BdMO National 2013: Secondary 6

BdMO
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BdMO National 2013: Secondary 6

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 $A$.)

asif e elahi
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Re: BdMO National 2013: Secondary 6

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.

*Mahi*
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Re: BdMO National 2013: Secondary 6

$\text{Number of boys}\times\text{Number of friends each of them has}$ $=\text{Number of friendships between boys and girls}$
$=\text{Number of girls}\times\text{Number of friends each of them has}$

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

Kiriti
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Re: BdMO National 2013: Secondary 6

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

*Mahi*
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Re: BdMO National 2013: Secondary 6

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