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.
as 
fir 
. Prove that 
for all 
.
+1 w
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 
.
+1 w
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
are positive real numbers.Prove that
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
with 
such that the equation 
has an integer solution 
.
such that
and 
(Adit Aggarwal, India.)
be positive reals such that 
. Prove that
be an isosceles trapezoid with 
. The inscribed circle 
of triangle 
meets 
at 
. Let 
be a point on the (internal) angle bisector of 
such that 
. Let the circumscribed circle of triangle 
meet line at 
and 
. Prove that the triangle 
is isosceles.
such that, for all positive integers 
and ,
be real numbers so that 
and 
Show that
inscribed in a circle 
with center 
. Let 
the incenter of triangle and 
the contact points of the incircle with sides 
of triangle respectively . Let also 
the foot of the perpendicular line from to the line 
.Prove that line 
passes from the antidiametric point 
of 
in the circle .( 
is a diametre of the circle ).
be a positive integer. Two players and 
play a game in which they take turns choosing positive integers 
. The rules of the game are: takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
be the feet of angle bisectors of 
,
,
thus 
is the intouch triangle of 
.
be the harmonic conjugates of wrt 
then 
is the intersection of the Apollonius circles of with radius 
,
,
(which gives a way to construct the point).
be the circumcenter, the incenter, the intouch triangle, 
harmonic conjugates of wrt 
. Then since 
, is the intersection of the Apollonius circles with diameter 
. If 
are midpoints of , then 
is the radical axis of the incircle and circumcircle and point , so 
are collinear (and perpendicular to ).
be the feet of perpendiculars from 
onto 
, and 
be the feet of perpendiculars from 
onto 
. Then since 
and are collinear, we have 
, thus 
bisects 
, so 
are collinear. Since 
and 
, we have 
. Thus is tangent to the incircle at 
and the result follows.
Prove that
Lines 
,
,
intersect the circumcircle of triangle 
,
,
.
have a common incircle.
hence