MathDB
Exists n numbers whose sum is equal to sum of its squares and arbitrary number

Source: Romanian District Olympiad 2002, Grade IX, Problem 4

October 7, 2018
number theoryalgebra

Problem Statement

Let n≥2 n\ge 2 be a natural number. Prove the following propositions:
a) a1,a2,…,an∈R∧a1+⋯+an=a12+⋯+an2  ⟹  a1+⋯+an≤an. a_1,a_2,\ldots ,a_n\in\mathbb{R}\wedge a_1+\cdots +a_n=a_1^2+\cdots +a_n^2\implies a_1+\cdots +a_n\le a_n. b) x\in [1,n]\implies\exists b_1,b_2,\ldots ,b_n\in\mathbb{R}_{\ge 0}  x=b_1+\cdots +b_n=b_1^2 +\cdots +b_n^2 .
Back to ProblemsView on AoPS