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Lithuanian TST 2014 #5

Source:

April 20, 2014

algebra proposedalgebra

Problem Statement

Given real numbers

x

and

y

. Let

s_{1}=x+y, s_{2}=x^2+y^2, s_{3}=x^3+y^3, s_{4}=x^4+y^4

and

t=xy

. a) Prove, that number

t

is rational, if

s_{2}, s_{3}

and

s_{4}

are rational numbers. b) Prove, that number

s_{1}

is rational, if

s_{2}, s_{3}

and

s_{4}

are rational numbers. c) Can number

s_{1}

be irrational, if

s_{2}

and

s_{3}

are rational numbers?

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