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 4 of 5: Determine the constant

In plain words

Combine the upper and lower bounds.

C=1/8

Detailed analysis

The previous inequality proves that $C=1/8$ works, while the test case proves no smaller constant works. Hence the least constant is $1/8$ .