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## Classical Inequalities

Your destination for questions of the form "do we know if this expression is always greater than this other expression?" Explore what humans know about mathematical inequalities.

# Level 2

$\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$

If $$a, b, c$$ and $$d$$ are any four positive real numbers, then find the minimum value of the expression above.

Let x be a positive real number. Find the minimum value of $8x^{5}+10x^{-4}.$

Positive reals $$a,b,c$$ satisfy $$abc=1$$. Find the minimum value of

$\dfrac{1+ab}{1+a}+\dfrac{1+bc}{1+b}+\dfrac{1+ca}{1+c}.$

Enter your answer correct to $$3$$ decimal places.

For $$a,b,c>0$$ and $$a+b+c=6$$. Find the minimum value of

$\large \left (a+\frac{1}{b} \right )^{2}+ \left (b+\frac{1}{c} \right )^{2}+\left (c+\frac{1}{a} \right )^{2}$

Find the minimum value of $$\sqrt{a^2+b^2}$$ when $$3a+4b=15$$.

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