IMO 2019/P3

Discussion on International Mathematical Olympiad (IMO)
tanmoy
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IMO 2019/P3

Unread post by tanmoy » Thu Jul 18, 2019 11:02 pm

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time:
  • Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged.
Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Proposed by Adrian Beker, Croatia
"Questions we can't answer are far better than answers we can't question"

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

Re: IMO 2019/P3

Unread post by Ragib Farhat Hasan » Thu Oct 17, 2019 3:05 am

GRAPH THEORY... is printed all over this problem as a hint, isn't it?

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

Re: IMO 2019/P3

Unread post by Ragib Farhat Hasan » Thu Oct 17, 2019 3:06 am

Anyone looking for a classic Graph Theory problem?

Look no further!!!

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

Re: IMO 2019/P3

Unread post by Ragib Farhat Hasan » Thu Oct 17, 2019 3:08 am

BTW, why haven't anyone responded with their solutions? :?

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

Re: IMO 2019/P3

Unread post by Ragib Farhat Hasan » Thu Oct 17, 2019 3:16 am

DAMN!!!!!!! This forum has got so dead...

It needs a serious waking call...

Maybe we should consider calling Thanos... to invade this forum. :twisted:

Maybe then the "Avengers" (admins, moderators and former-active members) might turn their focus to this forum, again. :lol:

(P.S: I don't know whether anyone will ever see this, but if in case anyone does, do leave a reply! :D )

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samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

Re: IMO 2019/P3

Unread post by samiul_samin » Tue Nov 12, 2019 8:27 am

Ragib Farhat Hasan wrote:
Thu Oct 17, 2019 3:16 am
DAMN!!!!!!! This forum has got so dead...

It needs a serious waking call...

Maybe we should consider calling Thanos... to invade this forum. :twisted:

Maybe then the "Avengers" (admins, moderators and former-active members) might turn their focus to this forum, again. :lol:

(P.S: I don't know whether anyone will ever see this, but if in case anyone does, do leave a reply! :D )
This forum can become alive by new active members .Former active members are now (mostly)engineers.

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