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9
SMT 2022 Algebra #9
SMT 2022 Algebra #9
Source:
March 27, 2023
Problem Statement
Let
P
(
x
)
=
8
x
3
+
a
x
+
b
+
1
P(x)=8x^3+ax+b+1
P
(
x
)
=
8
x
3
+
a
x
+
b
+
1
for
a
,
b
∈
Z
a,b\in\mathbb{Z}
a
,
b
∈
Z
. It is known that
P
P
P
has a root
x
0
=
p
+
q
+
r
3
x_0=p+\sqrt{q}+\sqrt[3]{r}
x
0
=
p
+
q
+
3
r
, where
p
,
q
,
r
∈
Q
p,q,r\in\mathbb{Q}
p
,
q
,
r
∈
Q
,
q
≥
0
q\ge0
q
≥
0
; however,
P
P
P
has no
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
r
a
t
i
o
n
a
l
<
/
s
p
a
n
>
<span class='latex-italic'>rational</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
r
a
t
i
o
na
l
<
/
s
p
an
>
roots. Find the smallest possible value of
a
+
b
a+b
a
+
b
.
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