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 3 of 5: Use AM–GM

In plain words

The total sum expands as $Q+2R$ .

(Q+2R)^2\ge8QR

Detailed analysis

By $Q+2R=\left(\sum_i x_i\right)^2$ and $Q,2R\ge0$ , AM–GM gives $(Q+2R)^2\ge4Q(2R)=8QR$ . Therefore the left side is at most $(\sum_i x_i)^4/8$ .