Problems on Number Theory and Alzebra (frm BdMO)

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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
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rakeen
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Problems on Number Theory and Alzebra (frm BdMO)

Unread post by rakeen » Sat Dec 11, 2010 3:59 pm

1. $2x^2 - 8x + 7$ ei rashi'r sorbocho maan koto?
My Attempt: I firstly turned into this: 2x(x-4)+7 . Then I placed different numbers instead of $x$. Then find that it is -1 . Is it true. Is there any smarter solution?


2. $x$ er $\Re$ maan ber koro jekhane x nicher equation satisfy kore:
\[3^x + 4^x = 5^x\]

3.$2^4 \times 5^5 \times 10^8$ ei rashite koyti onko ache?
MyAttempt: if we simplyfy $10^8$ to 5 and 2 , then we get : $10^12 \times 5$ And thus the ans is 13! is ti correct?

4. $3^a - 7^b - 1=0$ ei rashir koiti somadhan ase?(very difficult to me)

5.How many subset does this set have $A$ = $\{$ 1,2,3,.....,2003$\}$ .the sum of the elements containing in that subset must be $2007000$.

6. n er koyti purno dhonattok maaner jonno $n^2 - 14n + 24$ 1ti Prime number? (ami sudhu jani prime odd hoy(except 2))

7. n ke 3 diye bhag korle 1 thake, 4diye bhag kore 2 thake, 5 diye bhag korle 3 thake, 6 diye korle 4 thake. 7 diye bhag korle koto bhagsesh thakbe? (pattern dekhe mone hoy 5 hote pare. Kintu 5 diye bhag korle 3 thake. Tai mone hoche 7 diye bhag korle 0 thakbe, kintu prosner sathe milale tao hoy na. :evil: )

:?: :?: :?:
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by Moon » Sat Dec 11, 2010 9:19 pm

1. try to make a square and use the idea that the square of a real number is always $\geq 0$.
2. Already replied. But from next time please don't double post.
3. yes, correct!
4. I don't understand your problem 4. Have you written it correctly?
5. $\dfrac{2003\times (2003+1)}{2} = 2 007 006$. Does it help? ;)
6. I guess you should learn some number theory. Read my article on number theory http://www.kmcbd.org/Home/documents (Introduction to Number theory_KMC (By Tarik Adnan Moon).pdf)
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by rakeen » Sun Dec 12, 2010 7:37 pm

5. 0o0o0o0o. Very easy. Just 1. But if we consider proper subset, the result is 0. And would I know that 2003*(2003+1) / 2 = 200700. Jodi bolto sum hobe 200400! tahole?
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by rakeen » Sun Dec 12, 2010 8:00 pm

4. was set in BdMO(Div) Comilla 2008 #1.
it says " $a$ and $b$ $\Re$ hole nicher somikoroner koyti somadhan ache? $3^a -7^b -1 = 0$ "
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rakeen
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by rakeen » Sun Dec 12, 2010 8:09 pm

1. ultimately to amake oi 1,2,3..... bosiye dekhte hobe!
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by rakeen » Sun Dec 12, 2010 8:22 pm

hey, prob 2 kothay reply korso?
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by Moon » Sun Dec 12, 2010 10:30 pm

তুমিই তো পোস্ট করছিলা আলাদাভাবে :!:
viewtopic.php?f=8&t=65
এজন্যই তো বলছি:>>>Already replied. But from next time please don't double post.
"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.

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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by AntiviruShahriar » Fri Dec 17, 2010 6:06 pm

7no::
$3a+1=n=4b+2,theke n{1}=10,22,...........................[10+g12]..{g purnosongkha}$
$n{1}=5c+3$ er jonno 58,118.......{(5x-1)12+10}[n{1} hotee dhara toiri koree]$
${(5x-1)12+10}$ $x=natural number$
eita ki thik ase????????????????
Last edited by AntiviruShahriar on Sat Dec 18, 2010 11:39 am, edited 1 time in total.

AntiviruShahriar
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by AntiviruShahriar » Fri Dec 17, 2010 6:22 pm

Moon wrote:1. try to make a square and use the idea that the square of a real number is always $\geq 0$.
2. Already replied. But from next time please don't double post.
3. yes, correct!
4. I don't understand your problem 4. Have you written it correctly?
5. $\dfrac{2003\times (2003+1)}{2} = 2 007 006$. Does it help? ;)
6. I guess you should learn some number theory. Read my article on number theory http://www.kmcbd.org/Home/documents (Introduction to Number theory_KMC (By Tarik Adnan Moon).pdf)
1 no ta ki সর্বোচ্চ না সর্বনিম্ন হবে????????সর্বোচ্চ হলে তো অসীম হওয়ার কথা।

Dipan
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Re: Problems on Number Theory and Alzebra (frm BdMO)

Unread post by Dipan » Fri Dec 17, 2010 9:11 pm

Prob 2 er ans to 2.

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