inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Calculate the radius of convergence for the power series $$\sum_{n=1}^\infty (\frac{z-i} {z+2i})^n$$ using the Cauchy criterion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given an arc length of π cm with a corresponding central angle of $\frac{π}{4}$, the area of the sector containing this arc is __________ cm².
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2π \text{ cm}^2}
Compute the integral $\int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)}$ using residue theory from complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\pi\sqrt{2}$}
Consider the sequences \( \left(a_{n}\right) \) and \( \left(b_{n}\right) \) defined by
\[
a_{1}=3, \quad b_{1}=100, \quad a_{n+1}=3^{a_{n}}, \quad b_{n+1}=100^{b_{n}}.
\]
Find the smallest integer \( m \) for which \( b_{m} > a_{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
The maximum value of the function
$$
f(x)=\sqrt{3} \sin 2 x+2 \sin x+4 \sqrt{3} \cos x
$$
is .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{2}}
The base $MQ$ of trapezoid $MNPQ$ ($MQ \parallel NP, MQ > NP$) is the diameter of a circle which is tangent to line $MN$ at point $M$ and intersects side $PQ$ at point $K$. Additionally, $PQ = 4 \sqrt{3} KQ$. The radius of the circle is $R$, and $\angle NQM = 60^{\circ}$. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2R^2(5\sqrt{3} - 6)}
## Condition of the problem

Calculate the definite integral:

$$
\int_{0}^{\operatorname{arctg} \frac{1}{3}} \frac{8+\operatorname{tg} x}{18 \sin ^{2} x+2 \cos ^{2} x} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}+\frac{\ln2}{36}}
Given the circle equation x^2+y^2+2x-m=0, find the range of m such that point P(1,1) is outside the circle. Express your answer as an interval in standard interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,4)}
$\mathbf{F 1 7}$ (39-6, Bulgaria) Let $\mathbf{N}^{*}$ be the set of all positive integers, and the function $f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ satisfies: for any $s$ and $t$ in $\mathbf{N}^{*}$, we have
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2},
$$

Determine the smallest possible value of $f(1998)$ among all such functions $f$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given the function f(x) = ax^2 - c, satisfying the conditions: -4 ≤ f(1) ≤ -1 and -1 ≤ f(2) ≤ 5, determine the range for f(3). Express your answer as an interval (lower bound ≤ f(3) ≤ upper bound).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 ≤ f(3) ≤ 20}
What is the smallest three-digit number in Pascal's triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
How many different solutions does the congruence $x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}$ have? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If the graph of the function $y = \frac{kx^2}{k^2 + k - 2}$ is a hyperbola, and it is located in the second and fourth quadrants, then find the value of $k$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let \( \triangle ABC \) be a triangle with \( AB = 13 \), \( BC = 14 \), and \( CA = 15 \). Let \( O \) be the circumcenter of \( \triangle ABC \). Find the distance between the circumcenters of \( \triangle AOB \) and \( \triangle AOC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{91}{6}}
Find the limit of the sequence $y_n$ defined by:

$$y_n=\frac{1}{\sqrt{n^2-1}}+\frac{1}{\sqrt{n^2-2}}+\cdots+\frac{1}{\sqrt{(n-1)^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given sets $A=\{x|x\geq|x^2-2x|\}$, $B=\{x|x^2-2ax+a\leq0\}$, if $A\cap B=B$, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,1]}
Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
8. The planar region $S=\left\{(x, y) \mid x, y \in\left[0, \frac{\pi}{2}\right], \sin ^{2} x-\sin x \sin y+\sin ^{2} y \leqslant \frac{3}{4}\right\}$ has an area equal to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^{2}}{6}}
In front of you is a segment-digit display. To show time on a digital clock, each digit uses seven segments, each of which can be illuminated or not; the illuminated segments form a digit, as shown in the figure. For example, zero uses six segments, one uses two segments, and so on. The digital clock displays only hours and minutes. How many minutes throughout the day use more illuminated segments than the subsequent minute? (The day starts at 00:00 and ends at 23:59) (20 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630 \text{ minutes}}
Calculate the integral $\displaystyle \int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{8}\ln(2)\)}
How many acute angles can be drawn with a set of triangle rulers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let's find the smallest positive integers $x$ and $y$ for which the product of $180$ and $x$ is a fourth power and the product of $180$ and $y$ is a sixth power. Find the sum of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4054500}
Freddie and his team are collecting blankets for three days to be donated to the Children Shelter Organization. There are 15 people on the team on the first day. On the first day, each of them gave 2 blankets. For the second day, 5 new members joined the team, and they tripled the number of blankets they collected on the first day by asking door-to-door. On the second day, the new members contributed 4 blankets each, while the original members continued to contribute 2 blankets each. On the last day, they set up boxes at schools and got a total of 22 blankets, but they also received 30 more blankets through their online outreach efforts. How many blankets did they collect for the three days for donation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$, the number $d$ is a divisor of $\overline{aabbcc}$ if and only if $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. 

