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Four Reals (Posted on 2013-05-10) Difficulty: 2 of 5


If a, b, c, and d are real numbers, then prove that

(a + b + c + d) - (a2 + b2 + c2 + d2) ≤ 1.

See The Solution Submitted by Bractals    
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Easy way Comment 1 of 1
rewrite the lhs as
a-a + b-b + c-c + d-d ≤ 1

then it becomes obvious that we are dealing with the sum of four quadratics, each of which has a maximum of 1/4 when the variable is 1/2.  i.e. (1/2)-(1/2)=1/4.

The sum of the 4 numbers, each at most 1/4, is at most 1.

  Posted by Jer on 2013-05-10 12:05:59
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