MathDB

10.3

Part of VMEO III 2006

Problems(2)

A nica inequality

Source: VMEO III

10/15/2015
Prove that for all non negative real numbers a,b,ca,b,c we have a2+b2+c2≤(b2−bc+c2)(c2−ca+a2)+(c2−ca+a2)(a2−ab+b2)+(a2−ab+b2)(b2−bx+c2)a^2+b^2+c^2\leq\sqrt{(b^2-bc+c^2)(c^2-ca+a^2)}+\sqrt{(c^2-ca+a^2)(a^2-ab+b^2)}+\sqrt{(a^2-ab+b^2)(b^2-bx+c^2)}
inequalities
f(x^2 + f(y) - y) = (f(x))^2

Source: 2006 VMEO III Seniors 10.3 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

9/17/2021
Find all functions f:R→Rf : R \to R that satisfy f(x2+f(y)−y)=(f(x))2f(x^2 + f(y) - y) = (f(x))^2 for all x,y∈Rx,y \in R.
algebrafunctional