Minimum Number of Races to win the Formula 1 World Championship There are a few complications in the allocation of Formula 1 points through the season, such as half points for an incomplete race, and the fact that a team can change drivers mid-way through a season.
However, if we assume the same 20 drivers all start the 19 races; that at least ten drivers finish each race; that all races are completed; and points are awarded thus:


*

*1st : 25 points 2nd : 18 points 3rd : 15 points 4th : 12 points 5th :
10 points 6th : 8 points 7th : 6 points 8th : 4 points 9th : 2 points
10th : 1 point


... what is the minimum theoretical number of races that have to be run before we can say that an individual could be champion by that point. That is, that their points lead is unassailable.
The answer (I think) must be at least ten: until you've won half, there's no way you can guarantee being champion. But (given our constraints) other drivers must have been awarded some points by that point. 
It's at this point that I'm at a loss to work out what permutation of results minimises the points for second and below, and hence whether winning the championship in ten (or eleven, or twelve) races is a mathematical possibility.
 A: Andre covered the guaranteed case no matter what. if you are asking for the best case scenario (i.e. if the results in the first races are ideal for you) the answer is 11.
After 10 there cannot be any winner no matter what 
The most a driver can have after 10 is 250. 
The remaining drivers win on average 4 points per race (76 points for places 2-10, divided by 19 racers). This means that one average a driver will have 40 points after 10 races. As a racer can get another 225 points in the remaining 9 races, 250 are not enough.
After 11 there can be an winner under ideal circumstances 
If a driver wins the first 11 they have 275 points.
If every other driver has under 74 points, they can get at most 274 points. 
As the average of the points obtained by the 19 remaining drivers for the first 11 races is 44, is very easy to distribute the places 2-10 in the first 11 races in such a way that no one has over 74 points. If you want an actual example, I can create such a list. 
The list


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*Driver 1: 11 first place finishes, 275 points 

*Drivers 2-3: 4 second place finishes, no other point, 72 points. 

*Driver 4: 3 second place finishes, no other point, 54 points. 

*Drivers 5-6: 4 third place finishes, no other point, 60 points. 

*Driver 7: 3 third place finishes, no other point, 45 points. 

*Driver 8: 6 fourth place finishes, no other point, 72 points. 

*Driver 9: 5 fourth place finishes, no other point, 60 points. 

*Driver 10: 6 fifth place finishes, no other point, 60 points. 

*Driver 11: 5 fifth place finishes, no other point, 50 points. 

*Driver 12: 6 sixth place finishes, no other point, 48 points. 

*Driver 13: 5 sixth place finishes, no other point, 40 points. 

*Driver 14: 11 seventh place finishes, no other point, 66 points. 

*Driver 15: 11 eight place finishes, no other point, 44 points. 

*Driver 16: 11 ninth place finishes, no other point, 22 points. 

*Driver 17: 11 tenth place finishes, no other point, 11 points.


The list if far from ideal, if you want one ideal, you just move points from the last drivers to the ones at the top to make as many drivers as possible with 74 points.
A: I believe the earliest a season could ever be clinched is after 11 races.
11 wins gives the leader 275 pts.  There are 836 remaining pts from those first 11 races to be distributed among the other 19 drivers, so there is a lower limit of 44 pts for the next best cumulative score.  8 more races do not provide enough pts for 44 to catch 275.
Similarly, 10 wins gives the leader 250 pts and the lower limit for next in line would be 40.  9 more races offers a maximum of 225 pts for the chaser, for a total of 265.  Enough to pass the original leader if he or she was stuck on 250.  So 10 races isn't enough to create an insurmountable lead. 
A: If you win the first $15$, then you have $375$ points. The person in second place has at most $270$, and can get at most $100$ points in the last four races, so cannot catch up.
A similar calculation shows that winning the first $14$ is not enough to guarantee first place. For that produces $350$ points. The person in second place might have $252$, so could end with $377$.
A: I have an alternate solution to this, which I feel has a strong mathematical backing to it. I shall explain this below. If anything is wrong with my explanation, please let me know.
I am assuming that there are 20 races in the season, 20 drivers and no double points scheme. The standard points scheme 25-18-15-....-2-1 is followed.
Let's number the drivers from 1 to 20. Driver 1 wins all races, and is the title contender. He gets 25 points per race. There are other 19 drivers, and the points 18-15-12-...-4-2-1 must be equally distributed among them, so that no one driver is closer to driver 1.
So there is a set of 9 points per race, that must be equally distributed among 19 drivers. If we try to do some kind of summation followed by division, we end up with drivers having fractional points, which is not allowed. Hence, we should follow this technique. Drivers numbered 2-10 get points 18-1 in race 1. In race 2, 3-11 get points 18-1 and so on. When we hit the bottom, the points cyclicly wrap around. This image describes what I mean. It has drivers on rows and races as columns.
http://imgur.com/a/3hGh8
So I wrote a program that will do a summation and predict at which point the driver 1's lead is unattainable, and it showed 13. The program is here
http://ideone.com/aKpEWX
My predicted answer is 13. Please let me know is this is correct or not.
A: The minimum number of races for a guaranteed win is 16/20 when a 1st place is 25 points and 2nd is 18 points. 
For a guarantee we must assume that Driver 1 comes 1st until the guaranteed point and then gets zero points thereafter. And that driver 2 comes in 2nd place up until that point and then comes in 1st thereafter. 
If Driver 1 wins 15 races and thereafter gets zero points their total at season end is 375. With Driver 2 coming second up until that point and winning every race thereafter their total would be 395. 
If Driver 1 were to win 16 races and thereafter get zero points their total at season end would be 400. With Driver 2 coming second up until that point and winning every race thereafter their total would be 388. Therefore the minimum number of races a driver must win for a guaranteed championship win is 16. 
A: Take only two drivers.
Driver 1 wins minimum number of races for championship and no points for the others.
Driver 2 comes second each time driver one wins and wins the times driver one gets no points.

       Wins     Points(1st)     Points (2nd)    Total 

Driver 1    14      350                             350
Driver 2    5       125             252             377

        Wins    Points(1st)     Points (2nd)    Total 

Driver 1    15      375                             375
Driver 2    4       100             270             370
Minimum number of wins is 15 (375 points) without any possibility of being caught.
