Preparation Marathon

For discussing Olympiad Level Number Theory problems
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*Mahi*
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Re: Preparation Marathon

Unread post by *Mahi* » Wed Dec 28, 2011 10:05 am

Labib wrote:Would love to see the solution, Mahi.
Please, post it.
Try transforming it to $\frac {a}{b} \binom{4022}{2011}$
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Nadim Ul Abrar
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Re: Preparation Marathon

Unread post by Nadim Ul Abrar » Wed Dec 28, 2011 10:27 am

No 9 :
$\binom{4012}{2011}\equiv a mod2011$

then $\frac{4012!.2001!}{2011!.2001!}=\frac{4012!}{2011!}=2012.2013...4012 \equiv 2001!a mod2011$

Again $2012.2013...4012=2001!mod2011$

So that $2001!a \equiv 2001!mod2011$
or $a \equiv 1 \equiv \binom{4012}{2011} mod 2011$
Last edited by Nadim Ul Abrar on Wed Dec 28, 2011 8:31 pm, edited 2 times in total.
$\frac{1}{0}$

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Abdul Muntakim Rafi
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Re: Preparation Marathon

Unread post by Abdul Muntakim Rafi » Wed Dec 28, 2011 11:44 am

What's the answer of 1,2,3,5,7

My answer-
1. $00,21,42,63,84$
2. \[\binom{2015}{3}\] I mean 2015 C 3
3.$2^7-1$
5.$285$
7.none
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Nadim Ul Abrar
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Re: Preparation Marathon

Unread post by Nadim Ul Abrar » Wed Dec 28, 2011 12:52 pm

No 1 :
LEt S(n) donate the sum of digits of n .

If the number 0f digits be $\geq$ 4
then $S(n)_{max} =7.9.4=252 $ which is not a four digit number.
So number of digits will be $\leq$3

If the number 0f digits be 3
then $S(n)_{max} =7.9.3=189 $
so that $n=100+a.10+b=7+7a+7b$
or $93+3a=6b$
that cant be cz $7b_{max}=63$

If the number 0f digits be 2
then $10a+b=7a+7b$
or $a=2b$
from this property we have Four such number $21,42,63,84$

and If the number 0f digits be 1 then $a=7a$
it leads $a=0$

So all possible n is $0,21,42,63,84$ .
Last edited by Nadim Ul Abrar on Wed Dec 28, 2011 2:11 pm, edited 1 time in total.
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Nadim Ul Abrar
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Re: Preparation Marathon

Unread post by Nadim Ul Abrar » Wed Dec 28, 2011 1:04 pm

Abdul Muntakim Rafi wrote: 3.$2^7-1$
If you select m person's from a set of n persons then you can assign captain in m ways.

So that the answer will be

$\sum_{k=0}^{7}k \binom{7}{k}=7.2^6$
$\frac{1}{0}$

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Abdul Muntakim Rafi
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Re: Preparation Marathon

Unread post by Abdul Muntakim Rafi » Wed Dec 28, 2011 2:01 pm

Nadim ul Abrar:

1. Yeah I did it this way too... First I proved that it is not possible for numbers having more than 3 or 3 digits...
Then $10a+b=7a+7b$

2. I didn't understand the question's English version... The bangla translation is-
"ধর ৭ জন মানুষ আছে। একটা টিমে কমপক্ষে একজন থাকতে হবে। প্টিমে একজন ক্যাপটেন থাকবে। তাহলে কয়ভাবে টিম হতে পারে?"
I found out how many ways can the team be constituted... But I didn't understand the 'Captain' part... :( Damn...
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Nadim Ul Abrar
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Re: Preparation Marathon

Unread post by Nadim Ul Abrar » Wed Dec 28, 2011 3:08 pm

solution
No 2 : $\binom{2012+4-1}{4-1} =\binom{2015}{3}$
No 3 : $\sum_{k=0}^{7} k\binom{7}{k}$
No 5 : $1^2+2^2+...+9^2=285$
No 7 : $0$
No 8 : $(1+2+...+8)(1+2+...+9)=1620$
No 9 : $\binom{4012}{2011} \equiv 1 mod 2011$
No 10 :
2 3 6 14 30
1 3 8 16
2 5 8
3 3
$n^{th}$ term is= 2+(n-1)+(n-1)(n-2)+{(n-1)(n-2)(n-3)}/2

So Ans is 30
Last edited by Nadim Ul Abrar on Wed Dec 28, 2011 8:38 pm, edited 1 time in total.
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sm.joty
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Re: Preparation Marathon

Unread post by sm.joty » Wed Dec 28, 2011 3:31 pm

Preparation Marathon Ex-01 এর ৭ নং এর সমাধানটা কেউ এখনও আমারে বুঝায়া দিল না । :( :( :(

Nadim Ul Abrar wrote:
Abdul Muntakim Rafi wrote: 3. $2^7-1$
If you select m person's from a set of n persons then you can assign captain in m ways.

So that the answer will be
$\sum_{k=0}^{7}k \binom{7}{k}=7.2^6$
রাফি এবং আমি এই ফর্মুলা ব্যবহার করছি,
$_{1}^{n}\textrm{C}+_{2}^{n}\textrm{C}\cdots+_{n}^{n}\textrm{C}=2^n-1$

ভুল কেন হইল ????? :?:

আর ২ নং এর উত্তর এত সহজ ভাবে আসে কিভাবে :o আমি ফর্মুলা টা বুঝি নাই। :( :(

৯,১০ এর বিষয়টা বুঝলাম না । :( :(
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

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Nadim Ul Abrar
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Re: Preparation Marathon

Unread post by Nadim Ul Abrar » Wed Dec 28, 2011 3:43 pm

ঐ ক্যাপ্টেন বানাইবার কইসে না ?
খালি টিম গঠন করছস কা ?

১০ নাম্বার difference table দিয়ে করা
আর নাম্বার ৯ এর সোল উপরে দেয়া আসে :)
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protik
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Re: Preparation Marathon

Unread post by protik » Wed Dec 28, 2011 4:28 pm

@Nadim, 10 number ki apne differnce table dia korsen or masum bhaia bolse? or both??

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