We present the results of twenty-year observations of a complete sample of 68 flat-spectrum radio sources with flux densities S3.9GHz>200mJy carried out at centimeter wavelengths with the RATAN-600 radio telescope. Since 1995, we have observed simultaneously at six frequencies between 0.97 and 21.7GHz. Of the 56 sources identified with optical objects, 41 are quasars with redshifts between 0.293 and 3.263. Based on our analysis of the spectral shapes, we divide the sources into four classes. Changes of spectral class for individual sources are fairly rare. Based on the light curves and spectra, in most cases,a flare's evolution is in accordance with a model in which the variations result from the evolution of a shock in the radio jet. The main result of our study is that there is no redshift dependence for the true linear sizes of the radiating regions, the variability indices derived for all 20 years of data or for individual flares, or the peak frequencies of the spectra of the compact radio emission. We suggest that this testifies to an absence of cosmological evolution of the sample quasars, at least to z~3.