2008 ITAMO

Part of ITAMO

Subcontests

(3)

2

Italian Mathematical Olympiad 2008

Find all functions

f: Z \rightarrow R

that verify the folowing two conditions: (i) for each pair of integers

(m,n)

m<n

f(m)<f(n)

; (ii) for each pair of integers

(m,n)

there exists an integer

k

such that f(m)\minus{}f(n)\equal{}f(k).

Italian Mathematical Olympiad 2008

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?

1

Italian Mathematical Olympiad 2008

ABC

be a triangle, all of whose angles are greater than

45^{\circ}

and smaller than

90^{\circ}

. (a) Prove that one can fit three squares inside

ABC

in such a way that: (i) the three squares are equal (ii) the three squares have common vertex

K

inside the triangle (iii) any two squares have no common point but

K

(iv) each square has two opposite vertices onthe boundary of

ABC

, while all the other points of the square are inside

ABC

P

be the center of the square which has

AB

as a side and is outside

ABC

r_{C}

be the line symmetric to

CK

with respect to the bisector of

\angle BCA

P

r_{C}

2

Italian Mathematical Olympiad 2008

ABCDEFGHILMN

be a regular dodecagon, let

P

be the intersection point of the diagonals

AF

DH

S

be the circle which passes through

A

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

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

Italian Mathematical Olympiad 2008

Find all triples

(a,b,c)

of positive integers such that a^2\plus{}2^{b\plus{}1}\equal{}3^c.