inclusionAI/Ring-lite-rl-data huggingface dataset:
One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle.  A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced.  Tony fetches a tape measure from the garage and measures a distance of almost exactly  $12$  feet between a consecutive pair of trees.  Tony realizes the need to include the width of the trees in his measurements.  Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around  $15$  feet.  He then measures  $2$  feet to either side of the first and last trees in the row before the ends of the yard.  Tony uses these measurements to estimate the width of the yard.  If there are six trees in the row of trees, what is Tony's estimate in feet?

[asy]
size(400);
	defaultpen(linewidth(0.8));
	draw((0,-3)--(0,3));
	int d=8;
	for(int i=0;i<=5;i=i+1)
	{
	draw(circle(7/2+d*i,3/2));
	}
	draw((5*d+7,-3)--(5*d+7,3));
	draw((0,0)--(2,0),Arrows(size=7));
	draw((5,0)--(2+d,0),Arrows(size=7));
	draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7));
label(" $2$ ",(1,0),S);
label(" $12$ ",((7+d)/2,0),S);
label(" $15$ ",((7+3*d)/2,0),S);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82}
A number is guessed from 1 to 144. You are allowed to select one subset of the set of numbers from 1 to 144 and ask whether the guessed number belongs to it. For an answer of "yes," you have to pay 2 rubles, and for an answer of "no" - 1 ruble. What is the smallest amount of money needed to surely guess the number?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1, a_2, \dots, a_k$ such that $|a_n - a_{n-1}| = a_{n-2}$ for all $3 \leq n \leq k$. If $a_1 = a_2 = 1$, and $k = 18$, determine the number of elements of $\mathcal{A}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1597}
There are $10001$ students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:

1. Each pair of students is in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with $2m+1$ students ($m$ is a positive integer) is in exactly $m$ societies.

Find all possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5000}
Arrange the numbers \( 222^{2}, 22^{22}, 2^{222} \) in ascending order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222^{2} < 22^{22} < 2^{222}}
Given the real numbers \( a \geq b \geq c \geq d \) with \( a + b + c + d = 9 \) and \( a^2 + b^2 + c^2 + d^2 = 21 \), find the minimum possible value of \( \text{ab} - \text{cd} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In how many ways can we place a king and a rook on a chessboard so that neither attacks the other? (The squares of the chessboard are labeled in the usual way with letters and numbers; two positions are considered different if at least one of the pieces is on a different square in the two positions.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2940}
设$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$ $y _ { k } = \frac { 1 } { k } \sum _ { i = 1 } ^ { n } x _ { i } , k = 1 , 2 , \cdots , 2 0 0 1 .$求$\max \sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 } \vert .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Point E is the midpoint of side DC of square ABCD, where $\overrightarrow{AB}=a$ and $\overrightarrow{AD}=b$. Express $\overrightarrow{BE}$ in terms of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\overrightarrow{BE} = b - \frac{1}{2}a}
Given a right triangle \(ABC\), point \(D\) is located on the extension of hypotenuse \(BC\) such that line \(AD\) is tangent to the circumcircle \(\omega\) of triangle \(ABC\). Line \(AC\) intersects the circumcircle of triangle \(ABD\) at point \(E\). It turns out that the angle bisector of \(\angle ADE\) is tangent to circle \(\omega\). What is the ratio in which point \(C\) divides segment \(AE\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
A line parallel to leg \(AC\) of right triangle \(ABC\) intersects leg \(BC\) at point \(K\) and the hypotenuse \(AB\) at point \(N\). On leg \(AC\), a point \(M\) is chosen such that \(MK = MN\). Find the ratio \(\frac{AM}{MC}\) if \(\frac{BK}{BC} = 14\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
一根铁丝，第一次剪下(4/5)米，第二次剪下(3/5)米，剩下的与剪去的一样长，这根铁丝原来有多少米？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.8}
In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads; each head always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon's heads made the following statements:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
A platinum sphere weighs $p = 30$ grams in a vacuum. This sphere is immersed in mercury at $t = 100^{\circ}$C. The densities of both materials are known at $0^{\circ}$C, namely for platinum $d_{1} = 21.5$ and for mercury $d_{2} = 13.60$. The cubic thermal expansion coefficients are $a_{1} = 0.0000264$ for platinum and $a_{2} = 0.0001815$ for mercury. What is the weight of the platinum sphere in this mercury?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.310 \, \text{g}}
In a certain state, voters are registered as either Republicans, Democrats, or Independents. The ratio of registered Republicans to Democrats to Independents is 3:2:5. According to polling data, 70 percent of the Republicans, 30 percent of the Democrats, and 40 percent of the Independents are expected to vote for candidate X. Everyone else is expected to vote for candidate Y. Additionally, it is expected that 10 percent of the registered voters will not vote at all. Incorporating these factors, by what percent is candidate X expected to win the election?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\%}
the calendar of the year 2032 can be used again in the year ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2060}
A train 110 m long traveling at 60 kmph takes 20.99832013438925 seconds to cross a bridge of a certain length. How long is the bridge in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
$n \times n$ 矩阵 $A_{n}$ 的主对角元素为 n ，其余元素为 1. 已知多项式 $f(x)$ 满足 $f\left(A_{n}\right)=O$ 对任意 $1 \leqslant n \leqslant 100$ 均成立, 则 $\operatorname{deg} f$ 的最小值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149}
If
\[(1 + \tan 0^\circ)(1 + \tan 1^\circ)(1 + \tan 2^\circ) \dotsm (1 + \tan 30^\circ) = 2^m,\]
find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
5. We have cards with numbers 5, 6, 7, ..., 55 (each card has one number). What is the maximum number of cards we can select so that the sum of the numbers on any two selected cards is not a palindrome? (A palindrome is a number that reads the same backward as forward.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3 \times 3 \times 3$ cube such that each $1 \times 1 \times 3$ row or column contains exactly one white cube. Note that rotations are considered \textit{distinct}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
For each positive integer $n$, define the function $f(n)=\left\{\begin{array}{ll}0 & \text { when } n \text { is a perfect square, } \\ {\left[\frac{1}{\{\sqrt{n}\}}\right]} & \text { when } n \text { is not a perfect square. }\end{array}\right.$ (where $[x]$ denotes the greatest integer not exceeding $x$, and $\{x\}=x-[x])$, find: $\sum_{k=1}^{240} f(k)$.

