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Kvant 2023
M2754
M2754
Part of
Kvant 2023
Problems
(1)
Russian algebra with two polynomials
Source: Caucasus MO 2023, day 1, senior P2
3/10/2023
Given are reals
a
,
b
a, b
a
,
b
. Prove that at least one of the equations
x
4
−
2
b
3
x
+
a
4
=
0
x^4-2b^3x+a^4=0
x
4
−
2
b
3
x
+
a
4
=
0
and
x
4
−
2
a
3
x
+
b
4
=
0
x^4-2a^3x+b^4=0
x
4
−
2
a
3
x
+
b
4
=
0
has a real root.Proposed by N. Agakhanov
algebra
polynomial