Telescopic Sum
by P162008, Apr 24, 2025, 1:06 AM
Compute the value of 
is equal toNumber Theory
by fasttrust_12-mn, Aug 16, 2024, 10:21 AM
Find all integers 
for which 
is the square of an integer
for which 
is the square of an integer
PAMO Problem 4: Perpendicular lines
by DylanN, Apr 9, 2019, 6:54 AM
The tangents to the circumcircle of 
at 
and 
meet at 
. The circumcircle of 
meets sides 
and 
again at 
and 
respectively. Let 
be the circumcentre of . Show that 
is perpendicular to 
.
at 
and 
meet at 
. The circumcircle of 
meets sides 
and 
again at 
and 
respectively. Let 
be the circumcentre of . Show that 
is perpendicular to 
.A wizard kidnaps 101 people
by Leicich, Aug 29, 2014, 11:33 PM
A wizard kidnaps 
members from party 
, 
members from party , 
members from party , and 
members from party , keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with
or more people involved).
a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to
that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
members from party 
, 
members from party , 
members from party , and 
members from party , keeping them isolated in individual rooms in his castle, where he forces them to work.Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with
or more people involved).a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to
that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.IMO Shortlist 2013, Combinatorics #3
by lyukhson, Jul 9, 2014, 5:16 PM
A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.
(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
(ii) At any moment, he may double the whole family of imons in the lab by creating a copy
of each imon 
. During this procedure, the two copies and 
become entangled if and only if the original imons and 
are entangled, and each copy becomes entangled with its original imon ; no other entanglements occur or disappear at this moment.
Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.
(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
(ii) At any moment, he may double the whole family of imons in the lab by creating a copy
of each imon 
. During this procedure, the two copies and 
become entangled if and only if the original imons and 
are entangled, and each copy becomes entangled with its original imon ; no other entanglements occur or disappear at this moment.Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.
This post has been edited 1 time. Last edited by elitza, Jun 16, 2022, 7:49 AM
Reason: Fixed typo.
Reason: Fixed typo.
February 2010
January 2010
December 2009
November 2009
October 2009
September 2009
August 2009
July 2009