[/quote]
,
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
be a triangle, 
its circumcircle and 
the center of 
be a point on the segment 
the second intersection point of the circumcircles of triangles 
and 
.
and the tangent to 
intersect on the circumcircle of triangle 
.
,
,
.![\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]](http://latex.artofproblemsolving.com/d/1/8/d18152c5635ee2f2059a860bf2fa362fe525ebb3.png)
satisfying the following two conditions:
such that, for all indices 
,
.
and for any index ![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of 
and 
be two fixed points on the circle 
the intersection point of lines 
and 
,
.
be the circumcircle of triangle 
.
be the second intersection point of 
.
intersects 
.
intersects line 
at 
,
intersects 
the second intersection point of the circumcircle of triangle 
with 
into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube 
lie on, and the four planes that the tetrahedron 
lie on. How many regions do these ten planes split the space into?
such that 
.
on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
.
.
is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is 
and, for each of them, determine the minimum number of GCD operations Charlotte must use.
be an integer. We consider a square grid of size 
divided into 
unit squares. The grid is called balanced if:
,
or 
.
.
such that any balanced grid always contains an 
square whose absolute sum of the 
cells is less than or equal to 
.
,
.
and 
.
into three equal parts while the points 
and 
divide 
and 
intersect at 
and 
intersect at 
,
and 
are 
,
,
respectively, find the area of the triangle 
.
Y by
Let 
be a rectangle. Points 
and 
are chosen on 
and 
respectively such that there exists a point G such that quadrilateral 
is a rectangle. If 
, 
, and 
, then 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
be a rectangle. Points 
and 
are chosen on 
and 
respectively such that there exists a point G such that quadrilateral 
is a rectangle. If 
, 
, and 
, then 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
