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 acute triangle with 
. The angle bisector of 
intersects 
at 
, and the perpendicular to 
at intersects 
and 
at 
respectively. Suppose 
and 
intersect again at 
, prove that 
is the angle bisector of 
.
and 
Prove that
Let and 
Prove that
be complex numbers satisfying 
. Show that there exist some among the 
complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
and 
be three positive integers. Prove that there exist a positive integer 
and a set of 
positive integers 
, such that, for every 
, the 
-ary expansion of is a 
-digit palindrome, and the 
-ary expansion is exactly 
.
, s.t :
be an acute triangle with 
, midpoint 
of side , altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
satisfy,
Where {
} is a permutation of . Prove that 
has a unique solution in integers with 
and 
.
as 
fir 
. Prove that 
for all 
.
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at respectively. Point such that 
is a parallelogram. Prove that lines 
and are concurrent at a point on 
.
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
are positive real numbers.Prove that
, and points 
on it such that 
. The tangent at 
intersects the line 
at 
. The line 
intersects the arc of the semicircle at 
. Prove that 
.
and 
; if is the midpoint of , is A-median of the 
, right-angled at and, from 
, the angle bisector of 
intersects the semicircle at a point belonging to the arc 
, that point being the midpoint of the arc 
, done.
