Using a sample of $447.9times 10^6$ $psi(3686)$ events collected with the BESIII detector in 2009 and 2012, we study the decays $chi_{c0,2}to$ $etaeta$ and $etaeta$. The decays $chi_{c2}toetaeta$, $chi_{c0}toetaeta$ and $chi_{c2}toetaeta$ are observed for the first time with statistical significances of $9.6sigma$, $13.4sigma$ and $7.5sigma$, respectively. The branching fractions are determined to be $mathcal{B}(chi_{c0}toetaeta) =(2.19pm0.03pm0.14)times10^{-3}$, $mathcal{B}(chi_{c2}toetaeta) = (4.76pm0.56pm0.38)times10^{-5}$, $mathcal{B}(chi_{c0}toetaeta) = (8.92pm0.84pm0.65)times10^{-5}$ and $mathcal{B}(chi_{c2}toetaeta) = (2.27pm0.43pm0.25)times10^{-5}$, where the first uncertainties are statistical and the second systematic. The precision for the measurement of $mathcal{B}(chi_{c0}toetaeta)$ is significantly improved compared to previous measurements. Based on the measured branching fractions, the role played by the doubly and singly Okubo-Zweig-Iizuka disconnected transition amplitudes for $chi_{c0,2}$ decays into pseudoscalar meson pairs can be clarified.
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