  We present the North Galactic Cap sample of the Extremely Luminous Quasar
Survey (ELQS-N), which targets quasars with $M_{1450}<-27$ at $2.8 \leq z < 5$
in an area of $\sim7600\,\rm{deg}^2$ of the Sloan Digital Sky Survey (SDSS)
footprint with $90\text{\textdegree}<\rm{RA}<270\text{\textdegree}$. Based on a
near-infrared/infrared \textit{JKW2} color cut, the ELQS selection efficiently
uses random forest methods to classify quasars and to estimate photometric
redshifts; this scheme overcomes some of the difficulties of pure optical
quasar selection at $z\approx3$. As a result, we retain a completeness of
$>70\%$ over $z\sim3.0-5.0$ at $m_{i}\lesssim17.5$, limited toward fainter
magnitudes by the depth of the Two Micron All Sky Survey (2MASS). The presented
quasar catalog consists of a total of 270 objects, of which 39 are newly
identified in this work with spectroscopy obtained at the Vatican Advanced
Technology Telescope and the MMT $6.5\,\rm{m}$ telescope. In addition to the
high completeness, which allowed us to discover new quasars in the already
well-surveyed SDSS North Galactic Cap, the efficiency of our selection is
relatively high at $\sim79\%$. Using 120 objects of this quasar sample we are
able to extend the previously measured optical quasar luminosity function (QLF)
by one magnitude toward the bright end at $2.8 \leq z \leq 4.5$. A first
analysis of the QLF suggests a relatively steep bright-end slope of
$\beta\approx-4$ for this sample. This result contrasts with previous results
in the same redshift range, which find a much flatter slope around
$\beta\sim-2.5$, but agrees with recent measurements of the bright-end slope at
lower and higher redshifts. Our results constrain the bright-end slope at
$z=2.8-4.5$ to $\beta<-2.94$ with a 99\% confidence.
