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We construct a forward-in-time self-similar solution of the two-dimensional incompressible Euler equations, \\[ u(t,x)=t^{-\\frac{a}{1+a}} U\\left(\\frac{x}{t^{{\\frac{1}{1+a}}}}\\right), \\] with initial datum $u_0$. No smallness or sign assumption is imposed on the initial datum. The construction is a vanishing-dissipation limit of the hypodissipative self-similar profiles. 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