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.
such that
x>y>0
2f(x)x>0
x,y>0
and 
be touch points of the incenter of 
at 
and 
, respectively. Let 
and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
be a point inside a triangle 
. Let 
meet 
at 
, let 
meet 
at 
, and let 
meet at 
. Let 
be the point such that is the midpoint of 
, let 
be the point such that is the midpoint of 
, and let 
be the point such that is the midpoint of 
. Prove that points 
cannot all lie strictly inside the circumcircle of triangle .
, points 
and 
lie on the side , with 
. Also, 
is the midpoint of 
. Show that the centroid of lies on 
.
be an acute triangle such that 
with circumcircle 
and circumcentre 
. Let 
and 
be the tangents to at 
and 
respectively, which meet at 
. Let 
be the foot of the perpendicular from onto the line segment 
. The line through 
parallel to line meets at 
. Prove that the line 
passes through the midpoint of the line segment 
.
be a quadrilateral with 
. Diagonals and 
meet at . Let 
and 
denote the circumcircle and the circumcenter of triangle 
. Let 
and 
denote the circumcircle and circumcenter of triangle 
. Segment meets and again at 
and 
(other than and ), respectively. Let and 
be the midpoints of minor arcs 
(not including ) and 
(not including ). Prove that 
.
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle , and let be the midpoint of the side . Let be a point on the side and a point on the side such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
be a triangle with 
. Let be the orthocenter of . Point is the midpoint of and point lies on the side such that 
. Prove that
.
be an integer greater than 
. For a positive integer 
, let 
. Suppose that there exists a 
-element set such that is an -element subset of 
; shares at most one common element; is contained in exactly two elements of . in terms of .
lies inside triangle so that 
. The point 
is such that 
is a parallelogram. Prove that 
.
be a triangle with 
, and let be the foot of the altitude from . A point is chosen inside the triangle 
so that 
bisects . Let be the intersection point of the lines and . Let be the semicircle with diameter that meets the segment 
at an interior point. A line through is tangent to at . Prove that the lines 
and meet on .
and 
Show that
Let 
Show that
Let 
Show that
Prove that
Let 
Prove that
be the foot of the altitudes from 
.
,
.
from 
and subtracting 
,
and 
in terms of 
,
.