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BdMO National Secondary 2020 P1

Posted: Wed Feb 03, 2021 10:17 pm
by Mursalin
\(m\) 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.

Re: BdMO National Secondary 2020 P1

Posted: Thu Feb 04, 2021 8:50 pm
by Mehrab4226
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





Re: BdMO National Secondary 2020 P1

Posted: Sat Feb 20, 2021 2:54 pm
by anando
I think it's zero.

Re: BdMO National Secondary 2020 P1

Posted: Sat Feb 20, 2021 10:49 pm
by Mehrab4226
anando wrote:
Sat Feb 20, 2021 2:54 pm
I think it's zero.
Proof?

Re: BdMO National Secondary 2020 P1

Posted: Wed Sep 07, 2022 9:03 am
by Mathew23
Can no one solve this problem, fnf if it is 0, dordle can you explain in detail?