2005 Mediterranean Mathematics Olympiad

Part of Mediterranean Mathematics Olympiad

Subcontests

(4)

1

Mediterranian mathematics competition 2005, problem 4

A

be the set of all polynomials

f(x)

3

with integer coefficients and cubic coefficient

1

, so that for every

f(x)

there exists a prime number

p

which does not divide

2004

q

which is coprime to

p

2004

f(p)=2004

f(q)=0

. Prove that there exists a infinite subset

B\subset A

, so that the function graphs of the members of

B

are identical except of translations

1

Mediterranian mathematics competition 2005, problem 1

The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other. One of them says: You can't possibly guess my number. Then the other says: You are wrong, the number is 136. Which number did the professor tell them respectively? Give reasons for your claim.

1

Mediterranian mathematics competition 2005, problem 3

A_1,A_2,\ldots , A_n

(n\geq 3)

be finite sets of positive integers. Prove, that

\displaystyle \frac{1}{n} \left( \sum_{i=1}^n |A_i|\right) + \frac{1}{{{n}\choose{3}}}\sum_{1\leq i < j < k \leq n} |A_i \cap A_j \cap A_k| \geq \frac{2}{{{n}\choose{2}}}\sum_{1\leq i < j \leq n}|A_i \cap A_j|

|E|

is the cardinality of the set

E

1

Concurrent circles

k

k'

be concentric circles with center

O

R

R'

R<R'

holds. A line passing through

O

k

A

k'

B

O

A

B

. Another line passing through

O

and distict from

AB

k

E

k'

F

E

O

F

. Prove that the circumcircles of the triangles

OAE

OBF

, the circle with diameter

EF

and the circle with diameter

AB

are concurrent.