  Cool core clusters of galaxies require strong feedback from their central AGN
to offset cooling. 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 central
AGN jet power 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. 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 with, on average,
Lkin/Lnuc exceeding 200. Explanations of this puzzle carry significant
implications for the origin and operation of jets, as well as on establishing
the importance of kinetic feedback for the evolution of galaxies and their
surrounding medium.
