inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $[x]$ denote the greatest integer less than or equal to the real number $x$, and let $\{x\}=x-[x]$. Find the sum of the squares of all natural numbers \( m \) that satisfy
$$
[(2 m+1)\{\sqrt{2 m+1}\}]=m
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Seven fishermen stand in a circle. The fishermen have a professional habit of exaggerating numbers. Each fisherman has an exaggeration factor (unique integer) representing how many times the actual number is exaggerated. For example, if a fisherman with an exaggeration factor of 3 catches two fish, he will say he caught six fish. When asked, "How many fish did your left neighbor catch?", the responses (not necessarily in the order the fishermen are sitting) were $12, 12, 20, 24, 32, 42,$ and $56$. When asked, "How many fish did your right neighbor catch?", six of the fishermen responded $12, 14, 18, 32, 48,$ and $70$. What did the seventh fisherman respond?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A lock with three buttons opens when the buttons are pressed in a specific order immediately one after the other. What is the minimum number of button presses required to ensure that the lock opens? (Any series of button presses before the correct sequence does not affect the mechanism of the lock.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A local school is holding a food drive. Mark brings in 4 times as many cans as Jaydon. Jaydon brings in 5 more than twice the amount of cans that Rachel brought in. Additionally, Sophie brings in cans such that the ratio of the number of cans contributed by Mark, Jaydon, and Sophie is 4:3:2. If there are 225 cans total, how many cans did Mark bring in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
问：至多能从集合 \{1,2,\cdots,100\} 中取出多少个数，使得其中任意不同两数的和与积均不为 100 的倍数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Positive integers $a,b,$ and $c$ are chosen so that $a<b<c$, and the system of equations $2x + y = 2003 \quad$ and $\quad y = |x-a| + |x-b| + |x-c|$ has exactly one solution. What is the minimum value of $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1002$}
Select 5 apples from a pile randomly, and measure their masses (unit: grams) as: $125, 124, 121, 123, 127$. Find the standard deviation $s=$ ______ (grams) (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s \approx 2.24 \text{ grams}}
There is a cube placed on stands and six different paints. How many ways can you paint all the faces of the cube (one color for each face, not necessarily using all the paints) so that adjacent faces (sharing an edge) are painted in different colors? (16 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4080}
Given that $g\in C(\mathbb{R})$, $g(0) = 1$ and $g(1) = -1$, evaluate the following integral:

$$\int_{0}^1 \frac{g'(x) \cos(g(x))}{1 + g^2(x)}\mathrm{d}x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For a given real value of $x$, at what value of $n$ does the expression $x\ln x-n\ln(x!)$ change its behavior from increasing to decreasing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given \( a, b > 0 \) satisfying the equation \( \sqrt{|x|} + \sqrt{|x + a|} = b \) having exactly three distinct real solutions \( x_{1}, x_{2}, x_{3} \), with \( x_{1} < x_{2} < x_{3} = b \), find the value of \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Given $\lg 2=0.3010$ and $\lg 3=0.4771$, calculate the decimal place where the first non-zero digit of $\left(\frac{6}{25}\right)^{100}$ occurs. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
The most common dice is a six-sided die, which is a cube. The 6 faces have 1 to 6 points on them, with the sum of points on opposite faces equal to 7. From a single point in space, you can see multiple faces of the die. The minimum visible sum of points is 1, and the maximum is 15 (15 = 4 + 5 + 6). Determine which sums in the range from 1 to 15 are not possible to see.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
At an election, 2 candidates participated. One candidate got 20% of votes and was defeated by 500 votes. There were 10 invalid votes. How many total polled votes were there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{843}
