Where does snow survive the summer? It depends on how cold the air is at each height, how much falls, and which way the slope faces.
The science
Air cools with height at the lapse rate Γ (about 6.5 K per km): T(z) = T0 − Γ z. The 0 °C level is therefore at z0 = T0 / Γ. Precipitation falls as snow when the air is cold (all snow below −1 °C, all rain above +2 °C in this model) and increases with height (orographic lift).
Snow melts by the degree-day method: melt = DDF × max(0, T) in mm of water per day, with a degree-day factor about 4 mm per °C per day. Sun-facing slopes get more radiation, so their factor is up to 35 % larger, shady slopes 35 % smaller. The snowline is the lowest altitude above which at least 30 mm of snow water equivalent (SWE) lies on the ground. Each of 40 height bands (100 m) and 5 aspect classes keeps its own SWE.
Simplifications (honest list): no wind redistribution, avalanches, refreezing, sublimation, snow density or albedo changes; aspect classes are interpolated from the slope direction of the terrain; the air temperature is a sea-level yearly cosine plus a random daily anomaly with a fixed lapse rate; the same random weather is applied to all heights; the mountain is a synthetic peak, not a real one (heights up to about 3700 m, drawn 1.4 times taller). Numbers are illustrative of the mechanisms, not a forecast.
3D mountain modelled in Blender with a Cycles-baked terrain texture; the snow layer is computed live from the model; Three.js r160.