MathDB

JBMO Shortlist 2021 N2

Source: JBMO Shortlist 2021

July 2, 2022

JuniorBalkanshortlist2021number theorymaximumminimum

Problem Statement

The real numbers

x, y

and

z

are such that

x^2 + y^2 + z^2 = 1

. a) Determine the smallest and the largest possible values of

xy + yz - xz

. b) Prove that there does not exist a triple

(x, y, z)

of rational numbers, which attains any of the two values in a).