[/quote]
,
has incentre 
and incircle touching 
at 
.
be the antipode of 
in the circumcircle of 
is chosen on the internal angle bisector of 
such that 
.
be the midpoint of arc 
,
be the midpoint of 
.
for which:
for all 
.
particles on a circle situated at the vertices of a regular 
-
be the smallest number of collisions after which all particles return to their original positions. Find 
![$Q\in \{-1,1,0\}[x]$](http://latex.artofproblemsolving.com/b/4/0/b404c8d6415aaf3d4442cc5381f582c1f8f8b996.png)
with 
.![$\exists P \in \mathbb Z[x]$](http://latex.artofproblemsolving.com/a/2/e/a2e67dd74df9971ca90d1a5d81c1c8e1f3982276.png)
palindrom with 
?
for which 
,
denote the integer part.
such that 
is divisible by 
,
is divisible by 
,
is divisible by 
.
and 
Prove that
chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
such that there exists a set 
of 
integers satisfying the following condition: for each 
,
such that 
is divisible by 
.
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
be a family of functions from 
.
,
such that for all 
,![\[
y^2 \cdot f\left(\frac{g(x)}{y}\right) = h(xy)
\]](http://latex.artofproblemsolving.com/f/7/1/f7196164e0639695529b06788ea7cb1e50eddbe7.png)
and all 
,![\[
f\left(\frac{x}{f(x)}\right) = 1.
\]](http://latex.artofproblemsolving.com/1/5/b/15b363bf1754a4edd72423c3d0ea631173897beb.png)
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
and 
are given. Let 
be arbitrary points on 
.
.
of points in the plane is balanced if, for any two different points 
in 
in 
.
.
Y by Adventure10, Mango247
A triple 
of positive integers is called quasi-Pythagorean if there exists a triangle with lengths of the sides 
and the angle opposite to the side 
equal to 
. Prove that if is a quasi-Pythagorean triple then has a prime divisor bigger than 
.
of positive integers is called quasi-Pythagorean if there exists a triangle with lengths of the sides 
and the angle opposite to the side 
equal to 
. Prove that if is a quasi-Pythagorean triple then has a prime divisor bigger than 
.
.
.
!
!