Note: The numbers $a \ne 0$, $b$, and $c$ need not be different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Let  $a_1, a_2, a_3, a_4$  be integers with distinct absolute values. In the coordinate plane, let  $A_1=(a_1,a_1^2)$ ,  $A_2=(a_2,a_2^2)$ ,  $A_3=(a_3,a_3^2)$  and  $A_4=(a_4,a_4^2)$ . Assume that lines  $A_1A_2$  and  $A_3A_4$  intersect on the  $y$ -axis at an acute angle of  $\theta$ . The maximum possible value for  $\tan \theta$  can be expressed in the form  $\dfrac mn$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by James Lin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
From vertex \( B \) of parallelogram \( ABCD \), the perpendicular heights \( BK \) and \( BH \) are drawn. The segments \( KH = a \) and \( BD = b \) are known. Find the distance from point \( B \) to the orthocenter of triangle \( BKH \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{b^2 - a^2}}
The side edge of a regular tetrahedron \( S-ABC \) is 2, and the base is an equilateral triangle with side length 1. A section passing through \( AB \) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\sqrt{15}}}
设 a_1,a_2,\cdots,a_{20}\in\{1,2,\cdots,5\}，b_1, b_2, \cdots, b_{20} \in\{1,2,\cdots,10\}. 集合 X=\{(i,j)\ |\ 1\leq i<j\leq 20,(a_i-a_j)(b_i-b_j)<0\}, 求 X 的元素个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
How many natural numbers from 1 to 700, inclusive, contain the digit 6 at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{133}
Among the vertices of two unequal icosahedrons, six can be chosen to form the vertices of a regular octahedron. Find the ratio of the edges of the icosahedrons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}+1}{2}}
Given that the graph of the power function $f(x)$ passes through the point $\left( \frac {1}{2}, 8 \right),$ find the value of $f(3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{27}}
Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions. For example, he could write the fractions \(\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}, \frac{20}{1}\).

Antoine wants to have as many fractions with integer values as possible among the written fractions. In the previous example, he wrote three fractions with integer values: \(\frac{10}{2}=5\), \(\frac{15}{5}=3\), and \(\frac{20}{1}=20\). What is the maximum number of fractions that can have integer values?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left[1, \frac{\sqrt{5} + 1}{2}\right) }
Given the function $f\left(x\right)=(\frac{1}{x}-a)\ln \left(1+x\right)$, if the curve $y=f(\frac{1}{x})$ is symmetric about the line $x=b$, then the value of $a+b$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Sandy took some money for shopping. She needed to buy items from three categories: clothing, electronics, and food. Limited-time offers were available for all items, and Sandy had to prioritize purchasing those items first.

She spent 25% of her initial money on limited-stock clothing items, 15% on limited-time electronic offers, and 10% on discounted food items. After prioritizing those purchases, she spent an additional 20% of her remaining money on non-discounted items. Finally, she paid 8% sales tax on her total purchases. With $140 remaining after shopping, what was the original amount of money Sandy took along with her?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$397.73}
A circle passes through the vertices \( A \) and \( C \) of an isosceles triangle \( ABC \) (\( AB = BC \)) and intersects the sides \( AB \) and \( BC \) at points \( M \) and \( N \), respectively. The chord \( MK \) of this circle, which has a length of \( 2\sqrt{5} \), contains a point \( H \) that lies on \( AC \) and serves as the foot of the altitude of triangle \( ABC \). A line passing through point \( C \) and perpendicular to \( BC \) intersects line \( MN \) at point \( L \). Find the radius of the circle circumscribed around triangle \( MKL \) if \( \cos \angle ABK = \frac{2}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设A（2,0）为平面上的一个定点，$P ( \sin ( 2 t - 6 0 ^ { \circ } )$ cos(2t-$6 0 ^ { \circ } ) )$为动点，则当t由15°变到45°时，线段AP所扫过的图形的面积是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { \pi } { 6 }$
}
The h.c.f. of two numbers is 30 and the other two factors of their l.c.m. are some values. The larger of the two numbers is 450. What is one of the other factors of their l.c.m.?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A train of a certain length crosses an electric pole at a speed of 54 km/hr. It takes 6.666133375996587 seconds to cross the pole. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99.99}
Let there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random direction, and so forth until he leaps $15$ units in a random direction to celebrate PUMaC’s 15th year.
There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly $1/2$ after the $15$ leaps. The area of that circle can be written as $A\pi$. What is $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1240}
A bag contains an equal number of some coins, 50 paise and 25 paise coins respectively. The total value of the coins is 35 rupees. There are 20 coins of each type. What is the value of the first type of coin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
We only know that the password of a safe consists of  $7$  different digits. The safe will open if we enter  $7$  different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than  $7$  attempts?

