AoPS Academy
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.
lamps 
in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp 
and its neighbours (only one neighbour for 
or 
, two neighbours for other 
) are in the same state, then is switched off; – otherwise, is switched on.
Prove that there are infinitely many integers 
for which all the lamps will eventually be off.
Prove that there are infinitely many integers for which the lamps will never be all off.
integers 
, each less than or equal to 
, prove that at least one of them is divisible by some other member of the set.
be positive integers such that no one of them is divisible by any other member of the sequence. Then![\[
a_1 \geq 2^k,
\]](http://latex.artofproblemsolving.com/1/f/e/1fecb6d6811bfc4aff7ccefa20412f0a5b860066.png)
where 
is defined by the inequalities![\[
3^k < 2n < 3^{k+1}.
\]](http://latex.artofproblemsolving.com/2/8/f/28f12a76029727a828167f1cb8e118b84d710d62.png)
This estimate for 
is the best possible.
with 
, find the possible values of 
.
till 
tending to 0, and 
tending to 
, 
, where we can get arbitrarily close to 0.
, with equality holding at 
. the answer is ![$(0, 2^n]$](http://latex.artofproblemsolving.com/3/a/1/3a11781436fc3570bae811827eb3aa36fdc323dd.png)
.
can take is in the interval ![$[0,2^n]$](http://latex.artofproblemsolving.com/4/e/5/4e5b2571df54b4dd897e707cff51157836d40f1b.png)
. First we prove that ![$p(1) \in [0,2^n]$](http://latex.artofproblemsolving.com/c/2/6/c26f227e5dca80dcee5b2267d808aea5233058cd.png)
.
is 
, we can write
for some nonnegative values 
. We have
By AM-GM, we have![$$1+x_i = 1+\frac{x_i}{2}+\frac{x_i}{2}\ge 3\sqrt[3]{\frac{x^2_i}{4}}$$](http://latex.artofproblemsolving.com/e/6/d/e6d73567b7688111626a16b24afb9627e79534ae.png)
. Multiplying this for 
yields![$$3^n=(1+x_1)(1+x_2)\cdots(1+x_n) \ge 3^n\sqrt[3]{\frac{(x_1x_2\cdots x_n)^2}{4^n}}$$](http://latex.artofproblemsolving.com/3/c/9/3c9beefd4e441878728ce99ea96c938783ae64ce.png)
. Clearly 
, hence . Now we prove that every value in is achievable.
satisfying
and 
for all ![$p \in [0,4]$](http://latex.artofproblemsolving.com/2/7/2/2726d73386a79c578745d4732a9ea9a4499c41a0.png)
.
has a positive root. Indeed, just choose
. It follows that 
is nonnegative, also satisfies the required condition.
, it follows from the lemma that every value in is achievable
, choose 
for every 
. This reduces the case to , then it also follows from the lemma that every value in is achievable. As desired.
has exactly 
is 
,
.
with respect to the constraints 
and 
.
trivially; we have to find 
.
)
by AM-GM?
for every 
,
.
is, when all 
are equal.
![$3^n=(1+x_1)(1+x_2)\cdots(1+x_n) \ge (1+\sqrt[n]{x_1x_2 \cdots x_n})^n$](http://latex.artofproblemsolving.com/6/7/8/6785391efa8ebbad2aff41eca976f96491fc32e0.png)
.