If the graph of the function $f(x)$ is continuous and unbroken on the interval $(0, +\infty)$, and there is a unique irrational number root $x_0$ in the interval $(2,3)$, then to find the approximate value of $x_0$ with an accuracy of 0.001 using the "bisection method", the function value at the midpoint of the interval needs to be calculated \_\_\_\_\_\_ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Mrs. Hilt started by reading 2 books per day, but from the third day, she increased the number of books she read daily by 1. How many books did she read in one week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Person A and person B jog back and forth on a straight road that is 400 meters long, with speeds of 3 meters per second and 2.5 meters per second, respectively. They start simultaneously from opposite ends and meet a total of ____ times within 20 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
平面直角坐标系中, $\vec{e}$ 是单位向量, 向量 $\vec{a}$ 满足 $\vec{a} \cdot \vec{e}=2$, 且 $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数 $t$ 成立，求 $|\vec{a}|$ 的取值范围. 请给出该区间端点之和的整数部分。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The length of a rectangular plot is 20 meters more than its breadth. If the cost of fencing the plot is 26.50 per meter on flat ground, the price increases by an additional 10% for each meter of height the ground rises due to unevenness. If the total cost of fencing the plot is Rs. 5300, and the ground rises a total of 5 meters along the breadth of the plot, what is the length of the plot in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Find the two numbers such that their HCF is 23, the product is 98596, and the difference of these numbers is minimized. What is the greater number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{713}
A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) of positive integers satisfies \(a_{1} > 5\) and \(a_{n+1} = 5 + 6 + \cdots + a_{n}\) for all positive integers \(n\). Determine all prime numbers \(p\) such that, regardless of the value of \(a_{1}\), this sequence must contain a multiple of \(p\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n = n^2$, and in $\triangle ABC$, the ratio of the three sides is $a:b:c = a_2:a_3:a_4$, then the largest interior angle of $\triangle ABC$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630}
Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
Compare the values of $\sin 11^\circ$, $\sin 168^\circ$, and $\cos 10^\circ$. Express your answer as an inequality with the correct order of these values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin 11^\circ < \sin 168^\circ < \cos 10^\circ}
Two high school classes took the same test. One class of $20$ students made an average grade of $80\%$; the other class of $30$ students made an average grade of $70\%$. The average grade for all students in both classes is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$74\%$}
Find the least positive integer of the form $a\ b\ a\ a\ b\ a$, where $a$ and $b$ are distinct digits, such that the integer can be written as a product of six distinct primes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{282282}
A strip of size $1 \times 10$ is divided into unit squares. The numbers $1, 2, \ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one of the squares neighboring those already occupied, and so on (the choice of the first square and the choice of neighbor at each step are arbitrary). In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
On the first Saturday, Jessy told eleven jokes, while Alan told seven jokes. Meanwhile, Tom told five jokes and Emily told three jokes. The following week, on each day Sunday through Wednesday, each of them doubled the number of jokes they told from the previous day. How many jokes did they tell together after eight days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{806}
On the board, the natural number \( N \) was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers?

(I. Efremov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the minimum number of points in set M, given that circle C7 passes through exactly 7 points in M, circle C6 passes through exactly 6 points in M, and so on, with circle C1 passing through exactly 1 point in M?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the third side of an acute triangle if its two sides are \( a \) and \( b \) and it is known that the medians to these sides intersect at a right angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{a^2 + b^2}{5}}}
