# Questions tagged [a.m.-g.m.-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.

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### Minimum value of expression $\displaystyle \sqrt{16b^4+(b-33)^2}$

Finding point $P(a,b)$ on parabola $x=4y^2$ whose distance from the point $Q(0,33)$ is minimum and also find that minimum distance What I try : Let coordinate of point $P$ be $(4b^2,b)$ because point ...
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### How to prove this inequality $\sum_{r=1}^{n}a_{r}\sqrt{\frac{n-1}{1-a_{r}}}\ge\sum_{r=1}^{n}\sqrt{a_{r}}$

Let $({a_{r}})_{r=1}^{n}$ be a sequence of $n$ positive real numbers that sum to 1. Prove that for all $n>1$ :\begin{align} \sum_{r=1}^{n}a_{r}\sqrt{\frac{n-1}{1-a_{r}}}&\ge\sum_{r=1}^{n}\sqrt{...
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### Seeking Clarification on an Inequality Involving Roots and Products

Question If $a_j\gt 0$, $j=1,\ldots,m$ and $x_i\ge 0$ $i=1,\ldots,n.\;$ then prove that \begin{align} \sqrt[m]{\prod_{j=1}^{m}\left[a_j+\frac{1}{n}\sum_{i=1}^{n}x_i\right]}&\ge\frac{1}{n}\sum_{i=1}...
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### If $a,b,c > 0$ prove $\frac{(b+c-a)^2}{a}+\frac{(a+c-b)^2}{b}+\frac{(a+b-c)^2}{c}\ge\frac{b^2+c^2}{b+c}+\frac{a^2+c^2}{a+c}+\frac{a^2+b^2}{a+b}$

$a,b,c > 0$ $$\frac{(b+c-a)^2}{a}+\frac{(a+c-b)^2}{b}+\frac{(a+b-c)^2}{c}\ge\frac{b^2+c^2}{b+c}+\frac{a^2+c^2}{a+c}+\frac{a^2+b^2}{a+b}$$ I have used Tittu here with the second and the third ...
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### Prove $\frac{a+b}{c} + \frac{b+c}{a}+\frac{a+c}{b} + 6 \ge 2\sqrt2(\sqrt\frac{1-a}{a} + \sqrt\frac{1-b}{b} + \sqrt\frac{1-c}{c})$ when $a + b + c = 1$
$$\frac{a+b}{c} + \frac{b+c}{a}+\frac{a+c}{b} + 6 \ge 2\sqrt2(\sqrt\frac{1-a}{a} + \sqrt\frac{1-b}{b} + \sqrt\frac{1-c}{c})$$ The condition set for this is : $a + b + c = 1$ From that condition I ...