I see a lot of people here like to brag about their supposed intelligence and I want to put that to the test.
I have nothing fancy, nothing too niche or specific so that you don't need too much previous training to start. It is a problem completely about geometry and number theory, the literal pillars of elementary mathematics which means that if you can't solve this then you are not even close to being a genius. Lets go:
Lets set the stage. Consider the first quadrant of the plane (the one with only positive numbers) and consider just the integer points. Like (1,2) and (343,52) for example.
Definition 1: We say that a point is invisible if when you draw the line that goes from the origin to that point, that line crosses through at least another integer point.
Quick example: If you draw a line between the origin and the point (2,2) then you inevitably have to go through (1,1).
Theorem 1 (A little help you could use): If the numbers A and B are relatively prime then the point (A,B) is not invisible. And if a point (A,B) is not invisible then A and B are relatively prime.
Definition 2: We say that the point (A,B) forms an invisible square of length N if for each integer [math]i,j \in \{0,1,2,...,N \} [/math], the point (A+i,B+j) is invisible.
Quck example: The point (1308,1274) forms an invisible square of length 2. You can check for yourself that (1308,1274) is invisible, (1309,1274) is invisible, (1310,1274) is invisible, etc. if you want.
[math] \text{THE PROBLEM:} [/math]
If (A,B) is a point that forms an invisible square of length N, give your best lower bound for how big [math] min\{A,B\} [/math] has to be compared to N.
Easy example:
You can easily prove that [math] min\{A,B\} [/math] has to be at least bigger than N, by Bertrand's Postulate.
But you can give a much better lower bound. Go.