Let  $ABC$  be a right triangle with  $m(\widehat{A})=90^\circ$ . Let  $APQR$  be a square with area  $9$  such that  $P\in [AC]$ ,  $Q\in [BC]$ ,  $R\in [AB]$ . Let  $KLMN$  be a square with area  $8$  such that  $N,K\in [BC]$ ,  $M\in [AB]$ , and  $L\in [AC]$ . What is  $|AB|+|AC|$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
While waiting for customers, a watermelon seller weighed 20 watermelons (each weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) by balancing a watermelon on one side of the scale with one or two weights (possibly identical) on the other side of the scale. The seller recorded the weights he used on a piece of paper. What is the minimum number of different numbers that could appear in his notes if the weight of each weight is an integer number of kilograms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
2. Given are two circles $k_{1}$ and $k_{2}$ with centers $P$ and $R$ respectively, which touch each other externally at point $A$. Let $p$ be their common tangent, which does not pass through $A$, and touches $k_{1}$ and $k_{2}$ at points $B$ and $C$, respectively. The line $P R$ intersects the line $B C$ at point $E$, and the circle $k_{2}$ at points $A$ and $D$. If $\overline{A C}=\frac{\overline{A B}}{2}$, find the ratio $\frac{\overline{B C}}{\overline{D E}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Initially, there are 41 crayons and 26 pencils in the drawer. Rachel removes 8 crayons from the drawer, and then Sam adds 12 crayons back into the drawer. After that, the number of crayons in the drawer increases by 10%. How many crayons are in the drawer now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given the sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and $a_n= \frac {n\cdot 2^{n}-2^{n+1}}{(n+1)(n^{2}+2n)}\ (n\in\mathbb{N_+})$, find the expression for $S_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^{n+1}}{(n+1)(n+2)} - 1}
During Teacher's Day at a certain school, there are 200 teachers who each send a greeting message to exactly one other teacher. Determine the minimum value of \( k \) such that it is possible to select at least \( k \) representatives to attend a recognition ceremony, where no one in this group has sent a message to anyone else in the group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
A train of a certain length crosses an electric pole in 4.499640028797696 seconds, with its speed being 72 km/hr. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89.99280057595392}
Two differentiable functions $f(x)$ and $g(x)$ are defined on the interval $[0, 2]$. Given that $f''(x) - g''(x) = 0$, $f'(A) = 2$, $g'(A) = 4$, $f(B) = 3$, and $g(B) = 9$, find the value of $f(\frac{3}{2}) - g(\frac{3}{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Eliana walked 200 steps for her morning exercise, did some press-ups, then added some more steps to her count for the first day. The next day, she walked twice the number of steps she walked on the first day. And on the third day, Eliana walked an additional 100 steps. The total number of steps Eliana walked during these three days is 1600. How many additional steps did Eliana add to her count on the first day after doing press-ups?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given the proposition p: ∀x ∈ ℝ, 2^x > 0, what is the proposition ¬p? Express your answer as a logical proposition using correct logical notation and quantifiers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃x ∈ ℝ, 2^x ≤ 0}
Evaluate the integral
$$I = \int_0^{\infty} \frac{\log(x)}{x^2 - 1} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi^2}{4}\)}
In triangle \(ABC\) with a \(120^\circ\) angle at vertex \(A\), the bisectors \(AA_1\), \(BB_1\), and \(CC_1\) are drawn. Find the angle \(C_1 A_1 B_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that from any subset of \( S \) containing \( n \) elements, we can always select 10 numbers. No matter how we divide these 10 numbers into two groups, there will always be one group containing a number that is coprime with the other 4 numbers in its group, and another group containing a number that is not coprime with the other 4 numbers in its group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let $l$ be a tangent line at the point $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$ on the curve $C: y=\frac{x^2}{2}.$ A circle $S$ touches the curve $C$ and the $x$ axis.
