  One of the most interesting results from the OGLE-III study of the LMC
Cepheids is the large number of objects that pulsate simultaneously in the
first and second overtone (denoted 1O/2O). Double-mode Cepheids yield important
constraint on stellar evolution models. We show that great majority of the LMC
1O/2O Cepheids have masses M=3.0+/-0.5 Msun. According to current stellar
evolution calculations, these masses are lower than needed for the blue loop in
the helium burning phase to reach the instability strip. On the other hand, we
found most of these stars significantly overluminous if they are crossing the
instability before helium ignition. A possible solution of this discrepancy is
to allow for a large overshooting from the convective core in the main sequence
phase. We also discuss origin of double-mode pulsation. At the short period
range we find two types of resonances that are conducive to this form of
pulsation. However, at longer periods, it has a different (non-resonant)
origin.
