for any 
,
vertices is written a real number. Prove that
(with possibly repeating vertices) in which the sequence of numbers is increasing.
be positive integer such that ![\[ \text{LCM}(x+2, y+2)+\text{LCM}(x, y)=2\text{LCM}(x+1, y+1).\]](http://latex.artofproblemsolving.com/7/3/3/7338e4d6058ededa9b8cd9fcc945ae362db4bad4.png)
Prove that 
is divisible by 
.
distinct prime numbers such that such that 
Y by
f:Z+->Z+ satisfied:
i) x+2y | f(x)+2f(y) forall x, y \in Z+
ii) f(x)<=x^2/2 forall x>2024
a/ prove that exist N and k such that if p>N => f(p)=kp
b/ f(x)=kx forall x in Z+
i) x+2y | f(x)+2f(y) forall x, y \in Z+
ii) f(x)<=x^2/2 forall x>2024
a/ prove that exist N and k such that if p>N => f(p)=kp
b/ f(x)=kx forall x in Z+
