MathDB

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Part of 2019 Czech-Polish-Slovak Junior Match

Problems(2)

a+b and a^2+b^2 are integers , then rational a,b are integers

Source: Czech-Polish-Slovak Match Junior 2019, team p1 CPSJ

1/20/2020
Rational numbers a,ba, b are such that a+ba+b and a2+b2a^2+b^2 are integers. Prove that a,ba, b are integers.
number theoryrationalIntegers
radical diophantine \sqrt{a+2\sqrt{b}}=\sqrt{a-2\sqrt{b}}+\sqrt{b}

Source: Czech-Polish-Slovak Match Junior 2019, individual p1 CPSJ

1/20/2020
Find all pairs of positive integers a,ba, b such that a+2b=a−2b+b\sqrt{a+2\sqrt{b}}=\sqrt{a-2\sqrt{b}}+\sqrt{b} .
number theoryradicalequationDiophantine equationdiophantine