2

Part of 2006 Germany Team Selection Test

Problems(3)

Boxes without principle

Source: 1st German pre-TST 2006, problem 2

In a room, there are

2005

boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. a) Show that we can find

669

boxes, which altogether contain at least a third of all apples and at least a third of all bananas. b) Can we always find

669

boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?

inequalitiescombinatorics proposedcombinatorics

MOP [Points on the sides of the orthic triangle]

Source: also: Singapore TST (which year?)

A_{1}

B_{1}

C_{1}

be the feet of the altitudes of an acute-angled triangle

ABC

issuing from the vertices

A

B

C

, respectively. Let

K

M

be points on the segments

A_{1}C_{1}

B_{1}C_{1}

, respectively, such that

\measuredangle KAM = \measuredangle A_{1}AC

. Prove that the line

AK

is the angle bisector of the angle

C_{1}KM

trigonometrygeometryangle bisectorgeometry solved

Geo Ineq! [altitudes integers, inradius prime --> sides = ?]

Source: Contest 95'

The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number. Find the lengths of the sides of the triangle.

geometryinradiusgeometry unsolved