AoPS Academy
, find the smallest integer 
such that any graph with 
vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
{
} consisting 
distinct positive integers is 
if for every 
the number 
is divisible by the product of all numbers in except 
. Find all integers such that exists a set consisting of 
positive integers.
with 
and 
.
?
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
as the perpendicular bisector of 
thus we prove that the circle tangent to 
through 
touches the circle 
cut 
again at 
is P-isosceles 
is perpendicular bisector of 
cutting the arc 
of 
at its midpoint 
Since 
then 
C-excenter 
of 
is on 
By Sawayama's lemma, we conclude that the circle tangent to through is a Thebault circle of the cevian 
of externally tangent to its circumcircle 
circumcenter of 
.
tangents side 
and 
respectively and also tangent circumcircle of 
.
bisect segment 
,
is incenter of 