Eisatopon Math AI Challenges: ▪ Romania Team Selection Tests 2012

Παρασκευή 4 Μαΐου 2012

▪ Romania Team Selection Tests 2012

Ημέρα 1η

1. Let

,...,

be positive integers, and define

and

, for

where

denotes the greatest common divisor of the integers

,...,

. Prove that the sums

with

for

are mutually distinct

2. Let

be a cyclic quadrilateral such that the triangles

and

are not equilateral. Prove that if the Simson line of

with respect to

is perpendicular to the Euler line of

, then the Simson line of

with respect to

is perpendicular to the Euler line of

3. Let

and

be finite sets of real numbers and let

be an element of

. Prove that

where

and

4. Prove that a finite simple planar graph has an orientation so that every vertex has out-degree at most 3.

5. Let

and

be two given positive integers. A set of

real numbers

is said to be balanced iff

were an arithmetic progression with common difference

and

where an arithmetic progression with common difference

. Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection.

Comment: The intended problem also had "

and

are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Hopefully, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has

reals

so that

is an arithmetic progression with common difference

and

is an arithmetic progression with common differnce

Ημέρα 2η

1. Prove that for any positive integer

we have that

2. Let

be a convex circumscribed quadrilateral such that

and

. Prove that one of the diagonals of quadrilateral

passes through the other diagonals midpoint.

3. Find the maximum possible number of kings on a

chess table so that each king attacks exactly one of the other kings (a king attacks only the squares that have a common point with the square he sits on).