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IMO #1 1959 (Hardest IMO question ever)
Posted:
Wed Apr 18, 2007 11:34 pmby stupidityismygam
Prove that the fraction [unparseable or potentially dangerous latex formula] is irreducible for every natural number [unparseable or potentially dangerous latex formula]
Re: IMO #1 1959 (Hardest IMO question ever)
Posted:
Thu Apr 19, 2007 2:25 amby makashi
Posted:
Thu Apr 19, 2007 3:50 pmby Dragon of Light
How did you come up with that solution....
I never figure out what motivates these things...
Posted:
Thu Apr 19, 2007 3:56 pmby stupidityismygam
well if it is irreducible then the gcd=1
if the gcd = 1, then there is a well known theorem that states:
for positive integers a,b
they are two integers x,y such that
[unparseable or potentially dangerous latex formula]
Posted:
Thu Apr 19, 2007 4:08 pmby Dragon of Light
Darn, you're too smart. I quit.
Posted:
Thu Apr 19, 2007 4:10 pmby stupidityismygam
you are so right...
you know, since you mae MOP, and i dont even know what the AIME is...
Posted:
Thu Apr 19, 2007 5:34 pmby makashi
Actually, I just remembered the solution from when I first tried and failed it.
Posted:
Thu Apr 19, 2007 5:37 pmby stupidityismygam
cheater :)
Posted:
Thu Apr 19, 2007 10:22 pmby kashhustler
By the Euclidean Algorithm, GCD(21n+4, 14n+3) = GCD(14n+3, 7n+1) = GCD(7n+2, 7n+1) = GCD(7n+1, 1) which is obviously 1 so 21n+4 and 14n+3 are relatively prime