KöMaL 2021 C1648

King Arthur and Sir Lancelot are running a horse race. Sir Lancelot says, ``Since Your Majesty's speed is only $\frac{2}{3}$ of mine, I will give Your Majesty a handicap of $100$ metres, and then I would catch up within the length of the race track. Alternatively, if Your Majesty reduced speed by $2 \frac{\mathrm{m}}{\mathrm{s}}$ and I reduced mine by $5 \frac{\mathrm{m}}{\mathrm{s}}$, but I gave Your Majesty a $50$ metre handicap only, I would also be able to catch up within the length of the track. The sum of the two time intervals required to catch up is exactly $75$ seconds.'' Determine the speeds of King Arthur and of Sir Lancelot.