## BdMO National Higher Secondary 2020 P10

Mursalin
Posts:68
Joined:Thu Aug 22, 2013 9:11 pm
BdMO National Higher Secondary 2020 P10
বৃষ্টি একটা বিশেষ সেট $$A$$ বানাতে চায়। সে $$A=\{0, 42\}$$ দিয়ে শুরু করে। যেকোনো ধাপে সে একটা পূর্ণসংখ্যা $$x$$-কে $$A$$-তে ঢুকাতে পারবে যদি $$x$$, $$A$$-তে ইতোমধ্যে থাকা সংখ্যাগুলোকে সহগ হিসেবে ব্যবহার করে বানানো কোনো বহুপদীর মূল হয়। এভাবে সে $$A$$-এ নতুন নতুন পূর্ণসংখ্যা ঢুকাতেই থাকে। যখন সে $$A$$-এ আর ঢুকানোর মতো নতুন সংখ্যা খুঁজে পাবে না, তখন $$A$$-তে কয়টা সংখ্যা থাকবে?

Bristy wants to build a special set $$A$$. She starts with $$A=\{0, 42\}$$. At any step, she can add an integer $$x$$ to the set $$A$$ if it is a root of a polynomial that uses the already existing integers in $$A$$ as coefficients. She keeps doing this, adding more and more numbers to $$A$$. After she eventually runs out of numbers to add to $$A$$, how many numbers will be in $$A$$?
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Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

### Re: BdMO National Higher Secondary 2020 P10

Picking this up, as nationals are near, if there is any right time to post a solution to this problem, it must be now!
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

nimon
Posts:8
Joined:Sat Mar 27, 2021 7:30 am

### Re: BdMO National Higher Secondary 2020 P10laim

যেকোনো ধাপে সে একটা পূর্ণসংখ্যা x-কে A-তে ঢুকাতে পারবে যদি x, A-তে ইতোমধ্যে থাকা সংখ্যাগুলোকে সহগ হিসেবে ব্যবহার করে বানানো কোনো বহুপদীর মূল হয়।
Can you give more explaination to this line?

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

### Re: BdMO National Higher Secondary 2020 P10laim

nimon wrote:
Fri Apr 02, 2021 11:08 am
যেকোনো ধাপে সে একটা পূর্ণসংখ্যা x-কে A-তে ঢুকাতে পারবে যদি x, A-তে ইতোমধ্যে থাকা সংখ্যাগুলোকে সহগ হিসেবে ব্যবহার করে বানানো কোনো বহুপদীর মূল হয়।
Can you give more explaination to this line?
There are some integers in A. If you make a polynomial with coefficients are integers from A, and one of the roots of the polynomial is an integer $k \notin A$. Then you can add k in A.
For example,
$42x+42$is a polynomial with coefficients from A and root $-1$, but $-1$ is not in A, so we can add him in A. This process continues now $A=\{-1,0,42\}$. And other polynomials can also have degree more than 1.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Anindya Biswas
Posts:263
Joined:Fri Oct 02, 2020 8:51 pm
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### Re: BdMO National Higher Secondary 2020 P10

Does $\pm14$ and $\pm21$ belongs to $A$? That's where I am stuck!
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

### Re: BdMO National Higher Secondary 2020 P10

Anindya Biswas wrote:
Fri Apr 02, 2021 9:13 pm
Does $\pm14$ and $\pm21$ belongs to $A$? That's where I am stuck!
I think we can only get the factors of $42$.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Anindya Biswas
Posts:263
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### Re: BdMO National Higher Secondary 2020 P10

Mehrab4226 wrote:
Fri Apr 02, 2021 10:00 pm
Anindya Biswas wrote:
Fri Apr 02, 2021 9:13 pm
Does $\pm14$ and $\pm21$ belongs to $A$? That's where I am stuck!
I think we can only get the factors of $42$.
Yeah we can only get the factors of $42$ but I have no idea about this two factors, others somehow got into $A$.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Anindya Biswas
Posts:263
Joined:Fri Oct 02, 2020 8:51 pm
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### Re: BdMO National Higher Secondary 2020 P10

Sorry, but I could not resist posting the solution!
Last edited by Anindya Biswas on Mon Apr 05, 2021 1:00 am, edited 1 time in total.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm