Algebra · Inequalities · Arithmetic and harmonic mean · Double sums · Equality case
Let \(a_1, a_2, \ldots, a_n\) be positive real numbers with \(a_1 + a_2 + \cdots + a_n = 1\), and let
\[ S = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_i a_j}{a_i + a_j} \]
be the sum of all \(n^2\) expressions \(\dfrac{a_i a_j}{a_i + a_j}\) with \(1 \leqslant i, j \leqslant n\), the diagonal ones included (for \(i = j\) the expression is \(\dfrac{a_i^2}{2 a_i}\)). Prove that
\[ S \leqslant \frac{n}{2} . \]
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Serbian Republic Competition (Republicko takmicenje) 2000, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source