inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). A chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of the segment \( AB \) if \( BD \cdot BC = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Compute the sum of all positive integers $n<2048$ such that $n$ has an even number of 1's in its binary representation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1048064}
Find the resolvent of the Volterra integral equation with the kernel $K(x, t) \equiv 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\lambda (x - t)}}
Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ appears in $W$ if there are indices $k \le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called special if $Ua, Ub, aU, bU$ all appear in $W$. (The empty word is considered special.) You are given that there are no special words of length greater than 2015.

Let $N$ be the minimum possible number of special words. Find the remainder when $N$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{535}
Teacher Wang took 48 students from class 3 (5) for a spring outing. Each minibus can accommodate up to 8 passengers. How many minibuses does Teacher Wang need to rent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Consider a hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. The distance from the point on the hyperbola with an x-coordinate of $\frac{3a}{2}$ to the right focus is greater than its distance to the left directrix. Determine the range of acute angles formed by the two asymptotes of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0^\circ, 60^\circ)}
Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Determine the number of conjugacy classes of $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^3 + p^2 - p}
Given a circle $(x-3)^2+(y+1)^2=4$ with a moving point $P$ on it, and another moving point $Q$ on the line $x=-3$, find the minimum value of the distance $|PQ|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \text{ units}}
On a $11 \times 11$ grid, 22 cells are marked such that there are exactly two marked cells in each row and each column. Two configurations of marked cells are considered equivalent if one can be transformed into the other by permuting rows and columns any number of times. How many distinct configurations of marked cells exist, considering these equivalence conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In an acute triangle \(ABC\), heights \(CH\) and \(AH_1\) are drawn. It is known that \(AC = 2\), and the area of the circle circumscribed around triangle \(HBH_1\) is \(\pi / 3\). Find the angle between height \(CH\) and side \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
\( \triangle ABC \) is an isosceles triangle with \( AB = AC \). Point \( P \) is located inside \( \triangle ABC \) such that \( \angle BCP = 30^\circ \), \( \angle APB = 150^\circ \), and \( \angle CAP = 39^\circ \). Find \( \angle BAP \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find the number of strictly increasing sequences of nonnegative integers with the following properties:

1. The first term is $0$ and the last term is $12$. Therefore, the sequence has at least two terms.
2. Among any two consecutive terms, exactly one of them is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Given a proposition related to positive integers, if the proposition holds true when $n = k$ ($k \in \mathbb{N}^+$), then it can be inferred that the proposition also holds true when $n = k + 1$. Now, it is known that the proposition does not hold true when $n = 5$. What is the value of $n$ for which we can infer that the proposition does not hold true, based on the given information?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the part of the surface \(z = x^2 + y^2\), cut off by the plane \(z=2\). The normal vector is taken to be outward with respect to the region bounded by the paraboloid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
Let
$$
A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}}
$$

Find the largest integer less than or equal to \(\frac{1}{A}\).

The following decimal approximation might be useful: \(0.6931 < \ln (2) < 0.6932\), where \(\ln\) denotes the natural logarithm function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider a line of \( n \) soldiers. They are to be divided into several non-empty units, and a commander is chosen for each unit. Determine the number of ways this can be done.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{2n}}
In a $9 \times 9$ grid, each cell contains one beetle. Upon hearing a whistle, each beetle moves diagonally to an adjacent cell. Determine the minimum possible number of empty cells after the movement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
11.142. In a regular triangular prism, a plane is drawn through the side of the lower base and the opposite vertex of the upper base, forming an angle of $45^{\circ}$ with the plane of the lower base. The area of the section is $S$. Find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{S\sqrt{S}\cdot\sqrt[4]{6}}{2}}
Points \( A,B,C \) lie on a circle \( \omega \) such that \( {BC} \) is a diameter. \( {AB} \) is extended past \( B \) to point \( {B}^{\prime } \) and \( {AC} \) is extende
 past \( C \) to point \( {C}^{\prime } \) such that \( \overline{{B}^{\prime }{C}^{\prime }}\parallel {BC} \) and tangent to \( \omega \) at point \( D \) . If \( {B}^{\prime }D = 4 \) and \( {C}^{\prime }D = 6 \) , compute \( {BC} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{24}{5}$}
Determine the Löwenheim number of the extended First-Order Logic $L(Q^c)$, which includes Chang's quantifier $Q^c$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Find the derivative of the trace function Tr($A$) with respect to the matrix $B$ in the space $M(n)$ of $n \times n$ real matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Jenny wants to read 5 books in the next 15 days. She can read 60 words per minute, but she takes a 15-minute break after every hour of continuous reading. If the first book has 12,000 words, the second book has 18,000 words, the third book has 24,000 words, the fourth book has 15,000 words, and the fifth book has 21,000 words, how many minutes per day, on average, should she spend reading (including break time)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{124}
Determine the \(\hat{r}\) component of the following integral in cylindrical coordinates without performing the integration:

