[/quote]
,
of degree 
having 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
be a circle inscribed inside a rhombus 
.
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
and 
are taken such that the point 
and 
.
and 
,
and 
.
.
,
.
and 
be intersection points of lines 
and 
with circumcircle 
,
and 
intersection points of lines 
and 
with side 
.
,
with 
such that 
divides 
for all positive integers 
with 
.
such that 
Follows for all reals 
be a positive integer, and let 
denote the sum of the positive divisors of 
.
smallest positive integer relatively prime to 
-smooth if all of its prime factors are 
.
denote the number of 
.
be the least quadratic non-residue mod 
.
is convergent for some 
,
Y by
Let 
be a positive integer. Find the minimum 
, so that there exists 
satisfying:
(1)For every
or 
(2)For every
, there are at most 
indices 
with 
(3)For every
, there are at most indices with 
be a positive integer. Find the minimum 
, so that there exists 
satisfying:(1)For every
or 
(2)For every
, there are at most 
indices 
with 
(3)For every
, there are at most indices with 
This post has been edited 2 times. Last edited by mathematics2003, May 14, 2021, 12:46 AM
Reason: sorry for terrible typo
Reason: sorry for terrible typo
would not make sense if we restrict 
.
are pairwise distinct. Put the numbers on a grid.
subgrid not containing the first row and first column of the original grid. In this subgrid, each row contains at most 
X's and each column contains at most 
.
subgrid, call 
.
.
to show for all 
,
contradicting 
.
squares.
:
then mark them with both X and Y.
and 
or if 
and 
mark it with 
and 
For 
.
where 
.
for 
.
.
for 
.
for 
and 
.
for 
for 
.
