[/quote]
,
be a set of Natural numbers from 
to 
.
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the midpoint of 
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
times, that is, 
)
be an acute triangle with 
,
be the projection from 
.
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
,
,
of positive integers 
such that 
.
denotes component-wise function multiplication)
and 
Prove that
distinct points 
connected if the line 
points from 
such that there exists a set 
is connected with exactly 
other points from 
Y by CZRorz, Adventure10, Mango247
Let 
be a chord of a circle 
centered at 
, and let 
be a point on the segment such that 
. Two circles through , internally tangent to at 
and 
, respectively, meet again at 
. Let 
be one of the points of intersection of the line 
and the circle , and let the lines and 
meet at 
. Prove that the line 
is tangent to the circle .
Theoklitos Paragyiou (Cyprus)
be a chord of a circle 
centered at 
, and let 
be a point on the segment such that 
. Two circles through , internally tangent to at 
and 
, respectively, meet again at 
. Let 
be one of the points of intersection of the line 
and the circle , and let the lines and 
meet at 
. Prove that the line 
is tangent to the circle .Theoklitos Paragyiou (Cyprus)
This post has been edited 1 time. Last edited by parmenides51, Jul 20, 2020, 4:54 AM
Reason: latex
Reason: latex
and 
,
and 
are tangent to 
it's so easy note that: 
is cyclic, by simple angle chasing we get 
and 
are the internal bisector and external bisector of 
respectively.
and 
,
is the polar of 
is tangent to 
