2

Part of 2004 Bundeswettbewerb Mathematik

Problems(2)

A^2 + b^2 + c^2 is a perfect square

Source: BWM 2004, 1st round, problem 2

Consider a triangle whose sidelengths

a

b

c

are integers, and which has the property that one of its altitudes equals the sum of the two others. Then, prove that

a^2+b^2+c^2

is a perfect square.

Beautiful combinatorial geometry

Source: BWM 2004, 2nd round, problem 2

k

be a positive integer. In a circle with radius

1

, finitely many chords are drawn. You know that every diameter of the circle intersects at most

k

of these chords. Prove that the sum of the lengths of all these chords is less than

k \cdot \pi

geometryperimetergeometric transformationreflectionparameterizationfunctionintegration