$$\int \limits_0^{2\pi} \int \limits_0^a \frac{r^2}{(r^2+h^2)^{3/2}} \hat{r} \, dr \, d\theta$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are 4 stacks of coins on a table. The first stack has 9 coins, the second has 7, the third has 5, and the fourth has 10. In one move, you are allowed to add one coin to three different stacks. What is the minimum number of moves required so that all the stacks have an equal number of coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Calculate \( \lim_{a \rightarrow 0^+} \int_{D_a} \frac{|x|}{y(1+y^2)} \, dx \, dy \) where \( D_a=\{(x,y) \in \mathbb{R}^2: \frac{x^2}{2} \le y \le 2x^2 ; a \le \sqrt{x^2+y^2} \le \frac{1}{a} \} \) without using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4}}
A train of a certain length completely passes an electric pole in 8.999280057595392 seconds, if it travels at a speed of 80 kmph. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $M$ is the midpoint of $BC$ with $BM = 2$. $AM = c - b$. Find the maximum area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Find the remainder when $5^{5^{5^5}}$ is divided by 500.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both vehicles continue moving without stopping at point $C$. Determine the time at which the car and the truck meet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:40}
On a board, the 2014 positive integers from 1 to 2014 are written. The allowed operation is to choose two numbers \( a \) and \( b \), erase them, and write in their place the numbers \( \text{gcd}(a, b) \) (greatest common divisor) and \( \text{lcm}(a, b) \) (least common multiple). This operation can be performed with any two numbers on the board, including numbers that resulted from previous operations. Determine the largest number of 1's that we can leave on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1007}
In $\triangle ABC$, $\angle ACB = 90^{\circ}$, $CD$ and $CE$ are the heights to the base $AB$ and the angle bisector of $\angle C$, respectively. If $\triangle CED \sim \triangle ABC$, then find the measure of $\angle ECD$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.5}
The two brothers, without waiting for the bus, decided to walk to the next stop. After passing $\frac{1}{3}$ of the way, they looked back and saw a bus approaching the stop. One of the brothers ran backwards, and the other ran forward at the same speed. It turned out that everyone ran to their stop exactly at the moment when the bus approached it. Find the speed of the brothers, if the bus speed is $30$ km/h, neglecting the bus stop time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{330}
In a round-robin chess tournament organized at a school, every two students played one match against each other. Determine the minimum possible number of students in the school given the following conditions:

- Each girl student has at least 21 wins in matches against boy students.
- Each boy student has at least 12 wins in matches against girl students.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5399}
There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
In triangle $ABC$, $AB=\sqrt{30}$, $AC=\sqrt{6}$, and $BC=\sqrt{15}$. There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$, and $\angle ADB$ is a right angle. Find the value of the ratio
\[\dfrac{[ADB]}{[ABC]}.\]In other words, compute the ratio of the area of triangle $ADB$ to the area of triangle $ABC.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{38}}
Two identical cars are traveling in the same direction. The speed of one is $36 \kappa \mu / h$, and the other is catching up with a speed of $54 \mathrm{kм} / h$. It is known that the reaction time of the driver in the rear car to the stop signals of the preceding car is 2 seconds. What should be the distance between the cars to avoid a collision if the first driver suddenly brakes? For a car of this make, the braking distance is 40 meters from a speed of $72 \kappa м / h$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.5}
一只蚂蚁在正方体 $A B C D-E F G H$ 的顶点 $A$ 处，每次等概率地爬行到相邻三个顶点中的一个，那么六次爬行之后回到顶点 $A$ 的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{304}
Determine the radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a = 5$, $b = 10$, and $c = 20$ using the formula:
\[
\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.381}
When travelling from London to Edinburgh by train, you pass a sign saying 'Edinburgh 200 miles'. Then, \(3 \frac{1}{2}\) miles later, you pass another sign saying 'Half way between London and Edinburgh'. Calculate the total distance in miles from London to Edinburgh.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{393}
How many ordered triples of nonzero integers $(a, b, c)$ satisfy the equation $2abc = a + b + c + 4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.

