Ubiquitous asymmetric multi-core processors such as ARM big.LITTLE combine together cores with different power-performance characteristics on a single chip. Upcoming asymmetric many-core processors are expected to combine hundreds of cores belonging to different types. However, the accompanying task-to-core mapping schedules are the key to achieving the full potential of such processors. Run-time scheduling on asymmetric processors is a much harder problem to solve optimally than scheduling on symmetric processors with equivalent cores. We present the first-ever greedy scheduler to be proven theoretically optimal (under certain constraints) for asymmetric processors. The proposed scheduler, called A-Greedy, improves throughput by 26% and reduces average response time by up to 45% when compared to the default Linux scheduler on ARM big.LITTLE asymmetric multi-core.