Olympiad problems

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Found problems: 4

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2021 JBMO TST - Turkey, 4

Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$ Find the maximum value of the expression $$x^3+2y$$

2017 Iran MO (2nd Round), 4

Tags: algebra , 2-variable inequality , inequality , inequalities

Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that $$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$

2020 Australian Maths Olympiad, 1

Tags: algebra , inequality , 2-variable inequality

Determine all pairs $(a,b)$ of non-negative integers such that $$ \frac{a+b}{2}-\sqrt{ab}=1.$$

1968 IMO Shortlist, 12

Tags: inequalities , function , induction , inequality , 2-variable inequality , algebra

If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that \[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]

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