New Year Countdown - 9
Consider an inequality where a,b,c are real. Now for the above inequality the corresponding range of values of can be expressed in the form of where N is an Integer. Then find N mod 3.
The answer is 0.
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1 solution
The solution goes like this :-
( a + b ) 2 − a b − b c − c a + 1 > 0
= > a 2 + a ( b − c ) + ( b 2 − b c + 1 ) > 0
=> Now this is a quadratic inequality in terms of a. For this inequality to always give a positive value discriminant must be less than zero.(We can prove this by graph).
= > ( b − c ) 2 − 4 b 2 + 4 b c − 4 < 0
= > 3 b 2 − 2 b c + 4 − c 2 > 0
=> Now again applying the same logic as did above .......
= > 4 c 2 − 4 8 + 1 2 c 2 < 0
= > c 2 < 3
=> N = 3 and 3 mod 3 will be equal to 0.
That's it. As simple as that !!!!!!