olymmath

IMO 1999 Problem 2

  • Day 1

Let n≥2n \geq 2 be a fixed integer. Find the least constant CC such the inequality

∑i<jxixj(xi2+xj2)≤C(∑ixi)4\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4

holds for any x1,…,xn≥0x_{1}, \ldots ,x_{n} \geq 0 (the sum on the left consists of (n2)\binom{n}{2} summands). For this constant CC , characterize the instances of equality.

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