AoPS Academy
functions such that
for all real numbers 
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.
?
such that 
is a prime or a perfect square of an integer.
.
(prime).
But we know that 
. Contradiction.
.
(mod 5) . But 
(mod 5) 
Contradiction unless .
is the only solution.
. Viewing the equation modulo 
, we see that there are no possible solutions.
(
is a prime)
(for 
), we see that 
if an integer is always 
\for . works and thus it is the only possible solution.
, which clearly works. Then, we claim that no such 
exists.
(mod 4)
(mod 4)
to be a prime.
(mod 5), except .
and 
.
only, which clearly works.
worked. Then, clearly we must have 
.
This is not a QR mod 
,
.
for some 
,
For 
and 
are both greater than 
,
,
which is the product of two integers greater than 
,
which is again the product of two integers greater than 
: No such n exists such that the ![\[ \]](http://latex.artofproblemsolving.com/6/1/1/611ad9d590175c01c2041fe506310cb1c96ef993.png)
Say there is a ![\[ 5^n - 1 \equiv 3p \pmod{4} \]](http://latex.artofproblemsolving.com/2/1/5/215f60a1a41c19a4ca7625ba67fc0a839e73dd9c.png)
![\[ p \equiv 0 \pmod{4} \]](http://latex.artofproblemsolving.com/4/5/1/451b9434003ee6028beb13d1ef03e6c3bf2c7812.png)
then![\[ 5^n-1 \equiv 3x^2 \pmod{5} \]](http://latex.artofproblemsolving.com/e/b/4/eb43529ee1dbf00dc59d011b3d3a6376eb700eb4.png)
![\[ -1 \equiv 3x^2 \pmod{5} \]](http://latex.artofproblemsolving.com/b/9/5/b950a028d7661edaf24eacb5a35aad5f8663afcb.png)
Thats leads us with no sol for 
.
follows.
and 
and hence it cannot be a prime or a perfect square.
is an integer; therefore, we need 
to either be of the form 
for a natural number 
or of the form 
for a prime number 
,
is a quadratic non-residue 
- contradiction!
,
,
No such