Problem

Source: Romania National Olympiad 2023

Tags: irrational number, algebra



a) Show that there exist irrational numbers $a$, $b$, and $c$ such that the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational numbers. b) Show that if $a$, $b$, and $c$ are real numbers such that $a+b+c=1$, and the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational and non-zero, then $a$, $b$, and $c$ are rational numbers.