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 5 of 5: Characterize equality

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In plain words

Both inequalities must be sharp simultaneously.

x_i=x_j>0,\quad x_\ell=0\ (\ell\ne i,j)

Detailed analysis

For a nonzero tuple, equality in $x_i^2+x_j^2\le Q$ requires that all variables except the two members of the contributing pair vanish. Equality in AM–GM requires $Q=2R$ , which for the two remaining variables is $x_i=x_j$ . Conversely, that configuration gives equality. (The all-zero tuple is the trivial equality case.)