  We present a study of strong cool core, highly-luminous (most with L_x >
10^(45) erg/s), clusters of galaxies in which the mean jet power of the central
active galactic nucleus must be very high yet no central point X-ray source is
detected. Using the unique spatial resolution of Chandra, a sample of 13
clusters is analysed, including A1835, A2204, and one of the most massive cool
core clusters, RXCJ1504.1-0248. All of the central galaxies host a radio
source, indicating an active nucleus, and no obvious X-ray point source. For
all clusters in the sample, the nucleus has an X-ray bolometric luminosity
below 2 per cent of that of the entire cluster. Most have a nucleus 2 - 10 keV
X-ray luminosity less than about 10^(42) erg/s. We investigate how these
clusters can have such strong X-ray luminosities, short radiative cooling-times
of the inner intracluster gas requiring strong energy feedback to
counterbalance that cooling, and yet have such radiatively-inefficient cores.
If the central black holes have masses ~10^9M_sol then the power exceeds one
per cent of the Eddington luminosity, and they are expected to be
radiatively-efficient. Only if they are ultramassive (M_BH > 10^(10)M_sol),
would their behaviour resemble that of lower mass accreting black holes. Our
study focuses on deriving the nucleus X-ray properties of the clusters as
defined in the above question, while briefly addressing possible solutions.
