MathDB

2

Part of 2015 Vietnam National Olympiad

Problems(2)

Viet Nam National Math Olympiad 2015

Source:

1/8/2015
If a,b,ca,b,c are nonnegative real numbers, then 3(a2+b2+c2)≥(a+b+c)(ab+bc+ca)+(a−b)2+(b−c)2+(c−a)2≥(a+b+c)2.{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}
inequalitiesinductioncalculusderivativeinequalities proposed
VMO 2015 Problem 6

Source: VMO 2015

7/31/2016
For a,n∈Z+a,n\in\mathbb{Z}^+, consider the following equation: a^2x+6ay+36z=n  (1) where x,y,z∈Nx,y,z\in\mathbb{N}.
a) Find all aa such that for all n≥250n\geq 250, (1)(1) always has natural roots (x,y,z)(x,y,z).
b) Given that a>1a>1 and gcd⁡(a,6)=1\gcd (a,6)=1. Find the greatest value of nn in terms of aa such that (1)(1) doesn't have natural root (x,y,z)(x,y,z).
number theory