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Question Number 111275 by Aina Samuel Temidayo last updated on 03/Sep/20
In a tennis tournament n women and  2n men played. Each player played  exactly one match with every other  player. If there are no ties and the  number of the matches won by women  to the number of matches won by men  is 7:5, find n.
$$\mathrm{In}\:\mathrm{a}\:\mathrm{tennis}\:\mathrm{tournament}\:\mathrm{n}\:\mathrm{women}\:\mathrm{and} \\ $$$$\mathrm{2n}\:\mathrm{men}\:\mathrm{played}.\:\mathrm{Each}\:\mathrm{player}\:\mathrm{played} \\ $$$$\mathrm{exactly}\:\mathrm{one}\:\mathrm{match}\:\mathrm{with}\:\mathrm{every}\:\mathrm{other} \\ $$$$\mathrm{player}.\:\mathrm{If}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{ties}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{women} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{men} \\ $$$$\mathrm{is}\:\mathrm{7}:\mathrm{5},\:\mathrm{find}\:\mathrm{n}. \\ $$$$ \\ $$
Answered by 1549442205PVT last updated on 04/Sep/20
  Since every player played exactly   one match with every other player,  i)Case in every match one side is men  other side is women.Then  total number of matches of the tournamen  equal to n×2n=2n^2 .From the hypothesis  number of the matches won by women  to the number of matches won by men  is 7:5 and there are no ties we infer that  if denote 7k be  number of the matches won by women  then 5k be  number of the matches won by men  Therefore 2n^2 =7k+5k=12k⇔n^2 =6k  ⇒n^2 ⋮3 ⇒n=3q⇒(3q)^2 =6k  ⇒3q^2 =2k ⇒q⋮2,so q=2p  ⇒3(2p)^2 =2k⇒k=6p^2   ⇒n^2 =6.6p^(2 ) ⇒n=6p where p ∈N^∗   Thus,n=6,12,18,...  ii)Case in every match two players are  arbitrary sex.Total number of player  parting in tournament is 3n person  .Since every player played exactly one  match to 3n−1 other players ,total  number of the matches is ((3n(3n−1))/2)  Therefore we have  ((3n(3n−1))/2)=12k(∗)  We prove that capacity don′t ocurrs  Indeed,number of matches between  one side is female and the other side  is male is n×2n=2n^2 .Number of  matches between two female  players   is ((n(n−1))/2).Number of matches  between two male players is ((2n(2n−1))/2)  Hence, number of matches that involving  female players is A=2n^2 +((n(n−1))/2).   Number of matches that involving  male players is B=2n^2 +((2n(2n−1))/2)  It is easy to see that (A/B)<1.Therefore  don′t ocurrs case that A:B=7:5
$$\:\:\mathrm{Since}\:\mathrm{every}\:\mathrm{player}\:\mathrm{played}\:\mathrm{exactly} \\ $$$$\:\mathrm{one}\:\mathrm{match}\:\mathrm{with}\:\mathrm{every}\:\mathrm{other}\:\mathrm{player}, \\ $$$$\left.\mathrm{i}\right)\mathrm{Case}\:\mathrm{in}\:\mathrm{every}\:\mathrm{match}\:\mathrm{one}\:\mathrm{side}\:\mathrm{is}\:\mathrm{men} \\ $$$$\mathrm{other}\:\mathrm{side}\:\mathrm{is}\:\mathrm{women}.\mathrm{Then} \\ $$$$\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tournamen} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{n}×\mathrm{2n}=\mathrm{2n}^{\mathrm{2}} .