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Romania NMO 2023 Grade 7 P4

Source: Romania National Olympiad 2023

April 14, 2023

irrational numberalgebra

Problem Statement

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.

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