6

Part of 2017 Czech-Polish-Slovak Junior Match

Problems(1)

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

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

On the board are written

100

mutually different positive real numbers, such that for any three different numbers

a, b, c

a^2 + bc

is an integer. Prove that for any two numbers

x, y

from the board , number

\frac{x}{y}

Integerrationalnumber theory