\mathrm{From}\:\mathrm{the}\:\mathrm{hypothesis} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{women} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{men} \\ $$$$\mathrm{is}\:\mathrm{7}:\mathrm{5}\:\mathrm{and}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{ties}\:\mathrm{we}\:\mathrm{infer}\:\mathrm{that} \\ $$$$\mathrm{if}\:\mathrm{denote}\:\mathrm{7k}\:\mathrm{be} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{women} \\ $$$$\mathrm{then}\:\mathrm{5k}\:\mathrm{be} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{men} \\ $$$$\mathrm{Therefore}\:\mathrm{2n}^{\mathrm{2}} =\mathrm{7k}+\mathrm{5k}=\mathrm{12k}\Leftrightarrow\mathrm{n}^{\mathrm{2}} =\mathrm{6k} \\ $$$$\Rightarrow\mathrm{n}^{\mathrm{2}} \vdots\mathrm{3}\:\Rightarrow\mathrm{n}=\mathrm{3q}\Rightarrow\left(\mathrm{3q}\right)^{\mathrm{2}} =\mathrm{6k} \\ $$$$\Rightarrow\mathrm{3q}^{\mathrm{2}} =\mathrm{2k}\:\Rightarrow\mathrm{q}\vdots\mathrm{2},\mathrm{so}\:\mathrm{q}=\mathrm{2p} \\ $$$$\Rightarrow\mathrm{3}\left(\mathrm{2p}\right)^{\mathrm{2}} =\mathrm{2k}\Rightarrow\mathrm{k}=\mathrm{6p}^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{n}^{\mathrm{2}} =\mathrm{6}.\mathrm{6p}^{\mathrm{2}\:} \Rightarrow\mathrm{n}=\mathrm{6p}\:\mathrm{where}\:\mathrm{p}\:\in\mathbb{N}^{\ast} \\ $$$$\boldsymbol{\mathrm{Thus}},\mathrm{n}=\mathrm{6},\mathrm{12},\mathrm{18},… \\ $$$$\left.\mathrm{ii}\right)\mathrm{Case}\:\mathrm{in}\:\mathrm{every}\:\mathrm{match}\:\mathrm{two}\:\mathrm{players}\:\mathrm{are} \\ $$$$\mathrm{arbitrary}\:\mathrm{sex}.\mathrm{Total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{player} \\ $$$$\mathrm{parting}\:\mathrm{in}\:\mathrm{tournament}\:\mathrm{is}\:\mathrm{3n}\:\mathrm{person} \\ $$$$.\mathrm{Since}\:\mathrm{every}\:\mathrm{player}\:\mathrm{played}\:\mathrm{exactly}\:\mathrm{one} \\ $$$$\mathrm{match}\:\mathrm{to}\:\mathrm{3n}−\mathrm{1}\:\mathrm{other}\:\mathrm{players}\:,\mathrm{total} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{is}\:\frac{\mathrm{3n}\left(\mathrm{3n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\mathrm{Therefore}\:\mathrm{we}\:\mathrm{have}\:\:\frac{\mathrm{3n}\left(\mathrm{3n}−\mathrm{1}\right)}{\mathrm{2}}=\mathrm{12k}\left(\ast\right) \\ $$$$\mathrm{We}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{capacity}\:\mathrm{don}'\mathrm{t}\:\mathrm{ocurrs} \\ $$$$\mathrm{Indeed},\mathrm{number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{between} \\ $$$$\mathrm{one}\:\mathrm{side}\:\mathrm{is}\:\mathrm{female}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{side} \\ $$$$\mathrm{is}\:\mathrm{male}\:\mathrm{is}\:\mathrm{n}×\mathrm{2n}=\mathrm{2n}^{\mathrm{2}} .\mathrm{Number}\:\mathrm{of} \\ $$$$\mathrm{matches}\:\mathrm{between}\:\mathrm{two}\:\mathrm{female}\:\:\mathrm{players}\: \\ $$$$\mathrm{is}\:\frac{\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)}{\mathrm{2}}.\mathrm{Number}\:\mathrm{of}\:\mathrm{matches} \\ $$$$\mathrm{between}\:\mathrm{two}\:\mathrm{male}\:\mathrm{players}\:\mathrm{is}\:\frac{\mathrm{2n}\left(\mathrm{2n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\mathrm{Hence},\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{that}\:\mathrm{involving} \\ $$$$\mathrm{female}\:\mathrm{players}\:\mathrm{is}\:\mathrm{A}=\mathrm{2n}^{\mathrm{2}} +\frac{\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)}{\mathrm{2}}. \\ $$$$\:\mathrm{Number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{that}\:\mathrm{involving} \\ $$$$\mathrm{male}\:\mathrm{players}\:\mathrm{is}\:\mathrm{B}=\mathrm{2n}^{\mathrm{2}} +\frac{\mathrm{2n}\left(\mathrm{2n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{see}\:\mathrm{that}\:\frac{\mathrm{A}}{\mathrm{B}}<\mathrm{1}.\mathrm{Therefore} \\ $$$$\mathrm{don}'\mathrm{t}\:\mathrm{ocurrs}\:\mathrm{case}\:\mathrm{that}\:\mathrm{A}:\mathrm{B}=\mathrm{7}:\mathrm{5} \\ $$
Commented by 1549442205PVT last updated on 04/Sep/20
