IT-CESENATICO 2006

Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$ . At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$ , the other player has a card in his hand. What is it's value?

Let $A$ and $B$ be two distinct points on the circle $\Gamma$ , not diametrically opposite. The point $P$ , distinct from $A$ and $B$ , varies on $\Gamma$ . Find the locus of the orthocentre of triangle $ABP$ .

Consider the inequality $$(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).$$ a) Find all $n\ge 3$ such that the inequality is true for positive reals. b) Find all $n\ge 3$ such that the inequality is true for reals.

Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways:

1. take a coin from an arbitrary pile;
2. select a pile and divide it into two non-empty piles. The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.