3

Part of 2008 Romania Team Selection Test

Problems(5)

Convex hexagon with all sides of length 1

Source: Romanian TST 1 2008, Problem 3

ABCDEF

be a convex hexagon with all the sides of length 1. Prove that one of the radii of the circumcircles of triangles

ACE

BDF

geometrycircumcirclegeometry proposed

Inequality in a convex pentagon

Source: Romanian TST 2 2008, Problem 3

Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side. Note. If the pentagon is labeled

ABCDE

, the adjacent vertices of

A

B

E

B

A

C

inequalitiesgeometryvectortrigonometryanalytic geometrygeometry proposed

Impossible divisibility

Source: Romanian TST 3 2008, Problem 3

m,\ n \geq 3

be positive odd integers. Prove that 2^{m}\minus{}1 doesn't divide 3^{n}\minus{}1.

quadraticsmodular arithmeticnumber theoryprime factorizationnumber theory proposed

Subsets of \{1, 2,\ldots, n\}

Source: Romanian TST 4 2008, Problem 3

n \geq 3

be a positive integer and let m \geq 2^{n\minus{}1}\plus{}1. Prove that for each family of nonzero distinct subsets

(A_j)_{j \in \overline{1, m}}

\{1, 2, ..., n\}

i

j

k

such that A_i \cup A_j \equal{} A_k.

inductionpigeonhole principlecombinatorics proposedcombinatorics

Partitioning squares and an Erdos-type problem

Source: Romanian TST 5 2008, Problem 3

\mathcal{P}

be a square and let

n

be a nonzero positive integer for which we denote by

f(n)

the maximum number of elements of a partition of

\mathcal{P}

into rectangles such that each line which is parallel to some side of

\mathcal{P}

intersects at most

n

interiors (of rectangles). Prove that 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.

geometrysearchinductioncombinatorics proposedcombinatorics