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Balkan MO Shortlist
2020 Balkan MO Shortlist
A4
A4
Part of
2020 Balkan MO Shortlist
Problems
(1)
min length of EC and the polynomials for which this is attained wanted
Source: 2020 Balkan MO shortlist A4
9/14/2021
Let
P
(
x
)
=
x
3
+
a
x
2
+
b
x
+
1
P(x) = x^3 + ax^2 + bx + 1
P
(
x
)
=
x
3
+
a
x
2
+
b
x
+
1
be a polynomial with real coefficients and three real roots
ρ
1
\rho_1
ρ
1
,
ρ
2
\rho_2
ρ
2
,
ρ
3
\rho_3
ρ
3
such that
∣
ρ
1
∣
<
∣
ρ
2
∣
<
∣
ρ
3
∣
|\rho_1| < |\rho_2| < |\rho_3|
∣
ρ
1
∣
<
∣
ρ
2
∣
<
∣
ρ
3
∣
. Let
A
A
A
be the point where the graph of
P
(
x
)
P(x)
P
(
x
)
intersects
y
y
′
yy'
y
y
′
and the point
B
(
ρ
1
,
0
)
B(\rho_1, 0)
B
(
ρ
1
,
0
)
,
C
(
ρ
2
,
0
)
C(\rho_2, 0)
C
(
ρ
2
,
0
)
,
D
(
ρ
3
,
0
)
D(\rho_3, 0)
D
(
ρ
3
,
0
)
. If the circumcircle of
△
A
B
D
\vartriangle ABD
△
A
B
D
intersects
y
y
′
yy'
y
y
′
for a second time at
E
E
E
, find the minimum value of the length of the segment
E
C
EC
EC
and the polynomials for which this is attained.Brazitikos Silouanos, Greece
algebra
polynomial