Assuming that you have a TI-83/84...
Hit MODE, then change "FUNC" to "PAR", and "SEQUENTIAL" to "SIMUL".
Hit Y=, and then enter in your equations.
Hit WINDOW, then change the viewing dimensions to the following (these are my suggestions):
(I didn't mention the t-values; it's possible that you won't need to change them.)
Then hit GRAPH. Hopefully, you'll get a simulation.
I don't know if X1/Y1 wins the race or not, because on my calculator the graphs were rendered rather quickly. It might help if you figure out a set of parametric equations that will put a single, short line segment in the middle (at x = 50) to indicate the flag.
Don't forget to change the settings back in MODE when you're done.