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A graph $G$ is $(\\varepsilon,p,k,\\ell)$-pseudorandom if for all disjoint $X$ and $Y\\subset V(G)$ with $|X|\\ge\\varepsilon p^kn$ and $|Y|\\ge\\varepsilon p^\\ell n$ we have $e(X,Y)=(1\\pm\\varepsilon)p|X||Y|$. We prove that for all $\\beta>0$ there is an $\\varepsilon>0$ such that an $(\\varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $\\beta pn$ contains the square of a Hamilton cycle. 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