IT-CESENATICO 2008

Let $ABCDEFGHILMN$ be a regular dodecagon, let $P$ be the intersection point of the diagonals $AF$ and $DH$ . Let $S$ be the circle which passes through $A$ and $H$ , and which has the same radius of the circumcircle of the dodecagon, but is different from the circumcircle of the dodecagon. Prove that:

1. $P$ lies on $S$
2. the center of $S$ lies on the diagonal $HN$
3. the length of $PE$ equals the length of the side of the dodecagon

Find all functions $f: Z \rightarrow R$ that verify the folowing two conditions: (i) for each pair of integers $(m,n)$ with $m<n$ one has $f(m)<f(n)$ ; (ii) for each pair of integers $(m,n)$ there exists an integer $k$ such that $f(m)-f(n)=f(k)$ .

Find all triples $(a,b,c)$ of positive integers such that $a^2+2^{b+1}=3^c$ .

Francesca and Giorgia play the following game. On a table there are initially coins piled up in some stacks, possibly in different numbers in each stack, but with at least one coin. In turn, each player chooses exactly one move between the following: (i) she chooses a stack that has an even non-zero number of coins $2k$ and breaks it into two identical stacks of coins, i.e. each stack contains $k$ coins; (ii) she removes from the table the stacks with coins in an odd number, i.e. all such in odd number, not just those with a specific odd number. At each turn, a player necessarily moves: if one choice is not available, the she must take the other. Francesca moves first. The one who removes the last coin from the table wins.

1. If initially there is only one stack of coins on the table, and this stack contains $2008^{2008}$ coins, which of the players has a winning strategy?
2. For which initial configurations of stacks of coins does Francesca have a winning strategy?