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with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
.
board with a number written in each cell. Every two neighbour rows sum up to at least 
, and every two neighbour columns sum up to at most 
. Find the sum of all numbers on the board.
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
, 
and 
.
be complex numbers with 
. Find the greatest possible value of the expression 
be a positive root of the equation 
. Prove that the following sequence converges: 
(0<x<2π)
with domain ![$[0,2025]$](http://latex.artofproblemsolving.com/d/1/7/d17ac41d54c4b1888fb8bb43cf11b372691694a0.png)
satisfying 
and 
is strictly increasing on Compute smallest real M such that all functions in this set ,
.
have local extrema at 
respectively. find 
. Here 
are constants .
be the region in the complex plane enclosed by curve 
for 
. Compute perimeter of .
where 
are real numbers, not all equal to 
.
,
and 
. Then the left hand side is identically 0,
are solutions.
where are real numbers, not all 
equal to .
(so that 
)
for 
.
are solutions of 
.
are not equal 
, 
and 
.
and 
. takes the value at three different places 
, on the interval ![$[x_1,x_3]$](http://latex.artofproblemsolving.com/8/f/2/8f2a5cc84c4ff0e4409f0e7d5ca0d269ccce76ca.png)
;
must be identically , which implies 
.
are solutions ,then 
and 
can not be solution because both inequalities 
and 
ca not be true at same time
,
,
,
.
has 
as the two only solutions.
.
at 
for 
.
,
,
impossible.
,
,
impossible.
has at most two zeroes.