BdMO National Higher Secondary 2020 P6

Mursalin
Posts:68
Joined:Thu Aug 22, 2013 9:11 pm
BdMO National Higher Secondary 2020 P6
$$f$$ একটা এক-এক ফাংশন যার ডোমেইন আর কোডোমেইন উভয়ই ধনাত্মক পূর্ণসংখ্যার সেট এবং $$f(xy)=f(x)f(y)$$। $$f(2020)$$-এর সম্ভাব্য সর্বনিম্ন মান বের করো।

$$f$$ is a one-to-one function from the set of positive integers to itself such that $$f(xy) = f(x)f(y)$$. Find the minimum possible value of $$f(2020)$$.
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Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

Re: BdMO National Higher Secondary 2020 P6

I am not sure this solution is correct or not but meh,
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

Dustan
Posts:71
Joined:Mon Aug 17, 2020 10:02 pm

Re: BdMO National Higher Secondary 2020 P6

Mehrab4226 wrote:
Wed Feb 10, 2021 12:58 am
I am not sure this solution is correct or not but meh,
How did you get $f(2)=2$,$f(p)=p$

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

Re: BdMO National Higher Secondary 2020 P6

Dustan wrote:
Fri Feb 12, 2021 4:08 pm
Mehrab4226 wrote:
Wed Feb 10, 2021 12:58 am
I am not sure this solution is correct or not but meh,
How did you get $f(2)=2$,$f(p)=p$
We want the minimum value of $f(2020) = f(2)f(2)f(5)f(101)$,
So it is natural to assume that factors should have the minimum value possible.
$f(2) = 2$ since 1 is already been taken by $f(1)$ and $f$ is a one-one function. That's it.

My solution is not complete. If BDMO 2020 was a written exam then the marks I would get is 0. But it was not a written exam , so....
And keeping $f(p)=p$ for the rest of the primes gives us a valid function $f$ fulfilling the given criteria.
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:264
Joined:Fri Oct 02, 2020 8:51 pm
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Re: BdMO National Higher Secondary 2020 P6

Mehrab4226 wrote:
Wed Feb 10, 2021 12:58 am
I am not sure this solution is correct or not but meh,
Almost done, now showing that for all primes $p,q, r$,
$f(p)^2f(q)f(r)\geq 2^2\cdot3\cdot5$ should do the work, isn't it?
"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 P6

Anindya Biswas wrote:
Wed Mar 03, 2021 8:25 pm
Mehrab4226 wrote:
Wed Feb 10, 2021 12:58 am
I am not sure this solution is correct or not but meh,
Almost done, now showing that for all primes $p,q, r$,
$f(p)^2f(q)f(r)\geq 2^2\cdot3\cdot5$ should do the work, isn't it?
Hmm, but that should be obvious.
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:264
Joined:Fri Oct 02, 2020 8:51 pm
Contact:

Re: BdMO National Higher Secondary 2020 P6

Mehrab4226 wrote:
Wed Mar 03, 2021 11:29 pm
Anindya Biswas wrote:
Wed Mar 03, 2021 8:25 pm
Mehrab4226 wrote:
Wed Feb 10, 2021 12:58 am
I am not sure this solution is correct or not but meh,
Almost done, now showing that for all primes $p,q, r$,
$f(p)^2f(q)f(r)\geq 2^2\cdot3\cdot5$ should do the work, isn't it?
Hmm, but that should be obvious.
Yeah then we are done, you've constructed a function $f$ for which $f(2020)=60$ and also it can be shown that $60$ is the minimum. So, your solution is valid.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Naeem588
Posts:12
Joined:Sat Apr 03, 2021 1:41 am

Re: BdMO National Higher Secondary 2020 P6

What if we assume that f(n) = 1
For all positive integers.
??

Anindya Biswas
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Re: BdMO National Higher Secondary 2020 P6

Naeem588 wrote:
Tue Apr 06, 2021 12:48 pm
What if we assume that f(n) = 1
For all positive integers.
??
$f$ is one-to-one function.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Naeem588
Posts:12
Joined:Sat Apr 03, 2021 1:41 am

Re: BdMO National Higher Secondary 2020 P6

So that's why we made f(101)=5
So that is the lowest possible value.
I got it now.