MathDB

12.1

Part of VMEO IV 2015

Problems(2)

(a - b)/(a + b) is rational if a^3 + b^3 + c^3 - 3abc is rational

Source: 2015 VMEO IV Juniors 12.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/15/2021
Given a set S⊂R+S \subset R^+, S≠∅S \ne \emptyset such that for all a,b,c∈Sa, b, c \in S (not necessarily distinct) then a3+b3+c3−3abca^3 + b^3 + c^3 - 3abc is rational number. Prove that for all a,b∈Sa, b \in S then a−ba+b\frac{a - b}{a + b} is also rational.
rationalalgebra
a^2+b^2+c^2-ab-bc-ca>= k |(a^3-b^3)/(a+b)+(b^3-c^3)/(b+c)+({c^3-a^3)/(c+a)

Source: 2015 VMEO IV Seniors 12.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/19/2021
Find the largest constant kk such that the inequality a2+b2+c2−ab−bc−ca≥k∣a3−b3a+b+b3−c3b+c+c3−a3c+a∣a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right| holds for any for non negative real numbers a,b,ca,b,c with (a+b)(b+c)(c+a)>0(a+b)(b+c)(c+a)>0.
algebrainequalities