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Balkan MO Shortlist
2012 Balkan MO Shortlist
A3
A3
Part of
2012 Balkan MO Shortlist
Problems
(1)
Maximum possible number of distinct real roots
Source: 2012 Balkan Shortlist (BMO) A3
4/5/2020
Determine the maximum possible number of distinct real roots of a polynomial
P
(
x
)
P(x)
P
(
x
)
of degree
2012
2012
2012
with real coefficients satisfying the condition \begin{align*} P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*} for all real numbers
a
,
b
,
c
ā
R
a,b,c \in \mathbb{R}
a
,
b
,
c
ā
R
with
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0