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 an acute triangle (
) with circumcircle 
and orthocenter 
. Let 
be the midpoint of 
. Line 
intersects again at 
, and line 
intersects again at 
. Points 
and 
lie on 
and 
respectively, such that 
is a parallelogram.
is tangent to the circumcircle of triangle 
.
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
for which the inequality ![\[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]](http://latex.artofproblemsolving.com/4/d/b/4db42ebabc7309bba9629a9a7125be43abf113f0.png)
holds for all real numbers 
, 
, 
, and 
.
, 
be the incenter and the 
-excenter of a triangle ; , 
be the touching points of the incircle with , respectively; 
be the common point of 
and 
. The perpendicular to from meets 
at point 
. Prove that , , , are concyclic.
and 
are given. Let be the midpoint of the segment joining their centers, 
, 
be arbitrary points on , respectively such that 
. Find the locus of the midpoints of segments 
.
, be the circumcenter and the incenter of an acute-angled scalene triangle ; , , be the touching points of its excircle with the side and the extensions of , respectively. Prove that if the orthocenter of the triangle 
lies on the circumcircle of , then it is symmetric to the midpoint of the arc with respect to 
.
be a point inside a quadrilateral 
such that 
. Points 
, 
, 
are isogonally conjugated to with respect to the triangles 
, 
, 
, respectively. Prove that the diagonals of the quadrilaterals and 
concur. be a quadrilateral such that the excircles and of triangles and touching their sides and 
respectively touch the extension of at the same point . The segment 
meets at point 
, and the line meets at 
and 
. Prove that one of angles 
and 
is right
with the following properties 
If each positive integer is arbitrarily colored red or blue
distinct red positive integers 
satisfying
or there are 
distinct blue positive integers 
satisfiying
be reals such that 
.Prove that
Where 
such that 
Prove that 
2. Let 
be sidelengths of a triangle such that 
Prove that 
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
be a parallelogram such that 
Let 
and 
be the midpoints of and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
intersect at , 
and 
. I draw perpendicular to .
, thus 
and finally 
Prove that
Where 
.
,
.
.
,