# 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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### I want help knowing if my solution is correct

If $a, b, x, y$ are positive rational numbers such that $\frac 1x + \frac 1y = 1$ then prove that $\frac {a^{x}}{x}+ \frac {b^{y}}{y}$ $\ge ab$ This question is from Problems Plus in IIT Mathematics. ...
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### Given $A, B, C, D$ in $Oxyz$ space, find $M \in CD$ such that $MA + MB$ is smallest. Why can't I use AM-GM to solve this?

In the $Oxyz$ space, consider four points $A(-1, 1, 6),$ $B(-3,-2,-4),$ $C(1,2,-1),$ $D(2,-2,0).$ Find $M \in CD$ such that $△MAB$ has the smallest perimeter. As $AB$ is constant, the task is ...
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### Inequality $\forall a,b\in\mathbb{R}_{*}^{+}~~\text{then}~~ \dfrac{1}{a}+\dfrac{1}{b}+ab\geq 3$ [closed]

$\forall a,b\in\mathbb{R}_{*}^{+}\text{ then }\frac{1}{a}+\frac 1b+ab\geq 3$. Now this inequality: $a+\frac {1}{a}\geq 2$; $a+b\geq 2\sqrt{ab}$. But I can't use it!
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### A symmetric inequality involving product of three variables

Let $a, b,c \ge 0, ab + bc + ca + abc = 4$. Find the minimum of $S = \sqrt{a}+\sqrt{b}+\sqrt{c}$. My guess is that $S$ attends its minimum at $b = c = 2, a = 0$ and the other permutation of $(2, 2, 0)$...
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### Given $x_1^3+x_2^3+...+x_9^3=0$. Find the maximum value of $S=x_1+x_2+...+ x_9$.

Given 9 real numbers $x_1, x_2, ... , x_9\in [-1,1]$ such that $x_1^3+x_2^3+...+x_9^3=0$. Find the maximum value of $S=x_1+x_2+...+ x_9$. I have tried ordering the numbers from smallest to largest and ...
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### Finding Maximum Value using AM-GM Inequality

Let us have a set of natural numbers $S=\{x_1,x_2,...,x_n\}$ where $n≥4$, $n$ is even, such that all $(x_i\in S)≥0$ and $\sum_{i=1}^nx_i=1$.Find the maximum value of $\sum_{i=1}^{n-1}(x_i*x_{i+1})$.My ...
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### $\log_a 10 + \log_b 10 +\ log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$

Prove the following inequality for $a,b,c, \in (1,\infty)$ , such that $abc = 10$ $$\log_a 10 + \log_b 10 + \log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$$ I will transform logarithms to ...
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### $\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$

Prove the following inequality $$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$$ where $x,y,z \ge 0$ and $x+y+z=3$ I don't really know how to approach such an ...
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### Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$

the question Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$. the idea After i put some values to $a$, $b$, ...
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### Prove the inequality $\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right)$

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=2$. Prove the inequality: $\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right)$ I ...
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### Can anyone help prove or disprove that $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{N+1}$, where $x_i>0$
I was hoping the generalize this result: How to prove $x^{x/(1+x)}/(1+x)\geq1/2$ I believe that the following inequality holds: \$\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{...