---

The function $f(n)$ is defined as follows:
- $f(n) = 0$ if $n$ is a perfect square.
- $f(n) = \left[\frac{1}{\{\sqrt{n}\}}\right]$ if $n$ is not a perfect square.

Here, $[x]$ represents the greatest integer less than or equal to $x$, and $\{x\}$ represents the fractional part of $x$, which is $x - [x]$.

We need to find the value of $\sum_{k=1}^{240} f(k)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
Determine the number of solutions to the equation $e^{-z} = 3z-2$ in the annulus $\{z \in \Bbb C : 1 < |z| < 2\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The graph of the function $y=a^{x-1}+1 (a > 0 \;\text{and}\; a \neq 1)$ always has a fixed point at $\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). How many possible DNA sequences are there on Rthea?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28812}
If $x < 0$, then the range of values for $x + \frac{1}{x}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2]}
Find the smallest positive integer $n$ such that $\binom{2n}{n}$ is divisible by 200, where $200 = 2^3 \times 5^2$. Use Legendre's formula to solve the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
You have a flashlight that takes 2 working batteries. You have 12 batteries, but only 3 of them work. What is the fewest number of pairs you need to test to guarantee you can get the flashlight on?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In a right truncated quadrangular pyramid, a cross-section is made through the diagonals of the bases and another cross-section passes through a side of the lower base and the opposite side of the upper base. The angle between the cross-sectional planes is $\alpha$. Find the ratio of the areas of the cross-sections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cos \alpha}
Given a permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. Determine the maximum possible value of $k$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $ABC$ be a triangle with $\angle  A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9\sqrt{3}}{16}}
Let $k$ be an integer. If the equation $(x-1)\left|x+1\right|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4544}
Square ABCD has sides of length 3 cm. The points M and N lie on AD and AB so that CM and CN split the square into three pieces of the same area. Find the length of DM. Express your answer in centimeters (cm).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \text{ cm}}
Given positive real numbers $x$ and $y$ satisfying $x+y=1$, find the minimum value of $\dfrac{y}{x} + \dfrac{4}{y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that $a > b > c > 0$, find the minimum value of $2a^{2}+\frac{1}{ab}+\frac{1}{a(a-b)}-10ac+25c^{2}$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
A regular hexagon $K L M N O P$ is inscribed in an equilateral triangle $A B C$ such that points $K, L$ lie on side $A B$, points $M, N$ lie on side $B C$, and points $O, P$ lie on side $A C$. Calculate the area of the hexagon $K L M N O P$ if the area of triangle $A B C$ is $60 \mathrm{~cm}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 \, \text{cm}^2}
5. (ROM) Five points are given in the plane. Among the lines that connect these five points, no two coincide and no two are parallel or perpendicular. Through each point we construct an altitude to each of the other lines. What is the maximal number of intersection points of these altitudes (excluding the initial five points)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{310}
At a joint conference of the parties of liars and truth-lovers, 32 people were elected to the presidium and seated in four rows of 8 people each. During a break, each member of the presidium stated that among his neighbors, there are representatives of both parties. It is known that liars always lie, and truth-lovers always tell the truth. What is the minimum number of liars in the presidium under the described situation? (Two members of the presidium are considered neighbors if one of them is seated to the left, right, in front, or behind the other).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $[x]$ represent the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies $\left[\frac{1}{15}\right] + \left[\frac{2}{15}\right] + \left[\frac{3}{15}\right] + \cdots + \left[\frac{n-1}{15}\right] + \left[\frac{n}{15}\right] > 2011$, what is the smallest value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{253}
Eight positive integers are written on the faces of a square prism (two bases and a lateral surface consisting of four faces). Each vertex is labeled with the product of the three numbers on the faces sharing that vertex (two from the lateral faces and one from the bases). If the sum of the numbers at the vertices equals $1176$, determine the sum of the numbers written on the faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
The roots of the polynomial $x^4 - 4ix^3 + 3x^2 - 14ix - 44$ form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A man can row upstream at 20 kmph and downstream at 60 kmph. In addition, there is a water current flowing downstream at a constant speed of 5 kmph and a wind blowing upstream at a constant speed of 2.5 kmph. Determine the speed of the man in still water, assuming no wind or water current.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.5}
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that \[\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\] What is the maximum value of $P(a)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 - \sqrt{2}$}
