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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Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

Show that for any positive real number $x$ the inequality holds: $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$ where by $[a], \{a\}$ we mean the whole par and fractional ...
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Maximize $f(x)=(1-x)^5(1+x)(1+2x)^2$

For which value of $x$ is the product $(1-x)^5(1+x)(1+2x)^2$ a maximum, and what is this value? This is easy with calculus, but how would you do it without calculus? $f(x)=(1-x)^5(1+x)(1+2x)^2 \geq 0$ ...
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Regarding the specifity of the A.M.-G.M. inequality in finding maximum and minimum in Number Theory

Today, my math teacher solved a problem which asked to find the maximum value of the expression $x^2y^3$ when $x$ and $y$ are related as $3x+4y=5$. It was solved using the classic A.M.-G.M. inequality ...
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Finding all $x, y$ natural numbers

Find all natural numbers $x$ and $y$ such that $$\frac{\sqrt{x} + \sqrt{y}}{\sqrt[3]{x^2 - y^2}} \in \mathbb{N}.$$ My approach: $x>y$ because otherwise the radical would be negative and we can ...
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How to check that the arithmetic mean of any subset $S\ne\{1\}$ of $A={1,2,...,n}$ is at least $\frac 32$?

How to check that the arithmetic mean of any subset $S\ne\{1\}$ of $A=\{1,2,...,n\}$ is at least $\frac 32$? I don't have any idea where to start this. It is clear that this is right and I need to ...
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Prove $\sum\limits_{\mathrm{cyc}}\frac{a}{a^2+bc+4} \leq \frac{1}{2}$ for $a,b,c>0$ with $a+b+c=6$

Prove that for $a, b, c > 0$ where $a + b + c = 6$, the following inequality holds: $$\frac{a}{a^2+bc+4} + \frac{b}{b^2+ca+4} + \frac{c}{c^2+ab+4} \leq \frac{1}{2}.$$ From the AM-GM inequality, ...
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Proving Inequality Involving Sums and Products of Variables in a Probabilistic Analysis

I'm exploring a probabilistic analysis problem where I have variables (or, probabilities) $x_1,\ldots, x_n \in (0,1)$ in the range $(0,1)$ satisfying $\sum^n_{i=1} x_i \in (0,1)$. I aim to prove the ...
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This inequality is proposed by TATA box. Prove that for ${\forall}a,b,c \geq 0$ such that $ab+bc+ca=2$, prove the following inequality. $$\sum_{cyc}a^2 + abc \geq \frac{3}{8}\sum_{cyc}a^3 b +2$$
Prove that $a^2(b^2+4)+b^2(c^2+4)+c^2(a^2+4) \geq 15$
Let $$a, b, c$$ be real numbers with the property that $$a+b+c=3$$. Prove that $a^2(b^2+4)+b^2(c^2+4)+c^2(a^2+4) \geq 15$ Initially, I thought to use Cauchy-Schwarz Inequality and simplify. \$(a^2+b^...