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Sum of reciprocals maximum

Source: Festival of Young Mathematicians, Bulgaria 2013

September 17, 2013

inequalities proposedinequalities

Problem Statement

Let

a,b,c

be real numbers for which

a+b+c+d=19

and

a^2+b^2+c^2+d^2=91

. Find the maximal value of

\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}