MathDB

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Part of 2021 Austrian MO National Competition

Problems(2)

Simple inequality

Source: Austria Mathematical Olympiad 2021？

5/31/2021
Let a,b,c≥0a,b,c\geq 0 and a+b+c=1.a+b+c=1. Prove thata2a+1+b3b+1+c6c+1≤12.\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}. (Marian Dinca)
inequalitiesalgebra
(a^3 + b^3 + c^3)/3 >= abc + a + b + c for pairwise different naturals

Source: 2021 Austrian Federal Competition For Advanced Students, Part 2, p1

6/6/2021
Let a,ba, b and cc be pairwise different natural numbers. Prove a3+b3+c33≥abc+a+b+c\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c. When does equality holds?
(Karl Czakler)
inequalitiesalgebraAZE CMO TSTAZE EGMO TST