Let  $N$  be the number of integer sequences  $a_1, a_2, \dots, a_{2^{16}-1}$  satisfying \[0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1\] for all  $1 \le k \le 2^{15}-1$ . Find the number of positive integer divisors of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65537}
There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amount of grass)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。求第三个函数的表达式。答案的形式为 $y = a\varphi(bx)$，请给出 $a + b$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
There are 17 people at a party, and each has a reputation that is either $1,2,3,4$, or 5. Some of them split into pairs under the condition that within each pair, the two people's reputations differ by at most 1. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are able to form $k$ pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the integral \begin{equation} \int_0^\pi\frac{\sin nx}{a^2-2ab\cos x+b^2}\, dx \end{equation} assuming $a>b>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For each vertex of the triangle \(ABC\), the angle between the altitude and the angle bisector drawn from that vertex was determined. It turned out that these angles at vertices \(A\) and \(B\) are equal to each other and are less than the angle at vertex \(C\). What is the measure of angle \(C\) in the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^
^
}
In a box, there are 88 balls of three colors: red, yellow, and black. It is known that if 24 balls are randomly taken out, there will be at least 10 balls of the same color. The question is, under the condition mentioned above, how many balls must be taken out from the box at minimum to ensure that there are at least 20 balls of the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
A rectangular parallelepiped is inscribed around a sphere such that the diagonals of its base are equal to \(a\) and \(b\). Find the total surface area of the parallelepiped.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3ab}
Given the expansion of $\left(x-\frac{a}{x}\right)^{5}$, find the maximum value among the coefficients in the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
On each of the lines \( x=0 \) and \( x=2 \), there are 62 points marked with ordinate values \( 1, 2, 3, \ldots, 62 \). In how many ways can you choose three points from the marked 124 such that they form the vertices of a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7908}
Let's consider an alphabet of \( n \) letters, which we will use to form words. We say that a word contains a palindrome if a segment of that word, of more than one letter, reads the same forwards and backwards. For example, the word OLIMPIADAS contains the palindrome ADA. Given \( k \) as an integer greater than 2, determine how many words of length \( k \) can be formed, with our alphabet of \( n \) letters, that do not contain any palindrome of odd length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 (n-1)^{k-2}}
5. Arrange the 5 numbers $2,0,1,9,2019$ in any order in a row, to form an 8-digit number (the first digit cannot be 0), then the number of different 8-digit numbers that can be produced is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
In triangle \(ABC\), let the parabola with focus \(A\) and directrix \(BC\) intersect sides \(AB\) and \(AC\) at \(A_1\) and \(A_2\), respectively. Similarly, let the parabola with focus \(B\) and directrix \(CA\) intersect sides \(BC\) and \(BA\) at \(B_1\) and \(B_2\), respectively. Finally, let the parabola with focus \(C\) and directrix \(AB\) intersect sides \(CA\) and \(CB\) at \(C_1\) and \(C_2\), respectively. If triangle \(ABC\) has sides of length 5, 12, and 13, find the area of the triangle determined by lines \(A_1C_2\), \(B_1A_2\), and \(C_1B_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6728}{3375}}
Given that \( m \) and \( n \) are integers greater than 7, consider a rectangular grid formed by \( m \times n \) points. We color \( k \) of these points red such that no right-angled triangle with two sides parallel to the sides of the rectangle has all three vertices colored red. Find the maximum value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 2}
Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that is reached according to the original planned time. Calculate the time of arrival at point \( B \) in 12-hour time format (HH:MM).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11:40}
金字塔（正四棱锥）的底座为边长 \(200 \, \text{m}\) 的正方形。如果一个游客处于底面且距离底座中心 \(200 \, \text{m}\) 的圆周上，求游客可以同时看见金字塔两塔侧面的概率。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a triangle ABC with side lengths 6, 8, and 10, and a circle with radius 1 rolling inside the triangle, always touching at least one side, calculate the distance traveled by the center P of the circle when it returns to its initial position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given a point $M$ on the plane satisfying $M \in \{(x, y) \mid (x - 3\cos\alpha)^2 + (y - 3\sin\alpha)^2 = 25, \alpha \in \mathbb{R} \}$, find the area of the figure formed by all such points $M$ on the plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60\pi}
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying \( f(m+n) \geq f(m) + f(f(n)) - 1 \) for all \( m, n \in \mathbb{N} \). What values can \( f(2019) \) take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}