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 1 of 5: Find the sharp candidate

In plain words

A two-variable test gives the lower bound.

C\ge1/8

Detailed analysis

Take $x_1=x_2=1$ and all other variables zero. The left side is $2$ and $(\sum x_i)^4=16$ , so any valid constant satisfies $C\ge1/8$ .