[/quote]
,
be an equilateral triangle, and 
is a point on its incircle. Let 
be the circle tangent to 
passing through 
.
be the circle tangent to 
passing through 
,
be the circle tangent to 
passing through 
.
.
be a point on 
,
be a line passing through 
.
at 
and the circumcircle of triangle 
at 
)
and 
are tangent at 
and 
Prove that
Where 
with degree 
,
with product 
,
Suppose there exist positive reals 
with product 
Is it possible that the values 
,
,
,
be non-negative real numbers such that![\[2a+2b+2c+2d+ab+bc+cd+da+3=abcd.\]](http://latex.artofproblemsolving.com/d/e/8/de8ccd816cfe032a97d940982192b1abea21dc18.png)
prove that : ![\[\sqrt[4]{abc}+\sqrt[4]{bcd}+\sqrt[4]{cda}+\sqrt[4]{dab}\le\sqrt[4]{27(1+a)(1+b)(1+c)(1+d)}.\]](http://latex.artofproblemsolving.com/a/f/c/afc556cd8f5c7f4dedf153ffb92fb1f978c56823.png)
and 
.
,
,
and 
intersect the circumcircle of 
,
and 
respectively. Let 
,
be diameters in the circumcircles of triangles 
and 
respectively.
.
Where 
Where 
.
and 
are on 
respectively such that 
.
and 
meet at 
.
and 
is the orthocenter of 
.
and 
,
and 
.
.
.
satisfying:
and 
are externally tangent at a point 
be a line tangent to 
and 
.
and 
be diameters in 
and 
are tangent to 
and 
are tangent to 
.
,
times the radius of 
and 
are chosen on 
,
is the foot of the altitude from 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
and 
for which:
.
is a sequence of integers satisfying 
,
,
,
Prove that there exists 
such that 
Y by VicKmath7, Adventure10
The chess piece sick rook can move along rows and columns as a regular rook, but at most by 
fields. We can place sick rooks on a square board in such a way that no two of them attack each other and no field is attacked by more than one sick rook.
a) Prove that on
board, we cannot place more than 
sick rooks.
b) Find the maximum number of sick rooks which can be placed on
board.
c) Prove that on
board, we cannot place more than 
sick rooks.
fields. We can place sick rooks on a square board in such a way that no two of them attack each other and no field is attacked by more than one sick rook.a) Prove that on
board, we cannot place more than 
sick rooks.b) Find the maximum number of sick rooks which can be placed on
board.c) Prove that on
board, we cannot place more than 
sick rooks.This post has been edited 1 time. Last edited by parmenides51, Jan 20, 2020, 5:24 PM
