So there are boys (otherwise $n =0$ is a trivial answer )??? what is known about their number? Pierre.

in fact there are 2n boys and n girls playing at a tenning tournament. the problem is extremely simple: the boys won all the games they played between them namely n(2n-1) and the also won x games with the girls. the girls won the games between then, namely $\displaystyle \frac {n(n-1)}2 + 2n^2-x$, and all the games between them and the boys which the boys didn't win (obviously). we thus obtain $5n(2n-1) + 5x = \displaystyle 7\frac {n(n-1)}2 + 14n^2-7x$, and we get $x$ out of this and we obtain that $x$ is an integer iff $n$ gives residues 5 or 3 modulo 8 (or something like that). I see no combinatorics in this problem ... do you?

Valentin:Oh,no ,Miss "Can't understand it" is still bothering us with her infinite unsolved problems! Iris:Yeah, you're right,but i'm getting prepared for the 8th JBMO,so...I need some help because most of the problems are very difficult for me (Valentin:Really??!!I hadn't noticed that ) Irisi:I dont understand smth in the solution of the "extremly simple problem"(i'm usind the inverted commas for myself,becayse i know that it's very simple for you ) 1-What does it mean "the boys won all the matches beetwen them"?Does that mean every boy won every match he played with the other boys?That's impossible,because if A won with B than B lost with A(obviously),where A and B are boys. 2-Suppose i understand that when somebody will explain it,but how do you prove that?(this is may be absolutely easy but... ) 3-after working with the expresion you get x=n(5n-1)/8.How do you find what residues n should have,exept the case when 8 is a divisor of n?(the dump can't understan this too ) 4-Suppose we found the residues of n,then will have more than one solution,an infinity .But the problem said :"find n",not all the values of n,although by knowing that n is the number of girls in a tennis tornament,we can't have very high values of it. That's all. Valentin:Urraaaaaah!!!! Iris to Mr Know it all(that'sd not irony,you seem to know everything about everything in maths,lucky you

It's quite a shame for Serbia to propose at JBMO 2000 a problem that was given at Moldova NMO 1982, grade 10...

It's really easy. There are (9n^2-3n)/2 matches played and therefore (15n^2-5n)/8 matches won by girls. But those matches played by girls only must be won by girls. So (15n^2-5n)/8 ≥n(n-1)/2 , simplify and we get 11≥n . And the rest has posted by the others. May be this is too obvious so the solutions before did not state it .

@ stephencheng your inequality yields $ n\geq \frac{1}{11}$ and how about the rest ?

Iris Aliaj wrote: Irisi:I dont understand smth in the solution of the "extremly simple problem"(i'm usind the inverted commas for myself,becayse i know that it's very simple for you ) 1-What does it mean "the boys won all the matches beetwen them"?Does that mean every boy won every match he played with the other boys?That's impossible,because if A won with B than B lost with A(obviously),where A and B are boys. I think this means that : In all games that two players were boys,no girl won the game,this is obviously since the game will win one of the boys, or draw.(but there is no draw).So you should answer the questions : 1.In how many games both players was boys 2.In how many games both players was girls 3.In how many games a player was boy,and another girl. sorry for bad english !

Solution by Burhan Can Karaca: The match which was played a Boy vs. a Boy, has a Boy winner. It's same about Girl vs. Girl. So, Boy vs. Boy --> $\binom{2n}{2}=2n^2-n$ Girl vs. Girl --> $\binom{n}{2}=\dfrac{n^2-n}{2}$ Now let's assume that Boys won $x$ of the matches which was played a Boy vs. a Girl. So Girls won $2n^2 - x$ of them. $B$=Amount of the matches Boys won $=2n^2-n+x$ $G$=Amount of the matches Girls won $=\dfrac{n^2-n}{2} + 2n^2 - x$ $\dfrac{7}{5}=\dfrac{B}{G}=\dfrac{2n^2-n+x}{\dfrac{n^2-n}{2}+2n^2-x}$ From this, we have $8x=5n^2+n \Longrightarrow 8x=n(5n+1)$ Hence, $n$ must be equivalent $0$ or $3$ in $\mod8$