Recent data from the emph{Fermi} Large Area Telescope have revealed about a dozen distant hard-spectrum blazars that have very-high-energy (VHE; $gtrsim 100$ GeV) photons associated with them, but most of them have not yet been detected by imaging atmospheric Cherenkov telescopes. Most of these high-energy gamma-ray spectra, like those of other extreme high-frequency peaked BL Lac objects, can be well explained either by gamma rays emitted at the source or by cascades induced by ultra-high-energy cosmic rays, as we show specifically for KUV 00311$-$1938. We consider the prospects for detection of the VHE sources by the planned Cherenkov Telescope Array (CTA) and show how it can distinguish the two scenarios by measuring the integrated flux above $sim 500$ GeV (depending on source redshift) for several luminous sources with $z lesssim 1$ in the sample. Strong evidence for the origin of ultra-high-energy cosmic rays could be obtained from VHE observations with CTA. Depending on redshift, if the often quoted redshift of KUV 00311-1938 ($z = 0.61$) is believed, preliminary H.E.S.S. data favor cascades induced by ultra-high-energy cosmic rays. Accurate redshift measurements of hard-spectrum blazars are essential for this study.
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