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Roots of quadratic trinomials

Source: Tuymaada 2018 Senior League/Problem 1

July 20, 2018

quadraticsTuymaadaalgebra

Problem Statement

Do there exist three different quadratic trinomials

f(x), g(x), h(x)

such that the roots of the equation

f(x)=g(x)

are

1

and

4

, the roots of the equation

g(x)=h(x)

are

2

and

5

, and the roots of the equation

h(x)=f(x)

are

3

and

6

?
Proposed by A. Golovanov