[/quote]
,
be positive integers.Define injective function 
from 
to itself such that 
.
divides 
.
board that can be colored so that for each square in the board, at most one square adjacent to it is colored.
such that 
.
that divides 
for the positive integer 
.
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
of positive integers such that 
in all of this cases :
3-Every prime number less than 
can be represented in the form 
by taking an Appropriate pair 
and 
. The number 
is the least prime number that cannot be represented as a sum or an difference of a power of 
, 
, 
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
be real positive numbers such that: 
.
,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
satisfying
![\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
\]](http://latex.artofproblemsolving.com/9/5/9/9591f7c842d1b6afcba11b15b3ea3f9e01bc937a.png)
![\[
2^a + 3^b + 5^c = n!
\]](http://latex.artofproblemsolving.com/b/c/6/bc6b0754dd874f646fcab124c91583b620db273c.png)
of injective functions from the set of real numbers to itself satisfying
and 
.
is injective if 
for any distinct real numbers 
and 
.
,
such that any graph with 
Y by ImSh95, Amir Hossein
Let 
be a point in the plane. Prove that it is possible to draw three rays with origin in with the following property: for every circle with radius 
containing in its interior, if 
, 
and 
are the intersection points of the three rays with the circle, then ![\[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]](//latex.artofproblemsolving.com/d/a/4/da432234a66a9a2943a62460abde745fa750404e.png)
be a point in the plane. Prove that it is possible to draw three rays with origin in with the following property: for every circle with radius 
containing in its interior, if 
, 
and 
are the intersection points of the three rays with the circle, then ![\[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]](http://latex.artofproblemsolving.com/d/a/4/da432234a66a9a2943a62460abde745fa750404e.png)
we have that its circumradius is 
and its area is 
)
