AoPS Academy
be real numbers such that 
Prove that 
Where 
. Find all 
such that for all 
, 
, 
and 
be points on the same circumference with 
. Let 
and 
be the projections of onto 
and 
, respectively. Prove that 
cuts the segment 
into equal parts.
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
lies inside acute scalene triangle 
and satisfies 
and 
. Points 
and 
are similarly defined. Let 
and 
be the centroid and orthocentre, repsectively, of triangle . Prove that 
, and all lie on a common circle.
lies inside acute scalene triangle and satisfies and . Points and are similarly defined. Let and be the centroid and orthocentre, repsectively, of triangle . Prove that , and all lie on a common circle.
is perpendicular to 
that 
is the midpoint of 
and 
and 
tells and 
lies on a circle with diameter GH
was the intersection of circumcircle of ABC and AM then is on line AM and also 
are the humpty points so the rest is simple angle chasing .