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This article concerns collisionless gas flows over a flat cryogenic pump plate with a specific sticking probability. It presents exact density, velocity, temperature, and pressure solutions which are useful in determining the plate sticking probability. At any point off the plate, the local velocity distribution function (VDF) consists of several pieces of Maxwellian VDFs, one is characterized by free stream flow properties and the others by plate surface properties. Integrating the VDFs with different moments leads to these exact solutions, which are further validated by simulations with the direct simulation Monte Carlo (DSMC) method. We use these solutions to derive several useful expressions, based on special measured flow field properties at specific locations off the plate, to evaluate the pump sticking probability. Also, the exact solutions complement past studies of aerodynamic coefficients and heat transfer rate for collisionless gas flowing over a flat plate. They explicitly express the physical and geometrical factors. Evaluations of those solutions are much faster than DSMC simulations. This study can find other useful applications, e.g., cold gas spray for thin film depositions inside a vacuum chamber, and spacecraft plume impingement estimations.