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2021 Poland - Second Round
Problems
(1)
x^2+1/x^2+y^2+1/y^2 rational if x+1/x+y+1/y, x^3+1/x^3+y^3+1/y^3 rationals
Source: Polish Math Olympiad 2021 2nd round p1 day 2
5/31/2021
There are real numbers
x
,
y
x, y
x
,
y
such that
x
≠
0
x \ne 0
x
=
0
,
y
≠
0
y \ne 0
y
=
0
,
x
y
+
1
≠
0
xy + 1 \ne 0
x
y
+
1
=
0
and
x
+
y
≠
0
x + y \ne 0
x
+
y
=
0
. Suppose the numbers
x
+
1
x
+
y
+
1
y
x + \frac{1}{x} + y + \frac{1}{y}
x
+
x
1
+
y
+
y
1
and
x
3
+
1
x
3
+
y
3
+
1
y
3
x^3+\frac{1}{x^3} + y^3 + \frac{1}{y^3}
x
3
+
x
3
1
+
y
3
+
y
3
1
are rational. Prove that then the number
x
2
+
1
x
2
+
y
2
+
1
y
2
x^2+\frac{1}{x^2} + y^2 + \frac{1}{y^2}
x
2
+
x
2
1
+
y
2
+
y
2
1
is also rational.
algebra
retional