,
.
with integer coefficients such that for any positive integer 
,
is positive and 
be a triangle with incentre 
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
.
and the midpoint 
of 
lie on a line.
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
such that 
for all 
?
.
are positive real numbers satisfying the condition 
,![\[
P = \frac{1}{1+x^2} + \frac{1}{1+y^2} + \frac{z}{1+z^2}
\]](http://latex.artofproblemsolving.com/4/e/6/4e6a6e3d1c959f8d16f08f7a8f842c20749fcef0.png)
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
denote the set of all positive integers that can be expressed as a sum of three or fewer powers of 
.
be the smallest positive integer that cannot be expressed in the form 
,
.
.
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
fair coins are flipped. If 
.
for 
distinct integers , Then the number of integer values of 
for which 
Y by Adventure10, Mango247
The author of this posting is : nttu
____________________________________________________________________
Let a triangle ABC . M , N , P are the midpoints of BC, CA, AB .
a)
are lines throughing M, N, P and dividing the perimeter of triangle ABC into halves . Prove that : are concurrent at K .
b) Prove that : among the ratios :
, there exists at least one ratio 
.
____________________________________________________________________
Let a triangle ABC . M , N , P are the midpoints of BC, CA, AB .
a)
are lines throughing M, N, P and dividing the perimeter of triangle ABC into halves . Prove that : are concurrent at K .b) Prove that : among the ratios :
, there exists at least one ratio 
.
).
