We have developed a model predicting whether or not the magnetopause crosses geosynchronous orbit at a given location for given solar wind pressure Psw, Bz component of the interplanetary magnetic field (IMF), and geomagnetic conditions characterized by 1 min SYM-H index. The model is based on more than 300 geosynchronous magnetopause crossings (GMCs) and about 6000 min when geosynchronous satellites of GOES and Los Alamos National Laboratory (LANL) series are located in the magnetosheath (so-called MSh intervals) in 1994–2001. Minimizing of the Psw required for GMCs and MSh intervals at various locations, Bz, and SYM-H allows describing both an effect of magnetopause dawn-dusk asymmetry and saturation of Bz influence for very large southward IMF. The asymmetry is strong for large negative Bz and almost disappears when Bz is positive. We found that the larger the amplitude of negative SYM-H, the lower the solar wind pressure required for GMCs. We attribute this effect to a depletion of the dayside magnetic field by a storm time intensification of the cross-tail current. It is also found that the magnitude of threshold for Bz saturation increases with SYM-H index such that for small negative and positive SYM-H the effect of saturation diminishes. This supports an idea that enhanced thermal pressure of the magnetospheric plasma and ring current particles during magnetic storms results in the saturation of magnetic effect of the IMF Bz at the dayside magnetopause. A noticeable advantage of the model's prediction capabilities in comparison with other magnetopause models makes the model useful for space weather predictions.