Algebra

Classical Inequalities

Classical Inequality Statements: Level 2 Challenges

         

ab+bc+cd+da\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}

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

Let xx be a positive real number. Find the minimum value of 8x5+10x4.8x^{5}+10x^{-4}.

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

1+ab1+a+1+bc1+b+1+ca1+c.\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>0a,b,c>0 and a+b+c=6a+b+c=6. Find the minimum value of

(a+1b)2+(b+1c)2+(c+1a)2 \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 a2+b2\sqrt{a^2+b^2} when 3a+4b=153a+4b=15.

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