4

Part of 1999 Tuymaada Olympiad

Problems(2)

Tuymaada 1999, Q4

Source: Tuymaada 1999, Q4

Prove the inequality

{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,

2 < x, y, z < 4.

Proposed by A. Golovanov

inequalitiesinequalities unsolved

Tuymaada 1999, Q8

Source: Tuymaada 1999, Q8

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares.
Proposed by A. Golovanov

calculusintegrationanalytic geometrygeometryparallelogramnumber theorygreatest common divisor