[/quote]
,
of degree 
having 
distinct real roots 
)
: 
.
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
In each square of an 
board, we are to write one of the numbers 
, 
, or 
in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.
board, we are to write one of the numbers 
, 
, or 
in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.
.
be 
column and 
be 
which means 
