MathDB

Maximum Possible Number of Positive Numbers

Source: Bulgarian IMO TST 2005, Day 2, Problem 1

July 7, 2013

inequalitiesfunctionquadraticsinequalities proposed

Problem Statement

Let

a_{i}

and

b_{i}

, where

i \in \{1,2, \dots, 2005 \}

, be real numbers such that the inequality

(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})

holds for all

x \in \mathbb{R}

and all

i \in \{1,2, \dots, 2005 \}

. Find the maximum possible number of positive numbers amongst

a_{i}

and

b_{i}

i \in \{1,2, \dots, 2005 \}