---

The translation preserves the original text's formatting and structure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
设三角形的三边长分别是整数l，m，n，其中l&gt;m&gt;n.已知$\{ \frac { 3 ^ {l } } { 1 0 ^ { 4 } } \} = \{ \frac { 3 ^ { m } } { 1 0 ^ { 4 } } \} =$ $\{ \frac { 3 ^ { n } } { 1 0 ^ { 4 } } \}$，这里{x}=x-[x]，而[x]表示不超过x的最大整数．求这种三角形周长的最小值．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3003}
Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \( A \) such that, in any such set of numbers, each number does not exceed \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
In trapezoid \(ABCD \) with \(AB \parallel CD\) and \(AB + CD = b\), the diagonals \(AC\) and \(BD\) are related by \(5AC = 3BD\). Find the area of the trapezoid if \(\angle BAC = 2 \angle DBA\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 \sqrt{11}}{64} b^{2}}
【Question 25】
From 49 balls labeled with numbers 1 to 49, select several balls and arrange them in a circle so that the product of the numbers on any two adjacent balls is less than 100. How many balls can be selected at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
On a tram line, trams depart regularly every 10 minutes throughout the day. A tram takes one hour to travel from one end of the line to the other. A passenger gets on a tram at one end stop and travels to the final stop; having nothing else to do, they look out the window and count the trams coming from the opposite direction on the same route. How many trams will they count during the journey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
For which integer values of \( a \) is the polynomial \( x^{13} + x + 90 \) divisible by \( x^2 - x + a \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the interval in which the function $f(x)={(\dfrac{1}{2})}^{\sqrt{x-{{x}^{2}}}}$ is monotonically decreasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[0, \dfrac{1}{2}\right]}
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The second train is 280 m long. From the moment they meet, they will be clear of each other in 24.998 seconds. What is the length of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{219.96}
1. Given the quadratic function
$$
\begin{aligned}
f(x)= & a(3 a+2 c) x^{2}-2 b(2 a+c) x+ \\
& b^{2}+(c+a)^{2}(a, b, c \in \mathbf{R}).
\end{aligned}
$$

Assume that for any $x \in \mathbf{R}$, we have $f(x) \leqslant 1$. Find the maximum value of $|a b|$. (Yang Xiaoming, Zhao Bin)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{3\\sqrt{3}}{8}"]}
对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有
$f(X)=\frac{1}{|X|} \sum_{a \in X} a$.
集合对 $(A, B)$ 满足
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$,
且 $1 \leqslant|A| \leqslant 98$.
任取 $p \in B$, 令
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$.
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$, 求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{821}
Ali and Veli go hunting. The probability that each will successfully hit a duck is \(\frac{1}{2}\) on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. What is the probability that Veli hits more ducks than Ali? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form, and find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Use the bisection method to find an approximate zero of the function $f(x) = \log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\log 2.5 \approx 0.398$, $\log 2.75 \approx 0.439$, and $\log 2.5625 \approx 0.409$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.6}
In \(\triangle QRS\), point \(T\) is on \(QS\) with \(\angle QRT = \angle SRT\). Suppose that \(QT = m\) and \(TS = n\) for some integers \(m\) and \(n\) with \(n > m\) and for which \(n + m\) is a multiple of \(n - m\). Suppose also that the perimeter of \(\triangle QRS\) is \(p\) and that the number of possible integer values for \(p\) is \(m^2 + 2m - 1\). What is the value of \(n - m\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
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 $\overline{PQ}$. What is $AR+BR+CR+DR$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. What is the area of the region of the plane that is covered by the union of the five triangular regions? Express your answer in the form of $a\sqrt{b}$, where $a$ and $b$ are integers. Please provide the value of a + b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Five soccer teams held a tournament - each team played with each other once. For a win, 3 points were awarded, for a draw - 1 point, and no points were given for a loss. Four teams scored 1, 2, 5, and 7 points respectively. How many points did the fifth team score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}