MathDB

3

Part of 2014 Federal Competition For Advanced Students, P2

Problems(1)

triangle inequality =>a^n + b^n> c^n, b^n + c^n> a^n and a^n + c^n> b^n

Source: 45th Austrian Mathematical Olympiad National Competition (Final Round, part 2) 28 May 2014 p3

1/11/2020
(i) For which triangles with side lengths a,ba, b and cc apply besides the triangle inequalities a+b>c,b+c>aa + b> c, b + c> a and c+a>bc + a> b also the inequalities a2+b2>c2,b2+c2>a2a^2 + b^2> c^2, b^2 + c^2> a^2 and a2+c2>b2a^2 + c^2> b^2 ?
(ii) For which triangles with side lengths a,ba, b and cc apply besides the triangle inequalities a+b>c,b+c>aa + b> c, b + c> a and c+a>bc + a> b also for all positive natural nn the inequalities an+bn>cn,bn+cn>ana^n + b^n> c^n, b^n + c^n> a^n and an+cn>bna^n + c^n> b^n ?
inequalitiestriangle inequalitygeometric inequalityexponential