AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be an isosceles triangle with 
. Let 
and 
points over 
and 
, respectively, such that 
. Bisector of 
intersects line 
at 
. Show that 
passes through the midpoint of 
.
is acute triangle. 
is orthocenter, 
is the midpoint of , 
is the midpoint of smaller arc . Point is on 
such that, 
is perpendicular to 
. Prove that: 
is paralelogram(Synthetic sol needed).
, 
, 
and prime numbers 
and 
, such that:
be a cyclic quadrilateral with 
. Let be the midpoint of the arc 
not containing 
. Suppose there is a point 
inside such that 
and 
.
, and are concurrent.
. Prove that
in order to make it a perfect square?
is 
and 
. Prove that 
is 
degrees, where and 
are the midpoints of 
and 
, respectively.
and be positive real numbers such that 
. Find the minimum value of 
and 
be the incircle and incenter of triangle . Let 
, 
, 
be the tangency points of to , 
, 
and let 
be the reflection of about . Assume 
intersects the tangents to at and at points and 
. Show that 
.
and 
.
and 
Prove that
,
.
.
.
,
from the perpendiculars.
be the unit circle
to get that 
and 
angles sum up to 
we prove that 
and using the fact that 
we get that 
