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a^2 + bc is an integer, x/y is rational

Source: Czech-Polish-Slovak Junior Match 2017, Team p6 CPSJ

March 6, 2020
Integerrationalnumber theory

Problem Statement

On the board are written 100100 mutually different positive real numbers, such that for any three different numbers a,b,ca, b, c is a2+bca^2 + bc is an integer. Prove that for any two numbers x,yx, y from the board , number xy\frac{x}{y} is rational.
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