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of integers such that ![\[1+2^{x}+2^{2x+1}= y^{2}.\]](http://latex.artofproblemsolving.com/9/c/8/9c81ffa59a374578a1dba99ad772902ea17ebadb.png)
such that ![\[ f(xf(y))=f(x+y)\]](http://latex.artofproblemsolving.com/b/c/0/bc02441171db2c0bbbdf02f5739dc65e9f56c840.png)
for all positive reals 
and 
.
(consisting of one or more consecutive blocks of 
) that are perfect squares of positive integers.
such that there exist integers 
with ![\[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]](http://latex.artofproblemsolving.com/e/f/a/efaa98e77ac0273284ee3446e4335d8cc4f681cb.png)
is large enough, in every 
square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by 
.
is large enough, in every square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by .
, be the matrix corresponding to entries in the square.
, where 
.
over a sub-square, 
, equals the sum 
.
be a colouring of the lattice. If is sufficiently large, then by Gallai's theorem there exists 
monochromatic lattice points that are vertices of a square. These correspond to four entries of 
such that 
. This is exactly what we wanted to prove.
