BdMO 2017 Dhaka divitional

For students of class 9-10 (age 14-16)
soyeb pervez jim
Posts: 21
Joined: Sat Jan 28, 2017 11:06 pm

BdMO 2017 Dhaka divitional

Unread post by soyeb pervez jim » Wed Mar 28, 2018 8:02 pm

The $‘energy’$ of an ordered triple $(a,b,c)$ formed by three positive integers $a,b,c$ is said to be $n$, if the following ${a}\leq{b}\leq{c}, gcd(a,b,c)=1$ and $(a^n+b^n+c^n)$ is divisible by $(a+b+c)$. There are some possible ordered triple whose $‘energy’$ can be of all values of ${n}\geq{1}$ In that case, for what is the maximum value of $(a+b+c)$?

User avatar
samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

Re: BdMO 2017 Dhaka divitional

Unread post by samiul_samin » Fri Mar 30, 2018 11:32 am

This problem was posted before in this forum but no solution was given.

User avatar
samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

Re: BdMO 2017 Dhaka divitional

Unread post by samiul_samin » Sun Feb 10, 2019 3:49 pm

soyeb pervez jim wrote:
Wed Mar 28, 2018 8:02 pm
The energy of an ordered triple $(a,b,c)$ formed by three positive integers $a,b,c$ is said to be $n$, if the following ${a}\leq{b}\leq{c}, gcd(a,b,c)=1$ and $(a^n+b^n+c^n)$ is divisible by $(a+b+c)$. There are some possible ordered triple whose $‘energy’$ can be of all values of ${n}\geq{1}$ In that case, for what is the maximum value of $(a+b+c)$?
Answer:$3$
Only one solution and $a=b=c=1$

Post Reply