# 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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### How can i find minimum value of this inequality

$a_1,a_2,...,a_n$ are 8 distinct positive integers. $b_1,b_2,...,b_n$ are another 8 distinct positive integers ($a_i,b_j$ are not necessarily y distinct for $i, j = 1, 2, ...8$).Enter the smallest ...
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### To prove the inequality of positive rational numbers

Show that: $$\left(\frac{a+b}{a+b+c}\right)^{c} \left(\frac{b+c}{a+b+c}\right)^{a} \left(\frac{a+c}{a+b+c}\right)^{b}< \left(\frac{2}{3}\right)^{a+b+c} ,a\ne b\ne c$$ PS: I am supposed to use ...
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### Is there a solution for $(\frac{3}{x+y+z})^n+(\frac{3}{x+y+z})^{5-n}<2$?

Is there a solution for $$\left(\frac{3}{x+y+z}\right)^n+\left(\frac{3}{x+y+z}\right)^{5-n}<2$$ where $n\in\mathbb Z$ and $x,y,z>0, xyz=1$. This is the part of my attempts for my homework, that ...
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### How would I prove the following using the AGM inequality?

Question 17. Let $x,y\in\mathbb R$, $x,y\geq0$. Prove that $$(\sqrt x+\sqrt y)^2\geq2\sqrt{2(x+y)\sqrt{xy}}.$$ I believe I have to use AGM multiple times, but I am not exactly sure how
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### To prove the relation between n numbers and their AM - GM [duplicate]

Given $A$ and $G$ to be the arithmetic and geometric mean of n positive real numbers $a_1, a_2,...,a_n$ then for any $k > 0$ show that $$(k+A)^n \ge\ (k+a_1)...(k+a_n) \ge\ (k+G)^n .$$ I started ...
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