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 triangle with 
. The points 
and 
lie on the side 
such that 
is perpendicular to and 
. The line that bisects 
meets 
and 
at 
and 
, respectively. Prove that the quadrilateral 
is cyclic.
, 
and 
are nonzero real numberss satisfying 
. Prove that among the three numbers 
, 
and 
, at most two of them are greater than 
.
are constructed on the exterior of a triangle . The perpendiculars from the midpoints of 
to the respective lines 
are constructed. Prove that these three perpendiculars pass through a single point.
be positive real numbers such that : 
. Prove that : 
such that![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](http://latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers 
.
natural numbers around the perimeter of a circle. She wants the products of adjacent pairs of numbers to be exactly the set 
Can she accomplish this?
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least 
.
with incenter 
, circumcenter 
, and 
-altitude 
.
be the orthocenter of 
and let 
. Let be the point on such that 
and the line 
meets at . Prove that the line passing through and parallel to 
bisects 
.
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
and 
. Vasya made one big triangle out of all of them. Petya claims that he can make a different triangle from the same tiles (also all three). Is Petya right?
and . Vasya made one big triangle out of all of them. Petya claims that he can make a different triangle from the same tiles (also all three). Is Petya right?
and 
.
,
![$ a,b,c\in [0,1] $](http://latex.artofproblemsolving.com/2/1/b/21b329d0a92c18c6a2861742177969d5977d4558.png)
. Prove that
.
.
