BdMO National 2013: Secondary 6

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
BdMO
Posts: 134
Joined: Tue Jan 18, 2011 1:31 pm

BdMO National 2013: Secondary 6

Unread post by BdMO » Fri Jan 10, 2014 1:37 am

There are some boys and girls in a class. Every boy knows exactly $r$ girls, and every girl knows exactly $r$ boys. Show that there are an equal number of boys and girls in the class. (Assume that knowing is mutual, i.e. if $A$ knows $B$ then $B$ knows $A$.)

User avatar
asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm
Location: Sylhet,Bangladesh

Re: BdMO National 2013: Secondary 6

Unread post by asif e elahi » Tue Feb 11, 2014 9:56 pm

Sorry for my poor english

Let there are $m$ boys and $n$ girls.We draw $m+n$ points in a plane where every point indicates a boy or a girl.Let $M$ be the set of points which indicate the boys and $N$ indicate the set of points which indicate girls.If a boy $A$ knows a girl $B$,we join these two points.Every boy knows $r$ girls,so the number lines which is connected to the points of $A$ is $mr$.Again every girl knows $r$ boys,so the number lines which is connected to the points of $B$ is $nr$.But every line is connected to the points of $A$ and $B$.
So the number of lines connected to the points of $A$=the number of lines connected to the points of $B$
or $mr=nr$
or $m=n$
So there are equal number of boys and girls in the class. :mrgreen:

User avatar
*Mahi*
Posts: 1175
Joined: Wed Dec 29, 2010 12:46 pm
Location: 23.786228,90.354974
Contact:

Re: BdMO National 2013: Secondary 6

Unread post by *Mahi* » Tue Feb 11, 2014 10:13 pm

$\text{Number of boys}\times\text{Number of friends each of them has} $ $=\text{Number of friendships between boys and girls}$
$=\text{Number of girls}\times\text{Number of friends each of them has}$
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Kiriti
Posts: 27
Joined: Mon May 13, 2013 5:05 pm
Location: 401/1 South Paik Para, Kalyanpur, Mirpur, Dhaka-1216

Re: BdMO National 2013: Secondary 6

Unread post by Kiriti » Thu Feb 13, 2014 11:26 am

আমরা এটা প্রমাণ করেছি যে, যদি \(m ≠ n\) তাহলে উক্ত ঘটনা সম্ভব নয়, যেখানে \(m\) হলো ছেলেদের সংখ্যা আর \(n\) মেয়েদের সংখ্যা । তাহলে যদি \(m = n\) হয় তাহলে উক্ত ঘটনা সত্য হতেও পারে আবার না ও হতে পারে । এমনও হতে পারে যে , এই ঘটনা সবসময় সম্ভব না । এখন দেখাতে হবে যে, \(m=n\) হলে প্রতিটি ছেলে \(r\) টী মেয়েকে আবার প্রতিটি মেয়ে \(r\) টি ছেলেকে চিনে । তাই \(m ≠ n \) হলে উক্ত ঘটনা সম্ভব নয় দেখালেই যে \(m = n \) এর জন্য উক্ত ঘটনা সম্ভব হবে তার তো কোন মানে নেই ?? =D
"Education is the most powerful weapon which you can use to change the world"- Nelson Mandela

User avatar
*Mahi*
Posts: 1175
Joined: Wed Dec 29, 2010 12:46 pm
Location: 23.786228,90.354974
Contact:

Re: BdMO National 2013: Secondary 6

Unread post by *Mahi* » Thu Feb 13, 2014 12:48 pm

The question assumes that this incident is true. You can assume (like any other proof by contradiction) that $m \neq n$ and prove what it implies is a contradiction, which forces $m=n$. There isn't anything else needed in the proof.
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Post Reply