BdMO National Secondary 2020 P1

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
User avatar
Mursalin
Posts:68
Joined:Thu Aug 22, 2013 9:11 pm
Location:Dhaka, Bangladesh.
BdMO National Secondary 2020 P1

Unread post by Mursalin » Wed Feb 03, 2021 10:17 pm

\(m\) এমন একটি বাস্তব সংখ্যা যা \(3^m=4m\) সমীকরণ সিদ্ধ করে। \(\frac{3^{3^m}}{m^4}\) -এর সম্ভাব্য সকল মানের যোগফল বের করো।


Let \(m\) be a real number such that the following equation holds: \(3^m=4m\). Compute the sum of all possible distinct values that \(\frac{3^{3^m}}{m^4}\) can take.
This section is intentionally left blank.

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: BdMO National Secondary 2020 P1

Unread post by Mehrab4226 » Thu Feb 04, 2021 8:50 pm

Mursalin wrote:
Wed Feb 03, 2021 10:17 pm
\(m\) এমন একটি বাস্তব সংখ্যা যা \(3^m=4m\) সমীকরণ সিদ্ধ করে। \(\frac{3^{3^m}}{m^4}\) -এর সম্ভাব্য সকল মানের যোগফল বের করো।


Let \(m\) be a real number such that the following equation holds: \(3^m=4m\). Compute the sum of all possible distinct values that \(\frac{3^{3^m}}{m^4}\) can take.
Firstly,
$3^m = 4m$
$\therefore (3^m)^4=3^{4m}=(4m)^4=4^4 m^4 = 256m^4 \cdots (1)$
Now,
$\frac{3^{3^m}}{m^4} = \frac{3^{4m}}{m^4}$ [$\because 3^m=4m]$
$= \frac{256m^4}{m^4}$[Using (1)]
$= 256$
Thus the term in the question have only one value, which is 256. Therefore the required sum = 256




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

anando
Posts:3
Joined:Mon Sep 07, 2020 3:37 pm

Re: BdMO National Secondary 2020 P1

Unread post by anando » Sat Feb 20, 2021 2:54 pm

I think it's zero.

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: BdMO National Secondary 2020 P1

Unread post by Mehrab4226 » Sat Feb 20, 2021 10:49 pm

anando wrote:
Sat Feb 20, 2021 2:54 pm
I think it's zero.
Proof?
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

Post Reply