AoPS Academy
and 
take turns playing the following game: There is a pile of 
stones. In his first turn, selects a divisor of and removes this number of stones from the pile. then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.
denote the 
th Fibonacci number, defined by the recurrence:![\[
F_0 = 0,\quad F_1 = 1,\quad \text{and} \quad F_{n+2} = F_{n+1} + F_n \quad \text{for } n \geq 0.
\]](http://latex.artofproblemsolving.com/1/b/c/1bc4aa38be3c2c64832e25376a923df611807864.png)
For a fixed modulus 
, define the set![\[
S_m = \left\{ (i, j) \in \mathbb{N}^2 : F_i \cdot F_j \equiv 1 \pmod{m} \right\}.
\]](http://latex.artofproblemsolving.com/c/0/4/c04881682fdc91ae0a7a78a807044c3658e0412e.png)
Prove that 
if and only if 
.
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
that cannot be expressed as 
for some positive real number 
.
of cardboard, with 5 rectangular faces instead of 6 and there is no leftover cardboard. What is the largest volume this box can hold given that 
of cardboard is needed to cover the box?
can be written in the form![\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]](http://latex.artofproblemsolving.com/9/e/9/9e96c07ea3b0750af8bed344df6e8a86fcf0d772.png)
where 
and 
are positive integers. Find the value of 
.
you mentioned? Again, unclear.
on 
easily, 
are the solutions, now you can list all possible pairs.
chess board, finite games of two players, your P3 with 
and so on.
such that 
.
and 
Prove that
be real numbers . Prove that
Subject to the constraints:
Prove that
. Prove that
steps, can all the numbers on 6 vertices be equal?
individual tables arranged into 4 rows and 9 columns (all the rows are numbered from 1 to 4, all the columns are numbered from 1 to 9). Number of students in this class is 
.
row, 
column is numbered with an integer number 
.
.
having 
satisfy that: 