[37] - Algebraic Inequalities from and for Contests

Δίνονται οι αριθμοί 

$0 < x_1 < x_2 < x_3​$ και $0 < y_1 < y_2 < y_3$​.

a) Δείξτε ότι $$(x_1 + x_2)(y_1 + y_2) < 2(x_1 y_1 + x_2 y_2).$$

b) Δείξτε ότι $$\dfrac{x_1 (y_2 + y_3) + x_2 (y_1 + y_3) + x_3 (y_1 + y_2)}{x_1 y_1 + x_2 y_2 + x_3 y_3} < 2.$$