Olympiad problems

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Found problems: 4

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2006 Hanoi Open Mathematics Competitions, 9

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1$.Find the largest posible value of $$|x^3+y^3+z^3-xyz|$$

2009 Irish Math Olympiad, 5

Tags: inequalities , inequalities open

Hello. Suppose $a$, $b$, $c$ are real numbers such that $a+b+c = 0$ and $a^{2}+b^{2}+c^{2} = 1$. Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$ and determine the cases of equality.

2009 Hanoi Open Mathematics Competitions, 7

Tags: inequalities open , inequalities

Let $a,b,c,d$ be positive integers such that $a+b+c+d=99$. Find the maximum and minimum of product $abcd$

1980 Bulgaria National Olympiad, Problem 4

Tags: inequalities , inequalities open , Inequality

Let $a $, $b $, and $c $ be non-negative reals. Prove that $a^3+b^3+c^3+6abc\ge \frac{(a+b+c)^3}{4} $.

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