10th problem of BdMO'10

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
AntiviruShahriar
Posts: 125
Joined: Mon Dec 13, 2010 12:05 pm
Location: চট্রগ্রাম,Chittagong
Contact:

10th problem of BdMO'10

Unread post by AntiviruShahriar » Wed Dec 15, 2010 3:21 pm

$131$ টি ধনাত্নক পুর্ণ সংখ্যার এক্তি সেতে যেকনো সংখ্যার মৌলিক উত্পাদকগলো $42$ এর চেয়ে ছোট। দেখাও যে এই সেট থেকে এমন চারটি সংখ্যা নির্বাচন করা যাবে যেন তাদের গুণফল একটি পুর্ণবর্গ সংখ্যা হয়।
এইটা আমি National Math Olympiad (BdMO) এর পরীক্ষার দিন দেখার পর আর তাকাইও নাই। :shock: মনে হয় পারব না। প্রথমে কেউ হিন্তস দাও। তারপর solution টা hidden করে রাখো।

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: 10th problem of BdMO'10

Unread post by Moon » Wed Dec 15, 2010 4:11 pm

এইটা একটা আইএমও সমস্যার পরিবর্তিত রূপ।
Hint:
আমার পায়রাগুলো গুণে দেখা দরকার। ;)
"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.

kamrul2010
Posts: 120
Joined: Wed Dec 08, 2010 2:35 am
Location: Dhaka,Bangladesh
Contact:

Re: 10th problem of BdMO'10

Unread post by kamrul2010 » Thu Dec 16, 2010 1:26 am

Moon wrote:এইটা একটা আইএমও সমস্যার পরিবর্তিত রূপ।
Hint:
আমার পায়রাগুলো গুণে দেখা দরকার। ;)

হিহিহিহি! মজা পাইসি!
If computers have no doors or fences, who needs Windows and Gates?

AntiviruShahriar
Posts: 125
Joined: Mon Dec 13, 2010 12:05 pm
Location: চট্রগ্রাম,Chittagong
Contact:

Re: 10th problem of BdMO'10

Unread post by AntiviruShahriar » Thu Dec 16, 2010 1:47 am

কোন সালের??????আমি আই এম অ এর খুব কম ই করসি......।মুন ভাইয়ার হিন্ট তা সুন্দর হইসে............

tushar7
Posts: 101
Joined: Tue Dec 07, 2010 3:23 pm

Re: 10th problem of BdMO'10

Unread post by tushar7 » Thu Dec 16, 2010 2:52 pm

Moon wrote:এইটা একটা আইএমও সমস্যার পরিবর্তিত রূপ।
Hint:
আমার পায়রাগুলো গুণে দেখা দরকার। ;)
does that mean PHP ?!

Hasib
Posts: 238
Joined: Fri Dec 10, 2010 11:29 am
Location: খুলনা, বাংলাদেশ
Contact:

Re: 10th problem of BdMO'10

Unread post by Hasib » Thu Dec 16, 2010 8:57 pm

Pigone hole principal diya korte hoy. Problem solving strategies boi tate dekhci bole mone pore :lol:
A man is not finished when he's defeated, he's finished when he quits.

User avatar
sm.joty
Posts: 327
Joined: Thu Aug 18, 2011 12:42 am
Location: Dhaka

Re: 10th problem of BdMO'10

Unread post by sm.joty » Sat Dec 24, 2011 4:33 pm

PHP ব্যবহার করে এটা বোঝা যায় যে অন্তত ১১ টা সংখ্যা আছে যারা ৪২ এর চেয়ে ছোট ১৩ টা প্রাইমের একটা নির্দিষ্ট প্রাইম দ্বারা বিভাজ্য। এখন কি করবো সেটাই তো বুঝতে পারছি না। :? :? :? :?
Can any one help ??? :?:
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

sourav das
Posts: 461
Joined: Wed Dec 15, 2010 10:05 am
Location: Dhaka
Contact:

Re: 10th problem of BdMO'10

Unread post by sourav das » Sun Dec 25, 2011 11:10 am

[By Different letters consider different elements]

There are 13 primes less then 42.Let the set be $A$. Now define a function $f$ such that for any $a,b \in A$ ; $f(ab)=$Product of those $P_i < 42$ such that $ab$ has odd power of those $P_i$ in it's prime power factorization(Let denote it by PPF).Define $f(ab)=1$ if $ab$ is a perfect square. Now denote that if $f(ab)=f(cd)$ then $abcd$ is a perfect square[As $abcd$ will have all even parity of power of all $P_i$ ]....(i).And also denote that if $f(ab)=f(ac)$ then $f(bc)=1$...(ii)[As in PPF of $b$ and $c$ both have same parity of power in all of it's prime ]. Now we'll consider 4 cases:

** Case 1:
$f(ab)$is not equal to $1$ for all different $a,b$
Now there are $\binom{131}{2}$ choices for choosing $ab$. But there are at most $2^{13}-1$ choice of values for $f(ab)$ [REASON: there are 2 options for all $p_i<42$; either it divides $f(ab)$ or not. But the choice of all $p_i$ will not divide $f(ab)$ must be excluded as it is the case when $f(ab)=1$]

So there are $\binom{131}{2}-2^{13}+1$, $ab$ for which there exists different $cd$ such that $f(ab)=f(cd)$ [If $f(ab)=f(ac)$ then by (ii) contradiction] so by (i) we'll be done.
** Case 2:
For only 2 elements $f(ab)=1$ . Define $B=A \setminus {a}$ then all are same as Case 1 [Note $\binom{130}{2}>2^{13}-1$] to prove the given existence in $B$
** Case 3:
For only 3 different elements $f(ab)=f(ca)=f(bc)=1$. Consider $C=A \setminus \{b,c\}$ then all are same as Case 1 [Note $\binom{129}{2}>2^{13}-1$] to prove the given existence in $C$
**Case 4:
For existence of 4 or more than 4 elements such that $f(ab)=1$ then 2 cases are possible:
(i)If there exists different $a,b,c,d$ for which $f(ab)=f(cd)=1$ then we are done by (i).
(iii)If $f(ab)=f(ac)=f(ad)=1$ then $f(cd)=1$ (by (ii)) and we are done with $f(ab)=f(cd)=1$ using (i).


Please inform me if you find any bug.

And please inform me about the similar IMO problem.

And i still couldn't use the 'not equal sign ' :x
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

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

Re: 10th problem of BdMO'10

Unread post by *Mahi* » Thu Dec 29, 2011 12:11 am

There was a similar problem of IMO 1985-4, solved that months ago :)
And use \not tag. That works on most PCs. If that doesn't work , there are some special cases like $\neq$
Please read Forum Guide and Rules before you post.

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

Nur Muhammad Shafiullah | Mahi

User avatar
samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

Re: 10th problem of BdMO'10

Unread post by samiul_samin » Wed Mar 07, 2018 11:11 pm

*Mahi* wrote:
Thu Dec 29, 2011 12:11 am
There was a similar problem of IMO 1985-4, solved that months ago :)
And use \not tag. That works on most PCs. If that doesn't work , there are some special cases like $\neq$
The problemIMO 1985 #4 is harder than the national one.But,in that problem how many times do I need to use PHP?I have seen the solution from TAACOPS and AoPS but didn't understand the solution fully.

I also want to solve this problem by using PHP not by using function.

Post Reply