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 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 
.
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.
and 
, of equal radius intersect at different points 
and 
. Consider a circle externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
and be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:
in any cell of the grid.
. and .
be non-negative real numbers such that 
Prove that
be circumscribed about circle 
, and let be the orthocenter of 
. The circle touches line 
at . The tangent to the circle 
at meets at . Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
in the plane of 
such that 
Let 
be the points of intersection of the medians of the triangles 
, respectively. Prove that the segments 
and 
are equal and have a common point.
in the plane of such that Let be the points of intersection of the medians of the triangles , respectively. Prove that the segments and are equal and have a common point.
Prove that
