2 people start on opposite sides of a circular track. At the gun, they chase each other. Whoever catches the other person wins. A race could last from 2 minutes to days. Different divisions would be 50m, 100m, 500m, 1000m length of track. I imagine the strategies for this would be really interesting. Also lots of them would die of exhaustion.
Basketball, but nobody over 6 feet is allowed to play. Some stats: Michael Jordan was 6'6". What percent of people are that tall? less than 5%. That means that as far as his non-height abilities, there are at least 20 people who are as good or better than he was at shooting, speed, strategy, fakes, dribbling, in the clutch, etc. Plus, without 7 foot mutants hanging around, there would be some incredible passing and shooting.
Convert american football, basketball etc to soccer style time. For example in basketball just make one quarter last 20 minutes. The clock never stops except maybe on a technical foul. The only time anyone wastes in the NBA is when they're shooting free throws, and commercial breaks. If someone wastes time, the ref could just give him a technical foul. This would stop that whole end of the game stupidity of constantly fouling, since a foul would take longer than a normal play would, and you'd still be giving up the option for 2 points. Football would be almost the same, except it would suck slightly less.
Create a scale of how much help a bullfighter is allowed. At the easiest it'd be set up like now, with armored horses, tons of helpers whenever he wants, not even having to face the bull until it's bleeding everywhere. At hardest it's be bull vs man with sword. At the end of every season, if the bull has won less than 5% of the time, make the fight harder for the bullfighter in the next season. If the bull has won more than 5%, make it easier for the bullfighter. When a bull wins, it should be kept as a stud. All bulls should be descendants of the stud which won under the hardest conditions. Win = kill your opponent.
For the long jump, instead of there being a defined starting point, just have there be a 1m long takeoff area which is made of a material which shows steps. Long jumpers could start their jump from anywhere in this area, and the distance would be calculated from the place they actually started.
For sprints, instead of having a start, just let the runner run any distance, while filming them. After the run, choose an interval of the race distance from within their run and figure out how fast they did it. So they would be able to increase speed slowly for about 50m, then run the next 100m as fast as possible, then a bit more, and then stop. Using the video, the fastest internal interval would be chosen.
This could also allow competitions like the fastest 10m runner which would not be dominated by reaction time from the start.
You play a normal game like basketball, football, soccer etc, but without time. The game ends when one team leads by a certain amount. In basketball, it'd be 15 or so. In soccer, 1 or 2. This would stop stupid NBA games where they play even the whole game and it's just random who wins.
Play badminton with two birdies at once. It's not that hard if you can play badminton in the first place.
You play on a pool table. Two cueballs, no 8 ball. Simultaneously break, one person is solids, the other stripes. You can shoot whenever your cue ball stops. Scratching has no meaning. When one player is down to 1 ball, they call "UNO" and choose a pocket, market it with a post-it note. If their last ball goes in there, they win, if it goes in another pocket, they bring out 2 balls. Both players can be at uno simultaneously. Opposing player can try to win themselves or try to disrupt others win.
one person reads descriptions of movies from the tv guide. The other guesses the title.
Set up a tall platform and then a thin ledge on another building, slightly lower. It starts just 5 feet down.
Players jump off the building and have to catch themselves on the ledge. I don't think anyone would be able to catch themselves farther down than 10 feet.
Normal people chase each other in a warehouse area with varying head starts. People in movies are incredibly bad at hiding.
I think a 15 second head start is enough to get away and hide in a dark smoky warehouse area.