Problem 2

Let

n\ge2

be a fixed integer. (a) Find the least constant

C

such that for all nonnegative real numbers

x_1,\dots,x_n

\sum_{1\le i<j\le n}x_ix_j(x_i^2+x_j^2)\le C\left(\sum_{i=1}^n x_i\right)^4

. (b) Determine when equality occurs for this value of

C

Step 2 of 5: Bound the left-hand side

In plain words

Replace each pair’s square sum by the total square sum.

\sum_{i<j}x_ix_j(x_i^2+x_j^2)\le QR

Detailed analysis

Let $Q=\sum_i x_i^2$ and $R=\sum_{i<j}x_ix_j$ . For each pair, $x_i^2+x_j^2\le Q$ , so the left side is at most $QR$ .