Here question isn′t clear.If in every match one side is  women and other is men then total  number  of matches equal to n×2n=2n^2   If in one match two player are  arbitrary(both are female or both are  male or this side is male and other is  female)then total number of matches  will be ((3n(3n−1))/2)
$$\mathrm{Here}\:\mathrm{question}\:\mathrm{isn}'\mathrm{t}\:\mathrm{clear}.\mathrm{If}\:\mathrm{in}\:\mathrm{every}\:\mathrm{match}\:\mathrm{one}\:\mathrm{side}\:\mathrm{is} \\ $$$$\mathrm{women}\:\mathrm{and}\:\mathrm{other}\:\mathrm{is}\:\mathrm{men}\:\mathrm{then}\:\mathrm{total}\:\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{matches}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{n}×\mathrm{2n}=\mathrm{2n}^{\mathrm{2}} \\ $$$$\mathrm{If}\:\mathrm{in}\:\mathrm{one}\:\mathrm{match}\:\mathrm{two}\:\mathrm{player}\:\mathrm{are} \\ $$$$\mathrm{arbitrary}\left(\mathrm{both}\:\mathrm{are}\:\mathrm{female}\:\mathrm{or}\:\mathrm{both}\:\mathrm{are}\right. \\ $$$$\mathrm{male}\:\mathrm{or}\:\mathrm{this}\:\mathrm{side}\:\mathrm{is}\:\mathrm{male}\:\mathrm{and}\:\mathrm{other}\:\mathrm{is} \\ $$$$\left.\mathrm{female}\right)\mathrm{then}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches} \\ $$$$\mathrm{will}\:\mathrm{be}\:\frac{\mathrm{3n}\left(\mathrm{3n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$
Commented by Aina Samuel Temidayo last updated on 03/Sep/20
I thought the total number of matches  should be  (((3n)),(2) ) which is  (((3n)!)/((3n−2)!2!))=((3n(3n−1))/2)  ⇒ ((3n(3n−1))/2)=12k  3n(3n−1)=24k  3n^2 −n=8k
$$\mathrm{I}\:\mathrm{thought}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches} \\ $$$$\mathrm{should}\:\mathrm{be}\:\begin{pmatrix}{\mathrm{3n}}\\{\mathrm{2}}\end{pmatrix}\:\mathrm{which}\:\mathrm{is} \\ $$$$\frac{\left(\mathrm{3n}\right)!}{\left(\mathrm{3n}−\mathrm{2}\right)!\mathrm{2}!}=\frac{\mathrm{3n}\left(\mathrm{3n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\Rightarrow\:\frac{\mathrm{3n}\left(\mathrm{3n}−\mathrm{1}\right)}{\mathrm{2}}=\mathrm{12k} \\ $$$$\mathrm{3n}\left(\mathrm{3n}−\mathrm{1}\right)=\mathrm{24k} \\ $$$$\mathrm{3n}^{\mathrm{2}} −\mathrm{n}=\mathrm{8k} \\ $$
Commented by Aina Samuel Temidayo last updated on 03/Sep/20
They should mean the second case.  How did you get the first case  you stated here to be 2n^2 ?
$$\mathrm{They}\:\mathrm{should}\:\mathrm{mean}\:\mathrm{the}\:\mathrm{second}\:\mathrm{case}. \\ $$$$\mathrm{How}\:\mathrm{did}\:\mathrm{you}\:\mathrm{get}\:\mathrm{the}\:\mathrm{first}\:\mathrm{case} \\ $$$$\mathrm{you}\:\mathrm{stated}\:\mathrm{here}\:\mathrm{to}\:\mathrm{be}\:\mathrm{2n}^{\mathrm{2}} ? \\ $$
Commented by 1549442205PVT last updated on 04/Sep/20
Because every female played exactly  one match to one male player  Thus,here we need understand that  the tennis tournament is between  male and female.
$$\mathrm{Because}\:\mathrm{every}\:\mathrm{female}\:\mathrm{played}\:\mathrm{exactly} \\ $$$$\mathrm{one}\:\mathrm{match}\:\mathrm{to}\:\mathrm{one}\:\mathrm{male}\:\mathrm{player} \\ $$$$\mathrm{Thus},\mathrm{here}\:\mathrm{we}\:\mathrm{need}\:\mathrm{understand}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{tennis}\:\mathrm{tournament}\:\mathrm{is}\:\mathrm{between} \\ $$$$\mathrm{male}\:\mathrm{and}\:\mathrm{female}. \\ $$
Commented by Aina Samuel Temidayo last updated on 04/Sep/20
Ok.
$$\mathrm{Ok}. \\ $$

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