Problem 4

Part of 2007 Mongolian Mathematical Olympiad

Problems(2)

simple three variable

Source:

a,b,c>0

. Prove that \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}

inequalitiesinequalities unsolvedalgebra

syseq, sums of two squares

Source: Mongolian MO 2007 Teachers P4

x,y,z\in\mathbb N

xy=z^2+1

prove that there exists integers

a,b,c,d

x=a^2+b^2

y=c^2+d^2

z=ac+bd