National and Regional Contests Mongolia Contests Mongolian Mathematical Olympiad 2000 Mongolian Mathematical Olympiad Problem 2 inequality in vectors Problem Statement Let n ≥ 2 n\ge2 n ≥ 2 n n n x ⃗ = ( x 1 , … , x n ) \vec x=(x_1,\ldots,x_n) x = ( x 1 , … , x n ) y ⃗ = ( y 1 , … , y n ) \vec y=(y_1,\ldots,y_n) y = ( y 1 , … , y n ) f ( x ⃗ , y ⃗ ) = x 1 y 1 ‾ − ∑ i = 2 n x i y i ‾ . f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}. f ( x , y ) = x 1 y 1 − i = 2 ∑ n x i y i . f ( x ⃗ , x ⃗ ) ≥ 0 f\left(\vec x,\vec x\right)\ge0 f ( x , x ) ≥ 0 f ( y ⃗ , y ⃗ ) ≥ 0 f\left(\vec y,\vec y\right)\ge0 f ( y , y ) ≥ 0 ∣ f ( x ⃗ , y ⃗ ) ∣ 2 ≥ f ( x ⃗ , x ⃗ ) f ( y ⃗ , y ⃗ ) \left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right) ∣ f ( x , y ) ∣ 2 ≥ f ( x , x ) f ( y , y )