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such that for all 
is great if the following holds: for any point 
on the side 
, if 
and 
are the feet of the perpendiculars from to the lines 
and 
, respectively, then the reflection of in the line 
lies on the circumcircle of the triangle . Prove that triangle is great if and only if 
and 
.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a square colored like a chessboard.. Let be the foot of the altitude of triangle issuing from the vertex 
, and let 
be the circumcenter of triangle . Assume that 
. Prove that 
.
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
If a problem has not been solved within 
days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
and natural number 
,
are there such that 
and 
?
of a triangle to its sides 
are denoted 
respectively. Let 
be the inradius of the triangle. Prove that:
satisfied 
Find the minimum of 
satisfy that: 
we know that ![$[GBC]=[GAB]=[GCB]\implies g_a .a=\frac13 r(a+b+c) \iff g_a=\frac13 r. \frac{a+b+c}{a}$](http://latex.artofproblemsolving.com/5/7/5/575250aa3e20465e3866496d557087b3ae90d26f.png)
but 
then 
idem for the others
but since 
hence 