Narayanganj Higher Secondary 2014 P9
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
 samiul_samin
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Narayanganj Higher Secondary 2014 P9
At least how many numbers are needed to be taken to be sure that there are at least $11$ numbers among these numbers where the difference between any two is divisible by $7$?

 Posts: 62
 Joined: Sun Mar 30, 2014 10:40 pm
Re: Narayanganj Higher Secondary 2014 P9
At least $8$ numbers are needed to be taken to ensure that the difference between at least two numbers is divisible by $7$.
Now, if we take one more number, the difference between this and one of the "original" $8$ numbers is divisible by $7$.
Going on, we can see that at least $17$ numbers are required to be taken to ensure that there are at least $11$ numbers among them so that the difference between any two of those $11$ numbers is divisible by $7$.
Now, if we take one more number, the difference between this and one of the "original" $8$ numbers is divisible by $7$.
Going on, we can see that at least $17$ numbers are required to be taken to ensure that there are at least $11$ numbers among them so that the difference between any two of those $11$ numbers is divisible by $7$.

 Posts: 62
 Joined: Sun Mar 30, 2014 10:40 pm
Re: Narayanganj Higher Secondary 2014 P9
There is a much more formal solution to this problem using Modular Arithmetic.
But I'm too lazy to type that long a solution!
But I'm too lazy to type that long a solution!