4

Part of 2007 Moldova Team Selection Test

Problems(4)

golden ratio and pentagons: the flavor of ISL 2002 G5

Source: Moldova 2007 IMO-BMO TST II problem 4

Consider five points in the plane, no three collinear. The convex hull of this points has area

S

. Prove that there exist three points of them that form a triangle with area at most

\frac{5-\sqrt 5}{10}S

ratiogeometrycombinatorial geometrygeometry proposed

balanced polygon

Source: Moldova 2007 IMO-BMO TST I problem 4

Consider a convex polygon

A_{1}A_{2}\ldots A_{n}

M

inside it. The lines

A_{i}M

intersect the perimeter of the polygon second time in the points

B_{i}

. The polygon is called balanced if all sides of the polygon contain exactly one of points

B_{i}

(strictly inside). Find all balanced polygons. [Note: The problem originally asked for which

n

all convex polygons of

n

sides are balanced. A misunderstanding made this version of the problem appear at the contest]

geometryperimetergeometry proposed

Turan consequence

Source: Moldova 2007 IMO-BMO TST III problem 4

n

distinct points in the plane. Consider the number

\tau(n)

of segments of length 1 joining pairs of these points. Show that

\tau(n)\leq \frac{n^{2}}3

geometryrhombusPutnamcombinatorics proposedcombinatorics

p|n!+1, n does not divide p-1

Source: Moldova 2007 IMO-BMO TST IV problem 4

Show that there are infinitely many prime numbers

p

having the following property: there exists a natural number

n

p-1

p|n!+1

number theoryprime numbersnumber theory proposed