MathDB

Sequences

Source: St Petersburg Olympiad 2012, Grade 11, P4

September 29, 2017

algebra

Problem Statement

x_1,...,x_n

are reals and

x_1^2+...+x_n^2=1

Prove, that exists such

y_1,...,y_n

and

z_1,...,z_n

such that

|y_1|+...+|y_n| \leq 1

;

max(|z_1|,...,|z_n|) \leq 1

and

2x_i=y_i+z_i

for every

i