rattlesnakes compared to the Western rattlesnake's tail size.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}
If the distance from a point A on the parabola $C: y^2 = 4x$ to the focus of the parabola is 3, and $O$ is the origin, then the slope of line $OA$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{2}}
There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$.  

In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values?  

For example, we can choose $\{\Delta A1, \Delta B2, \odot C3\}$, but we cannot choose $\{\Delta A1, \square B2, \Delta C1\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1278}
Among the coordinates $(x,y)$ where $1 \leq x,y \leq 101$, choose some points such that there do not exist $4$ points which form an isosceles trapezium with its base parallel to either the $x$ or $y$ axis (including rectangles). Find the maximum number of coordinate points that can be chosen.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
Find the least positive integer $n$ such that 
$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For \(0 < m < 10\), let \(S(m, n)\) be the set of all positive integers with \(n\) 1s, \(n\) 2s, \(n\) 3s, \ldots, \(n\) \(m\)s. For a positive integer \(N\), let \(d(N)\) be the sum of the absolute differences between all pairs of adjacent digits. For example, \(d(122313) = 1 + 0 + 1 + 2 + 2 = 6\). Find the mean value of \(d(N)\) for \(N\) in \(S(m, n)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(m^2 - 1)}{3}}
Given that the number of rabbits in a farm increases such that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$, and the populations in the years $2001$, $2002$, and $2004$ were $50$, $80$, and $170$, respectively, determine the population in $2003$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
A space probe is to be launched from a space station 200 miles above Earth. Determine its escape velocity in miles/s. Take Earth's radius to be 3960 miles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\approx 6.76$ miles/s}
A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 4 : 13}
Let \(a_n = \frac{1}{n} \sin \frac{n\pi}{25}\) and \(S_n = a_1 + a_2 + \cdots + a_n\). Calculate the number of positive terms among \(S_1, S_2, \cdots, S_{100}\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Calculate the area enclosed by the curve defined by the equation $$\lfloor x + y\rfloor + \lfloor x - y\rfloor = 5$$ for $x\ge y$ and $x, y \ge 0$, where $\lfloor z\rfloor$ represents the greatest integer less than or equal to $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
A number, when divided by the sum of a certain number and 445, gives 2 times their difference as quotient and 25 as remainder. The number is 220025. What is the first number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{555}
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is marked on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\) and the ratio \(AK : KE = 3 : 7\). Find the ratio \(CL : AB\).

A regular pentagon is a pentagon with all sides equal and all angles equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7}
On a table, there are $100$ red and $k$ white buckets, all initially empty. In each move, a red and a white bucket are selected, and an equal amount of water is added to both. After a certain number of moves, no bucket is empty, and for every pair of buckets that were selected together at least once, the amount of water in these buckets is the same. Find all possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Let $x, y, z$ be positive real numbers that satisfy at least one of the inequalities: $2xy > 1$, $yz > 1$. Find the least possible value of:
$$xy^3z^2 + \frac{4z}{x} - 8yz - \frac{4}{yz}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
在ΔABC中，已知AC=3,sinC=ksinA(k≥2)，则ΔABC的面积的最大值为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are  $5$  accents in French, each applicable to only specific letters as follows:



- The cédille: ç
- The accent aigu: é
- The accent circonflexe: â, ê, î, ô, û
- The accent grave: à, è, ù
- The accent tréma: ë, ö, ü


Cédric needs to write down a phrase in French. He knows that there are  $3$  words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If  $n$  is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3 x^{2}+10 x y+3 y^{2}$ is the power of some prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
How many three-digit positive integers $N$ satisfy the following properties?  
The number $N$ is divisible by $7$.  
The number formed by reversing the digits of $N$ is divisible by $5$.  
Provide the number of such integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the function $f(x)= \frac{1}{3}|x^{3}|- \frac{a}{2}x^{2}+(3-a)|x|+b$ with six distinct monotonic intervals, determine the range of values for $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,3)}
The center of sphere $\alpha$ lies on the surface of sphere $\beta$. The ratio of the surface area of sphere $\beta$ that is inside sphere $\alpha$ to the entire surface area of sphere $\alpha$ is $1 / 5$. Find the ratio of the radii of spheres $\alpha$ and $\beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Let $k$ be a positive integer. Each cell of a $3 k \times 3 k$ grid is colored either red or blue. A triple of cells $\left(C_{1}, C_{2}, C_{3}\right)$ is called a good cell group if $C_{1}$ and $C_{2}$ are in the same row, $C_{2}$ and $C_{3}$ are in the same column, and $C_{1}, C_{3}$ are colored blue, while $C_{2}$ is colored red. Find the maximum possible number of good cell groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12k^{4}}
How many ways are there to place 31 knights in the cells of an $8 \times 8$ unit grid so that no two attack one another?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^\circ}
For which values of \( y \) does the equation \( x^{2}+2 y^{2}+8 z^{2}-2 x y z-9=0 \) have no solutions for any \( z \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{\sqrt{2}}<|y| \leq 2 \sqrt{2}}
Let point $P$ be a moving point on the parabola $x^2=4y$, and the distance from point $P$ to the $x$-axis is $d$. Point $P_1$ is a moving point on the circle $(x-2)^2+(y+1)^2=1$. When $d+|PP_1|$ is minimized, the coordinates of point $P$ are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2\sqrt{2}-2,3-2\sqrt{2})}
Let $\alpha \in \left(0, \frac{\pi}{4}\right)$, if $\tan\left(\alpha + \frac{\pi}{4}\right) = 2\cos2\alpha$, then $\alpha = $ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan(2 - \sqrt{3})}
Let $ABC$ be a triangle and $D$, $E$, and $F$ be the midpoints of sides $BC$, $AC$, and $BC$ respectively. The medians $AD$ and $BE$ are perpendicular, with $AD = 12$ and $BE = 9$. What is the value of $CF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In a rectangular parallelepiped, the sides of the base are \( a \) and \( b \), and the acute angle between them is \( 60^\circ \). The larger diagonal of the base is equal to the smaller diagonal of the parallelepiped. Find the volume of the parallelepiped.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6} ab \sqrt{ab}}{2}}
You are given a rectangular playing field of size  $13 \times 2$  and any number of dominoes of sizes  $2\times 1$  and  $3\times 1$ . The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?

(Walther Janous)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{257}
Given two circles described by the equations $x^2 + y^2 = m$ and $x^2 + y^2 + 6x - 8y + 21 = 0$, find the range of values for the real number $m$ such that the two circles intersect.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(9, 49)}
A paper square with an area equal to four times the area of a cell is placed on a sheet of grid paper. What is the minimum number of grid points that this square can cover? (A grid point is a point where the lines of the grid intersect; if a grid point lies on the edge of the square, it is considered covered.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
8. Fill $n^{2}(n \geqslant 3)$ integers into an $n \times n$ grid, with one number per cell, such that the product of the numbers in each row and each column is 30. Then there are $\qquad$ ways to fill the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{(n-1)^{2}}(n!)^{3}}
After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more than six dots on some faces). How many dots in total are needed for this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
In right triangle \( \triangle ACB \), \(\angle C = 90^{\circ}\), points \(D\) and \(E\) lie on \(BC\) and \(CA\) respectively, and \(\frac{BD}{AC} = \frac{AE}{CD} = K\). Let \(BE\) intersect \(AD\) at \(O\). If \(\angle BOD = 75^{\circ}\), what is \(K\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{3}}