The few tens of hypervelocity stars (HVSs) are the only stars moving more rapidly than the escape speed of our Galaxy. The conventional explanation is the disruption of a binary star by the black hole at the Milky Way's centre, which ejects one star at high speed (``Hills mechanism''). An alternative is the ejection of a ``runaway'' star from a binary system by the supernova explosion of the companion. These extreme scenarios make HVSs sensitive probes of exotic physics. If the HVSs come from our Galaxy's centre they should be evenly distributed across the sky, however 80% are near the constellation of Leo. We can explain the HVSs near Leo as the tip of a leading arm of thousands of HVSs ejected from the Large Magellanic Cloud, a galaxy passing the Milky Way at 378 km/s. Both the Hills mechanism and the runaway scenario can kick a star from this galaxy fast enough to make it an HVS. This leading arm will be discoverable with the European Space Agency Gaia space telescope in 2018.