MathDB

2

Part of 2001 Irish Math Olympiad

Problems(2)

perpendicular medians (old exercise)

Source: Ireland 2001

7/5/2009
Let ABC ABC be a triangle with sides BC\equal{}a, CA\equal{}b,AB\equal{}c and let D D and E E be the midpoints of AC AC and AB AB, respectively. Prove that the medians BD BD and CE CE are perpendicular to each other if and only if b^2\plus{}c^2\equal{}5a^2.
geometry proposedgeometry
hoops

Source: Ireland 2001

7/5/2009
Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with 20 20 beads, 10 10 of which are black and 10 10 are white. On each hoop the positions of the beads are labelled 1 1 through 20 20 as shown. We say there is a match at position i i if all three beads at position i i have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than 5 5 matches.
pigeonhole principlegeometrygeometric transformationrotationcombinatorics unsolvedcombinatorics