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: 2

2016 Korea USCM, 7
Tags: holder inequality , integral , inequalities , real analysis , function
$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that $$\int_0^\infty (1+x)f(x) dx<\infty$$ Then, prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$ (@below, Thank you. I fixed.)

2016 India Regional Mathematical Olympiad, 5
Tags: algebra , inequalities , holder inequality , symmetric inequality , am-gm
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$