MathDB

2

Part of 2012 JBMO ShortLists

Problems(3)

Inequality with abc=1

Source: JBMO Shortlist 2012 A2

2/4/2015
Let aa , bb , cc be positive real numbers such that abc=1abc=1 . Show that : 1a3+bc+1b3+ca+1c3+ab≤(ab+bc+ca)26\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}
inequalitiesinequalities proposed
Isosceles triangle !

Source: JBMO Shortlist 2012 G2

2/4/2015
Let ABCABC be an isosceles triangle with AB=ACAB=AC . Let also ω\omega be a circle of center KK tangent to the line ACAC at CC which intersects the segment BCBC again at HH . Prove that HK⊥ABHK \bot AB .
geometrycircumcirclegeometric transformationreflectionperpendicular bisector
Prime numbers !

Source:

2/4/2015
Do there exist prime numbers pp and qq such that p2(p3−1)=q(q+1)p^2(p^3-1)=q(q+1) ?
number theoryprime numbers