$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$Prove $sumlimits_i=1^nfraca_i^2b_i geq frac(sumlimits_i=1^na_i)^2sumlimits_i=1^nb_i$Harmonic mean: show $maxax,by ge frac1a+b(x+y)$, $a,b>1$, $x,yge 0$Proof of analogue of the Cauchy-Schwarz inequality for random variablesOlympiad inequality $fraca2a + b + fracb2b + c + fracc2c + a leq 1$.Prove Inequality with Cauchy SchwarzShow that $frac1x_1(x_1+1)+frac1x_2(x_2+1)+cdots+frac1x_n(x_n+1)gefracn2$ with $x_1x_2cdots x_n=1$Estimating the product of timelike vectors in $Bbb R^n_nu$Showing that two vectors are not orthogonal, using Cauchy-Schwarz$AM$- $QM$ inequalityCauchy-Schwarz inequality to prove $A$ is spd$x,yinBbb R,$ find the maxima of $fracx+2y+3sqrtx^2+y^2+1$
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