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2023 Brazil EGMO Team Selection Test, 3

Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

2016 Spain Mathematical Olympiad, 6

Tags: jensen , esp , rearrangement inequality , inequalities

Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds: $$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$

2014 Chile TST IMO, 1

Tags: jensen , inequalities

Given positive real numbers $a$, $b$, and $c$ such that $a+b+c \leq \frac{3}{2}$, find the minimum of \[ a+b+c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}. \]

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