MathDB

2

Part of 2012 Middle European Mathematical Olympiad

Problems(2)

3 variables inequality with abc=1

Source: Middle European Mathematical Olympiad 2012 - Team Compt. T-2

9/14/2012
Let a,b a,b and c c be positive real numbers with abc=1 abc = 1 . Prove that 9+16a2+9+16b2+9+16c2≥3+4(a+b+c) \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)
inequalitiesgeometryfunctionalgebrapolynomialinequalities proposed
The maximum number of 'allowed' set

Source: Middle European Mathematical Olympiad 2012 - Individuals I-2

9/14/2012
Let N N be a positive integer. A set S⊂{1,2,⋯ ,N} S \subset \{ 1, 2, \cdots, N \} is called allowed if it does not contain three distinct elements a,b,c a, b, c such that a a divides b b and b b divides cc. Determine the largest possible number of elements in an allowed set S S .
floor functioncombinatorics proposedcombinatorics