be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
.
be an integer. Rowan and Colin play a game on an 
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of 
is selected inside acute 
so that 
and 
.
is chosen on ray 
so that 
.
be the midpoint of 
.
is tangent to the circumcircle of triangle 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
and 
be positive integers. The cells of an 
grid are colored amber and bronze such that there are at least 
amber cells and at least 
bronze cells. Prove that it is possible to choose 
chosen cells lie in the same row or column.
and 
are erected outside an acute triangle 
Suppose that ![\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]](http://latex.artofproblemsolving.com/3/3/7/3378f0c4409d14bc0afddf2616992c13f624be3b.png)
Prove that lines 
and 
are concurrent.
-
of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let 
and 
?
of identical dominoes is a maximal grid-aligned configuration on the board if 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: 
be the number of distinct maximal grid-aligned configurations obtainable from 
Y by wu2481632, Davi-8191, samrocksnature, HWenslawski, math31415926535, jhu08, Adventure10, Mango247, ItsBesi, cubres, NicoN9
Find, with proof, the least integer 
such that if any 
elements are removed from the set 
, one can still find distinct numbers among the remaining elements with sum .
such that if any 
elements are removed from the set 
, one can still find distinct numbers among the remaining elements with sum .This post has been edited 1 time. Last edited by NormanWho, Apr 20, 2016, 9:31 PM
.
basically match up pairs to make 
and gg.
,
.
in fact works. Consider the pairs 
where each of the 
pairs have numbers that sum to 
usable pairs, so we are able to select the 
usable pairs, and their sum is 
,
.
causes any sum of 
numbers from 
are removed.
not removed numbers, giving a sum of at most 
.
,
under the worst case (least number of choices for 
pairs of 
at least, which is more than 
.
,
)
.
forces 
,
.
,
,
is achievable. There are 
.
.![\[(1,6048), (2,6047), \ldots, (3024,3025),\]](http://latex.artofproblemsolving.com/3/c/6/3c605115e0b235fc3d163eeca788e427f14c7c3e.png)
.
(hopefully I typed that right, it's supposed to be 
)![\[2017+2018+\dots+4032=1008*6049,\]](http://latex.artofproblemsolving.com/2/f/5/2f54abf69dc0c24431ebcb670c5a4184c60904ab.png)
which we need when we remove the positive integers from 
to 
,
,
,
,
.
numbers and are looking for 
,
to 
.
.
from the set and this is the smallest sum we can get from the remaining elements. Now we prove this satisfies the condition. We note that there are 
.
such pairs left in the set, which allows us to sum 
,
.
as always possible 
N = 1008.6049.
.
.
.
.
.
.
,
which is 
and so we are done..
are removed than the least sum is 
Just like we wanted.
works. To prove this consider the pairs: 
Regardless of which 
are deleted, at least 
.
clearly follows from removing 
.
works, take the pairs 
.
remaining numbers summing to 
