MathDB

3

Part of 2009 Mexico National Olympiad

Problems(2)

Cubes and 2s

Source: OMM 2009 3

7/15/2014
Let aa, bb, and cc be positive numbers satisfying abc=1abc=1. Show that
a3a3+2+b3b3+2+c3c3+2≥1 and 1a3+2+1b3+2+1c3+2≤1\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1
inequalitiesinequalities unsolved
Friends at a Party

Source: OMM 2009 6

7/15/2014
At a party with nn people, it is known that among any 44 people, there are either 33 people who all know one another or 33 people none of which knows another. Show that the nn people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.
combinatorics unsolvedcombinatorics