MathDB

2

Part of 1993 IberoAmerican

Problems(2)

8th ibmo - mexico 1993/q2.

Source: Spanish Communities

5/7/2006
Show that for every convex polygon whose area is less than or equal to 11, there exists a parallelogram with area 22 containing the polygon.
geometryparallelogramrotationgeometric transformationreflectioncombinatorics proposedcombinatorics
8th ibmo - mexico 1993/q5.

Source: Spanish Communities

5/7/2006
Let PP and QQ be two distinct points in the plane. Let us denote by m(PQ)m(PQ) the segment bisector of PQPQ. Let SS be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If PP and QQ are in SS, then m(PQ)m(PQ) intersects SS. (ii) If P1Q1,P2Q2,P3Q3P_1Q_1, P_2Q_2, P_3Q_3 are three diferent segments such that its endpoints are points of SS, then, there is non point in SS such that it intersects the three lines m(P1Q1)m(P_1Q_1), m(P2Q2)m(P_2Q_2), and m(P3Q3)m(P_3Q_3). Find the number of points that SS may contain.
pigeonhole principlegeometrycircumcirclerhombusperpendicular bisectorcombinatorics proposedcombinatorics