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The author obtains the classification of all invariant Einstein metrics on the following homogeneous spaces: $SU\left(3\right)/T_{max}$, $Sp\left(3\right)/Sp\left(1\right)\times Sp\left(1\right)\times Sp\left(1\right)$, $F_4/\text{spin}\left(8\right)$, $SU\left(5\right)/Sp\left(2\right)\times T^1$. Combining this with the results of other authors, the classification of all invariant Einstein metrics on all compact simply connected homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature is obtained.