This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 4

2020 Australian Maths Olympiad, 1
Determine all pairs $(a,b)$ of non-negative integers such that $$ \frac{a+b}{2}-\sqrt{ab}=1.$$
inequality 2-variable inequality algebra inequalities

1968 IMO Shortlist, 12
If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that \[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]
function inequalities induction inequality 2-variable inequality algebra AM-GM inequality Jensen's inequality mathematical induction

2017 Iran MO (2nd Round), 4
Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that $$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$
inequalities inequality 2-variable inequality algebra AM-GM inequality factorization

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 $$x^3+2y$$
inequalities 2-variable inequality algebra AM-GM inequality