[/quote]
,
and 
Prove that
such that 
and 
.
![\[ x^3+y^3+z^3=4^n\cdot{n^3} \]](http://latex.artofproblemsolving.com/d/8/3/d83c08bf50fee3d7783ee5fb0de27d9eaa5be17b.png)
be positive real numbers such that 
.
.
such that there exists a set 
of 
integers satisfying the following condition: for each 
,
such that 
is divisible by 
.
coins. Beginning with 
and remove 
coins from the stack. The player who takes the last coin wins. Prove that there are infinitely many integers 
cells are selected in a 
x
.
be the number of selected cells in the 
t
be the number of selected cells in the 
.
be real numbers such that 
Prove that 
Let 
Prove that 
such that 
,
.
,
. When we have equality ?
be postitive real numbers such that 
.
Y by Adventure10
Circle 
(with center 
) does not have common point with line 
. Draw 
perpendicular to , 
. Let 
be a point on ( is different from 
), 
and 
are tangent to circle , and intersect the circle at 
and 
respectively. 
intersects at 
. 
, 
are perpendicular to , , respectively, 
, 
. Prove that segment 
is bisected by line 
.
(with center 
) does not have common point with line 
. Draw 
perpendicular to , 
. Let 
be a point on ( is different from 
), 
and 
are tangent to circle , and intersect the circle at 
and 
respectively. 
intersects at 
. 
, 
are perpendicular to , , respectively, 
, 
. Prove that segment 
is bisected by line 
.
), which is easy to justify.
.
: 
be an isosceles triangle with 
,
as its orthocenter and 
as 
a
on 
.
.
be foot from 
intersects 
s
