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Part of 2003 Iran MO (2nd round)

Problems(2)

3-stratum numbers - Iran NMO 2003 (Second Round) - Problem1

Source:

10/4/2010
We call the positive integer nn a 3−3-stratum number if we can divide the set of its positive divisors into 33 subsets such that the sum of each subset is equal to the others. a)a) Find a 3−3-stratum number. b)b) Prove that there are infinitely many 3−3-stratum numbers.
number theory proposednumber theory
Easy Inequality - Iran NMO 2003 (Second Round) - Problem4

Source:

10/4/2010
Let x,y,z∈Rx,y,z\in\mathbb{R} and xyz=−1xyz=-1. Prove that: x4+y4+z4+3(x+y+z)≥x2y+x2z+y2x+y2z+z2x+z2y. x^4+y^4+z^4+3(x+y+z)\geq\frac{x^2}{y}+\frac{x^2}{z}+\frac{y^2}{x}+\frac{y^2}{z}+\frac{z^2}{x}+\frac{z^2}{y}.
inequalitiesinequalities proposed