[/quote]
,
by 
,
for every positive integer 
.
leave distinct remainders when divided by 
.
of distinct positive integers such that there exists a positive integer 
for which the number of divisors of 
and of 
are equal.
and 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
are coprime. Consider the sequence 
given by 
.
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
.
such that the n subsets 
are also disjoint.
s
.
satisfies 
.
,
.
such that for all irrational numbers 
and 
,
where 
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
with incenter 
,
touches sides 
and 
at points 
and 
,
and 
touches 
.
meets 
at 
.
is parallel to 
.
Y by Adventure10
Let 
be the least number of distinct points in the plane such that for each 
there exists a straight line containing exactly 
of these points. Find an explicit expression for 
Simplified version.
Show that![$f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$](//latex.artofproblemsolving.com/b/3/6/b368118b049f924ec0e0373b71b04729159a07b4.png)
Where ![$[x]$](//latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denoting the greatest integer not exceeding 
be the least number of distinct points in the plane such that for each 
there exists a straight line containing exactly 
of these points. Find an explicit expression for 
Simplified version.
Show that
![$f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$](http://latex.artofproblemsolving.com/b/3/6/b368118b049f924ec0e0373b71b04729159a07b4.png)
Where ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denoting the greatest integer not exceeding 
