MathDB

4

Part of 2004 Bundeswettbewerb Mathematik

Problems(2)

a cube is decomposed in a finite number of parallelepipeds

Source: German Olympiad (BWM) 2004, first stage, problem 4

6/15/2004
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.
geometry3D geometryspherecalculusinequalities solvedinequalities
Sqrt(x^2 + y^3) and sqrt(x^3 + y^2) are rational

Source: BWM 2004, 2nd round, problem 4

9/1/2004
Prove that there exist infinitely many pairs (x;  y)\left(x;\;y\right) of different positive rational numbers, such that the numbers x2+y3\sqrt{x^2+y^3} and x3+y2\sqrt{x^3+y^2} are both rational.
number theory proposednumber theory