  Very high-energy gamma-ray measurements of distant blazars can be well
explained by secondary gamma rays emitted by cascades induced by
ultra-high-energy cosmic rays. The secondary gamma rays will enable one to
detect a large number of blazars with future ground based gamma-ray telescopes
such as Cherenkov Telescope Array (CTA). We show that the secondary emission
process will allow CTA to detect 100, 130, 150, 87, and 8 blazars above 30 GeV,
100 GeV, 300 GeV, 1 TeV, and 10 TeV, respectively, up to $z\sim8$ assuming the
intergalactic magnetic field (IGMF) strength $B=10^{-17}$ G and an unbiased all
sky survey with 0.5 hr exposure at each Field of View, where total observing
time is $\sim540$ hr. These numbers will be 79, 96, 110, 63, and 6 up to
$z\sim5$ in the case of $B=10^{-15}$ G. This large statistics of sources will
be a clear evidence of the secondary gamma-ray scenarios and a new key to
studying the IGMF statistically. We also find that a wider and shallower survey
is favored to detect more and higher redshift sources even if we take into
account secondary gamma rays.
