National Competition

Question

Determine the largest constant C such that (x_1 + x_2 + + x_6)^2 C (x_1(x_2 + x_3) + x_2(x_3 + x_4) + + x_6(x_1 + x_2)) holds for all real numbers x_1, x_2, , x_6. For this C, determine all x_1, x_2, , x_6 such that equality holds.

Accepted Answer

We rewrite the right-hand side

Expanding yields

X^{2} + Y^{2} + Z^{2} \geq X Y + Y Z + Z X

This is equivalent to

(X - Y)^{2} + (Y - Z)^{2} + (Z - X)^{2} \geq 0

with equality for

X - Y = Y - Z = Z - X = 0

, i.e.,

X = Y = Z

, thus

x_{1} + x_{4} = x_{2} + x_{5} = x_{3} + x_{6}

.