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We consider the 1D compressible Euler equations of ideal gases, with nontrivial entropy. We introduce sets of initial data for which the unique locally well-posed classical solutions form shock singularities in finite time. For such initial data, the spacetime location of the first singularity is determined, and this first singularity is described as a cusp of order  . In particular, in the low-regularity setting we show that there exist solutions with  initial data which form cusps resembling −sgn (