AoPS Academy
from the set of non-negative integers to itself such that 
, whenever 
, are non-negative integers satisfying 
.
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
. Assume that the inequality is strict for some 
.
for every 
or 
for every .
be an integer. Ion draws a regular 
-gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the -gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.
, 
and the function 
defined by![\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]](http://latex.artofproblemsolving.com/8/f/b/8fb372df4b44c3d7c0690b310b12c684134fee74.png)
Prove that the following are equivalent:
is surjective;
, 
and 
.
, say {
}=
such that 
. We can make 
sufficiently small such that 
so that no integer is skipped. So in case of that , 
and done!
, and 
. Putting into 
we get,
I will do this by induction,
we have 
since .
which attains at 
, where 
is the maximum value for which 
.
,
Now 
. For equality case it's too trivial. For inequality, 
since was maximum and . And the induction is complete.
such that 
coprime, 
and 
.
