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2011 Austria Beginners' Competition
2
2
Part of
2011 Austria Beginners' Competition
Problems
(1)
4 integers wanted in a trinomial
Source: Austria Beginners' Competition 2011 p2
10/3/2021
Let
p
p
p
and
q
q
q
be real numbers. The quadratic equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
has the real solutions
x
1
x_1
x
1
and
x
2
x_2
x
2
. In addition, the following two conditions apply: (i) The numbers
x
1
x_1
x
1
and
x
2
x_2
x
2
differ from each other by exactly
1
1
1
. (ii) The numbers
p
p
p
and
q
q
q
differ from each other by exactly
1
1
1
. Show that then
p
p
p
,
q
q
q
,
x
1
x_1
x
1
and
x
2
x_2
x
2
are integers.(G. Kirchner, University of Innsbruck)
algebra
quadratics