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 a sequence of positive real numbers that satisfies 
and 
for every natural number 
. Prove that 
for every natural number .
be positive real numbers such that : 
. Prove that : 
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
+1 w
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and .
(ans- 
)
, with 
, and the circle 
with center and radius . Denote by 
the midpoint of side . The line 
intersects the circle for the second time in . Let be a point on the circle such that 
and 
. Show that 
.
satisfying 
Prove that 
When does equality hold?
be a point inside the triangle . 
intersects at 
. Points 
and 
are defined similarly. Line 
intersects 
at points 
, 
. The tangents at these points to intersect BC at points 
, 
. Points 
, 
, 
, 
are defined similarly. Prove that points , , , , , lie on one conic
be positive real numbers such that 
. Prove that:![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?
be the point of tangency of the excircle of a triangle (corrsponding to ) with the side . The line 
through is parallel to the bisector of 
. Lines 
and 
are analogously defined. Prove that 
have a common point.
.
and 
.
?
where 
denote the incentre of 
.
denote the second intersections of corresponding external bisectors of the angles of the triangle with the circumcircle 
.
we have 
with 
.
.
holds (as do the other analogous equalities). From here, we remark that the perpendiculars from 
,
and 
to lines 
concur. But now remark that 
and analogous perpendicularities imply that these perpendiculars are in fact just the lines 