## EGMO 2021 P1

For discussing Olympiad Level Number Theory problems
Asif Hossain
Posts: 169
Joined: Sat Jan 02, 2021 9:28 pm

### EGMO 2021 P1

The number $2021$ is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
Hmm..Hammer...Treat everything as nail

Mehrab4226
Posts: 208
Joined: Sat Jan 11, 2020 1:38 pm

### Re: EGMO 2021 P1

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

Asif Hossain
Posts: 169
Joined: Sat Jan 02, 2021 9:28 pm

### Re: EGMO 2021 P1

Mehrab4226 wrote:
Thu Apr 15, 2021 10:50 pm
Recheck your proof $2021^{2021}$ IS FANTABULOUS.
HINTS
Hmm..Hammer...Treat everything as nail

Mehrab4226
Posts: 208
Joined: Sat Jan 11, 2020 1:38 pm

### Re: EGMO 2021 P1

Mehrab4226 wrote:
Thu Apr 15, 2021 10:50 pm
Ok. There is a problem with the solution,
Note,
$m \to 3m \to 6m+1 \to 12m+3 \to 4m+1 \to 2m$
So if 2m is fantabulous so is m, and it is the same if 2m+1 is fantabulous.
Now,
$2021 \to 1010 \to 505 \to 252 \to 126 \to 63 \to 31 \to 15 \to 7 \to 3 \to 2 \to 1.$
So 1 is fantabulous.
If any number $k$ is not fantabulous, then we can decrease it to 1 and get 1 is not fantabulous which is not true. Contradiction. So all numbers are fantabulous.
$\therefore 2021^{2021}$ is fantabulous.
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