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- In one of the Bard's best thought of tragedies, our insistent hero, Hamlet, queries on two fronts about how life turns rotten.
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- That Queer Shake
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araver 3Fun with real numbers
For part I.
I get K=9.
The inequality holds for K=0.
Assume there is a largest K for which the inequality holds for all 0 <= a <= b <= c <= d. Then for some a=0, b=c=d > a= 0 we get (a + b + c + d)^2 >= K b c <=> (3d)^2 >=Kd^2. Since d>0, this means K<=9.
For all 0 <= a <= b <= c <= d the following holds true:
(a + b + c + d)^2 >= (0 + b + c + c)^2 = (b+2*c)^2 = b^2 + (2*c)^2 + 2*b*(2*c) = b^2 + c^2 + 3*c^2 + 2*b*c + 2*b*c =(b+c)^2 + 3*c^2 +2*b*c >= 4*b*c + 3*b*c + 2*b*c = 9*b*c.
Last step uses at once:
- (b+c)^2 >= 4bc which is equivalent to (b-c)^2 >=0
- 3*c^2 = 3*c * c >= 3*b*c since c>=b, c,b>=0
Equality holds for a=0, d=c, b=c.0
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