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

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

rewrite the lhs as
a-a² + b-b² + c-c² + d-d² ≤ 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)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