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We have discovered spectral signs of a strong simultaneous sublimation activity of main-belt primitive asteroids 779 Nina, 704 Interamnia, and 145 Adeona at their passages of perihelion in 2012. The main features is unusual shape of their reflectance spectra with a maximum near 0.50 μm as a manifestation of reflected light scattering in a coma of micron-size icy dust particles. The periodic nature of the process on the asteroids is confirmed at their next passages of perihelion in 2016-2018 with the same spectral and UBVRI-photometric methods. Due to eccentricity of the orbits, the asteroids undergo changes of subsolar temperatures with revolution around the Sun (~222÷257 K on Adeona, ~206÷240 K on Interamnia, and ~208÷262 K on Nina) and elevated water ice content in the matter is probably a source of the sublimation activity. We suggest that it may be a widespread phenomenon among primitive asteroids and is explained by their common origin. However, it contradicts previous notions that primitive asteroids include water only in a bound state. To verify the suggestion, as a first step we performed numeric modeling of migration of planetesimals from the outer Solar system and estimate probability of their collisions with asteroid parent bodies and delivery of ice material to main asteroid belt.