Problem 2:
In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament?
BdMO National Higher Secondary 2011/2
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Mehfuj Zahir
- Posts:78
- Joined:Thu Jan 20, 2011 10:46 am
Re: BdMO National Higher Secondary 2011/2
in any result of the match sum of the point of the participants is 2.so we have to play 90/2=45 matches.Now n(C)2=45.then solve the equation n=10(ans)
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FahimFerdous
- Posts:176
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- Location:Mymensingh, Bangladesh
Re: BdMO National Higher Secondary 2011/2
My solution is again the same. So, I don't have to post it. 
Your hot head might dominate your good heart!
Re: BdMO National Higher Secondary 2011/2
let the number of participants be $x$
therefore, total number of matches $x(x-1)/2$
total number of results is $x(x-1)$
if total number of wins is $y$,
total number of losses is also $y$.
then draw is $x(x-1)-2y$
now,we get, according to the question,
$x(x-1)-2y+y+y=90$
$x^2-x-90=0$
$(x-10)(x+9)=0$
since $x>0$,
therefore, $x=10$.
so,number of participants is $10$.(ANS)
therefore, total number of matches $x(x-1)/2$
total number of results is $x(x-1)$
if total number of wins is $y$,
total number of losses is also $y$.
then draw is $x(x-1)-2y$
now,we get, according to the question,
$x(x-1)-2y+y+y=90$
$x^2-x-90=0$
$(x-10)(x+9)=0$
since $x>0$,
therefore, $x=10$.
so,number of participants is $10$.(ANS)
Last edited by mahathir on Wed Feb 16, 2011 7:31 pm, edited 2 times in total.
Re: BdMO National Higher Secondary 2011/2
Mahathir, please learn how to use LaTeX (to write more readable equation). You can find a link below my post. Also writing simple equations is very easy. You just need to write your equation between two dollar sign.
For example if you write x(x-1)-2y+y+y=90 between two dollar signs, it will become $x(x-1)-2y+y+y=90$.
For example if you write x(x-1)-2y+y+y=90 between two dollar signs, it will become $x(x-1)-2y+y+y=90$.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: BdMO National Higher Secondary 2011/2
২. শরত অনুজায়ি, প্রতিটি খেলোয়ার প্রত্যেকের সাথে সুধু মাত্র একবার খেলবে।
অরথাত, n জন প্রতিজগির মধ্যে C(n,2) তি খেলা হবে। eita mara ber korte pari eibhabe: n ti point poroshporer sathe C(n,2) ti line create kore.
এখন,
কোনো ম্যাচে একজন জিতলে অন্য জন হারবে। তাই অই দুই জনের মোট পয়েন্ট ৩-১=২
আবার কোনো ম্যাচে দুই জন ড্র করলে তাদের পয়েন্ট ও ১+১=২ হয়।
এখন, C(n,2) ti match e total participate der point er sum = C(n,2) * 2
শরতো মতে,
C(n,2) * 2 = 90
Solving that we get, n^3 – n^2 = 180 . I cant solve it, but if u replace ‘n’ with different numbers then it is clear that n = 6. So, the number of participate should be 6.
অরথাত, n জন প্রতিজগির মধ্যে C(n,2) তি খেলা হবে। eita mara ber korte pari eibhabe: n ti point poroshporer sathe C(n,2) ti line create kore.
এখন,
কোনো ম্যাচে একজন জিতলে অন্য জন হারবে। তাই অই দুই জনের মোট পয়েন্ট ৩-১=২
আবার কোনো ম্যাচে দুই জন ড্র করলে তাদের পয়েন্ট ও ১+১=২ হয়।
এখন, C(n,2) ti match e total participate der point er sum = C(n,2) * 2
শরতো মতে,
C(n,2) * 2 = 90
Solving that we get, n^3 – n^2 = 180 . I cant solve it, but if u replace ‘n’ with different numbers then it is clear that n = 6. So, the number of participate should be 6.
r@k€€/|/
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Mehfuj Zahir
- Posts:78
- Joined:Thu Jan 20, 2011 10:46 am
Re: BdMO National Higher Secondary 2011/2
How you got\[n^{3}-n^{2}=180\]
Re: BdMO National Higher Secondary 2011/2
oops! mistake. it should be 10. and i just calculate this C(n,2)n=90 ! instead of that ^
r@k€€/|/