MathDB

3

Part of 1995 Nordic

Problems(1)

Σx_1^2=1, inequality with max{x_i}

Source: Nordic Mathematical Contest 1995 #3

10/4/2017
Let n≥2n \ge 2 and let x1,x2,...,xnx_1, x_2, ..., x_n be real numbers satisfying x1+x2+...+xn≥0x_1 +x_2 +...+x_n \ge 0 and x12+x22+...+xn2=1x_1^2+x_2^2+...+x_n^2=1. Let M=max{x1,x2,...,xn}M = max \{x_1, x_2,... , x_n\}. Show that M≥1n(n−1)M \ge \frac{1}{\sqrt{n(n-1)}} (1) .When does equality hold in (1)?
inequalitiesalgebramaximization