Denote by $(x(t),\ y(t))$ the center of the circle $S$. Find $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2} + 1}{30}}
4 and a certain number are factors of 60 n, and the minimum value of n is 8. What is the other factor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
25. How many even four-digit numbers can be formed from the digits $0,1,2,3,4,5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
A show debut and 200 people buy tickets. For the second showing, a certain number of people show up. If each ticket cost $25, the show made $20,000. What is the ratio of the number of people who attended the second showing to the number of people who attended the debut show?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:1}
The segment \( AB \) is the diameter of a circle. A second circle with center at point \( B \) has a radius of 2 and intersects the first circle at points \( C \) and \( D \). The chord \( CE \) of the second circle is part of a tangent to the first circle and is equal to 3. Find the radius of the first circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{\sqrt{7}}}
$\triangle ABC$ is inscribed in a circle of radius $r$ such that its base $AB$ coincides with the diameter $AB$. Point $C$ lies somewhere on the circle different from points $A$ and $B$. Let $s = AC + BC$. Determine the maximum possible value of $s^2$ given all permissible positions of $C$ on the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8r^2}
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each told the traveler whether the neighbor to their right was truthful. Based on these statements, the traveler was able to unequivocally determine the fraction of liars among the villagers. Determine what this fraction is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
Let $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and 
\[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \quad n \ge 4.\]
Determine $a_{14^{14}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In what base number systems can a three-digit number be equal to twice the number formed by reversing the order of its digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3k+2}
What is the negation of the proposition ∀x ∈ R, e^x - x - 1 ≥ 0? Express your answer as a logical statement using proper mathematical notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃x ∈ R, e^x - x - 1 < 0}
10.3. Lёsha colors cells inside a $6 \times 6$ square drawn on graph paper. Then he marks the nodes (intersections of the grid lines) to which the same number of colored and uncolored squares are adjacent. What is the maximum number of nodes that can be marked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
On an \(8 \times 8\) chessboard, 6 black rooks and \(k\) white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of \(k\). (Two rooks attack each other if they are in the same row or column and no rooks are between them.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $M = \{y | y = x^2, x \in \mathbb{Z}\}$ and $N = \{x | x^2 - 6x - 27 \geq 0, x \in \mathbb{R}\}$, with the universal set $U = \mathbb{R}$. Calculate the number of proper subsets of $M \cap (\complement_U N)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Seven little children sit in a circle. The teacher distributes pieces of candy to the children in such a way that the following conditions hold. - Every little child gets at least one piece of candy. - No two little children have the same number of pieces of candy. - The numbers of candy pieces given to any two adjacent little children have a common factor other than 1. - There is no prime dividing every little child's number of candy pieces. What is the smallest number of pieces of candy that the teacher must have ready for the little children?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
The sequence $x_1, x_2, x_3, \ldots$ is defined by $x_1 = 2022$ and $x_{n+1} = 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of $7$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{404}
Evaluate the line integral $\int_\mathcal{E} F*dr$ where $F(x,y,z) = x\,i+xy\,j+x^2yz\,k$ and $\mathcal{E}$ is the ellipse $x^2 + 4y^2 - 8y + 3 = 0$ in the xy-plane, without using parameterization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{2}$}
Determine the range of values for the real number $a$ such that the inequality $(a-2)x^{2}+2(a-2)x-4 < 0$ holds true for all real numbers $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2,2]}
Given \(3^{s} + 13^{t} = 17^{s}\) and \(5^{s} + 7^{t} = 11^{t}\), determine the relationship between \(s\) and \(t\) (fill in \(>, <\) or \(=\)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{<}
We draw a radius of a circle. We draw a second radius $23^\circ$ clockwise from the first radius. We draw a third radius $23^\circ$ clockwise from the second. This continues until we have drawn $40$ radii, each $23^\circ$ clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
"Pre-assembled" unit cube has a base and a top, as well as its sides. The top is rotated by an acute angle $\alpha$ around the line connecting the centers of the two faces, relative to the base. By how much does the top get closer to the base if the side edges remain the same length and the connection between the vertices also remains unchanged?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\sqrt{\cos\alpha}}
Given a triangle with sides of lengths 3, 4, and 5. Three circles are constructed with radii of 1, centered at each vertex of the triangle. Find the total area of the portions of the circles that lie within the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{2}"]}
Find the value of $\left(1+\cos \frac{\pi}{5}\right)\left(1+\cos \frac{3 \pi}{5}\right)$. The original answer is in the form of \frac{k}{m}, please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For a cone and a cylinder with the same base diameter and height, what is the ratio of their lateral surface areas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5} : 4}
Find the number of different ways to arrange all natural numbers from 1 to 9 inclusively, one in each cell of a 3x3 grid, such that the sums of the numbers in each row and each column are equal. The table cannot be rotated or reflected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}