MathDB

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Part of 2016 Azerbaijan JBMO TST

Problems(2)

inequalities

Source:

4/24/2023
Let a,b,c≥0a,b,c \ge 0 and a+b+c=3a+b+c=3. Prove that: 2(ab+bc+ca)−3abc≥∑cycab2+c222(ab+bc+ca)-3abc\ge \sum_{cyc}^{}a\sqrt{\frac{b^2+c^2}{2}}
Inequalitesinequalities
inequality involving triangle

Source: Romanian Dristict Olympiad 2015, Grade VIII, Problem 1

9/25/2018
If a,b,c a,b,c represent the lengths of the sides of a triangle, prove the inequality: 3≤∑cyca−a+b+c. 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} .
inequalities