*(5 points for Juniors and 4 points for Seniors)*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 6 }
To a natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, resulting in a power of ten. Find all such \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
If the inequality
$$
\left|a x^{2}+b x+a\right| \leqslant x
$$

holds for $x \in [1,2]$, then the maximum value of $3a + b$ is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
7.5. 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}
From the vertex of the obtuse angle \( A \) of triangle \( ABC \), a perpendicular \( AD \) is drawn. From point \( D \), a circle is drawn with radius equal to \( AD \), intersecting the sides \( AB \) and \( AC \) at points \( M \) and \( N \), respectively. Find the side \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{mc}{n}}
How many reorderings of \(2, 3, 4, 5, 6\) have the property that every pair of adjacent numbers are relatively prime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For distinct complex numbers $z_1,z_2,\cdots,z_{673}$, the polynomial  
$$(x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3$$
can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The sum 
$$\left| \sum_{1 \le j <k \le 673} z_jz_k \right|$$
can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352}
In a race on the same distance, two cars and a motorcycle participated. The second car took 1 minute longer to cover the entire distance than the first car. The first car moved 4 times faster than the motorcycle. What portion of the distance per minute did the second car cover if it covered $\frac{1}{6}$ of the distance more per minute than the motorcycle, and the motorcycle covered the distance in less than 10 minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Given sets \( A = \{(x, y) \mid y = a x + 2\} \) and \( B = \{(x, y) \mid y = |x+1| \} \), and that \( A \cap B \) is a singleton set, determine the range of values for the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1] \cup [1, +\infty)}
There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron, where you wait in a group with five other souls. Each of these souls also came through one of the entrances, and you are all waiting to get a ride across the river.

In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through the entrances does not matter, and the entrance you went through is fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Four students, named A, B, C, and D, are divided into two volunteer groups to participate in two off-campus activities. The probability that students B and C participate in the same activity is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Someone, when asked for the number of their ticket, replied: "If you add all the six two-digit numbers that can be made from the digits of the ticket number, half of the resulting sum will be exactly my ticket number." Determine the ticket number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
Arrange the 5 natural numbers from 1 to 5 in a row from left to right, in such a way that starting from the third number, each number is either the sum or difference of the previous two numbers. How many valid arrangements satisfy these conditions? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the equation $\frac{{x^2}}{9-k}+\frac{{y^2}}{k-4}=1$, which represents an ellipse with foci on the y-axis, determine the range of real number k that satisfies the conditions for such an ellipse. Express your answer as an interval or range of values for k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{13}{2}, 9)}
If \( x - 5 \) is a factor of \( cx^3 + 23x^2 - 5cx + 55 \), find the constant \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6.3}
Let the set
\[ S = \{m \mid m \in \mathbf{Z}_{+}, \text{each prime factor of } m \text{ is less than } 10\}. \]

Find the smallest positive integer \( n \) such that in any \( n \)-element subset of \( S \), there exist four distinct numbers whose product is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Through the edge \( BC \) of the triangular pyramid \( PABC \) and point \( M \), the midpoint of the edge \( PA \), a section \( BCM \) is drawn. The apex of a cone coincides with the apex \( P \) of the pyramid, and the base circle is inscribed in triangle \( BCM \) such that it touches the side \( BC \) at its midpoint. The points of tangency of the circle with segments \( BM \) and \( CM \) are the intersection points of the medians of faces \( APB \) and \( APC \). The height of the cone is twice the radius of the base. Find the ratio of the lateral surface area of the pyramid to the area of the base of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Five students and two teachers are arranging for a group photo in a row. Calculate the probability that the two teachers are not at the ends and not adjacent to each other. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{7}}
设 $M=1^{4}+2^{4}+\cdots+2013^{4}$. 则 $M$ 的个位数字为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), the value of \( x \) is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Given a set of points in space, a *jump* consists of taking two points,  $P$  and  $Q,$  and replacing  $P$  with the reflection of  $P$  over  $Q$ . Find the smallest number  $n$  such that for any set of  $n$  lattice points in  $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 1025 }
Vika and Masha decided to decorate a room with balloons. They bought 15 packs of balloons, each containing 1, 2, 3, ..., 15 balloons respectively (the number of balloons is indicated on each pack). To make inflating balloons more fun, they came up with a game: they take turns inflating one balloon at a time from any pack, opening a new pack if needed. The player who opens the last pack loses. Which of the girls can guarantee a win regardless of the opponent's strategy, if Vika takes the first balloon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Masha}
如果从数1,2，···，14中，按由小到大的顺序取出$a _ { 1 } , a _ { 2 } , a _ { 3 }$，使同时满足$a _ { 2 } -$ $a _ { 1 } \geq 3$与$a _ { 3 } - a _ { 2 } \geq 3$，那么所有符合上述要求的不同取法有 种.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120
}
From a stack of 20 one-hundred yuan banknotes, which includes 5 counterfeit notes, two notes are randomly drawn. If one of the drawn notes is tested on a money detector and found to be counterfeit, calculate the probability that the other note is also counterfeit. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {2}{17}}
设${{\left( {{x}^{6}}+x+1 \right)5}}={{a}_{0}}+{{a1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a50}}{{x30}}$，其中${{a}_{0}}$、${{a1}}$、${{a}_{2}}$、$\cdots $、${{a30}}$是各项的系数，则在${{a}_{0}}$、${{a1}}$、${{a}_{2}}$、$\cdots $、${{a30}}$这$31$个系数中，值为零的个数为______．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$10$}
In a certain singles table tennis tournament, it was originally planned that each pair of players would play one match, but three players quit after each playing two matches. As a result, a total of 50 matches were played. How many matches were played among the three players? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}